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PENNSYLVANIA ECONOMIC
REVIEW
Volume 18, Number 2 Fall 2011
Ralph E. Ancil Questioning the Margin: Using Averages and
an Extended Break-Even Analysis to Find
Optimal Solutions
Muhittin Kaplan Testing Aggregation Bias for the Impact of
Huseyin Kalyoncu Devaluation on the Trade Balance: An
Application to Turkey
Mitchell W. Derubis Access to Formal Credit and Rural Economic
Christian N. Minich Development: An Examination of Prevailing
Assumptions in the Land Rental Market of
Rural India
William K. Bellinger Retail and Population Density in Smaller U.S.
Jue Wang Cities
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PENNSYLVANIA ECONOMIC REVIEW
Volume 18, Number 2 Fall 2011
Co-Editors
Thomas Tolin & Orhan Kara, West Chester University
Editorial Board
Gerald Baumgardner, Pennsylvania College of Technology Michael Hannan, Edinboro University of Pennsylvania
Rocky Huang, Pennsylvania State University – Berks
James Jozefowicz, Indiana University of Pennsylvania Yaya Sissoko, Indiana University of Pennsylvania
Kenneth Smith, Millersville University of Pennsylvania
Roger White, Franklin and Marshall College
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PENNSYLVANIA ECONOMIC REVIEW
Volume 18, Number 2 Fall 2011
Table of Contents
Ralph E. Ancil
Questioning the Margin: Using Averages and an Extended Break-Even Analysis to Find Optimal
Solutions ……………………………………………………………..………….……………….…1
Muhittin Kaplan and Huseyin Kalyoncu
Testing Aggregation Bias for the Impact of Devaluation on the Trade Balance: An Application to
Turkey …...……………………………………….……………………………...…….…...…… 20
Mitchell W. Derubis and Christian N. Minich
Access to Formal Credit and Rural Economic Development: An Examination of Prevailing
Assumptions in the Land Rental Market of Rural India ………………………………...........35
William K. Bellinger and Jue Wang
Retail and Population Density in Smaller U.S. Cities ……………………………………..….….53
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Pennsylvania Economic Review
Volume 18, Number 2, Fall 2011
QUESTIONING THE MARGIN: USING AVERAGES AND AN EXTENDED BREAK-
EVEN ANALYSIS TO FIND OPTIMAL SOLUTIONS
Ralph E. Ancil
Geneva College
(The author wishes to thank James Jozefowicz, Brian O’Roark, and William O. Pearce for their
constructive comments on an earlier draft of this paper and to the anonymous reviewers for their
effort in this process. Naturally, the author retains full responsibility for any errors.)
ABSTRACT
This paper shows that for some economically significant cases marginal analysis can be replaced
with simple algebra, averages, and an extended version of break-even analysis. Two examples are
used to illustrate the point. The first is historical based on Friedrich von Wieser’s solution to the
imputation problem in Menger’s analysis for explaining factor prices. It is shown that factor
prices can be understood as averages with variations (or break-even points) instead of as marginal
values. In the second case, an extended version of break-even analysis in average revenue/
average cost functions using basic algebra illustrates how the profit-maximizing level of output
can be predicted and how the tangency solution slopes can be derived without the use of the
differential calculus or concepts of marginal revenue and marginal cost. It is concluded that
mainstream theory should recognize such cases where applicable and come to terms with their
implications.
INTRODUCTION
In his Economic Theory in Retrospect, Mark Blaug writes: “…there is a mutual
interaction between past and present economic thinking which illuminates both.” (Blaug, 1997, p.
134) In this spirit, two examples -- one from the past and one from the present -- are used to
illustrate the fact that there are important cases where a simpler mathematical technique can be
used to derive results usually reserved for some form of marginal analysis. The two cases
illustrate that there is often an understandable mixture of technique and ideology which obscures
the possibility of alternative interpretations of results or alternative pathways to those results.
Contrasting models or visions of how the economy works can bring a desirable clarity to present
practices and show that in some instances the method of technique or interpretation of results is a
matter of choice or preference rather than of analytical necessity.
In the first of the following two examples, the Austrian view of determining value
through the process of imputation is presented with special attention to Friedrich von Wieser’s
correction of that process. The second example shows that with an extended form of break-even
analysis and average cost and average revenue functions one can also derive the profit-
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maximizing level of output, which itself is seen as an average. In both cases, the solution of the
technical problem is the same but either the interpretation of the results or the pathway is different.
EXAMPLE ONE: VON WIESER’S IMPUTATION PROBLEM
Problem Statement and Literature Review
In the Austrian view the problem of value was treated by Carl Menger's method of
"imputation." According to this method, the satisfaction experienced from final consumer goods
is the source of economic value. Starting on the side of final demand, goods of first or lowest
order, entrepreneurs impute value to the upstream, intermediate goods, goods of higher order
(capital and labor). To determine the value of these intermediate goods, one determines the value
of an input variable to the total value by removing it from the product and determining the loss of
satisfaction that results. This approach, his famous “loss principle”, is often described as linear
and causal.
However, Wieser understood that this method in some cases could lead to an over-
valuation of the product where the sum of all the variables' values would be greater than that of the
total product (Natural Value, 1956, pp. 86-89). The problem can be put this way: Suppose a
hunter shoots a dangerous tiger. The success of the shot, 100%, is dependent on both the
cartridge and the rifle, so that x + y = 100. However, we do not know how to value each input
separately. If we follow Menger’s method, we must ask ourselves, what the loss of satisfaction is
if we removed the cartridge. The answer is a total loss or 100%. But if we remove the rifle, that,
too, is a 100% loss. Hence, by this line of reasoning, each input is valued at 100%, an obvious
error.
Wieser proposed to improve upon the matter by a method focused on the factor’s
“productive contribution.” He stressed that the valuation process must include a circular or
cooperative level of interdependence, as different factors are used in different industries, the least
profitable of which establishes their opportunity costs. The solutions for the values of these
interdependent factors were found in the use of the familiar method of simultaneous equations.
The historical literature on Wieser’s contribution gives a mixed evaluation of its merit.
Stigler’s work (1941, pp. 158- 178) is the classic study frequently cited. He gives a number of
criticisms including those attending the assumption of fixed production coefficients, infinite
elasticity of demand, and the limitations of a homogeneous equation of the first degree. He argues
that there is no final difference between Menger’s method (“loss principle”) and Wieser’s
(“productive contribution”) even though he admits that Menger’s method does over-determine the
value of a factor (p. 147 ). Ultimately, he finds the approach “ useless.” Schumpeter (1954, pp.
913-917) describes Wieser as one of the three great Austrian economists but the least technically
competent. He also points out that the difference between Menger’s “loss principle” and Wieser’s
“productive contribution” is fully removed by the method of infinitesimals, i.e., the differential
calculus. Wesley Mitchell (1969, pp. 345-374) gives a lengthy but uncritical exposition of
Wieser’s views including the valuation problem. Maurice Dobb (1973, pp. 193-197) is less
critical of Wieser than Stigler is. Although aware of the limitations of his analysis, Dobb believes
a number of the weaknesses are correctable. He especially emphasizes Wieser’s concern for
absolutely fixed proportions and indivisibilities deserves a respectful hearing. He recognizes that
though infinitesimals resolve the difference between continuous and fixed proportions, it does so
at a loss of realism. He finds most fault with Wieser’s assumption of what is given on the supply
side and its implications for the concept of capital goods. Pribram (1983, pp. 314-318) explains
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the solution verbally and speaks of it in a positive manner. Mark Blaug (1997, pp. 453-454)
explains the contribution in detail, noting its similarity to linear programming. For Blaug, who is
not exactly keen on Austrian thinking, the Austrian concern for indivisibilities, and with it
Wieser’s solution, is implicitly vindicated by the rise of linear programming techniques among
economists dealing with problems not solvable by the use of the calculus and the assumptions of
continuous differentiability. More recently, Ekelund and Hebert (2007, pp. 307-309) thoroughly
explain Wieser’s contribution and place it in a favorable light, arguing that it not only solves
Menger’s valuation problem, but also emphasizes the Austrian view of demand as the ultimate
source of economic value and the importance of the marginal unit.
In the following, Ekelund and Hebert’s perspective will be used, especially since the
criticisms of various aspects by others of Wieser’s approach are not relevant to the point of
concern here, namely, the mixture of technique and ideology in contrast to an alternative way of
thinking.
Analysis
As indicated above, Wieser deals with the valuation problem by using simultaneous
equations. He poses three industries with three factors, x, y and z, which are used in the
following ways:
x + y = 100 (1)
4y + 5z = 590 (2)
2x + 3z = 290 (3)
Each output value is the price of one unit of the product. The goal is to find the prices of the
factor inputs which exhaust the product.2
To explore the matter further it is convenient to drop the third dimension (z) and work in
the familiar Cartesian plane with two variables (x and y) while retaining three industries. To do
this we can simply unzip the third dimension, z, and place it in the third quadrant. This preserves
Wieser’s main point, interdependency of uses leads to the correct valuation of inputs, while
making a graphical analysis easier. It also allows us to visualize the linear process of imputation
simultaneously with that of circular interdependence.
His equations can be rewritten as:
x + y = 100 (4)
y + 5/4z = 147½ (5)
x + 3/2z = 145 (6)
And rearranging in terms of y (and without z) gives:
y(1) = 100 – x (7)
y(2) = 147 ½ - 5/4y (8)
y(3) = 96 2/3 – 2/3x (9)
Spreading them out over three quadrants with the appropriate sign changes gives:
y(1) = 100 – x (QI) (10)
y(2) = 147 ½ + 5/4x (QII) (11)
y(3) = -96 2/3 + 2/3x. (QIV) (12)
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Since this problem is being transformed to a two dimensional graph, the z function is
pictured in QIII connecting QII and QIV and this requires appropriate sign changes. In QIII, the z
variable from QI becomes input to both QII and QIV. Or, one can say the function in QIII merely
transforms the x output from QII to the y input to QIV and its slope is accordingly set as -1. It
operates, in other words, like a slack variable. Overall, in this entire process, since the output of
one function is the input of another, the dependent and independent variables alternate for QII and
QIV as explained above. With these points in mind, we have the following equations:
y1 = 100 – x1 (QI) (13)
x2 = -118 + 4/5y1 (QII) (14)
(y2 = -140 – x2) (QIII) (15)
x1 = 145 + 3/2y2 (QIV) (16)
The output value of y1 becomes the input value for the second equation and so on through all four
quadrants. (See Figure 1.)
In Figure 1, we can visualize the causal direction coming from the outlying field in each
quadrant with their product values. Moving through the production functions, entrepreneurs
impute the values to the input factors. At the same time we see the circular or interdependent
movement, say, from QI through QIV coordinating these input values among the industries to
achieve equilibrium. The solutions are: (40, 60) in QI; (-70, 60) in QII; (-70, -70) in QIII; and (-70,
40) in QIV. (See Figure 2.) In terms of the original three variables, the solutions are: x = 40, y =
60, and z = 70.
While, as indicated above, there have been a number of technical criticisms of his
solution from various economists, these points are not the subject matter of this paper.3 Rather, it
is the marginalist understanding given to the solution by Wieser himself and others that is of
concern here. Wieser writes:
In the case of production goods which are available, not individually but in stocks,
imputation of the productive contribution follows the marginal law. To each single item
or quantity is imputed the smallest contribution which, under the circumstances, can be
economically aimed at by the employment of this particular item or quantity…” (pp. 96-
97)
As Ekelund and Hebert (2007) put it:
Given that an input is used in the production of a number of final or consumer goods,
its value will be determined by the least valuable good it produces. This value is
determined at the margin, by the marginal utility of the last unit of the least valuable good
the input is producing. (original emphasis, p. 308)
They conclude that “…the marginal utility of final output is presented as the source of value by
Austrian economists.” (original emphasis, p. 309)
However, we don’t really see this marginality in the graphs or the equations. Instead, it
must be assumed or read into the technique. One could, alternatively, with equal validity, envision
a classical gloss on the matter and recast the solution in those terms. One could, for example, see
market prices and output in terms of averages with fluctuations around a central tendency.
There are at least two different ways of doing so. One is to transpose the solution lines
from Figure 2 into QI and find the simple averages for the variables x and y. Doing so for all four
quadrants, and using all positive numbers, gives a mean value of x* = 55, and standard deviation
of +/-15, and mean of y* = 65, with standard deviation of +/- 5. (See Figure 3, where A = (40,
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70); B = (70, 70); C = (70, 60); D = (40, 70); and the means are at point E = (55, 65); their ratio
also gives the slope of 1.1818.)
If we take Figure 2 and make some adjustments we can display these results over all four
quadrants. By connecting the solution points in QII and QIV, we have a diagonal line with slope
= -1.1818. Doing the same thing for QI and QIII, we have a diagonal line with slope = +1.1818.
Finding the intersection of these two lines, we have (-15, -5). This forms a solution box and
effectively moves the origin from (0, 0) to (-15, -5). In this case the means and standard
deviations reverse themselves: x* = -15 +/- 55 and -y* = -5 +/- 65. (See Figure 4.)
Another approach is to work the solution backwards, so to speak, by assuming an initial
knowledge of the total values of the input factors, x and y. (This is really a summing up of the
solution coordinates.) The total value of the x’s is 220 and that of the y’s is 260. The ratio of the
y’s to the x’s is:
260/220 = 1.1818. (17)
This ratio of total values is the same as the slope derived from finding the intersection of the two
diagonal lines in the solution box.
With this information one can predict the specific values in each quadrant in this way:
QI: x = -15 + 55 = 40 (18)
y = -5 + 65 = 60 (19)
QII: x = -15 – 55 = -70 (20)
y = - 5 + 65 = 60 (21)
QIII: x = -15 – 55 = -70 (22)
y = -5 – 65 = -70 (23)
QIV: x = -15 +55 = 40 (24)
y = -5 – 65 = -70. (25)
Using the total values, we are able to deduce the specific equilibrium values for each industry even
without knowing the production function of those industries. (See Figure 4.)
With simple averages, there is no assertion about the role of the margin, because all four
solution points are taken to be equally distant from the origin. The value of x = 40 is just as far
from x* = -15 as is x = -70, namely 55. Or, in terms of Figure 2, they are equally distant from
the mean of x* = 55, namely, 15. But there is no one least valuable product which can be argued
to cause or determine the value of the factors. It can be seen as a collective, interdependent,
statistical result where the industries participate in a common pool of factor values.
Weiser himself argues that the cost of production approach is proximately valid, for once
the marginal values are widely dispersed through the economy, the summing of factor prices is a
reasonable approach to their product values (Natural Value, pp. 171ff). From the point of view
of any one individual producer the value of the product must at least be equal to the sum of the
costs that go in to making it up. But, he insists, this is merely a relative assessment since the
ultimate value comes from demand via the theory of imputation.
However, if the results of his imputation analysis in the case discussed above are
reconfigured as a simple average, the sense of the margin is lost. A Smithian cost of production
sees the natural price as an average with no distinctive marginal value determining final value.
The interdependence is not only between industries with respect to factor prices, but also within
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them with respect to factor and product prices. The marginal is replaced with the central; the
individual addition is replaced by the collective mean.
So much for the historical example. The same principle can be applied to current
mainstream thought.
EXAMPLE TWO: STANDARD REVENUE AND COST CURVES
Problem Statement and Literature Review
Because entrepreneurs may make decisions in a manner different from marginal analysis,
say, by rules of thumb that involve simple averaging, it is useful to see if one can set that heuristic
approach on a sound theoretical basis. Therefore, in the previous paper, the problem of finding a
different pathway to the same marginal result, namely, profit-maximizing level of output, was
looked at more closely in terms of break-even points and averages (Ancil, 2010). In that paper it
was shown that an optimal solution can be found without using marginal analysis, and the
differential calculus, relying instead on simple algebra.1 The success of the examples given there
was due either to the symmetry of the total cost and revenue functions, (i.e., either both were
simple quadratic equations, or they were a combination of linear with quadratic equations), which
allowed one to average either the break-even points or the vertices to arrive at the profit-
maximizing level of output.2 In the present instance, we extend the technique to apply to the case
where this symmetry is not there, i.e., where one of the functions is cubic. It is found that there is
no problem in obtaining an algebraic solution using break-even points and averages and without
the use of marginal analysis.
The familiar break-even analysis, or cost-volume-profit analysis, is usually displayed as
a rough rule of thumb which has some validity in narrowly confined situations, that is, where the
assumptions of constant costs and prices are valid approximations, but which is not capable of
delivering a precise, theoretically sound result, an optimal result, because the concept of
diminishing returns is not considered. Consider, for example, Davies and Lam (2001, p. 190) who
compare the differences between standard economic theory and break-even analysis: “In the first
place, the economic model is an optimising model, which identifies the profit- and contribution-
maximising level of output and price. The break-even model shows no optimum as the levels of
profit and contribution simply increase with the level of output.” (emphasis added). In a similar
vein, the authors Keat and Young (2009, pp. 352-353) summarize the limitations of break-even
analysis, two of which are:
It assumes the existence of linear relationships, constant prices, and constant average
variable costs. However, when the effects of relatively small changes in quantity are
measured, linear revenues and variable costs are certainly good approximations of
reality…The analysis does not result in identification of an optimal point; it focuses on
evaluating the effect of changes in quantity on cost and profits.
What these authors conclude is in fact correct for the usual break-even analysis considered.3 But
if we extend the method, i.e., improve on it to account for the effect of diminishing returns, we
will be able to find optimal solutions after all.
Analysis
Consider the following total revenue and cost functions:
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TR = 2x – x2 and TC = x
3. See Figure 5. (26)
The relevant break-even points are at (0, 0) and (1, 1). A simple rule of thumb to take the break-
even points and divide by two yields the correct answer, that is, the correct profit-maximizing
level of output (PMO), for the case where TC = x2. However, it yields the same answer here,
where TC = x3. Obviously, this is not right. The solution from marginal analysis
4 is:
MR = 2 – 2x, and (27)
MC = 3x2 . (28)
Solving in the usual manner gives:
(x + 1/3)2 = 7/9, and (29)
x = (√7 – 1)/3
= .5486.
To correct this and arrive at this same answer by an alternative method, we begin with the
average revenue and average cost functions:
AR = 2 – x, and (30)
AC = x2. (31)
To find the break-even points for these, we set them equal to each in the familiar way and
solve:
2 – x = x2 (32)
0 = x2 + x – 2.
After completing the square this becomes:
(x + ½)2 = 9/4. (33)
Solving for x gives the break-even x-values:
x = (√9 – 1)/2 (34)
= 3/2 – ½
= 1 or -2.
The relevant break-even points are (1, 1) and (-2, 4). See Figure 6.
How can the optimal output level which maximizes profit (PMO) be found with the mere
use of break-even points and averages? We make the following observations:
The greatest difference between the AR and AC functions, 1.75, occurs at x = -1/2. This
can be determined by inspection as well as by comparing the slopes of the two functions. The
slope of the AR line is -1 and the slope of the line tangent to the AC curve at x = -1/2 is also -1.
(This is equivalent to shifting the AR line down until it is tangent to the AC curve at this point.)
The axis of symmetry (AS) value is given by the average of the x-values of the break-
even points (BEP):
AS = (BEP1 + BEP2)/2 (35)
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= - ½.
In the break-even equation, the combined AR/AC function, the slope at x = -1/2 is no
longer -1 but 0, at the vertex.
How do we arrive at the maximum total profit from the maximum average profit?5 Since
the vertex on the original average cost function is at x = 0 (or 0, 0) but the maximum difference
now is at the value x = -1/2 (y = ¼), they are obviously out of alignment with one another. The
maximum total profit can be found by bringing these two into harmony, by making the vertex
equal to 1.75 (7/4) at x = -1/2. This can be accomplished by either shifting the AR curve to the
left by the amount of the AS (x = -1/2) or by shifting the AC curve to the left by that amount.
This is also seen from the quadratic equation combining the AR/AC functions. The
completion of the square technique results in a constant (1/2) whose opposite is the axis of
symmetry (x = -1/2).
Working with the AR curve in Figure 6, we have:
AR = (2 – ½) – x (36)
= 1.5 – x.
This line passes through the point (-1/2, 2) which lies directly above (-1/2, 0). Solving for this
break-even equation:
1.5 – x = x2 (37)
(x + ½)2 = 7/4.
The break-even points at y = 0 are:
BEP1 = (.8229, .67716) (38)
BEP2 = (-1.8229, 3.32296). (39)
Re-graphing the equation, (x + ½)2 = 7/4, gives us Figure 7. The graph shows the break-
even points on the x-axis .8229 and -1.8229. It also shows a line passing with slope = 1/√7 +1,
through BEP2, passing through y = ½, and intersecting the main function at the PMO.
However, we can also achieve the same results from the equation (x + ½)2 = 9/4, by
solving for y = ½. (It is as if these two equations reduce the slope of AR to 0, shift the points to
the left and invert the function so that it opens downward. See Figure 8, where the B1 x-value is
.8229 and B2 is -1.8229.)
The profit-maximizing level of output is found along with the tangency solution and
other results by simple combinations of the break-even points. The formulas for doing so are:
1. PMO = │1/BEP2│ (40)
= 1/1.8229
= .5486.
2. PMO = 1 + BEP1/BEP2 (41)
= 1 + .8229/-1.8229
= 1 - 4514.
= .5486.
3. BEP1 - BEP2 = √7, and (42)
(BEP1 - BEP2)/2 = √vertex
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= distance between BEP1 and BEP2.
4. The slope, TS, of a line tangent to the TR and TC curves at
the PMO is found by:
TS = (BEP1/BEP2)/AS (43)
= (.8229/-1.8229)/-1/2
= .9028.
5. And finally, the profit-maximizing level of revenue (PMR)
is a line passing through the origin and passing through the
TR function at the PMO point is:
PMR = (1- BEP1/BEP2)x (44)
= (1-.8229/-1.8229)x
= 1.4514x.
= 1.4514(.5486)
= .7962.
The PMR can also be described as a linear revenue growth function which identifies the
maximum revenue at the PMO in terms of the break-even points as:
PMR = │1/BEP2│ + BEP1/BEP22
(45)
= │1/-1.8229│ + .8229/(-1.8229)2
= .7962.
One may reconfigure this information in a stylized graph so that it looks like Figure 9.
Here we modify the AR equation by using BEP2 as the slope:
AR’ = 2 + (BEP2)x (46)
= 2 – [(√7 + 1)/2]x
= 2 – 1.8229x.
This line intersects the x-axis at x = 1.0972. Dividing this in half gives the PMO of .5486. The
difference between this value (1.0972) and 2 is the tangency slope of .9028.
Finally, we can focus on quadrant one and break-even point1 (BEP1) exclusively and
derive all the relevant results. Reasoning in the manner explained above, we adjust the
intersection of the AR and AC from the nominal values to those resulting by shifting the AR line
down by ½. (Recall Figure 6). By drawing a line from the origin through BEP1 (.8229, .6771,
point A in Figure 10), we derive a new break-even line (BEL) which passes through this point and
intersects the original AR line at x = 1.0972 (point B). Dividing this in half gives .5486, the PMO.
(Another adjusted break-even line is the inverse of the value 1.0972 which is .9114, the slope of a
line passing through the PMO at y = ½. This is not on the graph.)
Alternatively, we can continue following the line until it intersects y = 1 at x = 1.21533
which is the inverse of .8229 (point C). Adding one to this figure gives the ratio of BEP2/BEP1 or
2.21533. Subtracting the inverse of this value from 1 gives the PMO, and dividing this into 2 we
get the slope of .9028 at the PMO. We have then the following results, for the adjusted break-
even line (ABEL):
ABEL = .8229x. (47)
Page 14
10
At ABEL = AR, x = 1.0972 and
y = 2-1.8229x (48)
PMO = 1.0972/2 (49)
= .5486.
Also,
PMO = 1 – BEP2/BEP1 (50)
= 1 – 1/2.21533
= .5486.
And,
TS = 2/2.21533 (51)
= .9028. See Figure 10.
We can also transpose the adjusted break-even line (ABEL) to the upper part of the graph
so that it intersects the AR line at point D in Figure 10 at the PMO.
Note also that the PMO is derived by multiplying the BEP1 by the ratio of the exponents:
PMO = 2/3 (.8229) (52)
= .5486.
For two other applications see the appendix.
CONCLUSION
The above examination adds further evidence that it is possible to consider some
economic models in terms of averages and break-even points and to see economic behavior as
revolving around central tendencies, fixed proportions, and constants rather than as small,
marginal changes. If entrepreneurs make price and output decisions based on such simple rules of
thumb, these decisions may have a more solid analytical foundation than has heretofore been
admitted. Recasting the results of these two cases into a non-marginal mode also emphasizes the
difference between the technical analysis and the verbal (ideological?) gloss and shows that the
marginal analysis or a corresponding marginalist interpretation, is optional, a matter of choice,
not a matter of analytic necessity. The alternative pathways to profit-maximization can be
motivated by a different theory or vision of how the economy works, more in the spirit of a
Smithian, classical view. If so, it may well be that Smith’s description of entrepreneurial
behavior still has a measure of merit. After explaining his view that the “ordinary and average”
rates of wages, profit and rent constitute the “natural price”, he writes:
The natural price, therefore, is, as it were, the central price, to which the prices of
all commodities are continually gravitating. Different accidents may sometimes
keep them suspended a good deal above, and sometimes force them down even
somewhat below it. But whatever may be the obstacles which hinder them from
settling in this center of repose and continuance, they are constantly tending towards it.
(Wealth of Nations, Bk I, vii, 15; emphasis added.)
The full implications of this are beyond the scope of this paper but will certainly depend
on how widely applicable the alternative techniques are. Clearly, the wider the application, the
Page 15
11
more it impacts research and education in economic theory and the more demands it will make
on the flexibility of the economics discipline to accommodate such a different model.
ENDNOTES
1 It was also shown in that paper that Boehm-Bawerk’s horse-trading example -- in which he
argued that the market clearing price was uniquely determined by a marginal pair -- was also
explicable as a simple average derivable from the total value of reservation prices, without
knowledge of any marginal pairs.
2 It is clear that for some aspects of this discussion one could refer to Rolle’s theorem or the mean
value theorem. However, as indicated in the main text, the point is to see how far one can go just
with simple algebra.
3Nonlinear or curvilinear break-even analysis is, of course, used when the costs and prices are not
constant. But, as Salvatore (2004, p. 300) writes, though this approach is similar to standard
marginal analysis of profit- maximization, the “objective shifts from the determination of the
optimum price and output in optimization analysis to the determination of the output levels at
which the firm breaks even or earns a target profit…” In other words, the goal of nonlinear break-
even analysis is the same as in the linear case, which is non-optimal.
4 Compare 2/(√7 + 1) in the solution of the combined AR/AC functions with the marginal result of
(√7 - 1)/3. Both of these expressions give the same answer for the profit-maximizing level of
output.
5 The familiar algebraic approach of the latus rectum with foci and directrix allows us to identify
the maximum, at the vertex, and the relevant break-even values and the mean as the axis of
symmetry.
REFERENCES
Ancil, Ralph E. (2010) “Using Averages and Break-even Points to Find Optimal
Solutions” in Proceedings of the Pennsylvania Economic Association 2010
Conference, June 3-5, 2010, pp. 26-37.
Blaug, Mark. (1997) Economic Theory in Retrospect. Cambridge University Press.
Cambridge.
Davies, Howard and Pun-Lee Lam. (2001) Managerial Economics (3rd
ed.). Prentice Hall.
Harlow, England.
Dobb, Maurice. (1973) Theories of Value and Distribution Since Adam Smith. Cambridge
University Press. Cambridge.
Ekelund, Jr., Robert B. and Robert Hebert. (2007) A History of Economic Theory and
Method. Waveland Press, Inc. Long Grove.
Keat, Paul G. and Philip K. Young. (2009) Managerial Economics (6th
ed.). Prentice Hall.
Upper Saddle River.
Page 16
12
Mitchell, Wesley C. (1969) Types of Economic Theory (vol. 2). Augustus M. Kelly
Publishers. New York.
Pribram, Karl. (1983) A History of Economic Reasoning. The Johns Hopkins University
Press. Baltimore.
Salvatore, Dominick. (2004) Managerial Economics in a Global Economy (5th
ed.).
Thomson/South-Western, Australia.
Schumpeter, Joseph. (1954) History of Economic Analysis. Oxford University Press. New
York.
Smith, Adam. (1981) An Inquiry into the Nature and Causes of the Wealth of Nations. Liberty
Fund. Indianapolis.
Stigler, George. (1941) Production and Distribution Theories : The Formative Period. The
Macmillan Company. New York.
Wieser, Friedrich von. (1956) Natural Value. Kelly and Millman, Inc. New York. [German ed.
1889, English ed., 1893]
Page 17
13
APPENDIX
One can also approach the solution by working with the y–axis as shown in Figure A-1.
One example, if the total cost were simply TC = x2, the y-value corresponding to the PMO is 1.5,
or one half of the distance between y = 2 and y = 1. In the present case where TC = x3 the
corresponding value is 1.4514. We can think of an average value centered at mean y = 1with
variations of +/- .4514, whose total distance equals the slope of the tangency lines of .9028. We
still have y = 1.4514 intersecting AR at point D, and y = .5486, intersecting the original break-
even line (BEL) at y = .5486.
We can also think of the solution being centered at mean y = 1.4514 (point D) with
variations of +/- .4514. The total vertical distance then goes from y = 1 to y = 1.9028. At y = 1,
this intersects AR and TR and TC at x =1.
An alternative way of visualizing the process is seen in Figure A-2. At zero costs the
center is at point (1, 1) where TR and AR are equal. As soon as costs are introduced a cost box
begins to expand around this point. In the present case, we take the ratio of the break-even points,
│.8229/-1.8229│ to derive the value of .4514 and apply it equally from the point (1, 1).
Travelling from this point leftward to the vertical line DF, we have the solution, since that line
intersects the original break-even line at y = x = .5486 (point F). It also insects the TR curve at its
point of maximum profit, at y = .7962.
We further observe that the distance from TR = .7962 to y = 1 is .2038 which is simply
the ratio of the break-even points squared, .45142. Adding this distance to y = 1 allows us to
interpret this portion of the y-axis as TR since 2 - 1.2038 = .7962. Shifting the AR line inward so
that it intersects at the point (.5486, 1.2038), the new slope for this line is the price of 1.4514. We
then have a budget graph showing that consumers’ budget is equal to the TR of .7962 since they
purchase quantity .5486 at a price 1.4514.
Lastly, we observe that 1.02038 is equal to 1.0972 squared.
Page 20
16
AR =
2 - x
TR =
2x - x2
TC
=
x3
AC
=
x2
AR
= 2-
x
AR' =
3/2 - x
Page 21
17
(x + 1/2) = 9/4
Figure 8
Slope = 1/ (7 + 1)
Figure 7
Page 24
20
Pennsylvania Economic Review
Volume 18, Number 2, Fall 2011
TESTING AGGREGATION BIAS FOR THE IMPACT OF DEVALUATION ON THE
TRADE BALANCE: AN APPLICATION TO TURKEY
Muhittin Kaplan and Huseyin Kalyoncu
Meliksah University, Turkey
ABSTRACT
The empirical literature on the relationship between devaluation and trade balance has mainly
concentrated on aggregate analysis and provides mixed results on the impact of devaluation on
trade balance. However, an analysis undertaken with aggregate data leads to the loss of valuable
information and econometric estimates of the relations at the aggregate level are subject to
aggregation bias. This is because an aggregate analysis of devaluation cannot fully accommodate
the information on the structural change, heterogeneities among trading partners of the economy
and the dynamics of the relationship. To deal with these deficiencies, the current study investigates
the impact of devaluation on trade balance employing data disaggregated by trading partners and
tests whether the parameters obtained from an aggregate trade balance model differ significantly
from the average of those of the corresponding disaggregate trade balance models using
aggregation bias tests of *
1q and *
2q developed and employed in a series of papers (Pesaran et al.
(1989), and Lee et al. (1990a, b), and Lee and Pesaran (1993) and Lee (1997)). This allows us to
find out whether aggregation biases are present in the previous studies of devaluation and to
discover the impact of devaluation on bilateral trade balances through a disaggregate analysis of
the Turkish trade data. The empirical results from our study indicate that devaluation has a
positive impact on both aggregate and bilateral trade balances of the Turkish economy. We also
find that the responsiveness of bilateral trade balances to devaluation differ significantly among
trading partners in terms of their contribution in this process. More importantly, the empirical
evidence in this paper suggests that the existing studies in the literature, carried out at the
aggregate level, underestimate the impact of devaluation on trade balance and, thus suffer from
aggregation bias.
INTRODUCTION
This study evaluates the impact of devaluation on trade balance using both aggregate and
disaggregated data by trading partner countries and to test whether aggregate analysis is subject to
aggregation bias. The relationship between devaluation and trade balance has attracted a lot of
interest in the recent empirical literature which provides mixed results on the effect of devaluation
on trade balance in the long-run. Most of the studies on the subject have mainly concentrated on
aggregate analysis and paid little attention to the links between devaluation and trade balance
operating at the disaggregate trading partner country level. However, this relationship is based on
complex structural relationships and an analysis undertaken with aggregate data is subject to two
deficiencies: The first, economic deficiency, concerns the loss of important information that can
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21
be obtained from an investigation of the dynamics of the devaluation-trade balance relationship at
the bilateral level since aggregate analysis disguises a lot of information. A disaggregate analysis
is better able to reflect these important dynamics and economic linkages. The second, statistical
deficiency, is closely related to the fact that there are complex interrelations among trading
partners and that the structure of the economy changes over time. These features mean that
econometric estimates of the relations at the aggregate level are subject to structural breaks and
aggregation bias.
For these reasons, the potential effect of devaluation on the trade balance cannot be fully
evaluated without taking the structural change, heterogeneities among trading partners and the
dynamics of the devaluation-trade balance relationship into account. Because this information
cannot be accommodated into an analysis carried out at the aggregate level, the results from
studies undertaken at aggregate level may be subject to bias. To avoid the potential for aggregation
bias in estimated aggregate relationships, researchers investigate the trade balance-devaluation
relationship at the disaggregate trading partner country level. However, to our best knowledge, no
study has statistically tested the aggregation bias in the empirical literature on the trade balance-
devaluation relationship.
Taking these discussions together, we evaluate the impact of devaluation on the trade
balance econometrically using both aggregate and disaggregate data. In this way, we identify
important aspects of the relationship between devaluation and trade balance which are not possible
in the aggregate analysis and improve the precision of our results by avoiding the aggregation
bias. Furthermore, using recently developed aggregation bias tests, we test for possible biases
which may have occurred in the previous studies on the subject carried out at the aggregate level.
Hence, we provide detailed analysis of the effects of devaluation on trade balance. The rest of the
study is organized as follows. Section 2 provides a brief survey of the empirical literature on the
relationship between devaluation and trade balance. Section 3 introduces the modeling strategy
and the data. Sections 4 presents the results obtained from estimating the trade balance equation at
aggregate and disaggregate level, provide aggregation bias tests results and comments on the
findings. Section 5 concludes.
THE IMPACT OF DEVALUATION ON TRADE BALANCE: A LITERATURE REVIEW
The question of the impact of devaluation on the trade balance has been studied in recent
years in a large literature. Since the publication of Magee’s paper (1973), two strands of the
empirical literature have developed in testing the impact of devaluation on the trade balance. The
first involves estimating the trade balance equation at the aggregate level. The second estimates
the same equation using the data disaggregated by countries. However, the results from these
literatures provide mixed results and hence no consensus view is reached on the impact of
devaluation on trade balance until now.
The aggregate approach considers the trade flow of a country with the rest of the world
using aggregate data. Several studies have examined the impact of devaluation on the trade
balance employing this methodology. Himarios (1985), in his study of 10 countries, and Himarios
(1989), in his study of 15 developing countries, show that devaluations, in general, improve the
trade balance. Recently, Gomes and Paz (2005) found the Marshall-Lerner condition held for the
Brazilian trade balance in the 1990s implying the positive effect of devaluation. In his empirical
analysis of the long-run response of 11 Middle Eastern countries’ trade balances to devaluation,
Bahmani-Oskooee (2001) also concluded that real depreciation has a favorable long-run effect on
the trade balances of most non-oil exporting Middle Eastern countries.
The findings of the studies by Narayan (2004), for New Zealand, and Buluswar et al
(1996), for India, however, indicate that devaluations have had no significant long-run effect on
the trade balance. In a similar study, Upadhyaya and Dhakal (1997) tested the effectiveness of
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22
devaluation on the trade balance in eight developing countries and found that devaluation, in
general, did not improve the trade balance in the long run. Bahmani-Oskooee and Alse (1994)
investigated the relationship between trade balance and exchange rate using the quarterly data
from 19 developed and 22 less developed countries, including Turkey, over the years 1971-1990.
Employing the Engle-Granger cointegration technique, the authors found that while devaluation
had no effect on trade balances of Canada, Denmark, Germany, Portugal, Spain, Sri Lanka, UK
and the USA, it had a positive effect for Costa Rica, Brazil, and Turkey, and a negative effect for
Ireland in the long-run. The list also includes Miles (1979) for 14 countries, Bahmani-Oskooee
(1985) for four developing countries, Brissimis and Levantakis (1989) for Greece, Felmingham
(1988) for Australia, Demirden and Pastine (1995) and Brada et al. (1997) for Turkey.
The disaggregate approach considers the bilateral trade flow of a country with the partner
country using the disaggregated trade data. Rose and Yellen (1989) take up bilateral approach on
the US’s bilateral trade with her six trading partners and report the lack of long run effect.
Marwah and Klein (1996) tried to test the phenomenon between Canada and its five largest trading
partners as well as the U.S. and her five trading partners and found mixed results. However,
Bahmani-Oskooee and Brooks (1999), in their study on the effect of devaluation on US’s bilateral
trade with her six trading partners found that real depreciation of the dollar has a favourable long-
run effect on US trade balance. Similarly, Shirvani and Wilbratte (1997) test the relationship
between the real exchange rate and the trade balance using monthly US bilateral trade data for the
flexible exchange rate regime, over the period 1973:1–1990:8. They focus on the USA as the
home country, and Canada, France, Germany, Italy, Japan and the UK as her trading partners.
They find that with the exception of Italy, there is a statistically significant association (in the
expected direction) between the real exchange rate and the trade balance in all cases.
Bahmani-Oskooee and Goswami (2003), investigates the effect of devaluation on Japan’s
bilateral trade with her trading partners. The long-run effect of currency depreciation is found to
be favourable. Ahmad and Yang (2004), investigates the effect of devaluation on Chins’s bilateral
trade with the G-7 countries. They found that real depreciation improves the trade balance with
some countries. Other studies that investigated the effect of devaluation on trade balance at
bilateral basis are: Arora et al (2003) for the India vis-à-vis her trading partners, Wilson (2001) for
Singapore, Malaysia and Korea vis-à-vis their trading partners, Bahmani-Oskooee and Kantipong
(2001) for Thailand vis-à-vis their trading partners, Bahmani-Oskooee et al (2006) for UK vis-à-
vis their trading partners and Halıcıoglu (2007) for Turkey’s bilateral trade with 13 trading
partners.
In summary, the review of the empirical literature shows that there are important
differences among trade balances with trading partners in terms of responsiveness of devaluation.
The level of development also seems to be important in determination of the impact of devaluation
on the trade balance. Taken together, these imply that the aggregate analysis of devaluation may
suffer from aggregation bias. However, the presence of aggregation bias is not tested statistically
in the empirical literature. To close this gap in the literature, we will estimate both aggregate and
disaggregate trade balance equations and test the aggregation bias statistically. The details of the
methodology employed in this study are given in the following section.
METHODOLOGY AND DATA
The modeling framework provides an adequate representation of trade balance model
which allows for structural change, heterogeneities among responsiveness of devaluation to trade
balances with trading partners, and the dynamics of trade balance-devaluation relationship;
namely, disaggregate analysis of devaluation-trade balance relationship. The trade balance
equations developed in this section will be used in the empirical analysis below.
Page 27
23
Many researchers, for example Bahmani-Oskooee (1991, 2001), Rose (1991) and Gomes
and Paz (2005) used a reduced form equation for the trade balance (TB) in order to study the effect
of devaluation on the trade balance. Kalyoncu and Kaplan (2007) and Kalyoncu et al (2009) stated
that there is an advantage of formulations that used in these studies. Without estimating the
structural parameters of the export and import functions this formulation allows a straightforward
test of the effect of devaluation on trade balance. In its natural logarithm form, the trade balance
(TB) is defined as the difference between the nominal imports )( *MEP in terms of domestic
currency units minus nominal exports )(PX such as )log()log()log( * PXMEPTB . We
can write TB equation using small case letters for the log of variables,
ttttttttt erxmppexmtb )( * (1)
In this equations pandpmx ,,, * represent the natural logarithm of the volume of exports and
imports, foreign import prices and domestic prices respectively. ter is the log of real exchange
rate. Import and exports functions can be defined as:
txtterbyax *
(2)
tmtt erdycm (3)
In these equations a and c are constants, b and d are foreign and domestic income elasticity, x
and m
are real exchange rate elasticity of exports and imports, respectively. After substituting the
equation (2) and (3) into equation (1), we obtain an expression that represents the long run TB,
given by equation 4.
tmxttt erdybyactb )1()( * (4)
By checking the coefficient of t
er variable, which gives the Marshall-Lerner (ML)
condition, one can easily test the effectiveness of devaluation. A negative coefficient on the t
er
variable indicates that the ML condition holds and devaluation improves the trade balance. To take
the dynamics of trade balance into account, the trade balance equation (4) is written as:
ititiitiitiitiitiitiitiiit tbereryyyytb 17165143
*
12
*
10 (5)
We estimate both bilateral trade balance equations (5) for each of the thirteen countries
using once-lagged values of explanatory variables and a lagged dependent variable. In the
aggregate counterpart of the trade balance equation of (5), the aggregate trade balance equation is
estimated. The lagged values variables are introduced into the equation to take short-run dynamics
into account following general to specific modeling approach developed by Davidson et.al. (1978,
p. 680) and Hendry (1979). Then a specification search was carried out on the OLS estimates of
the equations. This involved dropping those variables whose coefficients had t-values, which were
insignificant at a 10 percent level of significance (See Garderen and Lee and Pesaran (2000,
p.316); Pesaran and Pierse and Lee (1994, p.17); Lee and Pesaran (1993, p.39) for a similar
Page 28
24
specification search methodology). Moreover, since the main aim of the study is to investigate the
trade balance-devaluation relationship, we have always retained the real exchange rate variable
even when they were more insignificant than other variables of the equation. Finally, in our
empirical analysis, we compare the average elasticity from the disaggregate model with the
aggregate model using aggregation bias tests.
Aggregation bias test
As explained in the literature review, most studies on the subject were undertaken at the aggregate
level. However, from an economic and statistical point of view, it seems necessary to study the
relationship between devaluation and trade balance at the disaggregate level. From the economic
point of view, the responsiveness of bilateral trade balances to devaluation and the dynamics of
this relationship may be different across trading partners and aggregate analysis will obscure these
relationships. Furthermore, considering the fact that theories in economics are based on highly
disaggregated units (households, firms behaviour), one will see that any degree of aggregation
might incur a problem of coherence with the underlying theory. In this sense, even more detailed
model of economic process provides more accurate representation of reality.
From a statistical point of view, the empirical studies carried out at the aggregate level
may suffer from aggregation bias in the estimated parameters of the aggregate equation. This issue
is considered in a series of papers (Pesaran et al. (1989); Lee et al. (1990a, b); Lee and Pesaran
(1993); Lee (1997)), where general tests of aggregation bias are developed and employed. An
aggregate model may lead to biased estimates because valuable pieces of information about the
working of the trade balance simply cannot be characterized within an aggregate model. These
involve the change in composition of traded goods over time, heterogeneities among bilateral trade
balances in terms of responsiveness to devaluation, and differences in dynamics of bilateral trade
balance equations. For these reasons, the aggregate parameters will obscure the structural
relationships and it is unlikely that the aggregate model adequately represents the disaggregate
model.
We now turn our attention to comparing the estimates obtained from the disaggregate
model with its aggregate counterpart. For a simple trade balance function, considered to be a
function of only real exchange rate, say, the disaggregate model for m trade partners can be
written as follows:
,:iiiid
uerbH mi ,.....,2,1 (6)
where i
b is the 1nx vector of observations on the dependent variable, trade balance with the
thi trading partner, i
er is the 1nx matrix of observations on the regressor, bilateral real exchange
rate for the thi trading partner, i is the 11x vector of coefficient, real exchange rate elasticity,
associated with i
er , and iu is the 1nx vector of disturbances for the thi trading partner.
The corresponding aggregate equation is given by,
,b :aaaaa
verH (7)
where
m
i
ia
m
i
iaerbb
11
er ,
Page 29
25
and a
is the 11x aggregate real exchange rate elasticity, and a
v is the 1nx disturbance vector.
The problem of ‘aggregation bias’ is then defined as the deviations of macro parameters from the
averages of the corresponding micro parameters ( mm
i
id /1
). More specifically, a
measure of aggregation bias is defined as dab . A test of aggregation bias then involves
testing the null hypothesis of no aggregation bias ( 0:0 H ) under the assumption that the
disaggregate model holds. The potential for aggregation bias in estimated aggregate relationships
is addressed in Lee, Pesaran and Pierse (1990a,b) and they suggest two statistics, namely the *
1q and *
2q tests, which allow us to test whether the parameters obtained from an aggregate model
differ significantly from the average of those of the corresponding disaggregate models. However,
in practice, it is often the case that we are interested in long-run elasticities and we use a dynamic
model in our estimations. This can be shown in the context of the following simple trade balance
function with lag dependent variable,
ititiitiiituberb
1210 (8)
In this function, we are interested in testing statistically the bias in long-run real exchange rate
elasticity, which is equal to )1/()( 21 iiig . However, the average long-run real
exchange rate elasticity estimated from the disaggregate model can be written in alternative non-
linear and linear forms as shown below. When the average of long-run elasticity is defined in non-
linear form, an average of individual elasticities and the corresponding null hypothesis will follow
as,
m
i i
i
erm 1 2
11
1
1
0
1
1:
1 2
11
0
m
i i
i cm
H
When the average of long-run elasticity is based on average parameters (linear form), the average
elasticity and the corresponding null hypothesis will be,
m
i
i
m
i
i
rer
m
m
1
2
1
1
2
)/1(1
)/1(
011
:1
2
1
1
2
0
cm
cm
Hm
i
i
m
i
i
In a formal setting, the null hypothesis of no aggregation bias when the parameter of interest is
some function of estimated parameters (linear or non-linear) will be,
m
i
ig gm
bg1
)(1
)(
A test of aggregation bias then involves testing the hypothesis 0:0 gH . In testing the
hypothesis, two different aggregation bias statistics, *
1q and *
2q tests, are defined depending on
whether aggregate exchange rate elasticity )(bg is known a priori or not. If the aggregate model
is estimated with certainty (given a priori) and only the disaggregate model is estimated with
uncertainty, then the *
1q statistics will be used in testing the aggregation bias and defined as,
Page 30
26
m
i
n
m
i
imm
q1
1
1
*
1 )ˆ(1
)(ˆ)ˆ(1
)( iβgbgβgbg (9)
where
m
ji
jjiin Covm1,
2 ˆ)ˆ,ˆ(ˆˆ GββGΩ . On the null hypothesis of 0gη ,
*
1q is distributed
as )1(2 .
If it is assumed that both aggregate and disaggregate parameters are obtained from
estimation, the *
2q statistics is employed in the test of 0H based on the OLS estimate of ,
namely
gngnq ˆˆˆ 11*
2
(10)
where
m
ji
jiijnn
1,
1 ˆ PPΦ and '
iˆmˆ X)X(XGX)X(XGP
1
i
'
ii
1'
a
1
a
'
aai
in which )bG(Ga
ˆˆ and )β(Gii
ˆˆ . On the null hypothesis of 0ˆ , *
2q is distributed as
)1(2 .
In our empirical analysis, we employed both *
1q and *
2q tests because we want to test for
possible biases in real exchange rate elasticity. In this context, the use of aggregation bias tests is
very useful to characterize and summarize the overall properties of the model. If the test results
show evidence of aggregation bias, it is clear that important data are being lost or distorted
through the process of aggregation.
Data
Before undertaking the econometric analysis, this section introduces the data employed in the
applied work and provides a preliminary statistical analysis of it. Quarterly time series data are
used, and the sample period is from 1987: Q1 to 2005: Q4. All data has been taken from the
International Financial Statistics (IFS) and the Central Bank of the Republic of Turkey (CBRT)
electronic data delivery system. The series are seasonally adjusted using X-12 procedure.
Trade balance (bi) is defined as the Turkey’s trade balance with her trading partners. It is
defined as the ratio of the Turkish imports to exports with her trading partners. Logarithmic form
of trade balance is used. Domestic income (y) is defined as the logarithm of the Turkey’s real
GDP, foreign income (yi*) is defined as logarithm of the partner countries real GDP, eri is the
logarithm of the bilateral real exchange rates. Those countries which accounts for two percent or
more of total trade of Turkey are selected as trading partners of Turkey, namely Austria, France,
Germany, Iran, Italy, Japan, South Korea, Netherlands, Spain, Sweden, UK, and USA. The rest of
the trading partners are taken together and named as OTHERS. Hence, total number of trading
partners employed in the empirical analysis is equal to thirteen countries.
The data employed in estimation of the aggregate model is defined as follows: world
income (ya*) is defined as logarithm of Industrial production index, aggregate trade balance (ba) is
the natural logarithm of ratio of total imports over total exports and era is the natural logarithm of
the reel effective exchange rate.
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27
Unit root tests
As a first stage in the analysis of the data, it is useful to obtain a broad overview of the properties
of the output series involved. In particular, it is important to establish the level of integration of the
series before carrying out a regression analysis because non-stationary regressors invalidate most
of the standard empirical results. Such an analysis was undertaken for each of the variables of
interest considered at levels using the Augmented Dickey-Fuller (ADF) test. In each case, ADF
statistics were calculated for the bi , y, yi*, eri series and for the ba, ya* and era series in the
‘levels’ case. The selection of the number of lags is carried out using the Akaike Information
Criteria (AIC). The results of the ADF tests computed over the sample period for the levels and
first differences of variables. The 95% critical value for the ADF test statistic including an
intercept in the underlying Dickey-Fuller regression is -2.903 for 76 observations. The results of
ADF test show that there was evidence that the variables considered are I(1) in almost all cases
except trade balance variables, in where most of them are I(0).
It is well known that ADF tests have low power meaning that ADF tests do not reject the
null hypothesis of the series being I(1) often enough (Maddala and Kim, 2000). For this reason,
and because we have a panel of data here, in this work, we also applied the standardized "t bar"
test that proposed by Im, Pesaran and Shin (2003) for the unit root tests. The "t- bar" test increases
the power by exploiting the panel structure of the data, and is based on the average value of the
Augmented Dickey-Fuller statistics estimated for each sector. Using the AIC criterion to
determine the order of ADF regression ip , the test statistics on the data using standardized t-bar
formula were 0.781 for eri series and 0.931 for yi* series, which do not rejects the unit root null at
standard significance levels. For bi series, the null hypothesis of unit root is rejected with t-bar
value was -3.896 at standard significance levels. Taken these test results together, it seems
reasonable to proceed an empirical analysis of trade balance-devaluation relationship under the
assumption that eri and yi* series are I(1) and trade balance variables, bi, are I(0).
When the dependent variable is integrated of I(0) and independent variables are integrated of
I(1), it is still possible for these variables to be cointegrated if the linear combination of
independent variable have I(0) process. In this case error term will be stationary, since both the
dependent variable and the linear combination of independent variables are stationary (Charemza
and Deadman, 1997: 126-127). Considering the discussion given above, the stationarity of error
terms is tested for detecting the existence of long-run relationship.
EMPIRICAL RESULTS
The primary concern of this section is to identify the impact of devaluation on trade
balance based on the results obtained from estimating the bilateral and corresponding aggregate
trade balance equations. We then look at the results from the ‘aggregation bias’ tests. The trade
balance equations are estimated for thirteen bilateral trade balances in Turkey, using quarterly data
over the period 1987Q1-2005Q4 and employing the estimation and specification search procedure
described above. The rest of the section provides the results obtained from the disaggregate model,
the corresponding aggregate model and the aggregation bias tests.
Bilateral trade balance
In this sub-section, we will present and comment on the results obtained in estimating model (5)
for each of the thirteen bilateral trade partners of Turkey in turn. The results obtained in the
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underlying trade balance equations are presented in Table 1(a). The diagnostic statistics associated
with the trade balance equations are given in Table 1(b). These include the adjusted 2R , the
estimated standard error of the equation, , the value of maximum of log-likelihood (LLF), and
statistics for the tests of residual serial correlation2
.cs , functional form 2
ff , normality 2
n ,
heteroskedasticity 2
H and stationarity.
Table 1(a). The effect of devaluation on Turkish bilateral trade balance, 1987q1-2005q4.
i0 i1 i3 i4 i5 i6 i7
Long-run
er
elasticity
AUST 18.188 -0.387 2.043 -1.391 -0.549 0.370 -0.873 (1.02) (-0.78) (3.05) (-1.96) (-2.45) (3.372)
FR 12.635 -1.604 1.173 -0.319 0.589 -0.779 (0.80) (-4.13) (4.35) (-1.88) (7.64)
GER -14.097 -0.329 0.777 -0.086 0.513 -0.177 (-1.49) (-1.88) (3.51) (-0.98) (5.44)
IR -27.814 -0.075 1.287 -0.272 0.493 -0.538 (-0.55) (-0.17) (0.76) (-1.54) (4.77)
IT 6.218 -0.682 0.285 -1.002 0.746 0.886 -2.245 (0.48) (-1.65) (0.90) (-3.38) (2.37) (14.22)
JAP -84.970 0.884 1.593 -0.063 0.545 -0.139 (-2.06) (0.88) (2.54) (-0.31) (6.09)
KOR -13.895 0.359 0.993 -0.723 0.541 -1.579 (-0.31) (0.76) (0.70) (-1.31) (5.53)
NET -3.296 -0.510 0.421 -0.110 0.764 -0.468 (-0.24) (-2.52) (1.87) (-0.73) (10.51)
SPA 54.709 -1.331 2.391 -1.563 -0.827 0.628 -2.229 (2.76) (-4.35) (4.09) (-2.54) (-3.10) (7.73)
SWE -5.166 -1.163 0.908 -0.190 0.814 -1.026 (-0.19) (-2.39) (2.45) (-0.64) (14.03)
UK -16.915 -1.166 0.900 -0.082 0.673 -0.252 (-0.98) (-4.62) (3.50) (-0.50) (9.50)
USA 50.007 -2.676 1.235 -0.267 0.395 -0.442 (2.92) (-6.22) (3.57) (-1.70) (4.47)
OTHERS 24.783 0.039 0.649 -0.781 -0.325 0.648 -0.925 (1.56) (0.15) (1.89) (-2.26) (-2.03) (7.25)
Notes: Coefficients refer to the following estimated regression
ititiitiitiitiitiitiitiiit bereryyyyb 17165143
*
12
*
10. Values in
parentheses are t-statistics. Variable definitions are provided in the text. *, ** indicate significant
at 5% and 10% level respectively.
The estimated equations presented in Table 1(a) have generally acceptable diagnostics
and explain a considerable amount of variation in bilateral trade balances ranging from 0.46 for
Austria to 0.87 for USA. The diagnostic statistics given in Table 1(b) are generally acceptable,
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29
although normality is rejected in only the case of Iran. The last columns in these tables present the
results from the unit root tests applied to the error terms. The results indicate that there is no sign
of a unit root in the errors of the regressions.
Table 1(b). Summary and diagnostic statistics
2R LLF )1(2
sc )1(2
ff )2(2
N )1(2
H ADF
AUST 0.456 0.096 71.905 0.194 1.010 0.418
2.708 -2.763(2)*
FR 0.711 0.064 102.266 1.087 0.164 0.303 0.061 -8.079(0)*
GER 0.706 0.046 126.590 3.250 0.083 0.233 1.474 -5.112(3)*
IR 0.524 0.373 -29.936 1.036 3.957 1260* 0.003 -5.605(3)*
IT 0.823 0.068 97.945 0.080 0.016 1.252 1.055 -7.946(0)*
JAP 0.766 0.129 49.583 0.118
0.963 1.032 6.014 -8.921(0)*
KOR 0.597 0.302 -14.186 0.446 0.317 6.610 6.718 -5.986(3)*
NET 0.674 0.063 102.748 0.0003 0.645 0.385 0.746 -5.657(1)*
SPA 0.754 0.084 82.264 0.023 0.102 18.599* 0.139 -8.588(0)*
SWE 0.813 0.103 66.558 0.473 0.188 1.323 1.968 -4.427(3)*
UK 0.837 0.056 112.368 0.382 0.179 0.792 0.263 -7.667(0)*
USA
0.873 0.068 97.047 1.838 0.524 3.286 2.635 -9.156(0)*
REST 0.647 0.048 123.202 1.734 0.0005 2.993 0.0002 -7.612(0)*
Notes: ADF is the Augmented Dickey-Fuller test statistics carried out on the regression residuals.
The number given in parenthesis next to ADF test statistics are the number of augmentation in the
underlying Dickey-Fuller regression. (*) indicates that the null hypothesis is rejected at
conventional 5% significance level.
Taken together, these tables shed light on some important features of the process of trade
balance determination and the role of devaluation in this process. Firstly, there are considerable
differences among the bilateral trade balances of the economy in terms of their long-run responses
to the change in real exchange rates which range from -0.139 for Japan to -2.245 for Italy. The last
column in Table 1(a) provides the long-run exchange rate elasticity of trade balances. The
inspection of this table shows that the long-run impact of devaluation on the trade balance differs
significantly among bilateral trade balance equations. It seems that devaluation has relatively more
impact on the bilateral trade balances with those countries that are relatively similar level of
development with Turkey, such as Italy, Spain, South Korea than the trade balances with more
advanced countries, such as UK, USA, Germany (the long-run effect of devaluation on trade
balances with these countries are -0.252, -0.442, and -0.177 respectively).
Secondly, it is clear from the results tables that there are important dynamic adjustments
influencing bilateral trade balances. The lag dependent variables have statistically significant
coefficients in all of bilateral trade balance equations. In particular, relatively higher coefficients
on the lagged dependent variables (0.89 for Italy and 0.81 for Sweden) have important
implications for the impact of devaluation on trade balances. They imply that adjustment takes
time and only a small part of the long-run effects of devaluation is observed in the short-term. In
other words, the positive effects of devaluation can only be fully realised over time. In addition, it
can be argued that the differences in the extents of adjustment coefficients across bilateral trade
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30
balance equations show the importance of evaluation of effects of devaluation at disaggregate
level.
Thirdly, the size of the economic activity in trading partner countries seems to be closely
related to bilateral trade balances. The foreign income variable, y*, was negative as expected
(except in cases of Japan and Korea) and statistically significant in most of the cases. The results
indicate that the impact of change in income of trading partners on trade balance differ
significantly ranging from -0.075 for Iran to -2.676 for USA. This implies that Turkish exports to
France, Spain, Sweden, UK and USA is very sensitive to the economic activities in these
countries.
Fourthly, the results indicate that increase in domestic economic activity has a positive
and statistically significant impact on bilateral trade balance in most of the cases. While the short-
run coefficient on domestic income variable is higher than one only for France, Japan, Iran and
USA, it is less than one for the rest of the trading partners. Considering the sizes of coefficients on
eri variable, these findings imply that for the first group of countries, domestic economic activity
seems to be more important in determining the bilateral trade balances than the devaluation.
In summary, the estimation results clearly indicate that the estimates of bilateral trade
balance equations differ significantly from each other. While in some cases, devaluation plays an
important role in determining trade balances, in others, domestic income or foreign income seem
to be the main determinant of bilateral trade balances implying the presence of different trading
patters at the bilateral level. As mentioned earlier, the observed heterogeneities among estimates
and differences in dynamics in bilateral trade balance equations may lead to biases at the
aggregate level. In the following section, we first present the results from aggregate trade balance
model and then test statistically for the aggregation bias.
Aggregate trade balance equation
Having found that the long-run real exchange rate elasticities differ significantly among trading
partners of the economy and it plays an important role in determination of the level of trade
balance, we now turn our attention to the application of the tests of aggregation bias, which
compare the estimates of the disaggregated model with the corresponding estimates of the
aggregate trade balance equation for real exchange rate elasticity. To consider these issues, we
estimated the aggregate trade balance equation (5) and the results for the equation (5) are
presented in Table 2. It is worth noting that variables used in the estimation of the aggregate
equations are obtained by aggregating over the bilateral data employed in the disaggregated
analysis above.
Table 2. Aggregate trade balance equation, 1987q1-2005q4.
Dependent
Variable i0 i1 i2 i3 i4 i5 i6 i7
ba 2.072 -0.246 - 0.775 -0.527 -0.470 0.419 0.777 (t-values) (0.18) (-1.18) (3.00) (-1.88) (-2.97) (2.50) (9.50)
Summary and Diagnostic Statistics
2R 0.732 )1(2
sc 0.749
0.453
)1(2
ff 2.106
LLF -43.444 )2(2
N 1.928
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)0(ADF -8.697* )1(2
H 2.532
Note: see the notes in Table 1(a) and Table 1(b)
The results in Table 2 demonstrate that, while it is possible to obtain an aggregate
equation that is able to explain a very high proportion of the variability in aggregate trade balance,
it is not easy to accommodate the heterogeneities in coefficients and differences in dynamics that
exist in disaggregate models into the aggregate model. The diagnostic statistics related to these
results are acceptable and show that the aggregate equations are well specified. Since the
independent variables in the aggregate equation are I(1), we applied the ADF tests to the residuals
in the aggregate equation. The ADF test result, provided in Table 2, indicates that there are no unit
roots in the error terms.
The coefficient of real exchange rate has a correct sign and statistically significant. It is
equal to -0.231 which is smaller than the average of the disaggregate model. It is also worth noting
that the long-run coefficient of the domestic income and foreign income variables are almost equal
in size with -1.09 and 1.11 respectively. Taking these findings together it seems that domestic and
foreign economic activities seem to be more important in determining trade balance than
devaluation at the aggregate level in Turkey. It is also evident that the coefficient of the lag
dependent variable is relatively higher in the aggregate model with 0.776 than the average of
disaggregate model (0.604) implying that trade balance adjustments take a long time following a
devaluation.
Aggregation bias tests
In this section, we provide formal test results obtained in applying the aggregation bias tests, to
investigate whether the observed differences between aggregate and disaggregate long-run real
exchange rate elasticities are statistically significant.
The aggregation bias test results are presented in Table 3. At the risk of repetition, it
should be reiterated that we considered linear and non-linear cases corresponding to linear and
non-linear formulation of restrictions. Furthermore, each case involves two tests, one of which is
called the *
1q test and the other one the *
2q test. The first test of aggregation bias, the *
1q test,
contrasts average values of the disaggregate estimates with a 'consensus' view of the real exchange
rate elasticity of aggregate trade balance. In our study, we used the coefficient of real exchange
rate obtained from the aggregate model as a 'consensus' view elasticity, because the empirical
literature provides ambiguous results on the impact of devaluation on trade balance. The second
test, *
2q , compares the estimated elasticity based on the disaggregate model with the one obtained
from the aggregate model. Table 3 presents these elasticities. The test results that were given in
Table 3 show that there is a clear evidence to reject the null hypothesis of no aggregation bias in
both linear cases and in one of the non-linear case which is *
1q test. Failure to reject the null
hypothesis for the non-linear case (*
2q test) might be due to the imprecision of the estimated
coefficients in some of the disaggregate trade balance models.
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Table 3. The aggregation bias test results
Non-linear Case Linear Case
Aggregation bias
test
Q1-CHI-SQ(1) Q2-CHI-SQ(1) Q1-CHI-SQ(1) Q2-CHI-SQ(1)
4.49440** 1.17270 6.73213*
3.12174***
Average Coef. -0.898 (0.315) -0.453 (0.086)
Aggregate Coef. -0.231 (0.107)
Notes: Critical values of CHI-SQ(1) test are (2.706),(3.841) and (6.635) for 10%, 5% and 1%
significance level respectively. *,**,*** indicates the rejection of the null at 1%, 5% and 10%
significance level respectively. The number in parenthesis indicates standard errors of coefficients.
Taken together, the aggregation bias test results given in Table 3 clearly provide strong
evidence that empirical work on the aggregate level is subject to bias. The ‘aggregation bias’ test
results provided above have also important implications for the analysis of the devaluation. It
seems that the studies carried out at the aggregate level underestimate the impact of devaluation on
trade balance in Turkey. In this sense, the use of aggregate data for policy purposes may be
misleadingly in predicting the full impact of devaluation. It is also true that the knowledge on
bilateral responses to devaluation will help to exploit fully the potential positive effects of
devaluation and to achieve macroeconomic stability.
CONCLUSION
In this study, the effects of devaluation on trade balance of Turkey have been investigated
econometrically using the relevant statistical methods. Based on the information provided in the
literature, we identified the three important points that needs to be addressed in any study that
attempts to investigate fully the impact of devaluation on trade balance of Turkey; namely,
compositional change in traded goods, and the heterogeneities and differences in dynamics among
bilateral trade balances of the economy.
The brief overview of the empirical literature on trade balance-devaluation relationship
revealed that the most of the studies on the subject were carried out at aggregate level and
provided ambiguous results on the impact of devaluation on trade balance. We argued that because
the studies in the literature were carried out at the aggregate level they cannot accommodate the
information mentioned above satisfactorily and they may lead to biased results. In order to
accommodate this information into an empirical analysis, we set out our empirical analysis at the
disaggregate level.
The findings of the study indicated that there are significant differences in estimates and
adjustment coefficients among trading partner countries implying that it takes time to benefit from
devaluation and bilateral responsiveness of devaluation need to be taken into account in the
formulation of economic policies. These results together imply that important aspects of the trade
balance-devaluation relationship can only be accommodated within a disaggregate model. This is
confirmed in the statistical comparison of the estimated disaggregate model with its aggregate
counterpart, which indicates that trade balance is best modeled through a disaggregate model. The
aggregation bias test results also show that the aggregate model underestimates the effect of
devaluation in Turkey and may be misleading. Given the significant effects of devaluation on
trade balance determination, these results certainly indicate that the work undertaken at the
bilateral level is necessary for better policy formulation and for evaluation of the impact of
devaluation in Turkey.
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Pennsylvania Economic Review
Volume 18, Number 2, Fall 2011
The following article won the Best Undergraduate Student Paper Award at the 2011
Pennsylvania Economic Association Conference.
ACCESS TO FORMAL CREDIT AND RURAL ECONOMIC DEVELOPMENT: AN
EXAMINATION OF PREVAILING ASSUMPTIONS IN THE LAND RENTAL MARKET
OF RURAL INDIA
Mitchell W. DeRubis and Christian N. Minich*
Indiana University of Pennsylvania
(We would like to acknowledge Dr. James Jozefowicz and Jeff Moon for their help in clarifying
and strengthening this paper. Also, thanks to the International Crops Research Institute for the
Semi-Arid Tropic (ICRISAT) for giving us their Village Level Study data for 2001-2004
ABSTRACT
The effect of access to formal credit on the land rental market, a key input suggested by prior
studies as representative of production decisions overall, is analyzed econometrically. Using
pooled cross-sectional data from six Indian villages in semi-arid provinces of India over the years
2001-2004, a regression equation is estimated using logit. The dependent variable is the decision
to participate in the land rental market. Key types of independent variables are the log of formal
credit, the log of various kinds of informal credit, district, village, and individual characteristics.
Our results indicate that formal credit exerts neither a significant nor a substantial effect on the
decision to participate in the land rental market. The amount of land owned, the amount of rainfall
the year prior, and whether a farm was designated small all exert a negative, significant influence
on dependent variable; whether a farm is designated large and the value of draft animals exert a
positive, significant effect on the dependent variable.
INTRODUCTION
Background
In rural India, land is a scarce resource (Kochar, 1997b). This is particularly true in the
semi-arid climate of the villages in the sample of this paper. In the Indian economy, land is rented
in or rented out depending on each farmer's supply of other inputs, such as draft animals or farm
equipment. The land rental market is known to function well, and it is used to equilibrate
nonmarketable or imperfectly marketable goods, such as draft animals, farm equipment, or family
labor (Skoufias, 1995). Thus, the land rental market in India is an essential component in
examining development in rural Indian economies.
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As Kochar (1997b) notes, there are multiple reasons to cast doubt on the assumption that
lack of access to formal credit constrains agricultural production decisions in the case of
economies in which land is scarce. There are low levels of fixed capital and infrastructural
development, while fragmented landholdings keep the working capital required to operate a farm
low (Kochar, 1997b). Also, there could be substitutes for formal credit for rural farmers, such as
rental markets (Kochar, 1997b).
If access to formal credit does influence the production decisions of rural farmers, then
the farmers should equilibrate their access to formal credit by using the land rental market
(Kochar, 1997b). In other words, if access to formal credit is significant in the production
decisions of Indian farmers, then there should be a correlation between access to formal credit and
those farmers' use of the land rental market. In this way, analyzing how the land rental market
responds to formal loan amounts should provide a good indication of the impact of access to
formal credit on the land rental market. According to Kochar (1997b), little empirical research has
been conducted on this question, although the assumption has been widely held (see Jaynes
(1984); Eswaran and Kotwal (1989)). This paper is a response to that call for further empirical
investigation.
As Binswanger and Khandker (1992) point out, the benefits from a subsidized
agricultural formal credit program have affected primarily the farmers who currently receive these
subsidized loans. They note that if many of the farmers who receive these loans have larger
resource endowments, then this differential access will result in a selection bias that will
artificially make the program appear more beneficial than it actually is. This selection bias is
crucial, because one of Kochar’s (1997b) primary conclusions was that farmers’ production
decisions are more due to resource endowments, such as the amount of irrigated land, rather than
due to access to formal credit. Farmers who lack access to formal credit but borrow from informal
sources do have a higher rate of participation in the land rental market; however, the impact of the
informal credit sector appears to be less significant than the effect of owning other inputs, such as
irrigated land (Kochar, 1997b).
Kochar (1997b) has already challenged this assumption with an empirical study using the
All-India Debt and Investment Survey of 1981-1982 (1997b), and found that lack of access to
formal credit did not, in fact, exert a significant influence on the land rental market. However, her
study was limited, and there has been a lack of continued literature that surveys more modern data.
The goal of this paper is to provide an updated empirical study using data from 2001-2003,
implementing Kochar's (1997b) methodology in order to ascertain whether her conclusions hold
true across time, as well as to note the impact of economic development in India over the last few
decades.
One aspect of Kochar’s (1997b) original research that this paper should shed additional
light on is that the sample size is in a more arid locale than her sample in Uttar Pradesh. The
sample in this paper comes from six semi-arid villages across multiple provinces. If irrigated land
increases the likelihood to participate in the land rental market, then lack of irrigated land should
have a greater impact on the decision to participate. Kochar (1997b) also notes that her sample did
not include enough specific data on the type of informal credit being lent, whereas the data in this
paper contains more information on the nature of informal credit, enabling a more thorough
examination of the role of informal credit in the land rental market, in addition to the effect of
formal credit on the land rental market. Also, this study will use a pooled cross-sectional dataset,
rather than the cross-sectional analysis conducted by Kochar (1997b). Finally, the inclusion of
variables not present in Kochar’s study but suggested by other literature (e.g. Skoufias (1995))
should help to better examine the relationship between production decisions and their
determinants. These variables include a measure of the education of the members of a household,
dummy variables for the size of the farm, and an improved measure of draft animals.
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This paper will explore the effects of formal credit on the production decisions of Indian
farmers, which could have key policy implications for developing countries operating under the
assumption that better formal credit access is the highest priority for economic development. The
analysis in this study will corroborate or refute Kochar's (1997b) results, shedding further light on
prevailing assumptions on the role of formal credit in developing economies.
The second section will review the literature on the role of credit in developing
economies. The data and variables for this study are discussed in the third section. Section four
will explicate the methodology and the functional form selected for this analysis. The results will
be discussed in the fifth section. The conclusions and potential extensions of this research will be
discussed in the final section.
LITERATURE REVIEW
Kochar (1997b) examines the assumption that lack of access to formal, cheap credit is a
determining factor in rural farm rental markets in India. Other regressors considered are informal
credit, assets, land, the number of males and females in a household, irrigated land, how many
plots the land is divided into, and the ownership of draft animals. Kochar takes her data from the
All India Debt and Investment Survey from 1981-1982, and she uses the province of Uttar Pradesh.
Similar to the present study, Kochar uses a probit regression to analyze her data, and she divided
her full sample into borrowers and non-borrowers in order to see whether each group had different
determinants of their activity in the land rental market. Kochar concludes that her analysis of the
data suggests that credit constraints, whether formal or informal, do not determine the level of
input use in farming; rather, other inputs, such as the amount of irrigated land, were far more
significant. This study demonstrates the impact of formal credit on land tenancy, which in turn
impacts the overall income of these small rural farmers.
Krause et al. (1990) examine the determining factors of adoption of technology for small
farmers. Their data came from the National Agronomic Research Institute of Niger. They measure
the effect of interest charged for credit, the amount of equity a farmer has, credit repayment
insurance based on the weather, and the sharing of risk by suppliers and laborers on the adoption
of new technology. They find that risk aversion was a key variable in the decision to adopt
technology. The researchers conclude that low-interest credit is not an effective method for
encouraging technology adoption, since it does not address the risk-averse nature of farmers in
Niger. The method they found to be the most effective was having fertilizer-suppliers share in the
risk of the farmers.
Jamajou and Baker (1980) examine the effects of cheap credit on economic development
in Cameroon. They study the impact of cheap credit programs on the farmers' income, using the
credit limit and the rate of interest as the two regressors. Using a model to extrapolate the data
they received from small farmers in Cameroon, they conclude that the credit limit and flexibility
of the use of that credit is more effective for improving the small farmers’ income than a low
interest rate. The data is survey data, and the model is a projection based a series of assumptions
about the way in which small farmers will behave.
Binswanger (1980) conducts an experiment using game theory that subjected about 300
households in areas of India that are characterized as high in climatic volatility. He measures the
impact of income level, age, schooling, assets, net transfers, and “luck” (or experience) on a
farmer's risk aversion. The study concludes that at low levels of payoff, the risk aversion was
widely distributed; however, at higher levels of payoff, risk aversion is concentrated at
intermediate levels. Interestingly, risk aversion is not correlated with the level of wealth. This
article establishes a profile of the risk-averse nature of small rural farmers, particularly in India.
This corroborates the picture of farmers as risk-averse in the other articles (e.g. Krause et al.
(1990), Skoufias (1995)).
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Skoufias (1995) and Jodha (1981) provide further support for the use of the land rental
market as a proper input to use to measure the impact of formal credit on the production decisions
of rural farmers. Skoufias suggests that rural Indian farmers use the land rental market to
equilibrate nonmarketable and imperfectly-marketable inputs, such as draft power or family size.
Skoufias and Jodha also establish that the land rental market is a market that functions well.
Skoufias’ sample comes from the same villages as the sample for this paper, and a number of the
variables used in this study, such as education, percent of irrigated land, and the numbers of adult
males and females, come from his study. He also provides a precedent for using pooled cross
sectional data.
Binswanger and Khandker (1992) provide a 30-year panel analysis across 85 districts of
India, examining the benefit-cost ratio of India’s aggressive formal credit subsidy program. Their
specific attention is to the agricultural credit targets and their effect on the economic situation of
the rural farmers. They note that the program impacted farmers’ production decisions positively,
in general, but that wages did not increase. They found a selection bias of formal credit to favor
the farmers with larger resource endowments, and used variables for the investment of the
districts, agricultural variables, and banking variables.
Taylor and Adelman (2003) construct a model for agricultural households. While theirs is
not an empirical model, it is significant because it explores the nature of the household-farm
model, in which the demand of a household can also be supplied by itself, if the costs of obtaining
the good elsewhere are prohibitive. Household-farms are both producers and consumers in the
model. As the authors note, an agricultural-household perspective is implicit in models describing
or including access to credit, including Kochar (1997a and 1997b), and this paper. The authors
note that households in these rural economies face “missing markets” when the cost of
participating in the market are higher than the cost to produce the service itself. This is an
important insight for our present analysis, because it implies that our methodology of evaluating
the participation in the land rental market in terms of the costs or availability of the goods
(transaction costs) is a sound one.
Barslund and Tarp (2008) explore how the rural credit market operates in Vietnam by
constructing models of formal and informal credit demand as well as credit rationing. The authors
find, with the use of a probit regression, that land holdings, number of adults in a household, the
value of feed used, and the total value of the household’s assets are both statistically significant
and positively affect the probability of demanding formal loans. Age alone is deemed statistically
significant and negatively related to the probability of demanding formal loans. The value of feed,
poor credit history, and the number of dependents in a household is found to be statistically
significant and positively related to the probability of demanding informal loans. Age, educational
attainment level, and the value of total assets are found to be significant and negatively related to
the probability of demanding informal loans. Our paper explores very similar relationships and
uses many of the variables found to be significant by Barslund and Tarp (2008). The value of feed
variable was used as a proxy for livestock holdings and is improved upon in this paper by using
the value of livestock owned, which we believe will quantify both the number of livestock and
their respective conditions. It appears that the rural credit market in Vietnam operates in a very
similar fashion to the rural credit market in India.
Golait (2007) discusses trends in the agricultural credit market in India. This paper
describes the trends in formal lending and the market share of formal credit, trends in the market
share of co-op loans, and recent policy initiatives. It is important to note that while Golait (2007)
finds the market share of institutional lending to be generally increasing, to 66% in 1991 from 7%
in 1951, our sample shows informal lending to be much more prevalent than formal lending, in
line with the latest National Sample Survey Organization (NSSO) Survey as of 2007 (Golait
2007). Also of interest, Golait discusses the government’s recent finding that suicide among
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farmers is highly correlated with indebtedness. The paper concludes with recent policy changes in
rural agricultural lending, however, these changes did not take place during our sample years.
DATA
This study uses data provided by the International Crops Research Institute for the Semi-
Arid Tropics (ICRISAT), and the amount of rainfall was obtained from the data website India
Water Portal. Researchers at ICRISAT conducted a yearly study of six villages in the semi-arid
climate of India for the years 2001-2002, 2002-2003, 2003-2004, and 2004-2005. This is the
second generation of the Village Level Studies (VLS) conducted by ICRISAT on these villages.
ICRISAT conducted extensive surveys of households in six villages in rural India: Aurepalle,
Dokur, Shirapur, Kalman, Kanzara, and Kikheda. They took a representative sample of the
number of households in the villages at the beginning of 2000-2001, and then used the same
households throughout the study. These six villages are in three districts: Sholapur,
Mahaboobnagar, and Akola; these districts are in two states: Andhra Pradesh and Maharashtra.
The VLS collects agronomic, socioeconomic, and other data. The total number of cultivator
households surveyed per year is 334 in all six villages. This study uses data from the years 2000-
2001, 2002-2003, and 2003-2004, with 330 households which have all the variables of this study,
yielding 990 total observations. Due to missing values for some of the data, the full sample
analyzed in this study is 904.
This dataset is relevant for several reasons. It includes extensive documentation of a wide
range of sociological, economic, and agronomic data. Also, the survey data was collected by
individuals who know the local customs and speak the language of the village. The dataset
additionally categorizes informal credit by the identity of the lender—whether it is a moneylender,
a co-op, or friends and relatives. This allows for a more careful analysis of the impact of informal
credit than has been possible for other researchers (such as Kochar (1997b), in addition to the
impact of formal credit.
This study uses the dichotomous decision to participate in the land rental market as the
dependent variable (LEASED). The land leased out per household is not included, because
landlords often underreport their leasing-out of land in order to avoid taxes (Kochar, 1997b). Thus
leasing-in land is the dependent variable to measure the household's participation in the land rental
market.
Most of the variables in this paper have been either shown or predicted to have
significance in the literature (e.g. Skoufias (1995) and Kochar (1997b)). There are only a few
exceptions, which represent improvement on prior literature. First, we divided the informal loan
amount into three variables. The reason for this change is that the interest rates charged by co-ops
averaged less than 10% interest, friends and family loans were often lent at 0% interest, and the
money lenders often charged over 30%. There is a clear distinction there, and Kochar (1997b)
points out that further research could be conducted by examining more precisely the role of
informal credit on production decisions. The other deviation is including the education variable,
though it has been included in Binswanger (1980) and Skoufias (1995) for evaluating the risk-
averseness of farmers. We also added a net income variable, and a variable capturing farm size
variables in three groups.
Due to lack of data availability or integrity, we were unable to include a variable
capturing the infrastructural development of each village (or even the district). We were also
unable to obtain the number of land plots per household for a sufficiently large sample.
Variables that measure the impact of other inputs, such as draft power and family labor,
as well as the loan amounts, must be included. Additionally, some measure of the household's
assets at the start of each year is also necessary. In the literature, leasing decisions are assumed to
be an outcome of a given household's resources at the beginning of a crop year prior to the rental
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decision (Kochar, 1997b). These resources include the household's assets, measured as the values
of a household's farm equipment (EQUIP) and buildings owned by the household (BLDG). Family
labor is measured by the number of males (MALES) and females (FEMALES) over the age of 14.
Skoufias (1995) includes the family labor variables, household equipment and buildings.
The value of the draft animals within each household, measured in real 2001 rupees
(DRAFT), is included as part of the household’s resource endowment. This is an improvement
over Skoufias (1995), Kochar (1997b), and Barslund and Tarp (2008), who use the number of
draft animals or value of feed. Skoufias (1995) mentions that using the value of draft animals
would be preferable, but did not have access to that data. Another resource input is a household's
irrigated land holdings (IRR), measured as the percentage of the total land owned in acres, and the
household's land holdings, measured in acres (LAND). Kochar (1997b) uses the land irrigated in
acres, but this presents a collinearity issue with LAND. Skoufias (1995) uses the percentage of
irrigated land holdings. The formal loan amount (FORMAL) and informal loan amount are also
included as regressors. The informal loan is broken up by source: moneylenders (LENDERS),
friends and relatives (FRFAM), and co-ops (COOP). They are measured as the natural log of real
2001 rupees, as supported by Kochar (1997b), Barslund and Tarp (2008). Dummies are included
for each to measure any bias created by transforming values of “0” to “1” for the natural log to be
zero.
In order to capture the education within a household, which Binswanger (1980) linked as
a measure of the risk-averseness of farmers, we have included the individual in each household’s
maximum years of education (EDUC). We took two measures of this. First, we used a count
variable (EDUC), which took the number of years educated. We also took a dummy variable
(EDUCDUM) which has a value of “1” if there is an individual in the household who has any
education. The use of the dummy is supported by Skoufias (1995). We have also included
intercept dummy variables to capture the relative size of the farms, as defined within ICRISAT’s
measurement system, with one variable for farms designated “large” (LARGE), and farms
designated “small” (SMALL), with medium farms being the omitted condition.
RAIN1 and RAIN2 are the total amount of rainfall, in millimeters, in the province of the
village, in the year prior and two years prior, as supported by Kochar (1997a). WAGE is the
average wage rate in each village for that year, in real 2001 rupees, as used by Kochar (1997b).
INCOME is the net income of a household, in the natural log of 2001 rupees. POP is the total
number of households in each village in 2001.
Expected Signs
LAND is expected to have a negative sign, because as the household’s resource
endowment increases, the likelihood to participate in the market should decrease. The asset
variables (EQUIP, BLDG, and CONS) should have a negative impact, as increased assets will
lessen the need to engage in the leasing-in market. MALES and FEMALES are expected to be
positive factors, because family labor is assumed to be an imperfectly marketable good. DRAFT is
expected to be positive in sign, as an increase in the availability of draft animals should increase
the need to lease-in land. IRR is expected to have a positive sign. This is based on Skoufias
(1995). He reasons that the percent of irrigated land decreases the risk of cultivation. This should
increase the fruitfulness of self-cultivation, which would increase the returns from the land already
owned for less labor, ceteris paribus. The expected signs of the informal and formal loan variables
are ambiguous.
EDUC and EDUCDUM have ambiguous expected signs. Skoufias (1995) found
education to be negatively correlated with leasing-in land, but it was statistically insignificant.
LARGE has a negative expected sign, as the increased resource endowment of the household
should decrease the participation in the land rental market. SMALL should have a positive sign for
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the opposite reason. POP has a positive expected sign, as the larger the market, the more likely
individuals will be to participate in it. RAIN1 and RAIN2 both have negative expected signs, in
that a decrease in rainfall would indicate more risk for the farmers, who might respond by entering
into a fixed rental contract in order to hedge their risk. WAGE has a negative expected sign,
because as the wages for labor increases, the opportunity cost of working on the farm increases,
thus making renting land in less likely. INCOME has a negative expected sign, as increased
income should decrease the likelihood to participate in the land rental market. Many of the
expected signs come from Kochar’s (1997b) or Skoufias’ (1995) findings, in addition to economic
theory. The definitions of the variables, as well as their expected signs, are reported in Table 1.
Descriptive Statistics
The mean participation of land leased is 0.102 per household. The average amount of
formal credit borrowed is 7,244.56 rupees per household, the maximum being 400,000 and the
minimum being zero (pre-transformation for the logarithm). For money lenders, the maximum
was 10,875.29 rupees, and the minimum was zero. The mean amount of land owned is 5.76 acres,
with a maximum of 44 acres and a minimum of zero, indicating a wide variance. The average for
SMALL is 0.54, indicating that there are more small farmers than large (which has a mean of
0.14). The average number of years educated is 7. The mean population is 535 households, while
the minimum is 170 and the maximum is 649. This indicates that the average village population is
skewed toward the larger populations. Descriptive statistics for the dependent and all the
independent variables are in Table 2.
ECONOMETRIC MODEL
This study utilizes a pooled cross-sectional analysis to evaluate the effect of formal credit
and other factors on the likelihood of participation in the rental market. Skoufias (1995) analyzed a
similar pooled cross-sectional dataset, though he used Maximum Likelihood Estimation of the
amount of land leased, rather than the decision to participate. The initial equation was constructed
using variables that Kochar (1997b) and Skoufias (1995) used in their regression analyses. The
hypothesized logit model is:
(1)
where z= β1 + β2DRAFTi + β3MALESi + β4FEMALESi + β5IRRi +β6LANDi +β7LOGFORMALi +
β8LOGINCOMEi + β9LOGMONEYi + β10LOGFRFAMi + β11LOGCOOPi + β12LARGEi +
β13SMALLi + β14POPi + β15WAGEi + β16EQUIPi + β17BLDGi + β18RAIN1i +
β19RAIN2i + β20(EDUC or EDUCDUM) i + β212001dumi + β222002dumi (2)
RESULTS
In order to examine the relationship between formal credit and the decision to participate
in the land rental market, two sample groups were used. The first group is the full sample1, and the
second treats borrowers and non-borrowers as separate groups, under the assumption that the
determinants of those with access to formal credit may be different than those without access to
formal credit. This approach also allows for comparison with Kochar (1997b), who conducted this
form of analysis and found access to formal credit not to be significant in any of her regressions.
The results of logit regressions on the full sample are reported in Models 1-3 of Table 5. The
results are remarkably robust across the three models. The difference among the three is whether
the education variable used is the count variable EDUC or the dummy variable ECUCDUM, and
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whether or not SMALL is excluded. In all three models, LAND is negative and significant,
RAIN1 is negative and significant, and DRAFT is positive and significant, as expected. Other
variables had their expected signs but were insignificant: EQUIP and BLDG were robustly
negative, RAIN2 was negative, and IRR was positive. MALES and FEMALES were also always
positive, though insignificant.
In most of the regressions of this study, COOP has a negative sign, and it is almost
always significant. This result is complicated because the dummy measuring the transformation
necessary to convert the “0” values into a logarithm is also always negative and significant as
often as COOP is. The consistent significance of COOP probably has more to do with the large
number of 0’s that had to be transformed than with empirical significance.
LARGE (positive) and SMALL (negative) did not carry their expected signs, but were
significant at the 1% level. It is difficult to interpret these two variables, which are consistently
significant and robust across all the models tested in this paper. It is especially thorny because the
variable LAND is also consistently significant and negative. On the one hand, LAND would
suggest that, as the size of the farm increases, ceteris paribus, the likelihood to participate in the
land rental market decreases. On the other hand, LARGE suggests that larger farms, relative to the
medium-sized farms, are more likely to participate in the land rental market, and small farmers,
conversely, are less likely to participate.
It could be that LARGE and SMALL capture variation that is not simply limited to the
size of the farms. In other words, large farm households may enjoy advantages in markets that
small farmers do not enjoy. This sort of general selection bias of the land rental market for large
farm households over smaller farm households is suggested in the special case of selection bias in
the formal lending sector by Binswanger and Khandker (1992), but there has been little additional
evidence to suggest such a conclusion for the land rental market. Similar to the explanation offered
above, LARGE and SMALL could also capture comparative differences between larger and
smaller farm households relative to the medium size farm households, which the amount of owned
land does not reflect. A correlation matrix was run, and LARGE and SMALL were not collinear
with any other variable, including LAND. No one variable was substantially collinear with any
other variable.
LOGFORMAL is never significant in these models, and it always carries a positive
coefficient. This suggests that the access to formal credit does not significantly constrain a
household’s decision to participate in the land rental market. Kochar (1997b) found this variable to
be insignificant and positive in her probit of the full sample.
The education variable exhibits unusual behavior. When considered as a count variable, it
carries a negative sign and is not significant; however, when measured as a dummy variable, it
changes to a positive sign and is significant in one of the three models on Table 5, Model 3 (which
drops SMALL). When measured as a dummy (the same way Skoufias (1995) measures
education), the variable does not carry the negative sign Skoufias found. WAGE is only
significant in one model. This could be due to averaging male and female wage rates together.
Skoufias (1995) found that there is a difference between male and female wage rates’ impact of
the land rental market. The McFadden Pseudo R2 measures of the first two models are between
0.147 and .0153, with about 83% total correctly predicted. The model is superior to estimates
generated with just the constant term by over 2%.
Another way of evaluating whether or not access to formal credit influences the land
rental market is to treat those who did receive a loan and those who did not as distinct samples and
running separate regressions. Kochar (1997b) utilizes this technique, and it is one of the primary
reasons that she concludes that access to formal credit does not influence the land rental market.
Descriptive statistics for the borrowers and non-borrowers are reported in tables 3 and 4.
The results from these regressions are reported in Table 6. Models 1-3 use the sample of
formal credit borrowers, and Models 4-6 use the sample of non-borrowers. Similar to Kochar’s
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(1997b) findings, the sign on POP was negative for borrowers, but positive for non-borrowers.
Also, RAIN1, like Kochar’s (1997b) Monsoon80 variable, was negative in all cases, and
significant for non-borrowers, but not for borrowers. Assets for Kochar (1997b), and EQUIP and
BLDG for this study, were negative for borrowers and non-borrowers, though in neither study
were these variables significant. Like Kochar (1997b), MALES and FEMALES are positive across
borrowers and non-borrowers, though interestingly, MALES is significant at the 5% level for
borrowers, while FEMALES is only significant in one model at the 5% level for non-borrowers.
The COOP loan amount continued to be significant and negative for the non-borrowers, while it
was negative but insignificant for the borrowers.
LARGE and SMALL are robust across the models. LARGE is also significant in all but
one model for both borrowers and non-borrowers. In contrast, SMALL is only statistically
significant for non-borrowers, and then it is negative. The coefficients on SMALL are also much
larger in absolute value for the non-borrowers than for the borrowers, indicating that whether a
household is small relative to medium households is a more influential explanatory variable for
non-borrowers than for borrowers, all other things equal. Also, the coefficient on LARGE
decreases from borrowers to non-borrowers, indicating that larger farm households relative to
medium farm households who borrow are more likely to participate in the land rental market than
large farm households who do not borrow.
The number of draft animals also exhibited significant differences across borrowers and
non-borrowers. The signs on the value of draft animals in the borrowers equations are always
negative, although insignificant, but they are always positive and significant at the 1% level for
non-borrowers. The negative sign on DRAFT for borrowers disagrees with Kochar (1997b), who
found DRAFT to be positive. It also makes sense that DRAFT would be positive, since an
increase in the value of draft animals, which are an imperfectly marketable good, would increase
the likelihood to participate in the land rental market. Another key difference is in the education
variable. Both the dummy and the count variable are negative for borrowers, but positive for non-
borrowers. Also, the count variable is significant at the 10% level for borrowers but not the
dummy variable, and, conversely, the count variables are not significant for the non-borrowers but
the dummy variable is significant at the 5% level. This suggests that the level of education within
a household increases the likelihood to participate for non-borrowers, but decreases it for
borrowers, all things equal. The relevant differences between the borrowers and non-borrowers
that would explain this difference are unclear.
Crucially, LOGFORMAL is never significant in the borrower equations, at any level.
Also, variables such as LOGMONEY, which is significant at the 11% level, exhibit greater
influence. LOGMONEY has an approximate magnitude of 0.195 at the mean, while
LOGFORMAL has a magnitude of 0.05415 at the mean. MALES also has a greater magnitude
(and is significant), and IRR also has a greater magnitude, though it is insignificant in all the
models (in contrast with Kochar (1997b), who found similar magnitudes—0.276 at the mean
versus 0.225 at the mean in this study—and significance at the 5% level). The McFadden Pseudo
R2 measure increased for both borrowers and non-borrower equations (around 0.3 and 0.2,
respectively, depending on the regression), and the percent correctly predicted also increased (to
86.5% and 84.5%, respectively). The percentage increase over a prediction using just the constant
also increased, to around 4.5% and around 3.5% for borrowers and non-borrowers (depending on
the regression).
All of the above discussion must be taken with a caution. The small sample size used for
the estimations of borrowers could result in biased results. Additionally, this study was unable to
incorporate the number of plots per household, a variable which Kochar (1997b) found to be
significant, and a measure of infrastructural development. Finally, Kochar (1997b) and others note
that the credit markets (both formal and informal) are possibly endogenous. This study was unable
to correct for any potential endogeneity. Despite this potential endogeneity concern, there is doubt
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that it is a concern in this case. When Kochar (1997b) used a two-stage probit estimation to correct
for endogeneity of the formal loan amount within her sample, she found the signs and
significances to remain identical, except for the constant and wage variable. Also, the magnitudes
of the coefficients did not alter substantially from the probit to the two-stage probit. The primary
difference was the magnitude of the formal loan amount, which was overestimated in the initial
probit.
CONCLUSIONS
While there is still much room for analysis on this issue, this study has had a
corroborating effect on Kochar’s (1997b) analysis. The formal loan amount was never found to be
significant in any of the regressions, which is consistent with her findings. Second, it appears that
there are differences between borrowers and non-borrowers, differences that are inexplicable by
the formal loan amount itself. This supports Binswanger and Khandker’s (1992) finding that there
is a selection bias that occurs with respect to the formal credit sector, and that there is a bias
toward larger-endowed households. This overestimates the benefits of a low-interest subsidized
credit program, such as the one in which the government of India is aggressively engaged. This
paper is part of a growing body of literature, including Binswanger and Khandker (1992) and
Kochar (1997b), that supports the notion of selection bias in the formal credit sector, and that
differential access to a number of different inputs can be explained, not by access to formal credit,
but potentially by a selection bias favoring those with a larger resource endowment. This selection
bias is a compelling explanation behind the unexpected signs on the large and small farm dummy
variables, and it is perhaps some of that unobserved selection bias that is being reflected within
those two variables.
In summary, this study suggests that access to formal credit does not constrain
agricultural production decisions, insofar as the land rental market is a good production input.
There is also interesting potential evidence for selection bias toward households that are larger
relative to medium houses, and against households which are smaller than a medium house.
Further research should incorporate the number of plots per household, as well as a
measure of infrastructural development of each village. Also, the years of the study could be
increased, from three years to five, at least. A larger dataset would allow for more precise
estimation, and many of the variables, which were insignificant but theoretically relevant, could
become significant. As panel estimation techniques of logit regressions becomes more standard,
this dataset could be analyzed in that fashion. Finally, the endogeneity of the credit markets should
be addressed.
ENDNOTES
*Corresponding author: Department of Economics, Indiana University of Pennsylvania, Indiana,
PA 15705. Email: [email protected] .
1Following Studenmund (2011), we became concerned that only having 10% of 1’s and 90% 0’s
could present a potential bias to the coefficients. We took a subsample of the whole, using all of
the 1’s and 50% of the 0’s, leaving us with a sample of 500. This increased the participation rate in
our subsample to 20%, which was similar to Kochar’s (1997b) percent of 1’s. We were compelled
to use the full sample in order to be able to run regressions of borrowers and non-borrowers. Since
our sample only included 201 observations that used formal credit, we had to incorporate the
whole sample in order to be able to evaluate this. Our subsample coefficients are robust with those
in the full sample regressions, but note that the values of the coefficients are somewhat larger in
Page 49
45
the full sample. See Table 5 for a few examples. Further regressions are available from the authors
upon request.
Page 50
46
REFERENCES
Bell, C. (1990) Interactions between institutional and informal credit agencies in rural India.
World Bank Economic Review. 4(3): 297-327.
Binswanger, H.P. (1980) Attitudes toward risk: Experimental measurement in rural India.
American Journal of Agricultural Economics. 111(3): 395-407.
Binswanger, H.P. and Khandker, S.R. (1995) The Impact of formal finance on the rural
economy of India. Journal of Development Studies. 32(2): 234-262.
Golait, R. (2007) Current issues in agricultural credit in India: An assessment. Reserve Bank of
India Occupational Papers. 28(1)
Indiawaterportal.org
Kamajou, Francois and Baker, C.B. (1980) Reforming Cameroon’s government credit program:
Effects on liquidity management by small farm borrowers. American Journal of Agricultural
Economics. 62(4): 708-718.
Kochar, Anjini. (1997a) An empirical investigation of rationing constraints in rural credit
markets in India. Journal of Development Economics. 53: 339-371.
Kochar, Anjini. (1997b) Does lack of access to formal credit constrain agricultural production?
Evidence from the land tenancy market in rural India. American Journal of Agricultural
Economics. 79(3): 754-763.
Jodha, N.S. (1981) Agricultural tenancy: Fresh evidence from dryland areas in India.” Economic
and Political Weekly. December, 1981: A118-A128.
Krause, M.A., Deuson, R.R., Baker, T.G., Preckel, P.V., Lowenberg-DeBoer, K.C., Maliki,
R.K. (1990) Risk sharing versus low-cost credit systems for international development.
American Journal of Agricultural Economics. 72(4): 911-922.
Skoufias, E. (1995) Household resources, transaction costs and adjustment through land tenancy.
Land Economics. 37(8): 42-56.
Studenmund, A. (2011) Using econometrics: A practical guide. Boston, MA: Pearson Education
Inc. 6th
Ed.
Taylor, J., and Adelman, I. (2003) Agricultural household models: Genesis, evolution, and
extensions. Review of Economics of the Household. 1.1.
Village Level Studies. ICRISAT. Accessed Online.
Page 51
47
APPENDIX
Table 2: Descriptive Statistics, Full Sample
Variable Mean St. Dev Max Min
LEASED 0.102794411 0.30384182 1 0
FORMAL 7244.561876 27671.9715 400000 1
COOP 4258.199601 15174.9083 183000 1
FRFAM 1956.775449 10047.3749 200000 1
MONEY 10875.2984 24768.5349 360000 1
INCOME 44407.11341 49375.9278 416799.403 115
EQUIP 16611.41444 50844.9669 608526.108 0
BLDG 66366.07307 75625.5151 887052.8955 0
LAND 5.76501996 6.03798691 44.65 0
IRR (%) 0.373255 0.423989 1 0
IRR (acres) 2.105369261 3.619910077 33.15 0
LARGE 0.146706587 0.353989863 1 0
SMALL 0.54491018 0.49822767 1 0
POP 535.3203593 138.4703439 649 170
RAIN1 629.358007 109.5613781 846.326 449.221
RAIN2 663.4215569 132.4601462 996.938 449.221
WAGE 39.47716561 7.365419277 52.52631579 30.64150943
MALES 1.906906907 1.082907068 7 0
FEMALES 1.748748749 0.8862883 6 0
DRAFT 14186.90876 23375.51715 242059.7015 0
EDUC 7.18482906 5.304324194 18 0
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48
Table 4: Descriptive Statistics, Formal Borrowers Only
Variable Mean St. Dev Max Min
LEASED 0.099502 0.300083 1 0
FORMAL 36110.7 1000 52772.72 400000
COOP 2175.025 1 9888.029 100000
FRFAM 1831.184 1 6009.746 50000
MONEY 6998.547 1 25221.19 200000
INCOME 53694.04 1 59277.63 295431.6
EQUIP 26628.29 68875.64 578489.9 0
BLDG 73368.41 83694.35 691437.2 5000
LAND 7.142736 6.845727 44 0
IRR (%) 0.483232 0.441193 1 0
IRR (acres) 2.943632 3.461948 20.5 0
LARGE 0.154229 0.36207 1 0
SMALL 0.631841 0.483509 1 0
POP 477.9254 170.2703 649 170
RAIN1 597.3776 89.6332 846.326 449.221
RAIN2 682.0239 161.5571 996.938 449.221
WAGE 41.35359 6.626274 52.52632 30.64151
MALES 2.044776 1.050231 7 0
FEMALES 1.741294 0.918007 5 1
DRAFT 17715.46 29226.81 242059.7 0
EDUC 9.629442 4.230567 18 0
Table 5: Descriptive Statistics, Formal Non-Borrowers Only
Variable Mean St. Dev Max Min
LEASED 0.10362 0.304958 1 0
COOP 4780.944 1 16196.54 183000
FRFAM 1988.291 1 10829.55 200000
MONEY 11848.12 1 24573.7 360000
INCOME 41411.42 1 46199.78 416799.4
EQUIP 14097.82 44913.75 608526.1 0
BLDG 64608.93 73413.73 887052.9 0
LAND 5.419301 5.771015 44.65 0
IRR (%) 0.344842 0.415005 1 0
IRR (acres) 1.895019 3.630281 33.15 0
LARGE 0.144819 0.352138 1 0
SMALL 0.523096 0.499778 1 0
POP 549.7228 125.3322 649 170
RAIN1 637.383 112.6483 846.326 449.221
RAIN2 658.7535 123.7744 996.938 449.221
WAGE 39.0063 7.469676 52.52632 30.64151
MALES 1.87218 1.088875 7 0
FEMALES 1.750627 0.8787 6 0
DRAFT 13301.47 21592.51 219252.3 0
EDUC 6.533153 5.373318 18 0
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49
Table 3: Logit Results, Full Sample (1-3) and Subsample (4-6)
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
Variable
Coefficient
(St. Error)
Coefficient
(St. Error)
Coefficient
(Std.
Error)
Coefficient
(St. Error)
Coefficient
(St. Error)
Coefficient
(St. Error)
C 8.176188
b
(4.151907)
7.703657b
(4.244751)
4.45701
(3.89144)
12.00076b
(5.272179)
11.24298b
(5.275964)
10.27612b
(5.073243)
_2001dum 0.241842
(0.449458)
0.044364
(0.432537)
-0.03932
(0.41353)
-0.063413
(0.466175)
-0.178161
(0.456329)
-0.070588
(0.449205)
_2002dum 0.006398
(0.33799)
-0.038505
(0.331137)
-0.06988
(0.320523)
-0.101019
(0.361086)
-0.100599
(0.35849)
-0.194881
(0.349629)
LOGFORMAL .012386
(.0332)
.0010586
(.033088)
.009092
(.031779)
.025978
(.037585)
.021101
(.037340)
0.025599
(0.036218)
LOGCOOP -0.871921a
(0.361052)
-0.899578b
(0.372403)
-0.736442b
(0.342977)
-1.223365a
(0.483417)
-1.210927a
(0.481258)
-1.137295a
(0.466002)
COOPDUM -7.487433b
(3.269941)
-7.72921b
(3.373567)
-6.265732b
(3.106106)
-10.77699b
(4.429216)
-10.66868b
(4.408654)
-10.02301b
(4.264607)
LOGFRFAM 0.031582
(0.043513)
0.033716
(0.043224)
0.030819
(0.042287)
0.022811
(0.047454)
0.025881
(0.047085)
0.016859
(0.045846)
LOGMONEY 0.110434
(0.164944)
0.128637
(0.166565)
0.202349
(0.160103)
0.069078
(0.19035)
0.084579
(0.191808)
0.116
(0.184294)
MONEYDUM 0.941361
(1.618339)
1.015862
(1.626935)
1.526763
(1.559204)
0.416032
(1.841602)
0.491079
(1.849255)
0.707911
(1.775117)
LOGINCOME 0.06107
(0.162598)
0.035087
(0.161084)
0.054987
(0.158103)
0.07026
(0.16805)
0.067637
(0.165962)
EQUIP -7.14E-6
(5.82E-6)
-7.33E-6
(5.89E-6)
-6.24E-6
(5.03E-6)
-5.16E-6
(7.14E-6)
-5.51E-6
(7.22E-6)
-5.8E-6
(6.44E-6)
BLDG -2.79E-7
(2.54E-6)
-2.44E-6
(2.47E-6)
-1.68E-6
(2.2E-6)
-4.38E-6
(2.85E-6)
-4.14E-6
(2.82E-6)
-3.48E-6
(2.63E-6)
LAND -0.124364a
(0.032906)
-0.133985a
(0.033693)
-0.089841a
(0.030561)
-0.103622a
(0.033516)
-0.111106a
(0.034135)
-0.066599b
(0.030012)
IRR (%) 0.154443
(0.317724)
0.232577
(0.32075)
-0.009977
(0.299814)
0.191471
(0.349216)
0.288754
(0.355448)
-0.062812
(0.326269)
LARGE 1.375798a
(0.33017)
1.388423a
(0.332316)
1.710614a
(0.343143)
1.094157a
(0.375732)
1.131591a
(0.376602)
1.438139a
(0.372915)
SMALL -1.604909a
(0.311851)
-1.576405a
(0.315317)
-1.473875a
(0.333631)
-1.45102a
(0.33749)
POP -0.0000727
(0.001124)
-0.000187
(0.001078)
0.000179
(0.001018)
-0.000372
(0.001197)
-0.000433
(0.001179)
0.000427
(0.001131)
RAIN1 -0.004096b
(0.001801)
-0.00319c
(0.001738)
-0.002571
(0.001669)
-0.002832*
(0.001819)
-0.002252
(0.0018)
-0.00262
(0.001762)
RAIN2 -0.001332
(0.001039)
-0.001003
(0.001034)
-0.000633
(0.001017)
-0.001836*
(0.001144)
-0.001587
(0.001139)
-0.001581
(0.001116)
WAGE -0.008345
(0.023561)
-0.01654
(0.022454)
-0.037407c
(0.022256)
-0.001465
(0.026863)
-0.007795
(0.026334)
-0.023095
(0.026003)
MALES 0.110822
(0.120771)
0.106858
(0.120562)
0.083108
(0.11715)
0.170741
(0.130365)
0.174994
(0.129506)
0.129502
(0.126887)
FEMALES 0.170036 0.202482 0.17078 0.146774 0.171953 0.128152
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50
a denotes significance at the 1% level, b at the 5%, and c at the 10% level. * both variables were
significant at the 11% level
(0.142881) (0.14493) (0.138053) (0.158886) (0.161194) (0.150817)
DRAFT 9.28E-6b
(4.76E-6)
9.43E-6c
(4.66E-6)
1.06E-5b
(4.51E-6)
7.77E-6
(5.78E-6)
7.93E-6
(5.69E-6)
1.12E-6b
(05.5E-6)
EDUC -0.003377
(0.031316)
-0.001252
(0.034108)
0.009134
(0.033162)
EDUCDUM 0.59987
(0.33026)
0.734296b
(0.327848)
0.484717
(0.368493)
McFadden R2 0.147467 0.1530 0.1106 0.1537 0.157479 0.110972
Log Likelihood -259.04 -257.36 -270.24 -200.39 -199.5208 -210.5343
RLog
Likelihood -303.85 -303.85 -303.85 -236.81
-236.8140 -236.8140
%Corr Pred 83.41 83.5 71.26 70.8
n 904 904 904 454 454 454
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51
Table 6: Logit Results, Borrowers and Non-Borrowers
Borrowers Non-Borrowers
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
Variable
Coefficient
(St. Error)
Coefficient
(St. Error)
Coefficient
(St. Error)
Coefficient
(St. Error)
Coefficient
(St. Error)
Coefficient
(St. Error)
C 8.367749
(15.98079)
7.792326
(16.59159)
7.147672
(16.19137)
10.53778b
(4.420441)
9.686747b
(4.575896)
5.594368
(3.957521)
_2001dum -0.149731
(1.072603)
-0.005707
(1.092464)
-0.005603
(1.089259)
0.517233
(0.593371)
0.197032
(0.544822)
0.29257
(0.544538)
_2002dum -0.380262
(0.820677)
-0.318281
(0.834489)
-0.359838
(0.823511)
-0.105337
(0.453947)
-0.205063
(0.434713)
-0.108085
(0.427223)
LOGFORMAL 0.21658
(0.491731)
0.201745
(0.490095)
0.230667
(0.485552)
LOGCOOP -1.661119
(1.671929)
-1.699287
(1.700598)
-1.683194
(1.663499)
-0.88162b
(0.391305)
-0.941527b
(0.409209)
-0.615853c
(0.354662)
COOPDUM -14.67097
(14.67739)
-15.04046
(14.91145)
-14.92817
(14.59559)
-7.648318b
(3.53754)
-8.165826b
(3.700291)
-5.293439c
(3.205919)
LOGFRFAM 0.112134
(0.084629)
0.084091
(0.086272)
0.085049
(0.085956)
0.018362
(0.061744)
0.023877
(0.061364)
0.012078
(0.058739)
LOGMONEY 0.787652*
(0.493682)
0.794561*
(0.49287)
0.811696
(0.487155)
-0.005975
(0.191544)
0.010801
(0.194638)
0.10548
(0.179901)
MONEYDUM 7.62116*
(4.80717)
7.840714*
(4.831922)
7.939098c
(4.764944)
0.054141
(1.87454)
0.10813
(1.89593)
0.830415
(1.753283)
LOGINCOME -0.180128
(0.352935)
-0.182276
(0.364058)
-0.176794c
(0.36438)
EQUIP -2.63E-5
(2.04E-5)
-2.69E-5
(2.01E-5)
-2.58E-5
(1.96E-5)
-2.82E-6
(5.54E-6)
-3.44E-6
(5.7E-6)
-2.36E-6
(4.89E-6)
BLDG -6.78E-6
(6.89E-6)
-6.21E-6
(6.73E-6)
-5.87E-6
(6.62E-6)
-1.89E-6
(2.7E-6)
-1.42E-6
(2.6E-6)
-9.05E-7
(2.35E-6)
LAND -0.041988
(0.063776)
0.006522
(0.072259)
0.011076
(0.07126)
-0.252443a
(0.048767)
-0.269002a
(0.049566)
-0.17662a
(0.04393)
IRR (%) 0.898322
(0.838088)
1.149483
(0.866703)
1.04372
(0.82626)
0.065702
(0.366644)
0.175108
(0.369542)
-0.182083
(0.339452)
LARGE 2.448738b
(1.221021)
2.552727b
(1.249262)
2.641544b
(1.232353)
1.408668a
(0.375042)
1.441433a
(0.379192)
1.894807
(0.381245)
SMALL -0.49275
(0.816858)
-0.431977
(0.829874)
-2.132163a
(0.370693)
-2.078061a
(0.377113)
POP -0.001325
(0.003022)
-0.002885
(0.00308)
-0.002937
(0.003065)
0.000171
(0.001489)
-0.000428
(0.001388)
0.000479
(0.001366)
RAIN1 -0.00424
(0.003817)
-0.004316
(0.003889)
-0.004589
(0.003853)
-0.005046b
(0.002443)
-0.003435
(0.002272)
-0.003506*
(0.002254)
RAIN2 -0.004454
(0.002937)
-0.004581
(0.003007)
-0.004604
(0.003011)
-0.000236
(0.001301)
0.000223
(0.001288)
0.000363
(0.001242)
WAGE 0.034763
(0.073162)
0.082164
(0.0771)
0.081792
(0.076815)
-0.018902
(0.028167)
-0.024914
(0.027122)
-0.047686c
(0.026955)
MALES 0.839559b
(0.370698)
0.937853b
(0.389293)
0.926801b
(0.385506)
0.020281
(0.136484)
0.015089
(0.135257)
-0.007459
(0.131512)
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52
a denotes significance at the 1% level, b at the 5%, and c at the 10% level. *Significant at the 11%
level.
FEMALES 0.23377
(0.35272)
0.212324
(0.365729)
0.247387
(0.357526)
0.260506
(0.172579)
0.306754b
(0.175326)
0.173194
(0.159104)
DRAFT -6.86E-6
(1.47E-5)
-9.91E-6
(1.57E-5)
-8.48E-6
(1.49E-5)
2.06E-5a
(6.11E-6)
2.05E-5a
(6.06E-6)
1.92E-5a
(5.83E-6)
EDUC -0.191496c
(0.109575)
-0.195365c
(0.108752)
0.010788
(0.03622)
0.036025
(0.034575)
EDUCDUM -0.990826
(1.532128)
0.860049b
(0.364238)
McFadden R2 0.282695 0.304207 0.302150 0.201028 0.212807 0.125354
Log Likelihood
-46.25379 -44.86663 -44.99925 -192.1309
-189.2984 -210.3284
RLog
Likelihood -64.48272 -64.48272 -64.48272 -240.4727
-240.4727 -240.4727
%Corr Pred 86.32 86.79 84.61 84.8
n 195 195 195 719 719 719
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Pennsylvania Economic Review
Volume 18, Number 2, Fall 2011
RETAIL AND POPULATION DENSITY IN SMALLER U.S. CITIES
William K. Bellinger1 and Jue Wang
Dickinson College
ABSTRACT
In this article we test for patterns of retail density across three different dimensions of U.S. inner
cities--geographic centrality, low income levels, and ethnic minority status--using a two equation
recursive model with population density as a second dependent variable. The empirical analysis is
based on zip code level census data for 39 small to mid-sized U.S. cities. Our primary policy-
relevant finding is that neighborhoods with high percentages of African-Americans have
significantly lower retail density, all else equal, while the results for Hispanic, low income, or
otherwise centrally located neighborhoods are generally mixed or positive.
INTRODUCTION
The case has been made many times in several contexts that U.S. inner city
neighborhoods are significantly underserved by retail services. This hypothesized retail gap in
inner city neighborhoods has experienced revival as a research topic and a major component of
urban development strategy since the publication of Michael Porter’s path-altering article “The
Competitive Advantage of the Inner City” (1995). The concept of unrealized inner city retail
opportunity has been verified in large cities such as Denver (Weiler, et. al., 2003), Buffalo NY
(Rogers, et. al.), Newark NJ (ICICc), Brooklyn NY (ICICb), and Atlanta (Boston and Ross,
1997b). Most of these studies involved partnerships between research institutions and local
government development agencies, and all found a significant degree of unrealized development
opportunity in the retail sector.
However, this concept has not been explored with much theoretical depth or empirical
breadth in the urban economics or urban development literature. The only broadly based study we
discovered was conducted by the Department of Housing and Urban Development (U.S…., 1999).
It found that 48 of 539 cities had less estimated retail revenue than estimated purchasing power,
less than 9 percent of the total. Also, it did not separately analyze the inner city areas of these
cities. This combination of selected case studies and one predominantly negative large scale study
leaves open the question of whether the claims of a retail gap represent the tip of a systemic urban
market failure, conclusions with more limited applicability, or political arguments in favor of
increased largesse for urban American business.
Another often overlooked but crucial dimension of the issue is the interpretation of the
phrase “inner city”. Porter and the Institute for a Competitive Inner City have consistently defined
inner city neighborhoods in terms of alternative poverty or income and unemployment thresholds,
but the inner city can also be defined in terms of its relatively central location and its ethnic
characteristics (Mills, 1997, pp. 727-29). Separating these often overlapping and related concepts
may be an imperfect process, but these different dimensions of an inner city have different policy
implications, and should be actively explored.
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54
Suggested barriers to inner city retail development include a lack of large scale
commercial sites, high barriers to and costs of construction, high local costs for utilities and taxes
(Porter, pp. 62-65), biased or inadequate demand information (Weiler, et. al.; Weissbourd and
Berry) and discrimination by area or ethnicity (Blanchard, 2008). Biased demand estimates may
exist because of an undercounted population in low income areas2, under-estimated or
underreported income, unawareness of the negative relation between the average propensity to
consume and income level, or a lack of understanding of ethnic differences in taste. The very real
possibility of consumer discrimination based on neighborhood ethnic characteristics may also be a
factor.
Our empirical analysis measures patterns in retail density based on a two equation
recursive model. The first equation analyzes retail density using the inverse of a basic Lösch
market area model. This equation offers retail density hypotheses based on demand factors such
as population density, per capita income, and discrimination by consumers. Cost factors include
measures of crime and taxes, while agglomeration economies and retail land use in the central
business district are also included as location factors.
The second equation provides a multivariate estimator for gross population density based
on the monocentric city model. In addition to a relation between population density and radial
distance, hypotheses are tested regarding non-residential land use, housing market factors, and
ethnic characteristics. The final step in the empirical analysis is the testing of the reduced form
equation for this system, which allows us to test the roles of all three dimensions of the inner city
in determining retail density. In combination, these two equations and their reduced form allow
the consideration of retail density in a multi-dimensional format not previously considered in this
context.
This model is tested using zip code level data for a set of 39 small to mid-sized U.S. cities
from the population census (2000) and business census (2002). Retail density is specified in three
forms; stores per square mile, stores with over $1 million in annual revenue per square mile, and
estimated aggregate revenue per square mile. We find that there is evidence of a significant retail
gap for African-American neighborhoods in smaller U.S. cities, particularly for large stores.
However, no such gap is found for Latino or otherwise low income or centrally located areas.
Other results will be discussed in the empirical section to follow.
RETAIL DENSITY
The first step in developing the empirical model is to specify the model for the geographical
density of retail services. This is a surprisingly easy process since retail density (stores per unit of
area) is the inverse of an average store’s market area. Most hypotheses regarding retail density
can therefore be derived from a basic Lösch market area model in long run equilibrium. For a
typical individual consumer, the demand can be expressed in a linear form
x = a(I) – b(p + tr) (1)
where p is price, I is per capita income, and tr represents two way transportation cost for the
consumer at distance r from the store. To further simplify the model assumes that store price p is
fixed and the slope of the individual demand (b) is 1, a common simplification. In a market with
constant population density D, area A and market area radius R, aggregate demand (X) equals
)2( tRpaADX (2)
where the average commuting distance from the household to the firm is ρ ≈ .7071R. The total cost
function for the firm is:
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55
FXkC * (3)
where k is the average total operating cost, X is total output, and F is the firm’s fixed cost
associated with its location. Defining the firm’s profit function and profit maximizing price leads
to the following long run equilibrium market area:3
2
)222
(tka
D
FA
(4)
Since retail density per unit of geography is a direct inverse of the geographic area of the average
firm, equilibrium retail business density (B) equals.
F
tkaD
B
2)
222(
(5)
Equation (5) provides the basis for our hypotheses regarding retail density, while the logarithmic
form in (6) offers additional guidance for the empirical specification to follow.
Ftka
DB ln222
ln2lnln
(6)
The model’s predictions are generally straightforward. The relation between aggregate
demand and business density is expectedly positive. Both the individual demand (via endpoint a)
and population density D have positive relations to business density B.
0)222
(
tka
F
D
a
B (7)
0
2)
222(
F
tka
D
B
(8)
Assuming normal demand, per capita income will have a positive relation to the individual
demand endpoint a, and therefore to business density, all else equal. This relationship between
identifies the first dimension of the inner city to be tested in this paper.
The third component of demand, transportation cost, is a somewhat more complicated
variable, in part because of its long run connection to other components of the model. Within the
basic fixed price model, a rise in consumer transportation cost t lowers effective demand and
profit, increases the zero profit market area, and thereby lowers retail density.
0)222
(2
tka
F
DR
t
B (9)
Note that this result depends on the assumptions of perfect competition and fixed store prices
(Villegas, pp. 895-898). Additionally, the long run effect of transportation cost on the market area
also depends on its relation to population density. When one adds the assumption that population
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56
density D is a positive function of t, a common result of the monocentric city model, we find the
following relationship between retail density and transportation cost.
F
Rtka
D
F
tka
t
D
t
B)2)(
222(
2)
222(
(10)
With positive and the second addend of (10) negative, retail density B has an ambiguous
relation to transportation cost. The effect of alternative pricing assumptions noted in Villegas
reinforces this result.
The cost side of the model is unambiguous, with both operating cost k and fixed cost F being
negatively related to retail density.
0)222
(
tka
F
D
k
B (11)
0)222
(2
tka
F
D
F
B (12)
Another implication of equation (12) is that the long run marginal effect of F is likely to be
extremely small. Marginally higher location-related costs are therefore likely to be minor factors
in the long run. However, more absolute barriers related to regulation or zoning may represent
significant policy issues for inner city retail development. The same can be said for factors relating
to operating cost and neighborhood aggregate demand.
The Effect of Ethnic Discrimination on Firm Density
In the retail location context, both demand and cost-related sources of discrimination have
been identified in the literature. The most consistent and well-supported connection between
discrimination and firm density comes from consumer prejudice and its effect on demand.
Shoppers, especially white shoppers from outside a mixed or predominantly minority
neighborhood, may feel reluctant to shop there because of their prejudice or fear. Gallagher
(2003), among many others, shows that whites significantly exaggerate the percentage of
minorities living in mixed neighborhoods. Quillian and Pager (2001) show a positive correlation
between African-American population and crime, but also find that Caucasians systematically
overestimate the crime rate in neighborhoods with higher African-American populations. These
two facts imply that whites will significantly overestimate the danger of shopping in
predominantly Black neighborhoods. Given the generally higher average income for Caucasians,
this type of prejudice among white shoppers implies lower individual and aggregate retail demand
in Black neighborhoods. The same logic may apply to Latino neighborhoods.
The role of prejudice, amplified by the biased perceptions noted above, produces a negative
effect on demand. Following Becker, we will incorporate customer prejudice by specifying
individual demand endpoint (a) as a negative function of the minority population percentage (m)
in the neighborhood of the store.4 From equation (5), the effect of consumer discrimination arises
from its effect on individual demand endpoint a.
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57
0)222
(
1
m
atka
F
D
m
a
a
B
m
B (13)
where m1 refers to the effect on retail density of consumer-based ethnic discrimination. Therefore
demand side discrimination will lower retail density in minority neighborhoods, all else equal.
The role of racial and ethnic discrimination on operating cost is more mixed and less directly
related to neighborhood characteristics.5 Yet there is substantial evidence that racial discrimination
exists in capital markets, both in terms of neighborhood characteristics and the race of the
potential entrepreneur. There is more and better evidence of discrimination by lending institutions
against minority entrepreneurs (Blanchard and Zhao, 2008). The connection between this finding
and neighborhood characteristics is less well established, however (Bostic, et. al.). Such ethnic
redlining, if and where it exists, implies lower access to or higher cost for initial capital, and
therefore a higher value for fixed cost F. Therefore F can be assumed to be positively related to
the minority status of the neighborhood, m.
0)222
(2
2
m
Ftka
F
D
m
F
F
B
m
B (14)
where m2 refers to the effect of minority status on retail density caused by financial sector
discrimination. Combining the demand and fixed cost components of discrimination against
minority neighborhood retail produces the following relationship, which is also negative.
02
)]([
2)
222
)((
)()()
222
)(()(
mF
tkmaD
m
mF
m
matkmaDmF
m
B
(15)
Therefore racial discrimination lowers store density when these two hypothesized effects are
combined.
Combining equation (6) with the hypotheses noted above produces the following
estimating model for retail density, which in double log form predicts positive coefficients for
population density and per capita income, negative coefficients for minority status and all cost
related variables, and an ambiguous relationship between business density and consumer
transportation cost.
)(ln)222
),(ln(2lnln mF
tkImaDB
(16)
The first equation of our empirical model of retail density will be based on (16).
POPULATION DENSITY
While the roles of income and ethnicity in determining retail density are established in
the retail density analysis above, the role of location within the metropolitan area, and particularly
its inner city core, is only indirectly related to the previous analysis. Population density D
presents a direct connection to the monocentric city model of urban land use developed by Muth
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58
(1969), and others. As with the retail density function above, population density is derived from an
inverse of the model of residential land per person. The basic logic of the equilibrium population
density function is that land is more valuable near the CBD under the assumption of central city
employment, producing a lower level of average land per capita.6
The urban geography literature offers a diverse set of functional forms for the population
density gradient (Martori, McDonald). In preliminary tests we specified the population density-
distance function in two forms. The first form is a first degree negative exponential function with a
dummy variable for the zip code containing the central business district (CBD),
lnD = lnD0 + bu + cCBD dummy, (17)
where u is the radial distance to the city’s central point, and the second form is second degree
negative exponential function,
ln D = lnD0 + βu + δu2 (18)
These functions were tested nationally, for each of four regions, and for nine of the larger cities in
the data set. Overall and in most individual cases, the CBD dummy approach produced much
greater significance for distance and an equal or better overall fit. For example, no individual city
had a significant value for β and only one for δ in equation (18), while the first degree exponential
function with CBD dummy tests produced significant negative values for b in seven of the nine
cities and more mixed results for the CBD coefficient.7 Therefore the negative exponential
function with CBD dummy will provide the basis for the population density function tested in this
paper.
Other research has explored multivariate models of population density tested across
multiple cities. The independent variables in these models may be categorized as factors affecting
net population (residential) density and those that affect the relationship between gross and net
density.8 Among the main variables hypothesized to affect net density include income (Mills and
Price), the age of the city or its housing stock (Johnson and Kau), modal choice related
transportation cost variables such as the percent of commuters using public transit or autos per
capita (Muth, 1961), and the total population of the city (Muth 1969, Mills and Price). In this
equation we add ethnicity variables and additional housing market variables such as group housing
percentage and the housing vacancy rate.
The distinction between net and gross density involves measures of alternative land use.
Census data is limited in this context, but we can specify two variables related to this distinction.
The first is a dummy variable identifying the zip code containing the city’s CBD, which is also
included in the retail density specification. The second is the percentage of the zip code
population categorized as urban versus rural.
EMPIRICAL ANALYSIS
In this section we test the two equation recursive model developed above in two steps.
First we test each equation using OLS. In a two equation recursive system OLS is appropriate
when the residuals of the two equations are not correlated. Testing the residuals of the population
density equation against the three specifications of retail density produced insignificant
correlations in all cases at the 5 % level. We also test the reduced form of the model separately in
order to explore simultaneously all of the three dimensions of the hypothesized inner city retail
gap.
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Data
This study combines data from the 2000 population census with economic census data
from 2002. Given the recursive structure of our two equation model, this modest lag between
population characteristics and retail density seems appropriate. The data is compiled by zip code
for 39 small to mid-sized U.S. cities ranging in population from slightly over than 53,000 (Battle
Creek MI) to over 223,000 (Greensboro NC), with a mean population of just over 142,000.9 The
selected cities have relatively circular or semi-circular boundaries and are separated from other
urban clusters, although some are satellite or hub cities within larger metropolitan statistical areas.
Zip codes that lie partly or entirely within the city boundary are included. The total land area
covered for all zip codes in the 39 cities is about 7,637 square miles. Excluding two western zip
codes with over 200,000 acres each reduces this total to 6,737 square miles. The total land area
within the 39 city boundaries is about 1,704 square miles. Therefore the included zip codes cover
over 4.3 times the land area of the central cities after excluding the two major outliers.10
Therefore this analysis approximates metropolitan rather than central city retail and population
density patterns.
The Economic Census lists numbers of businesses in each zip code by industry subgroup
and revenue category. Our retail data includes two related categories in the North American
Industry Classification System (NAICS). These are Retail Trade (code 44) and General
Merchandise Stores (code 452). Geographical business patterns are tested using the total number
of firms in each zip code, firms reporting over 1 million in revenue, and an estimated total revenue
figure. The revenue estimate is derived by first assigning mid-point values to the all revenue
categories excluding the open ended category of over 1 million, estimating logarithmic
relationships between these size categories and the average rank of each category using a function
variously labeled as Zipf’s Law, the Pareto distribution, or the rank-size rule, converted to natural
logs.11
Total revenue for firms greater than 1 million was then calculated using the estimated
parameters.12
Finally, aggregate revenue for each zip code was estimated by multiplying the
number of firms in each revenue category by the estimated average revenue, then summing across
the categories. The results of this study are for the most part insensitive to the specific measure of
business activity.
Another essential variable required for this analysis is distance from the city’s central
point. Central point latitudes and longitudes for each zip code are reported by the Census Bureau.
The central point in the CBD was chosen through e-mail correspondence with local officials and
satellite images from Google Earth. Distance calculations require a variation of the Pythagorean
Theorem with parameters to convert degrees to miles and to adjust longitude in miles to the
corresponding latitude. This distance formula is seen in equation (19).
distance in miles = [((latitude2-latitude1)•69.1703234283616)2 + (69.1703234283616• (19)
(longitude2-longitude1)•COSINE(latitude 1 •.0174532925199433))2]
.5,
Testing Retail Density
In this section we test the proposed two equation recursive model using ordinary least
squares with robust standard errors. The market area model above predicts that the number of
firms in a given area will rise with population density D due to a smaller minimum market radius
and higher profits at the fixed maximizing radius, and with higher per capita income. Finally,
transportation cost has an ambiguous effect on business density.
While population density and per capita income are directly measurable, transportation
cost requires indirect estimation. Three variables relate to transportation cost. First, an indirect
estimate of transportation speed, a primary component of transportation cost, may be possible
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despite the limitations of census data. Average transportation time per worker in a given zip code
is easily calculated from census data. In order to approximate the marginal transportation cost per
geographic unit one must divide by a distance measure that might approximate, or at least vary
positively with, the average commuting distance in a given city. In this case we use mean radial
distance in each city as our geographic measure. This constructed average speed estimate produces
slower than expected average speeds, with a mean estimate of 10.2 m.p.h., but more importantly
behaves relatively well regarding common hypotheses. Regressing our estimate of travel time per
person-mile against population density (D), the percent of commuters using public transportation
(Trans), and two infrastructure related variables, median housing age (HAge) and a dummy
indicating the presence of an internal highway in the city (Hwy) produced the following results. t
statistics are in parentheses.
Time/mi = 5.27 - .012 Trans + .039HAge - .991Hwy + .0001D (20)
(12.56) (-0.19) (3.09) (-3.35) (2.67)
This variable for commuting speed will be combined with a variable measuring the percentage of
households that do not own an automobile. An alternative measure of modal choice and
transportation cost, cars per household, will also be tested.
The specification of the cost variables inevitably raises some questions, in part because
no direct measures of operating cost are available in economic census data. Cost-related variables
include the median property tax rate and the city-wide Uniform Crime Report index crime rate for
total reported crimes. The percentage of adults without a high school degree, a measure of local
human capital, and the labor force participation rate were dropped from the final results due to
collinearity problems and highly insignificant coefficients.
In addition to variables related to the market area model, other dimensions of firm
location decisions relate to agglomeration economies. Due to the tendency of retail firms to
cluster based on agglomeration economies, a dummy variable labeled Cluster takes the value of 1
when the total retail stores open all year are over 1 standard deviation above each city’s mean. A
similar dummy variable is included for the zip code containing each city’s central business district.
Of the 39 CBD zip codes only 6 contain retail clusters by this definition. Half of these are in the
western region with one in each of the other 3 regions. On average, zip codes containing retail
clusters are 3.76 miles from the city’s central point, as opposed to 3.29 miles for non-cluster zip
codes. Overall, areas with concentrated retail activity display a decentralized pattern.
The empirical model of retail density may now be specified as follows:
lnRetail = f(ln D, C,CBD, ln T, ln I, t ,ln Crime, %Black, %Latino,) (21)
where C is the retail cluster dummy, T is the median property tax, I is per capita income, t
represents various specifications of transportation cost, crime is total index crime, and D is the
population density.
The population density equation will be specified as follows, based on the discussion in the
previous section.
lnD = f(u, I, H, %Black, %Latino, Pop, V, G,) (22)
where D is population density, u equals radial distance from the city center, I is per capita income,
H is housing age, Pop is the total population of the City, V is the vacancy rate for housing units,
and G is the percentage of the population in group housing. Results are presented in Table 1 for 3
measures of retail density; total stores, estimated total retail revenue, and total stores with over 1
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million in revenue (Big Store Density), and in Table 2 for the population density equation with
various specifications of transportation cost.
Retail Density Results
Bruesch-Pagan tests for heteroscedasticity were mixed in the retail density specifications,
with insignificant Χ2 values for total stores and significant values for large stores and estimated
aggregate revenue. As a result the total store tests are HC2 robust standard errors while the others
use HC3 errors.13
Mean variance inflation factors, also reported in Table 1, suggest moderate
multicollinearity. Finally, given the existence of a variable that was fixed across each city’s zip
codes (crime rate), tests were also run using standard errors that are clustered by city.14
Results
are very similar to those reported below.
These empirical estimates of retail density offer relatively good overall fits and somewhat
mixed significance for individual variables. Among the demand variables, population density is
extremely significant, while per capita income fails to achieve significance. It is worth noting that
other household income and poverty measures included in preliminary tests were generally
insignificant as well. The retail cluster dummy is highly positive, if not particularly interesting,
while the CBD dummy borders on significance for total stores and is insignificant for large stores
and aggregate revenue. One interpretation of this mixed result is that small shops serving financial
district clients are likely to dominate the modern small city CBD.
The variables related to retail operating costs are of mixed significance. Median real
estate taxes are significantly negative at the .05 or .10 levels in all specifications, while the city
crime rate is a non-factor. It should also be noted that separate tests of violent and property crime
rates produced insignificant results similar to those for total reported crime. The unavailability of
crime data by neighborhood or zip code is clearly a limiting factor in this analysis. Since the cost
variables are relatively indirect measures of true costs, and are in some cases only available on a
city-wide basis, the hypothesis that cost disadvantages beyond taxes exist in the inner city should
not be rejected on the basis of this evidence alone.
The consumer transportation cost variables produce interesting results. The percent of
households without cars has a significant positive relationship to all retail measures, with higher
coefficients and t statistics for the total store count than for estimated revenue or large store
density. Similarly, the number of cars per household is significantly negative with a slightly
higher coefficient for total stores than other specifications. The estimated travel time per mile
variable highway variable has a negative relation to total store density but insignificant
coefficients for the other specifications. The main conclusion to be gathered from these findings is
that increased aggregate access to automobiles lowers retail density.
The final variables of interest are the ethnicity variables. In all specifications the
percentage of African-Americans is negatively correlated to retail density while the percentage of
Latinos is not significant. This result suggests that the possible existence of a retail gap in minority
neighborhoods is confirmed, but only for African-American neighborhoods. This finding will be
considered again in the reduced form tests to follow.
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Table 1: Retail Density Function
dependent
variable
(per sq. mile)
ln All
Store
Density
ln All
Store
Density
ln Big
Store
Density
ln Big
Store
Density
ln Est.
Revenue
ln Est.
Revenue
Equation 1-1 1-2 1-3 1-4 1-5 1-6
Constant
-9.11
(-3.03)
-4.94
(-1.73) -8.44
(-2.93)
-4.92
(-1.85) 6.25
(2.27)
9.90
(3.50)
Travel Time/
Mile -.0322
(-2.03)
-.0269
(-1.52)
-.025
(-1.76)
Pct. Without
Car .0368
(6.34)
.0303
(4.97)
.031
(4.93)
Cars per
household -1.74
(-6.66)
-1.57
(-5.08)
-1.55
(-5.08)
CBD dummy
.331
(1.89) .334
(2.08)
.2173
(1.33)
.191
(1.26)
.264
(1.48)
.249
(1.56)
Cluster
.929
(10.60)
.884
(9.73)
1.040
(11.50)
1.002
(10.76)
1.05
(9.55)
1.008
(10.51)
% African-
American -.0097
(-3.10)
-.011
(-3.50)
-.0097
(-3.02)
-.011
(-3.49)
-.009
(-3.42)
-.010
(-3.21)
% Latino
-.0045
(-0.94)
-.0003
(-0.08)
-.0038
(-0.74)
-.0004
(-0.08)
-.003
(-0.66)
-.0005
(-0.11)
ln Per Capita
Income (1,000s)
.516
(1.97)
.364
(1.46)
.4157
(1.58)
.319
(1.31)
.434
(1.96)
.319
(1.25)
ln Population
Density .9548
(22.95)
.934
(22.21)
.9221
(19.21)
.898
(19.34)
.944
(21.50)
.922
(19.39)
ln Total
Crime Rate
-.0416
(-0.35)
-.047
(-0.41)
-.0869
(-0.73)
-.099
(-0.85)
-.087
(-0.66)
-.100
(-0.82)
ln Median
Property Tax -.183
(-2.18)
-.241
(-2.90)
-.1586
(-1.79) -.219
(-2.55)
-.185
(-2.62)
-.238
(-2.71)
R2 .778 .786 .743 .755 .735 .744
Mean V.I.F. 1.89 1.76 1.89 1.76 1.89 1.76
t statistics are in parentheses. Boldfaced results are significant at the 5% level
Population Density Results
Results for the population density tests are reported in Table 2. Bruesch-Pagan tests
indicated significant heteroscedasticity, so HC3 robust standard errors are applied to all equations.
The vacancy and group housing rates are dropped in most reported specifications due to
insignificance, while housing age is omitted in some specifications in order to display the well-
known negative relation between housing age and radial distance in U.S. cities. As in the retail
density tests, alternative specifications of transportation cost are included.
There are few surprises in the population density estimates in Table 2, with population
density being positively related to the urban population percentage in the zip code, the total
population in the central city, and the median housing age. Population density is found to be
negatively related to radial distance in keeping with the predictions of the monocentric model.
Finally, the travel time per mile variable is more significant in the context of population density
than retail density (see Table 1), while the alternate variables measuring automobile access are less
significant.
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The omission of the median housing age variable from equations 2-3 and 2-5 increases
the significance of both the radial distance and CBD dummy variables. This is undoubtedly due to
the well-known tendency of city housing stock to grow outward over time, producing lower
housing age on the city’s periphery. It is also likely that older housing stock correlates negatively
with automobile access, thereby increasing transportation cost. Finally, the well-known filtering
model of housing use suggests a strong negative correlation between housing age and income,
which provides a reasonable explanation for the significantly negative income coefficient in 2-3.
Overall, while collinearity among the income, housing age, and radial distance variables produces
explainable variations in significance, this population density function provides reasonably strong
explanations of population patterns in smaller U.S. cities.
Table 2: Population Density Function
Dependent Variable: ln Population Density
Equation 2-1 2-2 2-3 2-4 2-5
Constant
3.15
(1.47)
2.22
(1.18) 5.51
(2.77)
1.69
(.90)
3.47
(1.73)
% African-American -.0012
(-.45)
-.0013
(-.55)
-.0025
(-1.06)
-.0027
(-1.37)
-.0029
(-1.39)
% Latino
.0102
(3.11)
.0114
(3.72)
.0083
(2.61)
.0084
(2.90)
.0054
(1.77)
Radial Distance
(100 foot units) -.0015
(-2.19)
-.0015
(-2.14)
-.0028
(-5.39)
-.0011
(-1.62) -.0023
(-4.20)
Median Housing Age
.0217
(4.94)
.0216
(4.87)
.0184
(4.28)
Central City
Population (1000s) .0017
(2.71)
.0016
(2.51)
.0019
(2.92)
.0020
(3.05)
.0025
(3.80)
% Urban Population .0613
(7.67)
.0620
(7.63)
.0669
(7.85)
.0618
(7.70)
.0659
(8.03)
ln Per Capita
Income (thousands)
-.236
(-1.22)
-.166
(-1.07) -.4036
(-2.57)
-.1186
(-.74)
-.21
(-1.58)
Travel Time/Mile .0327
(2.55)
.0456
(3.61)
Pct. Without Car .0071
(1.70) .0106
(2.39)
Cars Per Household .075
(.24)
.161
(.79)
-.146
(-.84)
CBD Dummy -.085
(-.62)
-.133
(1.07) -.326
(-2.86)
-.2067
(-1.77) -.3581
(-3.13)
Vacant Housing
Percentage
-.007
(-.54)
Group Population
Percentage
-.0052
(-.65)
R2 .700 .695 .661 .702 .677
Mean VIF 2.07 1.85 1.63 1.90 1.72
t statistics are in parentheses. Boldfaced results are significant at the 5% level
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Reduced Form Estimates
In this section we utilize the reduced form of this two equation system to consider more
broadly the possible causes of any systematic inner city retail gap. By combining the two
equations in this way all three of the dimensions of the inner city, central location, income, and
ethnicity, are tested in a single equation model. Combining the general specifications in equations
(21) and (22) produces the following reduced form equation for retail density:
lnRetail = f(C,CBD, ln T, ln I, t ,ln Crime, r, I, H, %Black, %Latino, Pop, %Black, %Latino,) (23)
where C is the retail cluster dummy, T is the median property tax, I is per capita income, t
represents various specifications of transportation cost, crime is total index crime, r equals radial
distance from the city center, I is per capita income, H is housing age, and Pop is the total
population of the City. Because of their insignificance, the vacancy and group housing variables
were dropped from the analysis.
Results are presented in Table 3 above. HC3 robust standard errors are applied to all
specifications. The substitution of multiple variables for population density increases the
collinearity of the model somewhat, as noted in the higher variance inflation factors. While two
retail density variables (property tax, CBD dummy) that were partially significant in Table 1 lose
their significance in the reduced form estimates, possibly due to increased collinearity, the main
contributors remain highly significant. These include African American percentage, automobile
access (an indicator of transportation cost), central city population, radial distance, the urban
population percentage, and the cluster dummy variable.
The primary purpose of the reduced form tests was to consider directly the roles of all
three dimensions of the inner city, central location, income, and ethnicity. Of these three
dimensions of the inner city discussed in the introduction, geographically central neighborhoods
display evidence of higher retail density, all else equal, primarily due to the higher population
density found in centrally located neighborhoods. The insignificant income coefficients also fail to
show a clear retail gap based on income. The third dimension of the inner city, ethnicity, indicates
a significant retail gap in store density for African-American communities but no such gap for
Latino neighborhoods. This relative lack of retail stores in African-American neighborhoods is the
primary social factor that verifies the existence of an inner city retail gap. Otherwise central, low-
income, or ethnically diverse neighborhoods show no such pattern in smaller U.S. cities.
CONCLUSIONS
Overall, the combination of a retail density equation based on an inverted market area
model and population density equation based on the monocentric city model proves useful in
explaining retail density in America’s smaller cities. For the most part, the hypothesized existence
of an inner city retail gap is not confirmed by these tests. Per capita income proves to have an
insignificant relationship to retail density, while geographically central neighborhoods have
greater retail density by all measures than those in the periphery. The third dimension of the inner
city, ethnicity, indicates a significant retail gap for African-American communities but no such
gap for Latino neighborhoods.
The identification of African American status as the lone source of a systematic retail gap
represents the most important policy issue in this study. Not surprisingly, a wide range of public
and private development efforts are available for addressing this issue. For example, new or
existing enterprise zones could focus their incentive programs on the development of retail
clusters in underserved areas. Mentoring and financial assistance programs for inner city
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Table 3: Reduced Form Estimates
Dependent
Variable
(per sq. mile)
ln All
Store
Density
ln All
Store
Density
ln Big
Store
Density
ln Big
Store
Density
ln Est.
Revenue
ln Est.
Revenue
Equation 3-1 3-2 3-3 3-4 3-5 3-6
Constant
-6.32
(-2.06)
-.173
(-.60) -6.10
(-2.18)
-1.86
(-.71) 8.73
(2.98)
12.83
(4.65)
Travel Time/
Mile
-.011
(-0.47)
.001
(.05)
-.002
(-.10)
Pct. Without
Car .040
(5.84)
.035
(5.89)
.347
(5.42)
Cars Per
Household -1.67
(-4.15)
-1.67
(-4.15)
-1.47
(-3.38)
CBD dummy
.081
(.43)
.229
(1.33)
-.085
(-.49)
.032
(.20)
-.012
(-.06)
.113
(.65)
Cluster
1.01
(9.81)
.973
(9.45)
1.09
(10.68)
1.05
(10.50)
1.11
(10.51)
1.08
(10.24)
% African-
American -.011
(-3.56)
-011
(-3.31)
-.012
(-4.07)
-.012
(-3.68)
-.011
(-3.45)
-.010
(-3.11)
% Latino
-.0007
(-.14)
.0047
(1.00)
-.0016
(-.33)
.003
(.72)
.0001
(.02)
.005
(1.08)
ln Per Capita
Income (1,000s)
.117
(.38)
-.171
(-.59)
.032
(.12)
-.232
(-.89)
.028
(.09)
-.232
(-.82)
Radial Distance
(100 foot units) -.0024
(-2.60)
-.0029
(-2.03)
-.0028
(-2.77)
-.002
(-2.56)
-.003
(-2.44)
-.002
(-2.14)
Median
Housing Age
.0087
(1.38)
.0093
(1.59)
.002
(.35)
.0028
(.45)
.006
(.91)
.007
(1.06)
Central City
Population (1000s) .0024
(2.54)
.0028
(3.17)
.0027
(2.97)
.0029
(3.50)
.002
(2.12)
.0023
(2.48)
% Urban
Population .064
(7.27)
.0636
(7.27)
.0655
(7.58)
.065
(7.61)
.065
(7.36)
.0644
(7.35)
ln Median
Property Tax
.018
(.17)
.049
(.48)
.008
(.08)
.049
(.53)
.013
(.13)
.050
(.49)
R2 .667 .665 .660 .660 .637 .636
Mean V.I.F. 2.12 2.03 2.12 2.03 2.12 2.03
t statistics are in parentheses. Boldfaced results are significant at the 5% level
entrepreneurs (Johnson, et. al.), infrastructure improvements, and effective (and widely advertised)
crime control efforts may also encourage greater retail opportunity in African-American
communities. More flexible access to transportation could provide another method of reducing the
spatial mismatch created by these retail gaps. The wide range of innovative programs particularly
concerned with urban food retail (USDA, pp. 104-111) provides evidence of the variety of policy
options associated with a particular product category.
However, the main point of this paper is not to promote specific policy solutions. Indeed
with such a diversity of opinion and range of proposals we would barely know where to begin.
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66
However, this paper adds two bits of information to our understanding of inner city retail
development. First, the combination of a retail density model based on a Löschian model of
market area and an empirical population density model based on the monocentric model of the city
provide a useful basis for analyzing the possible causes of a central city retail gap. Second, and
more importantly, this study provides evidence of a systematic retail gap in smaller inner cities
that weighs significantly on African-American neighborhoods, rather than on other poor or
centrally located areas. This finding offers additional support for the value of urban retail
development efforts in the African-American neighborhoods of our smaller cities.
ENDNOTES
1. William Bellinger is a Professor at Dickinson College, and Jue Wang is an undergraduate
economics and mathematics major. 2. The hypothesized population underestimate is based on at least two sources. First, there is some
evidence that census counts are understated in poor communities (U.S. Housing.., pp. 11-14).
Also, some business location decisions may not fully consider the implications of population
density when estimating potential demand (Porter, pp. 58-59).
3. Given the demand and cost functions in the text the firm’s profit is defined as
FR
tpakpAD )2
)((
Maximizing with regard to price gives us the profit maximizing price, p*
422
* tRakP
Plugging p* into the profit function gives us the maximum profit
FtRka
AD 2
)422
(
Given a zero profit long run competitive equilibrium the firm’s market area A is as presented in
the text.
4. From the earlier individual demand equation, individual demand with discrimination based on
neighborhood ethnicity will equal x = a(m) - P - tu where ∂a/∂m < 0.
5. For firms that hire predominantly local labor, wage discrimination by employers tends to lower
labor cost in minority neighborhoods. While average human capital may be lower in areas with
poorer schools or higher dropout rates, the likelihood of higher unemployment increases the
selectivity of employers’ choices among available workers. Porter (p. 61) includes low wage
moderate skill labor as a potential advantage of an inner city location.
6. In the monocentric model population density can be derived as follows: Assume an individual
maximizes a Cobb-Douglas utility function with two goods, land L and a composite C subject to a
budget that includes total round trip commuting costs of tr at distance r.
(a)Max LαC
1-α subject to y –tr = ρ(r)•L + Pc•C,
In (a) L = Land, C = the composite good, ρ is land rent, r is radial distance, and y = household
income. From (a) a simple Cobb-Douglas demand for land (b) can be derived, combined with the
land rent gradient (c and d) and differentiated twice with regard to distance to display the model’s
predictions for land demanded per person.
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67
(b) L*=α(y-tr)/ρ(r)
(c) ρ(d)= Rrty
try
/1
, where r and R are the distance and rent at the city’s boundary.
(d) Combining (2) and (3), L(u)=α(y-t r )1/α
(y-tr)(α-1)/α
/ρ(r)
(e) ∂L/∂r=t(1- α) •ρ(r)-1
>0
(f) ∂2L/∂r
2= -t(1- α) •ρ(r)
-2•∂ρ/∂r >0 (see Anas and Kim)
Equations (e) and (f) state that land per person increases at an increasing rate with distance from
the central point. Conversely, the population per unit of land will decrease at a decreasing rate.
Equation (f) is positive if ∂ρ/∂r is negative, a common finding noted in the text.
7. More specifically, the 9 cities tested individually were Springfield MA, Hartford CT, Rochester
NY, Akron OH, Grand Rapids MI, Greensboro NC, Shreveport LA, Salt Lake City UT and
Tacoma WA. The choices for these preliminary tests were based on the relatively large numbers of
zip codes and degrees of freedom available for these cities. Salt Lake City and Tacoma had
insignificant distance variables in the CBD dummy specification, while Akron had a significantly
negative coefficient for the square of distance in the second degree log specification. In the first
degree exponential with dummy variable specification, the zip code dummy was significantly
negative in Hartford and Akron, and was insignificantly negative for the other cities. The national
and regional tests were also far more significant for the zip code dummy specification.
8. Net population density describes the population density in residential areas only, while gross
density measures population density for an entire area regardless of primary land use.
9. The Northeastern cities are Springfield, Worcester and Lowell MA; Hartford and Waterbury
CT; Schenectady, Syracuse and Rochester NY; and Erie, Allentown and Reading PA. The
Midwestern cities are Rockford IL; Akron, Youngstown, and Canton OH; Ann Arbor, Flint,
Lansing, Battle Creek, Kalamazoo, and Grand Rapids MI; and Racine WI. The Southern cities are
Macon GA; High Point, Greensboro, and Wilmington NC, Greenville SC; Lake Charles and
Shreveport LA. The Western cities are Fort Collins CO; Boise ID; Billings MT; Salt Lake City
UT; Santa Clarita, Salinas, and Santa Rosa CA; Salem and Eugene OR, and Tacoma WA. The
South is somewhat underrepresented because of fewer circular cities, a lack of cooperation from
some local officials regarding the location of the city center, and a larger percent of the original set
of cities with fewer than 4 zip codes.
10. A related point is that zip code land areas are larger in more peripheral locations. A simple
regression indicated that for each mile of radial distance the average land area of a zip code
increased by 6.008 square miles, or 3,845 acres.
11. Zipf’s law is expressed as follows:
BRank
Kv Re , or ln Rev = ln K – β ln Rank.
12. More specifically, the highest and lowest ranks (one and the average number of firms above a
million, respectively), were then averaged using a geometric mean. For example, for the semi-log
specification of retail the estimated equation was Avg. Revenue = 5,712,622 – 1.285716.12 ln
Avg. Rank. With a mean of 28 firms per zip code reporting revenue above one million, the
revenue values were $5,712,622 for the largest firm and $1,035,719 for the 28th
firm. The
geometric mean of these is $2,432,420. These estimates are likely to quite conservative, as major
chains report revenue per store higher than the maximum estimates produced by this method. The
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double log specification produced a geometric mean of over $7 million. This and a mean of the
double log and semi-log estimates were tested without meaningful effect on the results.
13. Davidson and MacKinnon (pp. 553-554) recommend these choices. HC2 adjusts standard errors
via the formula 1/(1-hii), , while HC3 uses 1/(1-hii)2,
where hii= xi(X’X)-1
x’. According to Long
and Ervin (2000), who recommend the use of HC3 in all cases, there is little distinction between
robust standard error forms with over 250 observations, as is true in this study. The model was
also tested with standard errors clustered by city.
14. This model and data set sit squarely in an area of uncertainty regarding the appropriateness of
clustered standard errors. The natural choice of clustering by city involves somewhat less than the
ideal minimum number of clusters, which according to Kedzi (2004) is about 50. Also, the
resulting clusters are moderately unbalanced. The number of zip codes per city range from 4 to 21,
with three cities (Rochester NY, Salt Lake City UT, and Tacoma WA) having over 15 zip codes
each and two containing slightly over 5 percent of the total sample of 345 zip codes, which
violates the rule of thumb suggested by Rogers (1993). Finally, the explicit fixed effects in the
model are limited to one insignificant variable in the business density equation (crime) and a
somewhat more significant single variable in the population density equation (city population).
For all of these reasons we report unclustered estimates in the text.
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