This article suggests that the two current approaches of summarizing curvature conditions for certain flexible function forms are neither intuitive nor robust to skewed distribution of exogenous variables. They also ignore the possible correlation pattern between exogenous variables. Arguing that the purpose of summarizing curvature condition is to provide either the changes of aggregated dependent variable with respect to a change of explanatory variable, or curvature conditions of a representative individual, this article proposes two methods to address this issue.Research Methods/ Statistical Methods,

Chen, Adam Zhuo

Huffman, Wallace E.

English

Research Papers in Economics

  1
4-20-04 
 
 
SUMMARIZING CURVATURE CONDITIONS FOR FLEXIBLE 
FUNCTIONAL FORMS 
 
 
Adam Zhuo Chen 
Research Assistant, Department of Economics,   
Iowa State University 
zchen@iastate.edu 
Wallace E. Huffman 
C.F. Curtiss Distinguished Professor of Agriculture 
  and Professor of Economics, Iowa State University 
whuffman@iastate.edu 
 
Preliminary draft: Please do not quote without permission of the authors 
 
 
 
Paper prepared for presentation at the American Agricultural Economics Association 
Annual Meeting, Denver, U.S.A., August 1-3, 2004 
 
 
 
 
 
 
©Copyright 2004 by Adam Zhuo Chen, Wallace Huffman.  All rights reserved.  Readers may make 
verbatim copies of this document for non-commercial purposes by any means, provided that this 
copyright notice appears on all such copies. 
 
Authors thank an anonymous referee for his/her helpful comments. The usual disclaimer applies.   2
 
Summarizing Curvature Conditions for Flexible Functional Forms 
 
Introduction 
Curvature conditions of nonlinear function form usually vary across individual 
observations and may not be readily obtainable. In the practice of production function 
estimation, the marginal products and/or elasticities are calculated based on the parameter 
estimates of the flexible function form. In order to compare the curvature conditions, we 
need to summarize the information into readable dimensions. Even in a very restrictive 
case, where we compare estimates of different function forms using the same dataset, it is 
nearly impossible to compare the curvature conditions at all sample points if there are 
significant number of observations. To compare curvature conditions for function forms 
estimated with different datasets is more complicated. Ben-Akiva and Lerman (1985) 
discussed how the curvature conditions at different sample points should be summarized 
and presented in the context of discrete choice analysis. However, in the production 
economics literature, this topic has received inadequate attention. 
Literature provided two approaches to summarize marginal effects for production 
flexible function forms (Greene, 2003). The first is to calculate the marginal effects for 
individual and then present the summary statistics. The intuition underlying this approach 
is that these statistics will provide a picture of how the aggregated dependent variable 
will response to marginal aggregated changes of the explanatory variables. However, we   3
argue that this may not be true. The second approach is to evaluate the marginal effects/ 
marginal productions/ elasticities) at a sample point, e.g. mean/median/ geometric mean 
of explanatory variables. This approach has been widely used. Diewert and Wales (1987) 
compared three flexible functional forms by evaluating curvature conditions at the first 
and last sample points. Anderson & Newell (2003) proposed a method to simplify the 
calculation of marginal effects at a certain data point for discrete choice models. 
Meanwhile, it is noted that this approach hinges on strong distribution assumptions of 
explanatory variables. 
This article proposes two methods to address the issue of summarizing curvature 
conditions for flexible functional forms in the practice of production function estimation. 
The first approach is to improve the averaging approach by incorporating a weighting 
scheme according to the contribution of an individual observation. The second is to 
strengthen the representativeness of central points. We can either use central points that is 
more robust to outliers and non-normal distribution in providing a typical individual 
firm/household/person or to group the data points and summarize the curvature 
conditions for each group. The two new methods are more intuitive and robust to outliers 
and abnormal explanatory variable distribution. 
The rest of this article is organized as following. Section 2 critiques the usual 
practice of summarizing the curvature conditions in the context of production study. We 
propose our methods in Section 3. Section 4 concludes. 
   4
Critiques on the Common Practices 
Greene (2000) stated that: “For computing marginal effects, one can evaluate the 
expressions at the sample means of the data or evaluate the marginal effects at every 
observation and use the sample average of the individual marginal effects. The functions 
are continuous, so Slutsky theorem applies; in large sample they will give the same 
answer. But that is not so in small or moderate sized samples. Current practice favors 
averaging the individual marginal effects when it is possible to do so.” This statement is 
likely to be true when evaluating the marginal effects for discrete choice problems, where 
the exogenous variables are less correlated to each other and can be approximated as 
normal in large samples. We argue, however, that when we are evaluating the curvature 
conditions of flexible functional forms in the context of production, this result may not 
hold. There are two reasons underlie our argument: irregularity of the input quantities 
distribution; and possible correlation pattern between input usage. We describe them in 
the following subsections. 
 
Averaging Approach 
In the production economics context, it is not necessary that input quantities in large 
sample be normally distributed. U.S. Congress, Office of Technology Assessment (1984) 
claimed American farm size is distributed as a bi-model. In United States, there is an 
increasing trend that while average farm size is enlarged, the number of small farms (of 
which the purpose is for entertainment rather than income-generating) is increasing at the   5
same time. In developing countries, such trend exists as well due to the limitation of 
resource and restriction on farm size, i.e., the Household Responsibility System in China 
and Land to the Tillers Program in south Asia. The land ownership are consisted of large 
number of existing small farms and an increasing trend of land consolidation due to size 
economies and the labor migration from agriculture sector to manufacture and service 
sectors. Therefore large sample theory may not apply in the agricultural production 
context. 
Since the literature usually apply the averaging approach for marginal effects but not 
for elasticity, we focus our discussion on marginal effects henceforth in this subsection. 
One may argue that the purpose of averaging approach is to provide the sample mean, as 
well as standard deviation, of individual marginal effects. However, the mean values of 
marginal effects are not necessarily representative. Neither does the sample mean 
converge to the true value in large samples since the overall population may be 
non-normally distributed, e.g., bi-model. The average of individual marginal effects is not 
equal to the change of the aggregate dependent variable with respect to marginal change 
of an explanatory variable either. In fact, it is only a specific realization of the change of 
the aggregate dependent variable (e.g., the output in agricultural production function 
estimation) when the marginal changes of inputs of all observations in the sample are 
equally weighted. In the case of production function estimation, it measures the change of 
aggregate output when all individual observations have equal extent of change in inputs 
usage. However, in finite sample, small firms and large firms are likely to have different   6
levels of change of their inputs. The averaging approach fails to summarize the marginal 
effects of aggregate dependent variable as illustrated in the following. 
Denote output as y, input vector as x, while xji indicates the jth input of observation i. 
Thus we may define the averaging approach as: 
1
1 1()
j
n y
i nx i m
∂
∂ = = ∑ . 
The change of aggregated output with respect to the marginal change of the 
aggregated input is: 
1
1
11
() 0
11
1
0
1
()
lim
()
lim
n
ji i
n
ji i
nn
ii ii
nn
x
jij i ii
n
i i
n
x
ji i
yy
x x
y
x
=
=
==
∆→
==
=
∆→
=
∂∆
=
∂∆ ∑
∆
=
∆ ∑
∑∑
∑∑
∑
∑
 
However, since there are n independent control variables in the denominator, we 
cannot claim that: 
1
1
11 0 0
1
11
lim lim
n
ji ji i
n
nn i ii
n ii x x ji j ji i i
y y y
nx n x x =
=
== ∆→ ∆→
=
⎛⎞ ∆ ∆ ∂
== ⎜⎟ ⎜⎟ ∆∂ ∆ ∑ ⎝⎠
∑ ∑∑
∑
. 
Meanwhile, we can assign a weight wi to the input change of a single observation as 
its contribution to the aggregate changes such that 
1
ji
i n
ji i
x
w
x
=
∆
=
∆ ∑
, then the marginal 
effect of the aggregate amount is: 
1
11
() 0
11
1
0
1
()
lim
()
()
lim
n
ji i
nn
ii ii
nn
x
jij i ii
n
i i
n t
i i
yy
x x
y
wt
=
==
∆→
==
=
∆→
=
∂∆
=
∂∆ ∑
∆
=
∆
∑∑
∑∑
∑
∑
 
Denote 
1 ()
n
i i f y
= =∑ x , we know that when there is a change of  t ∆ , then the   7
amount of change of xi is  i wt ∆ , take first order Taylor series approximation of  () f x , the 
approximate change of 
1 ()
n
i i y
= ∆ ∑  is 
1
n
ii i wy
= ∆ ∑ , where  i y ∆  is  the  change 
corresponding to a change of  t ∆  for  xi, therefore we have: 
11 1 1
000
11 1
11 00
1
()
lim lim lim
lim lim
nn n n
i i ii ii ii i i
nn n ttt
ji i i ii i
nn ii i
i ii tt
n
i i
j i
yy w y w y
t xw t t w
wy y
w
tt
y
w
x
== = =
∆→ ∆→ ∆→
== =
== ∆→ ∆→
=
∂∆ ∆ ∆
===
∆ ∂∆ ∆
∆∆
==
∆∆
⎛⎞ ∂
= ⎜⎟ ⎜⎟ ∂ ⎝⎠
∑∑ ∑∑
∑∑ ∑
∑∑
∑
 
We can summarize this as Theorem 1. 
 
Theorem 1: Assume a weight wi denoting as the contribution of input change of 
observation i to the aggregate changes such that 
1
ji
i n
ji i
x
w
x
=
∆
=
∆ ∑
, then the marginal effect 
of the aggregate amount is: 
1
1
1
n
n i i
i n i
j ji i i
y y
w
x x
=
=
=
⎛⎞ ∂ ∂
= ⎜⎟ ⎜⎟ ∂ ∂ ⎝⎠
∑ ∑
∑
 
 
Lemma: Assuming the contributions to input change are the same across individuals, 
we have that 
1
1
1
1
n
n i i
n i
j ji i i
y y
nx x
=
=
=
⎛⎞ ∂ ∂
= ⎜⎟ ⎜⎟ ∂ ∂ ⎝⎠
∑ ∑
∑
. 
 
Greene (2002) argued that by applying large sample theory the averaged marginal 
effect converges to the marginal effects at a representative central points. However, this 
hinges on the assumption of normality in large samples. With the irregularity of input   8
distribution in the context of production function estimation, the result does not hold 
generally. Furthermore, as we have seen from above, the averaging algorithm is a special 
case of our method with equal weight for each individual observation. 
 
Representative Individual Approach 
The representative individual approach relies heavily on the multivariate normal 
distribution of inputs. It is facing two problems. First, if input distributions deviate from 
normal distribution, neither mean, geometric mean nor median is a good representative 
reference point. Median and geometric mean are more robust to the outliers but cannot 
handle the bi-model case. Second, many built-in functions in econometric computation 
packages ignore the potential correlation between the exogenous variables and evaluate 
the mean/median of the inputs respectively. The quantities of different inputs are possibly 
correlated to each other. The centroid calculated by averaging different input quantities 
(as calculated by many software packages) may not represent the whole sample well 
since it ignored the covariance structure. It is common that inputs are constrained or 
exhibit certain pattern of correlation between each other, especially when there are strong 
substitution effects. In the practicing of agricultural production function estimation, 
taking mean/median of inputs, i.e., land, labor, fertilizer, and capital, does not guarante to 
result a good representative farm. We argue that the usual representative individual may 
not be a realistic approach. We provide a simple example to illustrate the failure of 
representative point approach.   9
Suppose we have two inputs: labor L and capital K, without random disturbance, the 
production function is characterized as: 
22
12 3 4 5 ( , ) exp( ln ln (ln ) (ln ) (ln )(ln )) yf L K aL a K a L a K a L K == + + + +  
therefore we have the output-labor elasticity as: 
13 5
ln
2l n l n
ln
y
aa L a K
L
∂
=+ +
∂
, 
assume a correlation pattern between L and K is that: 
35 2l n l n aL aK C + = , C is a constant. 
Then we have the elasticity is a constant for all observations, but obviously when we 
evaluate at the respective means of the input; we will get a totally different result. In this 
case, geometric mean can be used and obtain the correct value. However, generally, since 
we do not have sufficient information during estimation, we cannot decide which central 
point to use. Furthermore, the non-linearity itself can be a source of the severe bias of the 
marginal effect estimates. Ben Akiva and Lerman (1985) discussed such bias in details in 
the context of discrete choice models. 
In summary, the representative individual approach may not produce appropriate 
results. 
 
New Ways to Summarize Curvature Conditions 
In this section we propose two new methods to summarize curvature conditions of 
flexible functional forms in the practice of production function estimation. 
   10
Method 1: 
We propose a simple solution to improve over the averaging approach. We can use a 
predetermined weight to adjust the contribution of curvature conditions. Either the ratio 
of individual input usage to the aggregate sample input usage or ratio of individual output 
to the aggregate output is potential good candidate. Yet no theory suggests a “best” 
weighting scheme exists. These firms with small input usage may have a marginal effect 
large in magnitude, however, giving the market imperfection in real world, their 
contribution to input change may be relatively small. Which force finally dominates the 
curvature condition change depends on which one is greater in magnitude. Assuming the 
individual contributes to the aggregation with a weight equal to the ratio of its own output 
to the aggregate output, then multiply the weight to the individual marginal effects and 
obtain the marginal effects of weighted aggregated mean. This is more intuitive and 
realistic than the approach of assigning equal weight for all individuals. This approach 
can be extended to the case of elasticities easily. Since the elasticites are unit-free, output 
percentage as a weight might be a good choice of the weighting scheme. 
 
Method 2: 
To improve the representative individual method, we need to consider how to reduce the 
dimension of the inputs thus to find an appropriate representative point. One way is to 
calculate the distance of individual observations to a reference point, i.e., the origin or the 
centroid, then locate the representative individual(s) using the usual mean, median, or   11
geometric mean. When the sample is severely clustered, the curvature conditions should 
be evaluated at multiple representative individuals for the existing clusters, respectively. 
The distance can be defined in various ways. Two commonly used distance 
measures are Euclid distance and Markov distance. 
Euclid distance is defined as:   
00 0 (, ) ( ) ' ( ) d =− − xx x x x x  
Markov distance is defined as:   
00 0 (, ) ( ) ' ( ) d =−Λ − xx x x x x , where 
1 S
− Λ= . 
Markov distance is more frequently used since it is invariant to the unit of the 
variables under study. 
 
A recipe of locating representative individual(s) can be described as: 
Step 1: Calculate the distance of individual points to a reference point (e.g., 
centroid); 
Step 2: Graph the histogram of the distance for the whole sample and decide 
whether there are clusters according to the graph (or clustering can be applied directly, 
then make the corresponding judgment whether the sample appears to be clustered or 
not); 
Step 3.1: If it appears to be a uni-model, then simple statistic procedure can be 
applied to locate the representative individual; 
Step 3.2: If it appears to be a clustered sample, then apply clustering algorithm, e.g.,   12
Hierarchical Clustering Methods, to group the observations into G groups, and compute 
the representative points within each group. The overall summary statistics can be a 
weighted average of these points or they can be presented directly since researchers may 
be interested in the marginal effects of different clusters. 
 
Note that we propose to use clustering rather than the classification procedure used 
in Ben-Akiva & Lerman (1986). The difference of classification and clustering is that 
classification “pertain to a known number of groups, and the operational objective is to 
assign new observations to one of these groups” while clustering is “a more primitive 
technique in that no assumptions are made concerning the number of groups or the group 
structure” (Johnson & Wichern, 2001). In most production studies, we do not have a 
predetermined G, therefore clustering is more applicable in these studies. Though we 
need to set a cut-off distance for the dendrogram (tree diagram) to decide how many 
clusters we keep, the number of clusters is ex post rather than ex ante pre-set in 
classification problem. 
Another clustering method may be used is non-hierarchical clustering method, e.g., 
K-means algorithm. It is computationally convenient but need a predetermined number of 
clusters, which is usually obtained from preliminary clustering, graphical observation, or 
simply intuition. 
 
   13
Conclusion 
In this article, we critiqued the usual practices of summarizing the curvature conditions of 
flexible functional forms. We also proposed two new methods to accomplish that goal. 
Theoretically, the new methods produce more robust and more accurate estimates for the 
curvature conditions of aggregated variables, as well as the curvature conditions of these 
variables at the representative points. Both methods provide policy makers a better 
picture of how the dependent variable may response to the marginal change of 
explanatory variables. 
Meanwhile, when applying the first approach, alternative weighting-schemes are 
possible with different interpretation. In clustering algorithm, not only we need to select a 
distance measure, but also need to choose which point the distance may refer to. With the 
importance of marginal effects in inferring policy implications, these works obviously 
deserve further exploration. 
   14
 
References: 
Anderson, S. and R. G. Newell (2003) “Simplified marginal effects in discrete choice models” 
Economic Letters 81 (2003) pp.321-326 
Ben-Akiva M., and Lerman (1985) Discrete Choice Analysis: Theory and Application to Travel 
Demand, Chapter 6: Aggregate Forecasting Techniques, The MIT Press, Cambridge MA 
Diewert W. E. and T. J. Wales (1987) “Flexible functional forms and global curvature conditions” 
Econometrica Vol. 55, No. 1 (1987) pp.43-68 
Greene, W.H. Econometric Analysis, Prentice Hall, Upper Saddle River NJ, 4
th ed. (2000) 
Johnson RA and DW Wichern, Applied Multivariate Statistical Analysis, Prentice Hall, New 
Jersey, 5th edition, 1995 
Stoker T. M. (1996) “Smoothing bias in the measurement of marginal effects” Journal of 
Econometrics 72 (1996) pp.49-84 
U.S. Congress, Office of Technology Assessment: “Technology, Public Policy and the Changing 
Structure of American Agriculture” Washington D.C., U.S. Government Printing Office, 
OTA-F-285 March 1986 
 

SUMMARIZING CURVATURE CONDITIONS FOR FLEXIBLE FUNCTIONAL FORMS

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