Labor productivity convergence in the Kansas farm sector:a three-stage procedure using data envelopment analysisand semiparametric regression analysis

Amin W. Mugera • Michael R. Langemeier •

Allen M. Featherstone

Published online: 17 June 2011

� Springer Science+Business Media, LLC 2011

Abstract This paper employs a three stage procedure to

investigate labor productivity growth and convergence in

the Kansas farm sector for a balanced panel of 564 farms for

the period 1993–2007. In the first stage, Data Envelopment

Analysis is used to compute technical efficiency indices.

In the second stage, labor productivity growth is decom-

posed into components attributable to efficiency change,

technical change, and factor intensity. The third stage

employs both parametric and semiparametric regression

analyses to investigate convergence in labor productivity

growth and the contribution of each of the three components

to the convergence process. Factor intensity and efficiency

change are found to be sources of labor productivity con-

vergence while technical change is found to be a source of

divergence. Policies that encourage investment in capital

goods may help to mitigate disparities in labor productivity

across the farm sector.

Keywords Labor productivity � Efficiency change �Technical change � Factor intensity � Convergence �Semiparametric regression

JEL Classification D24 � Q12

1 Introduction

The rise in US agricultural productivity over time has been

chronicled as the single most important source of economic

growth in the farm sector. This linkage between produc-

tivity and economic growth has stimulated interest in

empirical work trying to explain productivity changes (Ball

and Norton 2002). Numerous empirical studies have

examined labor productivity growth and the convergence

hypothesis for various countries and regions around the

world (Kumar and Russell 2002; Fare et al. 2006). How-

ever, fewer studies have examined convergence of agri-

cultural labor productivity around the world (e.g., Ball

et al. 2001; Gutierrez 2002) and total factor productivity

(TFP) in the United States (e.g., Ball et al. 1999). Previous

studies have not examined agricultural labor productivity

growth and convergence at the farm level.

Empirical work on productivity convergence in agri-

culture has generally used either cross-sectional or time-

series techniques. Some empirical studies have focused on

the decomposition of TFP into components attributed to

efficiency and technical change while others have focused

on convergence or divergence of TFP in agriculture (Ball

et al. 2001; Ball et al. 2004; Managi and Karemera 2004;

Liu et al. 2008).

Ball et al. (1999) decomposed aggregate productivity

growth for the US farm sector into its state-specific sources

and found that farm sector productivity growth is a func-

tion of productivity trends in the individual states. McCunn

and Huffman (2000) tested for both beta and sigma con-

vergence in state agricultural TFP growth rates and

examined the contributions of public and private research

A. W. Mugera (&)

Institute of Agriculture and School of Agriculture and Resource

Economics, Faculty of Agriculture and Natural Sciences,

The University of Western Australia M089, 35 Stirling Highway,

Crawley, WA 6009, Australia

e-mail: amin.mugera@uwa.edu.au

M. R. Langemeier � A. M. Featherstone

Department of Agricultural Economics, 342 Waters Hall,

Kansas State University, Manhattan, KS 66506, USA

123

J Prod Anal (2012) 38:63–79

DOI 10.1007/s11123-011-0235-1

and development (R&D) to convergence.1 The rate of beta

convergence was found to be variable and depended on

R&D state spillover, private R&D, and farmers’ education.

Ball et al. (2001) investigated the relative levels of farm

sector productivity for the United States and nine European

countries for the period 1973–1993. The study reported

convergence of TFP over the sample period and support for

the existence of a positive relationship between capital

accumulation and productivity growth. Ball et al. (2004)

investigated whether there was a tendency for TFP levels in

agriculture to converge across states and whether the

convergence rate could be explained by differences in the

rates of growth of factor intensities or by catching up.

The results were consistent with technological catch-up,

the range of TFP had narrowed over time and states with

lower initial levels of productivity grew more rapidly than

those with high initial levels of productivity. Rezitis (2005)

applied time series techniques to test for convergence in

agricultural factor productivity in the US and nine Euro-

pean countries. The study found evidence of convergence

among the sampled countries.

As noted above, previous studies have focused on pro-

ductivity at the state or country level. Agricultural pro-

ductivity growth at the national level emanates from

growth in productivity at the state levels that is primarily

driven by farm level growth. Ultimately, national level

productivity growth can be attributed to productivity

growth at the farm level. Likewise, productivity conver-

gence at the aggregate level is a result of convergence at

the micro level. Therefore, if convergence in productivity

levels at the national level is important, it is equally

important at the farm level.

The objective of this paper is to examine convergence of

labor productivity growth for a balanced panel of 564

farms in Kansas over the period 1993–2007 and two sub-

periods 1993–2002 and 1996–2005. This paper applies the

tripartite decomposition of labor productivity growth and

convergence analyses to the US farm sector. Labor pro-

ductivity growth is decomposed into efficiency change,

technical change, and factor intensity and the relative

contribution of each of those three components to pro-

ductivity growth convergence in the farm sector is inves-

tigated. A three stage procedure is used to test for

convergence. In the first stage, the nonparametric produc-

tion function approach is used to compute technical effi-

ciency indices. In the second stage, the approach by Kumar

and Russell (2002) is employed to decompose labor pro-

ductivity growth into three components: efficiency change,

technical change, and factor intensity. The third stage

regression analysis investigates whether there is conver-

gence in the growth rate of labor productivity, and the

contributions of each of the three components to this

process.

Convergence implies that labor productivity growth has

been particularly rapid on farms that had low initial pro-

ductivity levels. Identifying sources of productivity per

worker and their relative contribution to the narrowing of

the gap in productivity growth across the farm sector has

important policy implications. For example, policies to

promote the attraction of capital (factor intensity) are likely

to have different economic outcomes than policies to

improve efficiency (technological catch-up).

2 Methodology

A three-stage procedure is used to estimate productivity

change indices and test for convergence. The first stage

involves estimation of technical efficiency indices. The

second stage applies the Kumar and Russell (2002) to

decompose labor productivity growth into three compo-

nents: efficiency change, technical change, and capital

deepening. The third stage employs regression analysis to

explore catching-up and the relative contribution of each of

the three components to this process.

2.1 Computing efficiency indices using data

envelopment analysis

This article follows the approach by Henderson and

Zelenyuk (2007) to define the underling production tech-

nology, except that an input-orientation is used. For each

farm i (i = 1, 2,…, n), the period-t input vector is xti ¼

Kti ; L

ti

� �where Kt

i is physical capital and Lti is labor. Letting

yti be a single output for farm i in period t, the technology

for converting inputs for each farm i in each time period t

can be characterized by the technology set:

Tti � xt

i; yti

� �jcan produce yt

i

��: ð1Þ

The same technology represented in Eq. 1 can be

characterized by the following input requirement set:

Cti yt

i

� �� xt

ijxti can produce yt

i

� �; xt

i 2 <2þ: ð2Þ

We assume that the technology follows standard

regularity assumptions under which the Shephard (1970)

input-oriented distance function can be represented as:

Dti xt

i; yti

� �¼ sup

hh[ 0 : xt

i=h� �

2 C yð Þ� �

8yti 2 <1

þ: ð3Þ

This gives the complete characterization of the

technology for farm i in period t:

1 Beta convergence occurs when the partial correlation between

growth in productivity over time and its initial level is negative.

Sigma convergence occurs when the dispersion of productivity

growth across a group of firms, states, or economies falls over time.

64 J Prod Anal (2012) 38:63–79

123

Dti xt

i; ytijCt

i yti

� �� �� 1, xt

i 2 Cti yt

i

� �: ð4Þ

Equation 4 is simply the ratio of minimal (or potential)

input to actual input that can produce the same amount of

output. The Farrell input-oriented technical efficiency

measure can be defined as:

TEti � TEt

i xti; y

tijCt

i yti

� �� �

¼ supremum h[ 0jxti

�h 2 Ct

i yti

� �� �8yt

i 2 <1þ: ð5Þ

A farm is considered to be technically efficient when

TEti ¼ 1 and technically inefficient when 0\TEt

i\1: The

true technology and input sets are unknown and thus the

individual value of technical efficiency must be estimated

using either the nonparametric (data envelopment analysis)

or parametric (stochastic frontier analysis) techniques.

Given the production technology in Eq. 5, we use linear

programming to estimate the input distance function. The

Farrell input-based efficiency index for farm i at time t is

defined as:

e xti; y

ti

� �¼ min hj xt

i=h; yti

� �2 Tt

� �: ð6Þ

In the above equation, yit is output (gross farm income)

and xit, represent two inputs, capital (K), and labor (L). The

subscript i refers to an individual farm and the superscript t

represents the individual time period. The efficiency index

value for each farm is found using the following linear

program:

Minimizeh;z1;...;z j

h

subject to

Yi�P

k

zkYtk

hKi�P

k

zkKtk

hLi�P

k

zkLtk

zk� 08k:

8>>>>>><

>>>>>>:

9>>>>>>=

>>>>>>;

;ð7Þ

where h is the efficiency measure to be calculated for each

farm i at time t, and zk is the intensity variable for farm i.

The above model assumes constant return to scale (CRS).

Constant returns to scale suggest that all firms operate at an

optimal scale. However, imperfect competition and finan-

cial constraints may cause farms to operate below optimal

scale (Coelli et al. 2005). AddingPk

k¼1 Zk ¼ 1 to the

constraints in the above model imposes variable returns to

scale (VRS) while the equationPk

k¼1 Zk\1 imposes

decreasing returns to scale (DRS), respectively.

2.2 Tripartite decomposition of labor productivity

Following Kumar and Russell (2002), labor productivity

growth is decomposed into (1) efficiency change (movement

towards or away from the best-practice frontier), (2) tech-

nical change (shift of the best-practice frontier), and (3)

factor intensity (movement along the best-practice frontier).

The key relation of the tripartite decomposition is:

Dy ¼ DEFF� DTECH� DKACC ð8Þ

where Dy ¼ yc=yb is the relative change in output per

worker between the current period c and the base period b.

Data envelopment analysis is used to estimate the best-

practice frontier and the associated efficiency levels for

individual farms assuming a common technology for all

farms in each period. The nonparametric deterministic

methods require no specification of the functional form

of the technology or any assumption about the market

structure or absence of market imperfections. Malmquist

productivity indexes are used to compute the relative per-

formance of each farm to the best practice frontier by

decomposing productivity into efficiency change and

technical change before decomposing labor productivity

growth into the tripartite components. Details about this

approach are documented in Kumar and Russell (2002) and

Fare et al. (2006).

2.3 Convergence hypothesis analyses using regression

analysis

Recent efforts to understand economic growth have

focused on the tendency for productivity growth rates to

converge or diverge across countries or regions. Based on

the tripartite breakdown, this study quantifies the contri-

bution of efficiency change, technical change, and capital

accumulation to the convergence of labor productivity

growth. The classical analytical method of b-convergence

analysis is employed by running separate cross-sectional

regressions of labor productivity growth and the growth of

each of the three components on the logarithm of the initial

labor productivity level for each farm.

In the context of this study, the concept of b-conver-

gence builds on the notion that average farms that are

initially low in productivity will experience faster pro-

ductivity growth. The empirical test thus builds on a

regression of productivity growth on initial productivity. A

negative correlation between the two periods provides an

indication of convergence, because it suggests that farms

with relatively low initial labor productivity levels catch-up

to those with high initial productivity. This is based on the

idea that imitation is easier than innovation and farms that

initially lag behind should enjoy a more rapid growth rate

than advanced farms.

Linear regression is the most widely used tool in the

analysis of convergence. However, despite the popularity

of linear regression models, a more robust specification is

called for in some situations where there is a linear

relationship between (the mean of) the dependent and

some of the independent variables (denoted here as X) but

J Prod Anal (2012) 38:63–79 65

123

the effect of other independent variables (denoted here as

Z) on the dependent variable is unknown, or not necessarily

linear in Z. The semiparametric specification allows for a

regression function that maintains linearity in some of the

independent variables (X) but the functional form of the

parameters with respect to the other variables (Z) is not

known. Two semiparametric models are considered: the

partial linear model (PLM) and the smooth coefficient

model (SCM). In both cases, farm size is nonparametric

and enters the semiparametric model as a categorical var-

iable (very small farms, small farms, medium sized farms,

and large farms).

In empirical work, it is important to test whether the

parametric model or the semiparametric model is an ade-

quate description of the data. Varieties of methods exist for

testing for correct specification of parametric regression

models (Hardle and Mammen 1993; Horowitz and Hardle

1994; Horowitz and Spokoiny 2001; Hsiao et al. 2007). This

study follows Hsiao et al. (2007) that allow for the mix of

both continuous and categorical data types often encoun-

tered in applied settings (for details, see Hsiao et al. 2007).

2.3.1 Linear regression model

The linear regression model summarizes the relationship

between an outcome variable y and vectors X and Z through

a linear mean regression where the mean of y is modeled as

a linear function of both X and Z. In this paper, X is a vector

of continuous variables and Z is a vector of categorical

variables that measure the effect of farm size. The ordinary

least square (OLS) linear regression models the relation-

ship among the dependent variable and the explanatory

variables as follows:

yi ¼ Xibx þ Zibz þ ei ð9Þ

where y is a vector of a dependent variable, X and Z are

matrices of the levels of independent variables, b’s are

vectors of the regression coefficients, and e is a vector of

random errors that are assumed to follow a normal distri-

bution with zero mean and constant variance.

From the general regression model in Eq. 9, the fol-

lowing specific regressions are used to estimate conver-

gence/divergence2:

yi ¼ aþ b ln yio þ bzZ þ ui ð10Þ

yEFFi ¼ aEFF þ bEFF ln yio þ bzZ þ uEFFi ð11Þ

yTECHi ¼ aTECH þ bTECH ln yio þ bzZ þ uTECHi ð12Þ

yKACCi ¼ aKACC þ bKACC ln yio þ bzZ þ uKACCi: ð13Þ

The independent variables in the above equations are

the natural logarithm of initial labor productivity (ln yio)

and binary variables to control for farm size. In this

case, farm size is allowed to affect the model’s intercept

only. The dependent variables for Eqs. 10–13 are annual

rate of growth of labor productivity, annual contribution

of efficiency change, annual contribution of technical

change, and annual contribution of capital accumulation,

respectively. A significant negative coefficient on initial

labor productivity would indicate convergence and a

positive coefficient would indicate divergence. As shown

by Delgado-Rodriguez and Alvarez-Ayuso (2008), the

total convergence parameter is the sum of parameters of

convergence of the individual estimates:

b ¼ bEFF þ bTECH þ bKACC: ð14Þ

2.3.2 The partial linear model

Following Li and Racine (2007), a partial linear model

(PLM) consists of two additive components, a linear

parametric and a nonparametric part:

Yi ¼ a Zið Þ þ X0ibþ ei: ð15Þ

In the above model, X0ib is the parametric component,

a Zið Þ is the nonparametric component whose functional

form is not specified, and ei denotes an error term with zero

mean and common variance. Taking the expectation of

(15) conditional on Zi, yields:

E YijZið Þ ¼ E XijZið Þ0bþ a Zið Þ: ð16Þ

Subtracting (16) from (15) to eliminate the unknown

function a Zið Þ yields:

Yi � E YijZið Þ ¼ Xi � E XijZið Þð Þ0bþ ei: ð17Þ

Using the shorthand notation, ~Yi ¼ Yi � E YijZið Þ and~Xi ¼ Xi � E XijZið Þ; and applying the least square methods

to (17), the following estimator of b is obtained:

binf ¼Xn

i¼1

~Xi~X0i

" #�1Xn

i¼1

~Xi~Yi: ð18Þ

Note that subtracting (16) from (15) introduces two new

unknown functions, E YijZið Þ and E XijZið Þ; and therefore

the above estimator of binf is infeasible. This differencing

allows inference to be made on b as if there were no

nonparametric components in the model. The unknown

conditional expectations can be consistently estimated

2 When the average annual growth rate of labor productivity is

regressed against initial labor productivity, the fit of this regression

tests for absolute (unconditional) b-convergence. When other control

variables are included in the regression, the regression tests for

conditional (relative) b-convergence. Kansas farms are heterogeneous

and binary variables are included in the regression models to control

for farm size.

66 J Prod Anal (2012) 38:63–79

123

using kernel methods. Replacing the unknown conditional

expectations that appear in binf (i.e., ~Y and ~X) with their

kernel estimators enables a feasible estimator of b to be

obtained.

To do this, let:

Yi � E YijZið Þ def n�1Xn

j¼1

YjKh Zi; Zj

� �=f Zið Þ;

Xi � E XijZið Þ def n�1Xn

j¼1

XjKh Zi; Zj

� �=f Zið Þ; where

f Zið Þ ¼ n�1Xn

j¼1

Kh Zi; Zj

� �and Kh Zi; Zj

� �

¼Yq

s¼1

h�1s k

zis � zjs

hs

ð19Þ

The term k (.) is a univariate kernel function and hs a

smoothing parameter. Therefore, ~Yi and ~Xi in binf can be

replaced by Yi � Yi and Xi � Xi and the problem solved

using the least squares method. The detailed derivation of

the asymptotic distribution of the feasible estimator of b, is

documented in Yatchew (2003) and Li and Racine (2007).

It is important to note that in the PLM, the intercept cannot

be identified separately from the unknown function a Zið Þbecause the functional form of the unknown function is not

specified.

2.3.3 Smooth coefficient model

Following Li et al. (2002) and Li and Racine (2007), a

smooth coefficient model (SCM) nests a PLM and is

specified as:

yi ¼ a Zið Þ þ X0

ib Zið Þ þ ei: ð20Þ

In the above equation, b (Zi) is a vector of unspecified

smooth functions of Zi. When b (Zi) = b, the model

reduces to a PLM. The main difference between the PLM

and the SCM is that the PLM assumes the slope

coefficients b are invariant to the nonparametric com-

ponent, Zi. In contrast, the SCM allows the nonparametric

variable to affect the slope coefficient b. The semi-

parametric SCM allows more flexibility in functional

forms than the PLM or the parametric OLS models.

At the same time, it avoids much of the ‘curse of

dimensionality’ problem, as the nonparametric functions

are restricted only to part of the variable Z. In other

words, the SCM lets the marginal effect of a given

variable be represented as an unknown function of an

observed covariate. Instead of restricting the marginal

effect of y with respect to X to be constant and equal to a

parameter b, the SCM writes this marginal effect as an

unknown of some explanatory variable, say Z (Koop and

Tobias 2006). The model in Eq. 20 can be expressed more

compactly as:

yi ¼ a Zið Þ þ X0

ib Zið Þ þ ei ¼ ð1; x0

iÞa Zið Þb Zið Þ

!

þ ei

� X0

id Zið Þ þ ei; ð21Þ

where d Zið Þ ¼ a Zið Þ; bðZiÞð Þ0� �0

and d Zið Þ is a vector of

smooth but unknown functions of Zi. Li et al. (2002)

proposed the following local least squares method to

estimate d Zið Þ:

d Zið Þ ¼ nhqð Þ�1Xn

j¼1

XjX0

jKZj � Z

h

" #�1

� nhqð Þ�1Xn

j¼1

XjyjKZj � Z

h

( )

� Dn Zð Þ½ ��1An Zð Þ

ð22Þ

where Dn Zð Þ ¼ nhqð Þ�1Pn

j¼1

XjX0jK

Zj�Zh

� �;An Zð Þ ¼

nhqð Þ�1Pn

j¼1

XjyjKZj�Z

h

� �( )

; K (.) is a kernel function, and

h ¼ hn: The theorem that establishes the consistency and

asymptotic normality of d Zið Þ are proven in Li et al. (2002)

and Li and Racine (2007).

2.4 Sigma convergence

An alternative approach to convergence analysis is the

Sigma convergence (r-convergence) test which examines

changes over time in the dispersion of the variable of

interest, in our case, labor productivity growth. Sigma

convergence (r-convergence) is considered to have

occurred when the dispersion of labor productivity growth

among farms diminishes over time. The b-convergence is a

necessary but not sufficient condition for r-convergence

because b-convergence could occur due to farms with

initial lower productivity growing at such rapid rates that

they end up having higher productivity growth at a later

period than farms with higher initial productivity growth.

To test for r-convergence, we use changes in the variance

of productivity growth across farms to measure changes in

labor productivity growth dispersion. Following McCunn

and Huffman (2000), the basic model is defined as follows:

Variance(LPG)t¼/1 þ /2Trendþ et ð23Þ

where LPG is labor productivity growth across farms in

period t, /1 and /2 are parameters to be estimated, and et is

a zero-mean random disturbance term. A significantly

negative coefficient associated with the time variable,

Trend, implies r-convergence.

J Prod Anal (2012) 38:63–79 67

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3 Description of data

Data for this study are from the Kansas Farm Management

Association database (Langemeier 2010). We used a bal-

anced panel of 564 farm households for the 1993–2007

period and two sub-periods: 1993–2002 and 1996–2005.

The period 1993–2007 captures labor productivity

dynamics from the beginning and the end of sample. The

1993–2002 period represents a 10 year window when real

output was low, relative to the entire sample, while the

1996–2005 period represents a window when real output

was high (see Appendix Fig. 5). The sub-periods also

reflect the different policy environments producers oper-

ated in. The passage of the 1996 Federal Agricultural

Improvement and Reform (FAIR) Act introduced sub-

stantial changes in the overall farm policy environment in

that price support measures and deficiency payments were

reduced in favour of production flexibility contract pay-

ments. As noted by Serra et al. (2005), the Act had a sig-

nificant effect, not only on farm output price and subsidy

levels, but also on farm income variability and household

wealth. This could have altered the off-farm work deci-

sions3 that have implications on labor productivity.

Therefore, the 1996–2005 period depicts productivity

changes post 1996 FAIR Act while the 1993–2002 depicts

the pre and transition period. A 10 year window is chosen

to capture any effects of technological innovation because

productivity response to new technology unfolds over time.

The 1997–2006 period was not chosen because both rep-

resent unusual years; 1997 was followed by a slump in

output while 2006 was followed by a remarkable increase

in real output.4 Given the large change in productivity from

2006 to 2007, this will be illustrated later in this paper, the

change in output from 2006 to 2007 warrants further

explanation. Wheat and soybean yields were similar in

these two years; however, corn and grain sorghum yields

were 20 and 36% higher, respectively, in 2007 (USDA-

NASS 2010). Also, it is important to note that crop prices

were in general higher in 2007 than they were in 2006.5

In the first stage, input and output data are used to

compute the technical efficiency indices. One output is

used, gross farm income, which is the summation of gross

crop income and gross livestock income. Nominal GFI is

deflated by the Personal Consumption Expenditure Index,

with 2007 as the base year. Two inputs are used, capital

and labor. Real capital is calculated in the following

manner. First, total capital is calculated as the sum of

annualized asset charges and purchased inputs.6 Second, a

deflator is constructed using the price indices for purchased

inputs (Purinp) and annualized asset charges (Capp) by

farm and year, with 2007 as the base year7:

deflator =Purchased Inputs

Total Capital

� Purinp

þ Assets Charges

Total Capital

� Capp ð24Þ

Third, estimates of real capital by farm and year are

computed by dividing the nominal capital by the deflator:

Real Capital ¼ Assets Chargesþ Purchased Inputs

deflator

¼ Nominal Capital

deflatorð25Þ

Labor is measured as the number of farm workers per

farm per year. To obtain this value, we first deflate the total

annual cost of labor (includes hired and unpaid labor) by

the labor price index with 2007 as the base year. This value

is then divided by the average annual salary of a farm

worker assuming a 40 h work week, 48 work weeks in a

year, and average hourly wages. As with other studies of

productivity decomposition, no attempt is made to account

for quality differences in inputs because of lack of such

information from the data. For reporting results, the

observations in the sample are grouped according to farm

size. Distinguishing farm sizes provides additional insight

on differential productivity growth across the farm sector.

Farms are disaggregated into four groups: (1) very small

farms (VSF) are farms with less than $100,000 in GFI; (2)

small farms (SF) include farms with GFI between $100,000

and $250,000; (3) medium farms (MF) include farms with

GFI between $250,000 and $500,000; and (4) large farms

(LF) are farms with GFI above $500,000. The sample sizes3 Serra et al. (2005) found that the introduction of fixed decoupled

payments in 1996 may have reduced the likelihood of off-farm labor

participation. However, government payments that reduce farm

household participation to non-farm markets increased after 1996

thus motivating a higher participation in on-farm labour markets.4 The period 1997–1998 were El Nino years. El Nino winters are

expected to be wet in the Great Plains but this period was very dry.

The November-January period was one of the driest on record in

Kansas and winter wheat deteriorated badly with the worsening

drought.5 Comparing 2006–2007, corn prices increased from 2.41 to 3.63 per

bushel, grain sorghum prices increased from 4.14 to 6.29, soybean

prices increased from 5.51 to 7.84, and wheat prices increased from

4.38 to 5.92.

6 Asset charges include repairs, rental charges for land and machin-

ery, auto and conservation expense, cash interest, real estate and

property taxes, general farm insurance, depreciation, and opportunity

interest charged on owned equity. Purchased inputs include fuel and

oil, seed, fertilizer and lime, chemicals, feed, utilities, and crop

insurance.7 The purchased input price (Purinp) is a weighted average of USDA

agricultural price indices for fuel, seed, fertilizer, services, chemicals,

and feed. The asset charges (Capp) price is a weighted average of

USDA agricultural price indices for repairs, services, rent, and

interest prices. The weights are determined using cost shares.

68 J Prod Anal (2012) 38:63–79

123

within each group are not constant over time; some farms

moved from one class to another as farm income increased

or decreased.

The data are used in the first and second stages to

compute annual growth rates of labor productivity, effi-

ciency change, technical change, and factor intensity for

the period 1993–2007 and two sub-periods, 1993–2002 and

1996–2005. The third stage uses the productivity measures

to test for convergence. Three binary variables for very

small farms, small farms, and medium sized farms are used

to control for farm sizes in the parametric model. Large

farms serve as the reference farms.

4 Empirical results

4.1 Production frontier and efficiency

The best-practice production frontiers for the sampled

farms, from the beginning (1993) and end of sample period

(2007), along with the scatter plots for the output per

worker versus capital use per worker, are presented in

Fig. 1. The 2007 best-practice frontier, indicated by long

dash lines, is superimposed on the 1993 best-practice

frontier, indicated by the solid lines. The upward shift of

the frontier between the two periods indicates technical

change and the kinks on each frontier represent technically

efficient farms for the specified year. It is evident from

these frontiers that technology change is non-neutral. The

best-practice frontiers shifted upwards between the two

periods but not by the same proportion for every value of

the factor-labor ratio. This result is consistent with the

observations by Managi and Karemera (2004) that rejected

Hicks neural technological change in the US agriculture.

To assess the efficiency of each farm, we examined their

location relative to the frontiers for the entire sampled

period. The mean efficiency scores for each year under

CRS, VRS, and DRS are reported in Table 1. On average,

the sample farms moved further from the best practice

0

500000

1000000

1500000

2000000

Out

put p

er W

orke

r

0 500000 1000000 1500000

Factor use per Worker

1993 Best Practice Frontier2007 Best Practice Frontier

Fig. 1 Empirical best-practice

frontiers for sampled farms,

1993 and 2007

Table 1 Technical efficiency scores under VRS, DRS and CRS

models

Years Eff. score Eff. score Eff. score

VRS DRS CRS

1993 0.6250 0.6151 0.5833

1994 0.6242 0.6072 0.5844

1995 0.5770 0.5627 0.5326

1996 0.6096 0.5790 0.5545

1997 0.6223 0.6119 0.5928

1998 0.6122 0.6022 0.5858

1999 0.5628 0.5344 0.4936

2000 0.6386 0.6243 0.5991

2001 0.6447 0.6167 0.6008

2002 0.5768 0.5529 0.5463

2003 0.5297 0.5154 0.4767

2004 0.6232 0.5956 0.5748

2005 0.5159 0.4993 0.4657

2006 0.5563 0.5307 0.5032

2007 0.5699 0.5434 0.5229

Mean 0.5925 0.5727 0.5478

Technical efficiency scores under variable returns to scale (VRS),

non-increasing returns to scale (DRS) and constant returns to scale

(CRS) technologies

J Prod Anal (2012) 38:63–79 69

123

frontiers over the 1993–2007 period. The largest efficiency

gains were achieved in the 2001/2002 period and the

lowest gains in 2005. We do not know whether the failure

to catch-up is because the frontier is shifting over time or if

most producers are not able to access the existing tech-

nologies, or a combination of both.

However, our results are consistent with those that Serra

et al. (2008) observed that for Kansas farms, an increase in

decoupled payments would increase farms’ technical

inefficiencies because decoupled payments are government

transfers not linked to production or yield. Because higher

production yields are not receiving premiums, producers

may not have the incentives to produce the maximum

attainable output. They also observe that farmers may

respond to a decline in price supports by reducing the

efficiency with which they operate.

4.2 Labor productivity decomposition

Table 2 reports the mean growth rates of labor productivity

and its three components by farm size—efficiency change,

technical change, and factor intensity—for the three peri-

ods 1993–2007, 1993–2002, and 1996–2005. Comparing

the results for each sub-period, it is evident that produc-

tivity growth was high in the 1996–2005 sub-period

(2.09%) due to the high rate of advancement in technology

(1.79%) and the high rate of factor intensity (2.18%). In

contrast, productivity growth was low (1.34%) from 1993

to 2002 mainly due to the slow rate of technological

advancement (0.29%) and factor intensity (1.91%). The

results by farm size provide strong evidence that produc-

tivity growth varies by farm size. While deterioration in

efficiency was the norm in the three periods, medium sized

and large farms achieved efficiency gains in the low pro-

ductivity period, 1993–2002. The annual rate of growth in

productivity was high from 1993 to 2007 (4.67%) com-

pared to 1993–2002 (1.34%) and 1996–2005 (2.09%). This

suggests that rapid productivity growth occurred between

2003 and 2007 to drive up the average annual growth rates

from the average of 1.34% in the 1993–2002 period to

4.67% in the 1993–2007 period. Very small farms

achieved higher growth rates in technical change from

1993 to 2002 and 1996 to 2005 (0.35 and 1.97%) compared

to small farms (0.14 and 1.86%) and medium sized farms

(0.25 and 1.64%). For the 1993–2007 period, there is a

positive correlation between the rates of growth in factor

intensity and labor productivity (0.823), factor intensity

and technical change (0.321), and factor intensity and

efficiency change (0.100). Those relationships lend support

to the hypotheses that technological innovation and

improvement in efficiency are embodied in capital deep-

ening (factor intensity). Ta

ble

2G

row

tho

fla

bo

rp

rod

uct

ivit

yan

dth

etr

ipar

tite

dec

om

po

siti

on

com

po

nen

tsfo

rse

lect

edp

erio

ds

Per

iod

Pro

du

ctiv

ity

gro

wth

(gY

)E

ffici

ency

chan

ge

(gE

FF)

Tec

hn

ical

chan

ge

(gT

EC

H)

Fac

tor

inte

nsi

ty(g

KA

CC)

19

93

–2

00

71

99

3–

20

02

19

96

–2

00

51

99

3–

20

07

19

93

–2

00

21

99

6–

20

05

19

93

–2

00

71

99

3–

20

02

19

96

–2

00

51

99

3–

20

07

19

93

–2

00

21

99

6–

20

05

All

4.6

71

.34

2.0

9-

0.9

2-

0.7

9-

1.8

72

.59

0.2

91

.79

3.0

01

.91

2.1

8

VS

F0

.92

-1

.20

-0

.27

-2

.98

-2

.35

-4

.04

2.1

40

.35

1.9

71

.75

0.8

01

.81

SF

3.8

11

.52

1.8

5-

1.1

1-

0.6

9-

2.0

62

.20

0.1

41

.86

2.7

22

.07

2.0

6

MF

5.6

33

.23

3.0

1-

0.3

40

.32

-0

.97

2.6

30

.25

1.6

43

.34

2.6

62

.33

LF

7.0

34

.01

4.0

7-

0.0

71

.17

-0

.28

3.3

60

.42

1.6

13

.75

2.4

22

.74

All

all

farm

s,V

SF

ver

ysm

all

farm

s,S

Fsm

all

farm

s,M

Fm

ediu

msi

zed

farm

s,an

dL

Fla

rge

farm

s.T

he

rep

ort

edes

tim

ates

are

gro

wth

rate

san

dth

efo

llo

win

geq

ual

ity

ho

lds:

gY

=g

EF

F?

gT

EC

H?

gK

AC

C

70 J Prod Anal (2012) 38:63–79

123

4.3 Regression analysis results

This section uses three empirical models (OLS, PLM, and

SCM) to explore the relationship between initial levels of

labor productivity and the annual growth rates of labor

productivity, efficiency change, technical change, and

factor intensity. Table 3 presents the estimated results from

1993 to 2007. The columns marked (1) show the results

when the dependent variable is the average annual growth

rate of labor productivity for the OLS, PLM, and SCM

models, respectively. The coefficient on initial labor pro-

ductivity is negative (slope) and comparable across the

three models (-5.230, -5.224, and -5.067). This suggests

that, on average, farms that had lower initial labor pro-

ductivity levels achieved higher annual growth rates in

labor productivity relative to those that had higher initial

labor productivity levels. The slope and dummy variable

coefficients for the OLS model are all statistically signifi-

cant at the 1% significance level. The large farms are the

reference farms so coefficients on the binary variables

provide estimates of the difference in the speed of con-

vergence relative to large farms. Values of the binary

coefficients indicate that the speed of convergence is

inversely correlated with farm size. The speed of con-

vergence8 for very small farms (-0.155), small farms

(-0.122), and medium sized farms (-0.080) is faster

relative to large farms (Appendix Table 7 provides a

summary of speed of convergence for all parameters).

Comparing the in-sample fit across the three models, the

SCM had a higher adjusted R-squared (R2 = 0.501) com-

pared to the PLM (R2 = 0.497) and the OLS model

(R2 = 0.494). The SCM also had narrower residual stan-

dard errors (8.712) compared to the PLM (8.781) and the

OLS model (8.847). Therefore, the additional flexibility

offered by allowing the initial productivity parameter to

vary with respect to farm size increased the fit of the

model.

The columns marked (2) show the results when the

dependent variable is the average annual growth rate of

the efficiency indices. The coefficient on initial labor

Ta

ble

3R

egre

ssio

nre

sult

sfo

rg

row

thin

ou

tpu

tp

erw

ork

er,

effi

cien

cych

ang

e,te

chn

ical

chan

ge,

and

fact

or

inte

nsi

tyo

nin

itia

lg

row

thin

ou

tpu

tp

erw

ork

erfo

r1

99

3–

20

07

Ord

inar

yle

ast

squar

em

odel

Par

tial

linea

rm

odel

Sm

ooth

coef

fici

ent

model

(1)

gY

(2)

gE

FF

(3)

gT

EC

H

(4)

gK

AC

C

(1)

gY

(2)

gE

FF

(3)

gT

EC

H

(4)

gK

AC

C

(1)

gY

(2)

gE

FF

(3)

gT

EC

H

(4)

gK

AC

C

Inte

rcep

t69.2

70***

(3.4

52)

32.2

37***

(2.1

86)

-7.7

39***

(0.9

71)

44.7

72***

(2.8

92)

63.5

30

30.0

62

-9.0

98

41.5

49

Slo

pe

-5.2

30***

(0.2

99)

-2.7

15***

(0.1

83)

0.9

33***

(0.0

81)

-3.4

48***

(0.2

42)

-5.2

24

(0.2

98)

-2.7

12

(0.1

83)

0.9

33

(0.0

81)

-3.4

40

(0.2

42)

-5.0

67

-2.6

70

0.9

99

-3.3

00

D-V

SF

-9.2

94***

(0.4

55)

-4.5

53***

(0.2

79)

-0.6

51***

(0.1

24)

-4.0

90***

(0.3

69)

D-S

F-

5.2

54***

(0.3

60)

-2.0

91***

(0.2

20)

-0.7

95***

(0.0

98)

-2.3

69***

(0.2

92)

D-M

F-

2.3

34***

(0.3

51)

-0.7

53***

(0.2

15)

-0.5

66***

(0.0

96)

-1.0

15***

(0.2

85)

Sd(R

esid

)8.8

47

3.3

21

0.6

55

5.8

15

8.7

81

3.2

95

0.6

49

5.7

87

8.7

12

3.2

39

0.6

31

5.7

27

Adj.

R2

0.4

94

0.4

00

0.3

58

0.2

98

0.4

97

0.3

39

0.3

62

0.3

00

0.5

01

0.4

09

0.3

80

0.3

08

JnT

est

-0.7

41

-0.7

12

2.4

51**

-0.6

86

564

obse

rvat

ions

are

use

din

the

regre

ssio

ns.

Fig

ure

sin

par

enth

esis

repre

sent

the

robust

stan

dar

der

rors

.T

he

ast

eris

ks*,

**,

and

***

mea

ns

the

corr

espondin

gco

effi

cien

tis

signifi

cant

atth

e10,

5,

and

1%

level

,

resp

ecti

vel

y.

The

Jnte

stis

the

Hsi

aoet

al.

(2007

)te

stst

atis

tic

for

the

null

hypoth

esis

of

corr

ect

par

amet

ric

model

spec

ifica

tion.

Dbin

ary

var

iable

,V

SF

ver

ysm

all

farm

s,SF

smal

lfa

rms,

and

MF

med

ium

size

dfa

rms

8 Test of b-convergence is obtained by estimating the following

equation:

ð1=TÞ � logðyiT=yi;0Þ ¼ aþ b � logðyi;0Þ þ aZi;T þ li0;T where yiT

and yi0 are labor productivity at the beginning and the end of time

interval (T), Z represent binary variables that control for farm size,

and li0;T is random error term. If we interpret this as a the transition

toward a steady state growth rate, then b is a measure of the speed of

convergence—the fraction of gap between current labor productivity

growth and the long-run value that is reduced each period. The actual

speed of convergence, b, varies directly with the estimated parameter,

b, and are computed as follows: b ¼ � log 1þ b� ��

T . Significant

estimates of bare obtained if b is significant.

J Prod Anal (2012) 38:63–79 71

123

productivity (slope) is negative across the three models:

OLS, PLM, and SCM (-2.715, -2.712, and -2.670). All

the coefficients for the OLS model are statistically sig-

nificant at the 1% level. This suggests that, on average, the

improvement in efficiency was higher on farms with lower

initial productivity levels. This supports convergence in

productivity growth for Kansas farms. The speed of con-

vergence varies inversely by farm size with very small

farms achieving a higher speed of convergence (-0.114)

compared to small farms (-0.075) and medium farms

(-0.037). In terms of the in-sample fit, the adjusted

R-squared for the SCM model (R2 = 0.409) was higher

than that for the PLM (R2 = 0.339) and the OLS models

(R2 = 0.400).

The regression results when the dependent variable is

the annual growth rate in technical change are

reported in the columns marked (3). All three models

(OLS, PLM, and SCM) indicate a positive relationship

between the growth rate of technical change and initial

labor productivity (0.933, 0.933, and 0.999). This sug-

gests that technical change contributed to productivity

disparity rather than convergence during the 1993–2007

period. Farms with a high level of productivity at the

beginning of the period benefited more from technolog-

ical innovation relative to those that started with lower

levels of productivity. All estimated parameters in the

OLS model are statistically significant at the 1% level

and the speed of divergence varies inversely with farm

size.

The columns marked (4) present the estimated results

when the dependent variable is the annual growth rate of

factor intensity. All three models (OLS, PLM, and SCM)

show an inverse relationship between the annual growth

rate of factor intensity and the initial labor productivity

levels (-3.448, -3.440, and -3.300). This indicates that,

on average, farms with lower initial labor productivity

levels acquired capital at a higher rate than farms that

started with higher productivity levels. The speed of con-

vergence varied inversely with farm size with smaller

farms converging at a faster rate than small, medium, and

large farms.

Although the estimated results are comparable across

the three models, the parametric linear model is adequate in

explaining the relationship between the initial levels of

labor productivity and the annual growth rates in output per

worker, efficiency change, and factor intensity. The Hsiao

et al. (2007) test indicates that the null hypothesis of cor-

rect parametric model specification is not rejected (test

results are reported on the last row of Table 3). However,

the parametric linear model is rejected for the relationship

between initial labor productivity and the annual growth

rate in technical change (p value for the null of correct

specification is \0.05). This result implies that the

semiparametric models may be a more appropriate speci-

fication for this relationship.9

Figure 2 summarizes the partial regression functions for

the four growth rates (labor productivity and its three

components) using the logarithm of initial labor produc-

tivity levels on the horizontal axis. The broken lines in each

panel give point-wise 95% confidence envelopes around

the fit. Panels A, B, C, and D show the relationship

between the initial productivity levels and the growth rates

of labor productivity, efficiency change, technical change,

and factor intensity, respectively. The slopes of the

regression lines in panels A, B, and D are negative while

that of panel C is positive. This indicates that, on average,

there has been convergence in the growth rate of labor

productivity, efficiency change, and factor intensity, and

divergence in technical change. Panels A and D are

remarkably similar suggesting that the pattern of produc-

tivity growth attributable to factor intensity is similar to the

pattern of growth in labor productivity. This lends support

to the previous conclusions that factor intensity has been

the major driving force of labor productivity growth in the

1993–2007 period.

4.3.1 Sub-period 1993–2002

The estimated parametric and semiparametric results for

the period 1993–2002 are reported in Table 4. All three

models indicate an inverse relation between the initial

levels of labor productivity and the four growth rates. This

is in contrast to the 1993–2007 period where a positive

relationship between the average annual growth of tech-

nical change and initial labor productivity was found.

Comparisons across the three models (OLS model, PLM

and SCM) indicate convergence in the annual growth rates

of labor productivity (-7.001, -7.001, and -6.916), effi-

ciency change (-2.616, -2.613, and -2.485), technical

change (-0.179, -0.179, and -0.195), and factor intensity

(-4.213, -4.208, and -4.198). The farm size binary

variables in the OLS model indicate that the speed of

convergence is inversely correlated with farm size for the

growth rates in labor productivity, efficiency change, and

factor intensity. However, the speed of convergence in the

annual growth of technical change is higher for small farms

9 The semi-parametric regression models (PLM and SCM) are

appropriate when the parametric model (OLS) fails the test of correct

model specification, as indicated by the parametric misspecification

test. The models are used as a benchmark for the parametric model

because the true functional form and data generation process of the

parametric regression that fails the specification test is unknown. The

added value of semi-parametric techniques consists in their ability to

deliver estimators and inference procedures that are less dependent on

functional form assumptions.

72 J Prod Anal (2012) 38:63–79

123

Panel (A): Output Growth

10.0 10.5 11.0 11.5 12.0 12.5 13.0

-4-2

02

4

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in O

utpu

t per

Wor

ker,

199

3-07 Panel (B): Efficiency Growth

-4-2

02

46

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in E

ffici

ency

Cha

nge,

199

3-07

Panel (C): Technology Growth

-2-1

01

2

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in T

echn

ical

Cha

nge,

199

3-07 Panel (D): Capital Growth

-20

24

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in F

acto

r In

tens

ity, 1

993-

07

10.0 10.5 11.0 11.5 12.0 12.5 13.0

10.0 10.5 11.0 11.5 12.0 12.5 13.0 10.0 10.5 11.0 11.5 12.0 12.5 13.0

Fig. 2 Growth Rates in Output

per Worker and the Three

Decomposition Components

plotted against the 1993 Output

per Worker for the period

1993–2007. Dotted lines show a

95% confidence envelope

around the fit

Table 4 Regression results for growth in output per worker, efficiency change, technical change, and factor intensity on initial growth in output

per worker for 1993–2002

Ordinary least square model Partial linear model Smooth coefficient model

(1)

gY

(2)

gEFF

(3)

gTECH

(4)

gKACC

(1)

gY

(2)

gEFF

(3)

gTECH

(4)

gKACC

(1)

gY

(2)

gEFF

(3)

gTECH

(4)

gKACC

Intercept 88.125**

(4.505)

32.567**

(3.243)

2.572**

(0.494)

52.987**

(3.401)

81.741 28.039 2.462 50.796

Slope -7.001**

(0.369)

-2.616**

(0.265)

-0.179**

(0.040)

-4.213**

(0.278)

-7.001

(0.368)

-2.613

(0.265)

-0.179

(0.040)

-4.208

(0.278)

-6.916 -2.485 -0.195 -4.198

D-VSF -10.122**

(0.932)

-5.351**

(0.671)

-0.197**

(0.102)

-4.574**

(0.703)

D-SF -5.020**

(0.882)

-2.799**

(0.635)

-0.348**

(0.097)

-1.872**

(0.666)

D-MF -1.534**

(0.896)

-1.128**

(0.645)

-0.196**

(0.098)

-0.210**

(0.676)

Sd(Resid) 12.788 6.626 0.154 7.290 12.702 6.575 0.153 7.238 12.488 6.530 0.146 7.081

Adj. R2 0.460 0.245 0.078 0.317 0.463 0.250 0.083 0.321 0.472 0.255 0.124 0.336

Jn Test -0.490 0.105 6.952*** 0.367*

564 observations are used in the regressions. Figures in parenthesis represent the robust standard errors. The asterisks *, **, and *** means the

corresponding coefficient is significant at the 10, 5, and 1% level. The Jn Test is the Hsiao et al. (2007) test statistic for the null of correct

parametric model specification. D binary variable, VSF very small farms, SF small farms, and MF medium sized farms

J Prod Anal (2012) 38:63–79 73

123

(-0.030) compared to very small farms (-0.018) and

medium sized farms (-0.018).

Overall, all three models produce parameter estimates

that are comparable in magnitude, although the estimates

for the SCM are slightly higher than those from the other

two models when the dependent variables are growth rates

in labor productivity, efficiency change, and capital deep-

ening. The SCM produced a slightly lower estimate when

the dependent variable was the growth rate in technical

change. The SCM also performed slightly better in terms of

the in-sample fit for all the four growth rates compared to

the other two models. Using the Hsiao et al. (2007) test, the

null hypothesis of correct parametric specification is

rejected for the relationship between the initial labor pro-

ductivity and the growth rates of technical change and

factor intensity (p values are\0.01 and\0.1, respectively).

Hence, the semiparametric models are appropriate in

making inferences for those two relationships.

Figure 3 summarizes the above results by plotting the

partial regression functions for the four growth rates (labor

productivity and its three components) on the logarithm of

initial labor productivity levels. Panels A and D suggest that

farms that had lower initial labor productivity achieved

higher growth rates in labor productivity and capital deep-

ening than those that started with higher labor productivity.

The plots indicate that the farms that started with the highest

levels of productivity experienced declining growth rates.

Panel B shows that the decrease in the growth rate of effi-

ciency has been disproportionate. Farms that started with

lower initial productivity levels experienced a rapid decline

in efficiency while others experienced a gradual decline in

efficiency. A few farms experienced gains in efficiency.

Panel C suggests that growth in technical change was

positive for many farms, although some farms that started

with lower productivity levels experienced almost negligi-

ble growth in technical change. Other farms that started

with moderate productivity levels experienced a rapid

decline in technical change while those that started with

high productivity levels had a very rapid growth in technical

change. This observation lends support to the notion that

technological innovation and adoption was correlated with

a very high initial level of labor productivity.

Panel (A): Output Growth

10.0 10.5 11.0 11.5 12.0 12.5 13.0

-4-2

02

4

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in O

utpu

t per

Wor

ker,

199

3-02 Panel (B): Efficiency Growth

-4-2

02

46

8

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in E

ffici

ency

Cha

nge,

199

3-02

Panel (C): Technology Growth

-0.5

0.0

0.5

1.0

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in T

echn

ical

Cha

nge,

199

3-02

Panel (D): Capital Growth

-2-4

02

4

Initial level of Output per Worker in Logs, 1993

Ave

rage

Gro

wth

in F

acto

r In

tens

ity, 1

993-

02

10.0 10.5 11.0 11.5 12.0 12.5 13.0

10.0 10.5 11.0 11.5 12.0 12.5 13.0 10.0 10.5 11.0 11.5 12.0 12.5 13.0

Fig. 3 Growth Rates of Output

per Worker and the Three

Decomposition Components

plotted against 1993 Output per

Worker for the period

1993–2002. Note: Dotted linesshow a 95% confidence

envelope around the fit

74 J Prod Anal (2012) 38:63–79

123

4.3.2 Sub-period 1996–2005

Table 5 provides the regression results for the sub-period

1996–2005. All three models (OLS, PLM, and SCM) indi-

cate an inverse relation between the four growth rates (labor

productivity, efficiency change, technical change, and cap-

ital deepening) and the initial level of labor productivity. The

estimated parameters are comparable across the three mod-

els. The models show convergence in the growth rates of

labor productivity (-7.010, -7.003, and -6.879), efficiency

change (-3.849, -3.846, and -3.597), technical change

(-0.337, -0.337, and -0.394), and factor intensity (-2.823,

-2.818, and -2.756). All estimated parameters in the OLS

model are statistically significant at the 1% significance

level. The parameter estimates of the binary variables indi-

cate that the speed of convergence varies inversely with farm

size for the growth rates in labor productivity, efficiency

change, and factor intensity. However, the binary variable

parameter estimates for very small farms and small farms

when the dependent variable is annual growth rate in tech-

nical change are positive and close to zero. This indicates

that although there is convergence in the growth of technical

change, the convergence is driven by medium farms while

large and small farms contribute to divergence. The Hsiao

et al. (2007) test rejects the null hypothesis of correct para-

metric specification of the relationship between initial labor

productivity and the growth rates of efficiency change and

technical change (p values are \0.001). Hence, the semi-

parametric models are appropriate in making inferences for

those two relationships.

Figure 4 provides a summary of the results for the sub-

period 1996–2005. Panels A and D indicate that conver-

gence in the growth rates of output per worker and capital

deepening follow the same pattern. Farms that started with

low levels of output per worker experienced a rapid growth

in labor productivity and factor intensity relative to those

that started with high levels of output per worker. This

suggests that farms with low initial levels of output per

worker increased their capital per worker intensity rapidly

to improve productivity. Panel B indicates that convergence

in the growth rate of efficiency change was proportionate

across all the farms. Panel C presents a mixed picture on the

relationship between the growth rate in technical change

and initial levels of output per worker. A majority of the

farms experienced convergence in the growth of technical

change while some farms experienced divergence. Hence,

technical change is both a source of convergence and

divergence in the growth of labor productivity.

4.4 Comparison across periods

A comparison of the results obtained for the 1993–2007,

1993–2002, and 1996–2005 periods shows variation in the

role played by each component. The rate of convergence of

productivity growth was rapid in the two sub-periods

(1993–2002 and 1996–2005) compared to the entire sam-

ple. With regard to the existence of technological catch-up,

all periods show a trend towards convergence although the

speed of convergence is not uniform for the 1993–2007 and

1993–2002 periods. A possible explanation for this is that

Table 5 Regression results for growth in output per worker, efficiency change, technical change, and factor intensity on initial growth in output

per worker for 1996–2005

Ordinary least square model Partial linear model Smooth coefficient model

(1)

gY

(2)

gEFF

(3)

gTECH

(4)

gKACC

(1)

gY

(2)

gEFF

(3)

gTECH

(4)

gKACC

(1)

gY

(2)

gEFF

(3)

gTECH

(4)

gKACC

Intercept 90.101**

(4.196)

46.964**

(2.985)

5.747**

(0.444)

37.391**

(2.982)

83.401 40.495 6.472 34.882

Slope -7.010**

(0.340)

-3.849**

(0.242)

-0.337**

(0.036)

-2.823**

(0.242)

-7.003

(0.340)

-3.846

(0.242)

-0.337

(0.036)

-2.818

(0.242)

-6.879 -3.597 -0.394 -2.756

D-VSF -10.551**

(0.626)

-7.172**

(0.445)

0.057**

(0.066)

-3.436**

(0.445)

D-SF -5.847**

(0.508)

-3.775**

(0.361)

0.075**

(0.054)

-2.147**

(0.361)

D-MF -2.655**

(0.509)

-1.561**

(0.362)

-0.045**

(0.054)

-1.049**

(0.362)

Sd(Resid) 12.624 6.386 0.141 6.376 12.548 6.340 0.140 6.342 12.452 6.176 0.132 6.251

Adj. R2 0.470 0.398 0.215 0.198 0.472 0.402 0.221 0.201 0.476 0.417 0.270 0.213

Jn Test -1.101 1.579* 27.798* -0.968

564 observations are used in the regressions. Figures in parenthesis represent the robust standard errors. The asterisks *, **, and *** means the

corresponding coefficient is significant at the 10, 5, and 1% level. The Jn Test is the Hsiao et al. (2007) test statistic for the null of correct

parametric model specification. D binary variable, VSF very small farms, SF small farms, and MF medium sized farms

J Prod Anal (2012) 38:63–79 75

123

some farms that started with initial low productivity

achieved a remarkably high growth in efficiency compared

to those that started with high levels of productivity. It

could also reflect changes in the farm structure following

the 1996 FAIR Act.

The tendency towards convergence in labor productivity

and factor intensity followed an identical pattern across the

three periods. The results with respect to the effect of

technical change are mixed. Technical change has been a

significant source of divergence for the 15-year period

(1993–2007). However, analyses of the 10-year sub-peri-

ods show both tendencies of convergence and divergence,

with convergence playing a dominant role. The process of

convergence is rapid for farms that had initial output per

worker levels between $59,900 and $162,800. Farms with

initial output per worker above $162,800 exhibit tendencies

towards divergence.10 The implication of this phenomenon

is that technological innovation hinges strongly on high

labor productivity.

4.5 Sigma convergence

The sigma convergence test for labor productivity growth

was conducted for the entire sample and for a breakdown

of each farm size category. Results for the sigma conver-

gence test are reported in Table 6.

The estimated coefficient for the Trend for the entire

sample is negative, the expected sign for convergence, but

not statistically significant from zero. Evidence for r-con-

vergence for each farm size groups is no different; the trend

variables are negative but not statistically significant from

zero. These results suggest that the dispersion (variance) of

productivity growth in the entire sample and within the farm

size groups has not narrowed over the 15 years.

To pursue this issue further, an alternative Likelihood

ratio test proposed by Carree and Klomp (1997) is used.

The test statistic is as follows:

u ¼ N � 2:5ð Þ ln 1þ 0:25r2

1 � r2T

� �

r21r

2T � r2

1T

� �

!

ð26Þ

where N is the number of farms, r21 and r2

T are the vari-

ances of initial and final year of variables to be tested, and

Panel (A): Output Growth

10 11 12 13

-50

510

15

Initial level of Output per Worker in Logs, 1996

Ave

rage

Gro

wth

in O

utpu

t per

Wor

ker,

199

3-05 Panel (B): Efficiency Growth

-20

24

68

10

Initial level of Output per Worker in Logs, 1996

Ave

rage

Gro

wth

in E

ffici

ency

Cha

nge,

199

3-05

Panel (C): Technology Growth

-0.6

-0.4

-0.2

0.0

0.2

004

0.6

Initial level of Output per Worker in Logs, 1996 Ave

rage

Gro

wth

in T

echn

ical

Cha

nge,

199

3-05

Panel (D): Factor Intensity Growth

-20

24

Initial level of Output per Worker in Logs, 1996

Ave

rage

Gro

wth

in F

acto

r In

tens

ity, 1

993-

05

10 11 12 13

10 11 12 13 10 11 12 13

Fig. 4 Growth Rates of Output

per Worker and the Three

Decomposition Components

plotted against 1993 Output per

Worker for the period

1996–2005. Note: Dotted linesshow a 95% confidence

envelope around the fit

10 Those figures are computed by taking the antilog of initial labor

productivity as depicted on the graphs.

76 J Prod Anal (2012) 38:63–79

123

r21T is the covariance between the cross sectional growth

rates in years 1 and T. Under the null hypothesis of con-

stant variance r21 ¼ r2

T , the statistics follows the v2 distri-

bution with 1 degree of freedom. The constant of variance

of labor productivity growth for the three periods is not

rejected at the 5% critical value of 3.84 (u1993�07 ¼�0:035; u1993�02 ¼ 0:043; u1996�05 ¼ 0:051) confirming

the sigma convergence test that there has been no variance

narrowing in labor productivity growth.

5 Summary and conclusions

This paper employed a three-stage procedure to investigate

labor productivity and convergence in the Kansas farm

sector. The relationships between the initial labor produc-

tivity and growth rates in labor productivity, efficiency

change, technical change, and factor intensity for

1993–2007, 1993–2002, and 1996–2005 were explored.

Departing from the previous convergence literature, the

speed of convergence was allowed to vary by farm size.

Specifically, convergence was modeled using a general

function estimated via semiparametric regression tech-

niques. The results from the semiparametric models are

contrasted to those from a parametric model that assumes

the speed of convergence to be invariant to farm size.

Overall, we find evidence of b-convergence but no r-

convergence in labor productivity growth. The analysis

for the 1993–2007 period found an inverse relationship

between labor productivity at the beginning of the period

and annual growth rates in labor productivity, efficiency

change, and factor intensity. This lends support to the

‘‘catching-up’’ hypotheses that farms that lagged behind

the productivity leaders in 1993 exhibited rapid rates of

growth in output per worker driven by efficiency change

and factor intensity. However, the hypothesis is rejected

for the growth of technical change indicating that farms

with higher initial labor productivity experienced greater

rates of technical change relative to farms with initial low

productivity. This implies that farms that were produc-

tivity leaders at the beginning of the period benefited

more from technological innovation relative to those

farms that where followers. It is noteworthy to clarify that

although capital deepening is the main source of conver-

gence in productivity, it also contributes to the growth in

efficiency improvement and technological progress. In

general, the results lend support to the observation by Ball

et al. (2001) that there is a positive interaction between

capital accumulation and productivity growth in the farm

sector.

For the sub-periods 1993–2002 and 1996–2005, the

main conclusions are similar to the above with one

exception. There is an inverse relationship between growth

rates of technical change and labor productivity at the

beginning of the periods (1993 and 1996). This indicates

that technical change was a source of both convergence and

divergence in the sub-periods, although convergence

dominated divergence.

Given that factor intensity is a major driving force of

convergence in labor productivity, agricultural sector pol-

icies that encourage farms to invest in capital goods may

help to mitigate wide disparities in labor productivity

across the farm sector. Accumulation of physical capital is

one of the necessary conditions for sustained productivity

growth. Although prior results indicate that farms have

lagged behind rather than caught-up in the sample period,

convergence tests indicate that efficiency deterioration was

a source of labor productivity convergence. Therefore,

policies that promote the diffusion of new production ideas

and techniques would improve the productivity of indi-

vidual farms. A key policy question is whether the best

available technology is also implementable. Policies that

focus on making technology available to a majority of

farms would lead to the convergence of labor productivity.

The accumulation of human capital is equally necessary, as

the effective use of new technology requires higher levels

of training and education. Policies that focus on training,

education, and improved extension services may induce

efficiency catch-up.

Overall, from a policy perspective, the results imply

reduction in the inequality in labor productivity growth

across Kansas farms. However, further research is needed to

investigate what led to convergence in labor productivity.

This further work could investigate whether convergence

took place due to the slowdown of the most productive farms

Table 6 Test for r-convergence of labor productivity growth, 1993–1994 to 2006–2007

Variables All farms Very small farms Small farms Medium farms Large farms

/1 1158.140*** 1547.036*** 964.757*** 1019.820*** 1013.510***

/2 -12.48 -4.713 -3.508 -22.900 -11.960

R2 0.077 0.004 0.012 0.130 0.029

Estimated regression is Variance(LPG)t¼/1 þ /2Trendþ et . Significant coefficients are identified by asterisk: *** indicates significant at 1%

level

J Prod Anal (2012) 38:63–79 77

123

to match the growth performance of the less productive

farms rather than the latter group catching-up. Further work

could also investigate why technological change contributed

to labor productivity growth divergence rather than con-

vergence. Following Henderson and Russell (2005), this

study could be extended by taking into account human

capital and thus decompose labor productivity growth in

four elements: efficiency change, technical change, capital

accumulation, and human capital accumulation.

Appendix

See Fig. 5 and Table 7.

Table 7 Speed of convergence/

divergence

Larger farms are the reference

point for farm sizes. The speed

of convergence is higher when

the computed value is negative

and approaches 1 and slower as

the value approaches zero.

Positive values indicate

divergence

1993–2007 1993–2002 1996–2005

Labor productivity

All farms -0.122 -0.208 -0.208

Very small farms -0.155 -0.241 -0.241

Small farms -0.122 -0.180 -0.192

Medium farms -0.080 -0.093 -0.130

Efficiency change

All farms -0.087 -0.129 -0.158

Very small farms -0.114 -0.185 -0.210

Small farms -0.075 -0.133 -0.156

Medium farms -0.037 -0.076 -0.094

Technical change

All farms 0.180 -0.016 -0.029

Very small farms -0.033 -0.018 0.006

Small farms -0.039 -0.030 0.008

Medium farms -0.030 -0.018 -0.004

Factor intensity

All farms -0.099 -0.165 -0.134

Very small farms -0.108 -0.172 -0.149

Small farms -0.081 -0.106 -0.115

Medium farms -0.047 -0.019 -0.072

1993 1994 1995

1996

1998

19992000 2001

2002

2003

2004 2005

2006

2007

1997

150000

200000

250000

300000

350000

400000

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Year

Gro

ss F

arm

Inco

me

($)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Co

effi

cien

t o

f V

aria

tio

n

Mean CV

Fig. 5 Mean and coefficient of

variation for gross farm income

(in 2007 prices)

78 J Prod Anal (2012) 38:63–79

123

References

Ball VE, Norton GW (2002) Agricultural Productivity: measurement

and Sources of Growth. Kluwer Academic Publishers, Boston

Ball VE, Gollop FM, Kelly-Hawke A, Swinand G (1999) Patterns of

state productivity growth in the US farm sector: linking state and

aggregate models. Am J Agric Econ 81:164–179

Ball VE, Bureau JC, Butault JP, Nehring R (2001) Levels of farm

sector productivity: an international comparison. J Prod Anal

15:5–29

Ball VE, Hallhan C, Nehring R (2004) Convergence of productivity:

an analysis of the catchup hypothesis within a panel of states.

Am J Agric Econ 86:1315–1321

Carree MA, Klomp L (1997) Testing the convergence hypothesis: a

comment. Rev Econ Stat 79:683–686

Coelli T, SP Rao D, O’Donnell CJ, Battese GE (2005) An

introduction to efficiency and productivity analysis, 2nd edn.

Springer, New York

Delgado-Rodriguez MJ, Alvarez-Ayuso I (2008) Economic growth

and convergence of EU member states: an empirical investiga-

tion. Rev Dev Econ 12(3):486–497

Fare R, Grosskopf S, Margaritis D (2006) Productivity growth and

convergence in the European Union. J Prod Anal 25:111–141

Gutierrez L (2002) Why is agricultural labor productivity higher in

some countries than others? Agric Econ Rev 3(1):58–72

Hardle W, Mammen E (1993) Comparing nonparametric versus

parametric regression fits. Ann Stat 21:1926–1947

Henderson JD, Russell RR (2005) Human capital and convergence: a

production-frontier approach. Int Econ Rev 46:1167–1205

Henderson DJ, Zelenyuk V (2007) Testing for (efficiency) catching-

up. South Econ J 73:1003–1019

Horowitz JL, Hardle W (1994) Testing a parametric model against a

semiparametric alternative. Economet Theory 10:821–848

Horowitz JL, Spokoiny VG (2001) An adaptive, rate-optimal test of a

parametric mean- regression model against a nonparametric

alternative. Econometrica 69:599–631

Hsiao C, Li Q, Racine JS (2007) A consistent model specification test

with mixed discrete and continuous data. J Economet 140:

802–826

Koop G, Tobias JL (2006) Semiparametric Bayesian inference in

smooth coefficient models. J Economet 134:283–315

Kumar S, Russell RR (2002) Technological change, technological

catch-up, and capital deepening relative contributions to growth

and convergence. Am Econ Rev 92:527–548

Langemeier M (2010) Kansas farm management SAS data bank

documentation. Staff Paper No. 11–01. Department of Agricul-

tural Economics, Kansas State University, USA

Li Q, Racine JS (2007) Nonparametric econometrics: theory and

practice. Princeton University Press, New Jersey

Li Q, Huang CJ, Li D, Fu T (2002) Semiparametric smooth

coefficient models. J Bus Econ Stat 20:412–422

Liu Y, Shumway CR, Rosenman R, Ball VE (2008) Productivity

growth and convergence in US agriculture: new cointergration

panel data results. Working Paper No. 2008-4. School of

Economic Sciences, Washington State University

Managi S, Karemera D (2004) Input and output biased technological

change in the US agriculture. Appl Econ Lett 11:283–286

McCunn A, Huffman WE (2000) Convergence in US productivity

growth for agriculture: implications of interstate research

spillovers for funding agricultural research. Am J Agric Econ

82:370–388

Rezitis AN (2005) Agricultural productivity convergence across

Europe and the United States of America. Appl Econ Lett

12:443–446

Serra T, Goodwin BT, Featherstone AM (2005) Agricultural policy

reform and off-farm labour decisions. J Agric Econ 56:271–285

Serra T, Zilberman D, Gil JM (2008) Farm’s technical Inefficiencies

in the presence of government programs. Aust J Agric Resour

Econ 52:57–76

Shephard RW (1970) Theory of cost and production functions.

Princeton University Press, Princeton

United States Department of Agriculture, National Agricultural

Statistics Service, Kansas Field Office. (2010). Kansas Farm

Facts 2010

Yatchew A (2003) Semiparametric regression for the applied

econometrician. Cambridge University Press, New York

J Prod Anal (2012) 38:63–79 79

123