Comparative Statics Made Simple

8/14/2019 Comparative Statics Made Simple

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I. Introduction

Comparative statics are much like the before and after shots one sees in

advertisements for weight reduction, hair replacement, or silverware polishing. One looks

at the decided level of a variable before the change in a parameter and then compares it with

the level after the change. But unlike advertisements, the comparative statics that

economists consider are theoretical. They are predictions that need to be confirmed by real

world evidence.

It is often the case, however, that such confirmation is not forthcoming.

Predictions are usually all policy-makers have available before they are required to act. In

such situations, it becomes critical to understand the particular assumptions comparative

static predictions are based upon. Such assumptions can then themselves become the heart

of policy debates, with evidence regarding their reasonableness brought to the fore.

Formal economic theory enters the fold to lay bare the critical assumptions that

drive the conflicting predictions that arise in less formal discussions. As an example of

these less formal discussions, consider the Rio Climate Change Treaty that committed

Australia to a reduction in the level of Greenhouse Gases to 1990 levels by the year 2000.

This goal gives rise to several comparative statics questions regarding what instruments

might be effective in achieving this change. Specifically, would the addition of a carbon

tax reduce the quantity of Greenhouse gases generated by Australian industry? What about

a subsidy to producers to switch to less Greenhouse gas intensive technologies (i.e., based

on solar or wind power)? Or an education campaign on consumption?

Each of these comparative statics questions asks: if there is a change in some

parameter out of the control of individual decision-makers, how does this change the levels

of variables they decide upon? Normally, there is an assumption that decision-makers act

rationally, but this need not be the case (see Milgrom and Roberts, 1994). Also,

sometimes the question is phrased in terms of variables arising out of market interactions of

agents, most obviously prices, rather than a less aggregated measure.


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In the past, however, formal theory has suffered from two critical problems in

analysing comparative statics. First, the predictions derived are often based on

assumptions that are divorced from everyday presumptions on peoples tastes or firms

technology. These assumptions seem to arise out of the technical need for a mathematically

precise construction rather than the substantive issues at hand. But, more importantly,

these formal predictions have been relatively inaccessible even to those trained in

undergraduate economics. Hence, these predictions are far removed from the intuitions

commonly used in economic analysis. They add little to debates that ought to focus on the

key relevant issues turning predictions one way or another.

Those days, however, are in the past. The purpose of this paper is to lay out a new

set of tools that are the results of recent formal theory. These tools provide a general

method of using formal theory to understand the key substantive assumptions underlying

economic intuitions. But, more importantly, these tools are accessible. With just a little

calculus and even less algebra they allow economists to formalise arguments themselves

and focus on the critical assumptions that drive a comparative static prediction. This can be

done because the mathematics of the new results are very closely aligned with the intuitions

economists and others use in everyday discussions of policy.

II. A Little Notation

Although these new tools for comparative statics analysis are not technically

difficult to use, some notation and reacquaintence with what constitutes optimising

behaviour is required. When choosing rationally, economic agents are generally assumed

to be choosing some variable from a choice set to maximise a given payoff function.

Therefore, suppose that an agent makes a choicex from a set of feasible choicesX. Each

element x ofX can be considered to be made up of many dimensions. To make this

simple, one can imagine that x is a vector of I component choice variables, i.e.,

x x xI ( ,..., )1 . So x may represent a bundle of commodities in consumer theory or an


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allocation of factors as in the theory of the firm. In turn, the choice set, X, can represent a

budget set (given prices and income) in consumer theory or the feasible technological

alternatives in the theory of the firm.

For any choice of x, it is assumed that the agent receives a payoff

( ) ( ,..., )x x xI 1 . This is termed the agents objective function and it might be a utility

function, a profit function, or expectations of these. A rational economic agent is assumed

to choose variables from the choice set that maximises their objective function. In notation,

the agent solves:

Maximisex X x ( ) (P).

An element of the choice set that solves this problem is called an optimum and is denoted

by x* . Note that there may be more than one x* that solves this problem.

But just observing that a particular choice is an optimum is not very interesting in

and of itself. It becomes interesting when the context in which it is determined is

considered. That is, there are a variety ofparameters, not under the control of the agent,

that influence the agents choices. For instance, relative prices determine the mix of goods

a consumer purchases and government taxes may determine the level of factors a firm

employs. These external or exogenous parameters are denoted here by which again can

be of many dimensions, i.e., ( ,..., )1 J . So may be a list of prices, policies or even

the actions of other agents. And like choice variables, not all parameters can possibly arise.

Thus, it is assumed that such parameters come from a set .

When one studies comparative statics, one is interested in the qualitative properties

of the optimum (or optimal set) that the agent chooses. That is, one is interested in how the

optimal choices, x* , vary with : What are the properties of ( ; )x andX that mean that

the level of a good demanded falls with a rise in its price, that labour supplied rises with the

wage, or that output will fall in the face of an increase in taxes? Determining such

qualitative relationships is the heart of economic theory. Of particular interest, however,


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Before turning to these theorems, it is worthwhile to digress upon the

methodological premises underlying this approach. If one could actually solve the

optimising problem, such as (P), for the function x*( ) , then it would be easy to check

whether (P) displayed monotone comparative statics or not. However, it is rare to have

enough information about objective functions and choice sets to be able to do this. Indeed,

many objective functions, such as utility functions, are very artificial constructs and not

something one would expect to have precise information about. It is a more reasonable

expectation that qualitative conditions are available about the structure of these problems.

So while one may not know the magnitude of the marginal returns to changing a choice

variable, one may know that these returns will be greater or smaller if a parameter is higher

or lower. And the tools for comparative statics tell us that this is the only sort of

information one needs. So, as economists, we can back away from restrictive assumptions

on the forms of functions and sets. When one is deriving comparative statics -- itself, in

essence, a qualitative exercise looking at the directions rather than magnitudes of change --

one need only have qualitative information.

III. Optimisation with One Choice Variable

In order to build intuition about what drives a comparative static result, it is a good

starting point to consider situations in which there is one choice variable. In notation, the

agent chooses a single variablex from X . Also, for simplicity, suppose that the agent

is maximising an objective function that is composed of a benefit minus a cost:

( ; ) ( ; ) ( )x B x C x= . 2

The intuition that guides comparative statics is a familiar one to economists. When

the marginal benefit to an activity is increased and there is no change in marginal cost, more

2 In principle, of course, the cost function C(.) could be a function of the parameter, or both benefits and

costs could be influenced by other parameters. In the application below this is the case. What is important

for the intuition here is that the parameter under examination interacts with benefits or costs but not both.

The theorems below will, for the most part, hold for more general payoff functions.


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of that activity will be undertaken by an optimising agent. In the above case, the benefit to

increasingx is: (for x x> )

B x B x B( ; ) ( ; ) ( ) .

Note that marginal cost ( C x C x( ) ( ) ) does not change as changes. So if for any higher

, B( ) has a higher value, then x*( ) will be monotone nondecreasing. In this case,

raising the parameter directly raises the marginal net return to doing x. Hence, after the

parameter change, doing more ofx will result in higher payoffs.

It is instructive to compare this monotone reasoning with more traditional methods

of conducting comparative statics. Suppose that B and C are twice continuously

differentiable and that Bxx < 0 and Cxx > 0 ,3 so that the objective function is globally

concave. Then an interior optimal choice is the solution to the first order condition:

B x C xx x( ; ) ( )* * = .

One could be tempted to reason then as follows: suppose that was increased. Then if

Bx 0, the left hand side of the above equation would be higher then the right hand side.

However, since the cost function is convex, raisingx actually results in an increase in

marginal cost. Hence, after the parameter change, the optimum, x* , is higher.4

This traditional reasoning is unsatisfactory because it focuses us on the convexity of

the cost function and away from the assumption that marginal benefits are higher as the

parameter is raised. The convexity assumption was made to justify the use of

manipulations of the first order condition. For an interior optimum to exist, the first order

condition must hold and will do so provided the appropriate boundary and continuity

assumptions are made. Local concavity of benefits and convexity of costs, ensures that the

condition is indeed a maximum (rather than a minimum or inflection point). The convexity

and concavity assumptions, therefore, seem innocuous and, hence, at first blush there

seems little reason to worry about using it.

3 Subscripts here denote partial derivatives.4

Use of the implicit function theorem leads to similar reasoning, as is demonstrated in the Appendix.


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But, to emphasise, these conditions play no actual role in guaranteeing the existence

of an optimum -- the continuity and boundary conditions are sufficient for that (Milgrom

and Shannon, 1994). They are only necessary to validate the traditional analysis, but with

it raises the importance of such assumptions in our intuition. Moreover, from a formal

point of view, the traditional approach cannot deal with situations in which optima are not

characterised by local concavity of benefits or convexity of costs. While this might not

seem restrictive when dealing with a single variable case, as I will demonstrate later for a

simple two variable case, concavity assumptions become unnecessarily restrictive and

traditional analyses based on them quite cumbersome. In contrast, monotone reasoning

does not require any assumptions on the convexity of costs or the concavity of benefits

making the methods easier to apply and interpret. Moreover, this reasoning does not

require that the objective function be continuous or differentiable, or that the choice variable

x be divisible.

To formalise the monotone methods approach to comparative statics, I will maintain

the assumption that choice variables are divisible and that objective functions are

continuously differentiable. The monotone methods approach involves assumptions

regarding the qualitative relationship between a parameter and the objective function or

constraint sets. As discussed above, these often amount to assumptions on the interaction

between the parameter and the marginal return to an activity. To see this, I need to

introduce the following terms. A continuously differentiable function f x( ; ) is said to

have nondecreasing (increasing) differences if fx 0 (fx > 0) and f x( ; ) has

nonincreasing (decreasing) differences if fx 0 (fx < 0).5

Suppose then that the

benefit function,B(.), above has nondecreasing differences in ( ; )x . Then it is also true

that the payoff function, (.) has nondecreasing differences in ( ; )x .6 This means that

5 The conditions here imply that the function f has nondecreasing (increasing) differences according to the

precise definition of Milgrom and Shannon (1994). That is, let f y f x y f x y( ) ( , ) ( , ) . Then f has

nondecreasing (increasing) differences in (x;y) if, for any >x x and >y y , f y f y( ) ( ) ( ) > .6 This is because

x x

B= .


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raising , raises the marginal return to doing more ofx. Intuitively, one might suppose

that this is sufficient to ensure that the optimal choice would not fall with . So long as the

constraint set does not change it turns out that this reasoning is correct and it holds for

conditions on general payoff functions such as (.).

Theorem 1 (Topkis). Suppose that X= . If (i) (.) has nondecreasingdifferences in ( ; )x and (ii) X does not vary with , then x* is monotonenondecreasing in .

If the solution is guaranteed to be an interior one, then increasing differences implies that

x* is monotone increasing in . Conversely, when has nonincreasing (decreasing)

differences, x

*

will be monotone decreasing in . .7

Thus, the theorem confirms that it is

marginal returns rather than assumptions of concavity or convexity that matter for

monotone comparative statics. Note, however, the condition on the constraint set. If the

constraint set moves in response to a change in an exogenous parameter, then it is critical

that its movement reinforces the conditions on the objective function. If changing raised

the marginal returns to x but shifted X downwards so that higher levels ofx became

infeasible, then x*

might be forced to fall despite the change in marginal returns. Such

situations are possible in consumer theory when a change in relative prices shifts the budget

set. To keep things simple for this paper, I have ruled out such possibilities.8

Before showing how this theorem is applied, it is useful to note an extension that

deals explicitly with payoff functions that take the form of benefits minus costs, as above.

7 The theorem can be generalised to require only ordinal conditions on payoffs. Therefore, it holds for

situations when an increase in a parameter raises the ratio of marginal benefits to marginal costs as opposed

to their difference. For the complete ordinal theory of comparative statics see Milgrom and Shannon (1994)

and Milgrom (1994).8

See Milgrom (1994) for a clear discussion of such issues.


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Corollary 1 (Milgrom and Shannon). Suppose that the conditions of Theorem 1hold and that ( ; ) ( ; ) ( )x B x C x= . Then the following two statements areequivalent:

(i) B has nondecreasing (increasing) differences in ( ; )x 9,10

(ii) x*

is monotone nondecreasing (increasing) in, for all C.

So while Theorem 1 provides a sufficient condition for monotone comparative statics, its

corollary provides a much stronger result that nondecreasing differences is the least

restrictive assumption we can place to generate monotone comparative statics. No weaker

assumption will do the trick.11

Application: Environmental Policy

To see how these results can be used, consider the situation of a government trying

to regulate pollution at the level of the firm. Suppose a firm produces two outputs -- its

intended good,y, and a pollutant,p. The firm has available two inputs -- oil, z, and other

physical inputs (e.g., capital and labour),x. The government has several instruments at its

disposal: (i) it can tax production ofp at a rate of; (ii) it can tax the use of oil at rate ; (iii)

it can tax final output at a rate of ; or (iv) it can promote the use of environmentally

friendly technologies,by raising (the efficiency of oil in production) and (an

abatement technology) Under these assumptions a typical firm maximises:

max ( ) ( ) ( ) ( , ; ) ( ; )( , )x z z x z xy P z P x p f z x P z P x g z1 1 1 + =

where Pz and Px are the prices of oil and the other input respectively, f is the production

function ofy and g is the production function ofp. It is assumed that marginal products are

non-negative (f fz x, 0 ) and that using more oil raises pollution levels ( gz > 0). The

9 If ( ; ) ( ) ( ; )x B x C x= then the corollary holds when (i) is replaced with a condition that C havenonincreasing (decreasing) differences in ( ; )x .10 The equivalence holds here because of our previous stated assumptions on the choice set and the

differentiability of the objective function. Then a non-negative cross partial derivative is necessary and

sufficient for nondecreasing differences to hold in the sense used by Milgrom and Shannon (1994).11 Condition (i) in the corollary is what is termed by Milgrom (1994) as a critical sufficient condition . It

is not a necessary condition because there could exist particular functional forms for the objective function

so that the comparative static may still hold for some changes in parameters even when this condition is


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technologies are assumed to be environmentally friendly. That means that they improve the

marginal productivity of oil in producing y (fz > 0), reduce the absolute level of p

produced ( g < 0), and reduce the marginal impact of oil on emissions ( gz < 0).

The key comparative statics questions regard what happens to the use ofz and x

and the level of pollution when there are changes in the governmental variables. Since only

single variable maximisation has been discussed thus far, suppose that for the moment,x is

fixed (i.e., it can only be changed in the long-run). In this case, to provide sufficient

conditions for comparative statics using Theorem 1 and necessary conditions using

Corollary 1, amounts to the same thing -- looking at the signs of mixed partial derivatives.

That is,

Raising Comparative Static Relevant Condition

Output Tax Rate, decreases z* z zf= < 0

Oil Tax Rate, decreases z* z zP= < 0

Emission Tax Rate, decreases z* z zg= < 0

Production Technology, increases z*

z zf= >( )1 0

Abatement Technology, increases z* z zg= > 0

Therefore, quite natural assumptions lead to the results that the taxation variables reduce the

quantity of oil used in production and hence, reduce both intended output and pollution.

However, the use of environmentally friendly technologies encourages the use of oil, while

having an ambiguous effect on the level of pollution.12

The point here is that the mathematics of these results make transparent the intuition

behind them. Each of the taxation variables has the effect of lowering the marginal return

to using more oil, or raising its marginal cost and hence, one would expect less oil to be

not satisfied. This is demonstrated in the Appendix. Nonetheless, it is the least restrictive qualitative

assumption that could be made.


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used by the firm (all other factors remaining equal). On the other hand, the technology

variables have the opposite effect. One () improves the marginal product of oil (i.e.,

fz > 0) while the other () reduces the marginal costs of oil in generating pollution (i.e.,

gz < 0). This latter variable becomes relevant precisely because pollution is being taxed.

If it were not for this, then the firm would not change its oil usage in response to the

technological change. Hence, there would be no effect on pollution as opposed to a

negative effect.13

Nonetheless, these results depend critically on the fact that x has been fixed.

Allowingx to be chosen makes the comparative statics analysis more complicated and in

the next section I will show how monotone methods make such analysis quite simple. For

the moment, some additional things can be noted. First, one can use the concepts of

nondecreasing differences and the like to demonstrate precisely what it means for two

actions to be complements or substitutes. Suppose thatx were changed as if it were an

exogenous parameter. Then what would happen to the optimal amount of oil used?

Theorem 1 shows that x and z will be complements (i.e., z x*( ) is nondecreasing) if

raising x does not lower the marginal product of oil (i.e., fzx 0) and that they will be

substitutes (i.e., z x*( ) is nonincreasing) if raisingx does not raise the marginal product of

oil (i.e., fzx 0). Ifx were also a choice variable and z were fixed, then x z*( ) is

nondecreasing (nonincreasing), if fzx 0 (fzx 0).

12 The form of the profit function here meets the conditions required for Corollary 1 to hold. Therefore, it

can be seen that each relevant condition is in fact the weakest that can be imposed to generate each of the

stated comparative static results.13 Note that these parameters can represent any policy instrument that increases the incentives of firms to

adopt technologies with these effects. These range from mandated use to subsidisation of research and

development. These conclusions on the introduction of environmentally friendly technologies and the

resulting level of pollution have not to my knowledge been explored or emphasised in the environmental

economics literature. One recent discussion by Rosenberg (1989) informally talks of this possibility,

however, no general theoretical exploration has been conducted. It should also be noted that if pollution

permits were in place, then the total level of pollution could be fixed. Such policies could reverse some of

the effects discussed here. However, for a more detailed assessment of this issue see Gans (1995).


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Using the precise definitions of complements and substitutes, it is possible to

analyse a long-standing question regarding the short and long-run demand for oil in

response to a price shock. Suppose there is an oil price shock (i.e., Pz rises) and suppose

thatx is fixed in the short-run. The question posed by Samuelson (1947) and answered

affirmatively was whether the long-run adjustment in oil use in response to the oil shock

would be greater than the short-run response? (See also Heal and Chichilnisky, 1991)

Note first that withx fixed, Theorem 1 tells us that the demand for oil will fall as its price

rises (because zPz = < 0). Ifz and x are complements, then z P xz*( , ) is decreasing in

Pz and increasing inx. Therefore, the price shock lowers z which in turn causes the firm

to wish to lowerx.14 This second effect causes a further reduction in the level of oil used.

So in the long-run, the demand for oil falls to a lower level than the short-run. But the

same conclusion also holds when z and x are substitutes. In this case, z P xz*( , ) is

decreasing in Pz andx. Therefore, while the price shock still lowersz, this leads to a long-

run increase inx. This, in turn, reduces the long-run demand for oil even further.

The point here is that for the cases of complements and substitutes, the long-run

adjustment after the price shock exceeds the short-run adjustment. But note that this

analysis also indicates that such conclusions are not possible if the production function is

such thatx andz are sometimes complements and sometimes substitutes. Milgrom and

Roberts (1994) show that this can lead to short-run adjustment exceeding long-run

adjustments. This case was neglected by standard treatments of this problem (e.g., Varian,

1992) until its analysis using the methods described in this paper.15

14 There is no first order effect of P

zonx since

xPz

= 0.15 It is also useful to note here that all the comparative statics results here are global as opposed to

local. That is, they do not rely on very small changes in the parameters leading to small changes in

choice variables. Thus, the price change here can be a true price shock (i.e., a large discrete change in the

price of oil) and the policy changes may represent radical changes in policies such as those that are

associated with changes in technology. See Milgrom and Roberts (1994) for more on this distinction.


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IV. Optimisation with Many Choice Variables

The traditional analysis of comparative statics when agents choose many variables

is very complicated. Keeping track of effects and ensuring that objective functions and

constraint sets are concave is extremely difficult and computationally complex. This is the

reason why most analyses confine themselves to the analysis of one or two endogenous

variables.

It turns out that an application of the methods of comparative statics introduced in

the last section can be readily applied to situations when there are many choice variables at

the disposal of agents. While it is true that this is more complicated than the single variable

case and that it is harder to generate monotone comparative statics, the gains in simplicity

and ease of application over traditional analyses are even more pronounced.

Suppose now that the agent can select many dimensions of an action, i.e.,

x x xI1 2, , ..., . The choice set, X Xi I i , now has I dimensions with Xi denoting the

choice set for variable i. These correspond to the familiar cases when consumers choose a

basket of goods, firms choose the levels of many factors, and players use multiple

instruments in games. Intuitively, when a parameter changes it might interact with more

than one choice variable. But these variables also interact with one another. Therefore, to

be able to get a monotone comparative statics all of these effects must reinforce each other.

Countervailing forces will mean that a monotone result is unavailable. Nonetheless, the

approach here can make precise the reasons for a failure of monotonicity, by showing the

analyst where the countervailing effects arise. Then it becomes an empirical matter whether

one effect is likely to be larger than another.

As in the previous section, what is critical is the impact parameter changes and

changes in actions have on the marginal returns to other actions. The concept of

nondecreasing differences formalised the notion that a parameter determined the marginal

return to an action. Here, however, for movements in choice variables to reinforce one

another, a related concept of how each affects the marginal return of the other is required.


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Basically, what is required is nondecreasing differences in both directions between every

pair of variables. A function f(x,y) is said to be (strictly) supermodular iff has

nondecreasing (increasing) differences in (x;y) and f has nondecreasing (increasing)

differences in (y;x). Nonetheless, when f is continuously differentiable this amounts to

assuming that fxy >( )0 for a function to be (strictly) supermodular. Therefore, in

practice, supermodularity turns out to be very similar to nondecreasing differences because

one focuses attention on the signs of mixed partial derivatives. Observe too that

supermodularity is very similar to the concept of complementarity discussed earlier. When

two choice variables are supermodular in an objective function, raising one raises the

marginal return to the other. This is another way of saying that two actions are

complementary.16

In problems considered in this section, if there is a change in an exogenous

parameter, that change might alter the marginal returns to one or more choice variables.

Suppose, however, that it raised (or at least did not lower) the marginal returns to every

choice variable. Then unless all those choice variables are complementary, there might be

countervailing effects that end up lowering the marginal returns to actions. In such cases

the comparative statics derived would be non-monotonic and hence, ambiguous.

Therefore, to generate unambiguous comparative statics predictions, all the interactive

effects must reinforce one another.

Theorem 2 (Topkis). Suppose that Xi , for all i. If (i) has nondecreasingdifferences in each ( ; )xi ; (ii) is supermodular in each ( ; ),x x i ji j ; and (iii) X

does not vary with , then each xi*

is monotone nondecreasing in .

Once again this holds for the increasing cases if the solution is an interior one. The reverse

holds for nonincreasing (decreasing) differences, so long as remains supermodular in the

choice variables.

16 Note that this is a slightly different use of the word complementarity to its price theoretic

interpretation. Often it is said that two inputs are complementary if raising the price of one leads to more

use of the other. The definition of complementarity here subsumes this commonly used price theoretic

interpretation. (See Milgrom and Shannon, 1994)


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Theorem 2 simplifies considerably the analysis of comparative statics. Previous

analyses based on the implicit function theorem involved the construction and inversion of

anIdimensional matrix and the addition of restrictive and wholly unnecessary conditions

to ensure concavity. Here all that need be done is to examine the pairwise interactions

between choice variables and parameters. Thus, using the methods here, one need only

find the appropriate mixed partial derivatives and sign them. This makes it possible to

analyse complex environments and obtain clean and intuitive results.

A stronger theorem is available for situations when payoff functions take the form

of a benefit minus costs.

Corollary 2 (Athey, Milgrom and Roberts). Suppose that conditions of Theorem2 hold. In addition, suppose that,

( ,..., ) ( ,..., ; ) ( )x x B x x C xI I i ii

I

1 1

1

= =

.Then the following two statements are equivalent:

(i) B is supermodular in each ( ; ),x x i ji j ; and has nondecreasing differencesin each ( ; )xi

(ii) Each xi*

is monotone nondecreasing in , for all Ci.

Once again, it can be seen that restrictions such as supermodularity and nondecreasing

differences are the minimal assumptions that must be imposed in order to generate

monotone comparative statics.

Application: Environmental Policy Again

Now one can return to the effects of environmental policy on firm behaviour but

this time relaxing the condition thatx is fixed. Looking to the influence of most parameters

the issue of whetherx andz are complements and substitutes is critical. This depends on

the sign of:

zx zxzx

zx

ff

f=

( )1

0

0

0

0if .

The following results can now be derived:


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Raising Comparative Static Relevant Conditions

Output Tax Rate, decreases z

*

decreases x*

z zf= < 0,

xf= < 0,

fzx 0

Oil Tax Rate, decreases z*

decreases (increases) x*

z zP= < 0 , x = 0 ,fzx ( )0

Emission Tax Rate, decreases z*

decreases (increases) x* z zg= < 0, x = 0,

fzx ( )0

Production Technology, increases z*

increases (decreases) x* z zf= >( )1 0 ,

x x zx

f f= ( ) ( ) , ( )1 0 0

Abatement Technology, increases z*

increases (decreases) x* z zg= > 0, x = 0,

fzx ( )0

Observe that the assumptions required to generate monotone comparative statics are more

restrictive than in the single variable case.17 However, in the case ofz at least, the

directions of the possible comparative statics are unchanged. Nonetheless, if the additional

assumptions underlying those results do not hold, it is possible that the optimal choice ofz

in response to a parameter change could move in the opposite direction. Consider the case

of a change in . This still causes a first order decrease inz andx. However, ifz andx

are substitutes (that is, fzx < 0, contrary to the condition stated in the table), then the

decrease in each variable will induce an increase in the other. Such second order effects

could outweigh the first and hence, the ultimate result of higher taxes on intended output

could be an increase in the use ofz and higher unintended pollution.

It is worth emphasising here that this additional complexity is relatively small

relative to the complexity of conducting similar analyses using traditional methods. In the

appendix, I demonstrate the use of traditional analysis so that its limitations can become

more transparent. That traditional analysis requires more assumptions and also more

algebra and cannot generalise to the non-differentiable, non-convex cases.

17 The table should be read carefully. The comparative statics results on the optimal z as a result of

changes in all the parameters but and , do not require the assumptions on complementarity or

substitutability ofz andx inf(.). These are only necessary for the results on the optimalx.


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In constrast to the analysis in the appendix, the assumptions underlying certain

results are laid bare using the methods of this paper (e.g., the criticality of the complements

case for changes in ). This focuses debate on the relevant conditions that determine

policy conclusions. More importantly, however, these methods allow a careful

consideration of how the mix of policies contributes to the achievement of the overall goals

of policy. As has been demonstrated in this paper in a very simple context, taxes on

pollutants and the subsidisation or encouragement of the use of environmentally friendly

technologies might be substitutes in achieving a reduction in pollution levels. Therefore,

governments should use one or the other policy instrument but not both. Nonetheless,

some other policies, such as the multiple use of tax instruments might be complements and

such policies ought to be used in conjunction with one another.18

V. Conclusions -- Monotone Thinking as Economics

This paper has barely scratched the surface on the available new theorems on

comparative statics. The single agent optimisation problems have been extended to

problems under certainty when agents maximise expected utility (Athey, 1995), to multi-

person decision contexts such as games (Milgrom and Roberts, 1990; Milgrom and

Shannon, 1994) and to other equilibrium notions (Milgrom and Roberts, 1994). Finally,

some of the methods presented here have proved useful beyond simple comparative statics

to problems of existence of social choice rules (Gans and Smart, 1995).

All of these theorems and the applications of them are indicative of a new focus on

monotone thinking in economics. Informal analysis has often described economic

predictions in terms of causes and effects. That is, one hears of an earthquake in Japan

causing construction damage, which requires funds for repairs, which will come from

Japanese capital overseas, which will flow into Japan pushing up exchange rates etc. The

methods described here provide a formal way of understanding such complex webs of

18 For an analysis of the interactions between policy instruments in generating a transition to a market


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18

cause and effect. But they do so in a familiar way, by focusing attention on marginal

returns rather than unnecessary conditions such as convexity and divisibility. This paper

introduces the barest essentials to apply the new tools. The promised fruit will come in

terms of new applications and results on previously unanalysable topics.

economy see Gates, Milgrom and Roberts (1995).


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Appendix: Traditional Comparative Statics with Many Variables

Looking at the environmental policy problem, using the implicit function theorem,

here I will analyse the effects on the optimal choices of oil, z* , and the other input, x* , of a

rise in output takes, . This method of comparative statics assumes that the conditions for

an interior optimum exist and then looks at the local behaviour of the optimal choices -- that

is, how these optimal choices move for small changes in the exogenous parameter.

Consider then the first order necessary conditions for an interior optimum:

z f P gz z z= =( )1 0

xf Px x= =( )1 0.

In order to apply the implicit function theorem, it must be assumed that the determinant of

the Jacobian does not equal zero anywhere. That determinant is:

zz zx

xz xx

zz zz zx

xz xx

zz zz xx zx

f g f

f ff g f f =

= ( )

( ) ( )

( ) ( )( ) ( )

1 1

1 11 2 2 .

For this to be non-zero, ( )f g f f zz zz xx zx >2 or ( )f g f f zz zz xx zx


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20

Given the concavity assumptions, complementarity betweenz andx is a sufficient but not

necessary condition for the signs of these derivatives to be negative. However, such

conditions are the easiest to interpret, especially as one moves to three endogenous

variables or more.

In summary, therefore, to ensure that the optimal choices ofz andx fall, locally,

with , the traditional analysis required that (i) fzx 0; (ii) f gzz zz 2 . In contrast, the monotone methods approach required only (i) to

provide a sufficient condition for a global comparative static result (using Theorem 2) and

tells us that this condition is a necessary condition (using Corollary 2) if we do not wish to

impose any additional assumptions such as (ii) to (iv). So monotone methods yields a

stronger result with fewer assumptions. Finally, we know from Milgrom and Shannon

(1994) that these results will also generalise easily to non-differentiable cases when choice

sets are not convex, something that cannot be said for the traditional analysis.


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References

Athey, S. (1995), Characterising Properties of Stochastic Objective Functions,

Discussion Paper, MIT.

Athey, S., P. Milgrom and J. Roberts (1996), Monotone Methods for ComparativeStatics Analysis, Princeton: Princeton University Press (forthcoming).

Gans, J.S. (1995), The Promotion of Environmentally Friendly Technologies, mimeo.,University of New South Wales.

Gans, J.S., and M. Smart (1995), Majority Voting with Single-Crossing Preferences, Discussion Paper, No.95/1, School of Economics, University of New SouthWales &Journal of Public Economics (forthcoming).

Gates, S., P. Milgrom and J. Roberts (1995), Complementarities in the Transition fromSocialism: A Firm-Level Analysis, in J. McMillan and B. Noughton (eds.),

Reforming Asian Socialism: The Growth of Market Institutions, Ann Arbor:University of Michigan Press.

Heal, G., and G. Chichilnisky (1991), Oil and the International Economy, Oxford:Oxford University Press.

Milgrom, P. (1994), Comparing Optima: Do Simplifying Assumptions AffectConclusions?Journal of Political Economy, 102 (3), pp.607-615.

Milgrom, P., and J. Roberts (1990), Rationalizability, Learning, and Equilibrium in

Games with Strategic Complementarities,Econometrica, 58 (6), pp.1255-1277.

Milgrom, P., and J. Roberts (1994), Comparing Equilibria, American EconomicReview, 84 (3), pp.441-459.

Milgrom, P., and C. Shannon (1994), Monotone Comparative Statics, Econometrica,62 (1), pp.157-180.

Rosenberg, N. (1989), Energy-Efficient Technologies: Past and Future Perspectives,reprinted in Exploring the Black Box: Technology, Economics, and History,Cambridge: Cambridge University Press, 1994, pp.161-189.

Samuelson, P.A. (1947), Foundations of Economic Analysis, Cambridge (MA): HarvardUniversity Press.

Shannon, C. (1992), Complementarities, Comparative Statics and Nonconvexities inMarket Economies, unpublished Ph.D. Thesis, Stanford University.

Topkis, D.M. (1978), Minimizing a Submodular Function on a Lattice, OperationsResearch, 26 (2), pp.305-321.

Varian, H. (1992),Microeconomic Analysis, 3rd ed., New York: Norton.

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