metadata: authors: 0: author: Butzer,Rita
1: author: Larson,Donald F.
2: author: Mundlak,Yair
count: World
docna: 0: docna: Heterogeneous technology and panel data :
            the case of the agricultural production function
docty: Policy Research Working Paper
subtopic: Climate Change and Agriculture,Construction Industry,Common Carriers Industry,Food & Beverage Industry,General Manufacturing,Pulp & Paper Industry,Plastics & Rubber Industry,Crops and Crop Management Systems,Fertilizers,Hydrology,International Trade and Trade Rules,Food Security,Business Cycles and Stabilization Policies
teratopic: Rural Development,Macroeconomics and Economic Growth,Social Protections and Labor,Private Sector Development
abstracts: cdata!: The paper presents empirical analysis of
            a panel of countries to estimate an agricultural production
            function using a measure of capital in agriculture absent
            from most studies. The authors employ a heterogeneous
            technology framework where implemented technology is chosen
            jointly with inputs to interpret information obtained in the
            empirical analysis of panel data. The paper discusses the
            scope for replacing country and time effects by observed
            variables and the limitations of instrumental variables. The
            empirical results differ from those reported in the
            literature for cross-country studies, largely in augmenting
            the role of capital, in combination with productivity gains,
            as a driver of agricultural growth. The results indicate
            that total factor productivity increased at an average rate
            of 3.2 percent, accounting for 59 percent of overall growth.
            Most of the remaining gains stem from large inflows of fixed
            capital into agriculture. The results also suggest possible
            constraints to fertilizer use.
repnme: repnme: Heterogeneous technology and panel data :
            the case of the agricultural production function
display_title: Heterogeneous technology and panel
            data : the case of the agricultural production function
listing_relative_url: /research/2008/02/9041917/heterogeneous-technology-panel-data-case-agricultural-production-function
topicv3: Agriculture and Food Security m1326205 435,Manufacturing Sectors m1326521 1281,Water m1326522 137,Trade m1326924 698,Fertilizers m1327354 175
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Rural Development,Macroeconomics and Economic Growth,Social Protections and Labor,Private Sector Development
no_of_pages: 48
author_keys: 0: author_key: 782658
1: author_key: 92635
2: author_key: 782657
model_extractions: WPS4536 Policy ReseaRch WoRking PaPeR 4536 Heterogeneous Technology and Panel Data: The Case of the Agricultural Production Function _Yair Mundlak Rita Butzer Donald F. Larson_ The World Bank Development Research Group Sustainable Rural and Urban Development Team February 2008
Policy ReseaRch WoRking PaPeR 4536

# **Abstract**

The paper presents empirical analysis of a panel of countries to estimate an agricultural production function using a measure of capital in agriculture absent from most studies. The authors employ a heterogeneous technology framework where implemented technology is chosen jointly with inputs to interpret information obtained in the empirical analysis of panel data. The paper discusses the scope for replacing country and time effects by observed variables and the limitations of instrumental variables. The empirical results differ from those reported in the literature for cross-country studies, largely in augmenting the role of capital, in combination with productivity gains, as a driver of agricultural growth. The results indicate that total factor productivity increased at an average rate of 3.2 percent, accounting for 59 percent of overall growth. Most of the remaining gains stem from large inflows of fixed capital into agriculture. The results also suggest possible constraints to fertilizer use.

This paper—a product of the Sustainable Rural and Urban Development Team, Development Research Group—is part of a larger effort in the department to examine the determinants of agricultural growth. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at DLarson@worldbank.org. This study was supported by the Bank’s Research Support Budget under the research project, “The Contributions of Governance to Growth in Agriculture" (RP0 94759).

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Heterogeneous technology and panel data: The case of the agricultural production function**

Yair Mundlak<sup>a</sup> , Rita Butzer<sup>b</sup> and Donald F. Larson<sup>c</sup>

**________________________________________________________________________**

**JEL classifications:** C230, D240, Q110, O130

**Keywords** : Agriculture, development, economic growth, panel data, productivity

## **Acknowledgment**

We thank Yacov Tsur for helpful comments. This study was funded by the Bank’s Research Support Budget under the research project, “The Contributions of Governance to Growth in Agriculture" (RP0 94759).

a Corresponding author. E-mail address: mundlak@agri.huji.ac.il. Professor Emeritus, Hebrew University of Jerusalem.

b Ph.D. Candidate, Economics, The University of Chicago.

c Senior Research Economist, World Bank.
# Heterogeneous technology and panel data: The case of the agricultural production function

# **<u>Introduction</u>**

The focus of empirical analysis of agricultural production functions, similar to that of production functions in general, has changed over the years.<sup>1</sup> Following the work of Cobb and Douglas (1928), research centered on questions about the efficiency of the factor markets. Research interest subsequently shifted to issues related to changes in factor demand, leading to interest in the elasticity of substitution, to factor augmentation, to a search for the proper algebraic form, to issues related to aggregation, and more recently to issues related to the variability in income and in productivity growth. The statistical aspects of the analysis were affected by the recognition of the endogeneity of the inputs, raised by Marschak and Andrews (1944), and by the methods of accommodating this complication in the estimation. This led to the use of panel data, to the dual approach to the estimation, and to the use of instrumental variables.

A common assumption in much of the work was that of a homogeneous technology, implying that a common production function generated observations used in the analysis. In reality, firms face the practical problem of choosing which technology to employ jointly with inputs so that technology is heterogeneous in that there is more than one function associated with the data. In this paper we examine the consequences of this extension in estimating an aggregate agricultural production function using panel data, where the underlying functions are all of the Cobb-Douglas form. We model the technology choice as conditional on predetermined variables referred to as state variables. This approach provides a different view of the empirical results of most of the aforementioned subjects. The root of the difference is the recognition that the observations in the sample represent moves between functions as well as movements along a given function. The empirical formulation allows for the dependence of parameters of the 1 For the development of the work on agricultural production functions, see Mundlak (2001).
function, just as the inputs, on the state variables, and in this sense the observed, or implemented, technology is endogenous.

The paper is oriented toward the understanding of the role of inputs and technology in agriculture. Although the subject has a long history, it is still relevant, and is related, among other things, to the interest in structural changes that take place in the process of economic development. This topic of inquiry has real world implications because most of the poor households in developing countries live in rural areas and depend on agriculture for their livelihood.<sup>2</sup> An increase in agricultural productivity directly contributes to the welfare of the rural area. Changes in factor demand due to changes in technology affect the intersectoral flow of resources, primarily labor and capital, which constitutes the essence of the structural changes in the process of development. Understanding this process has policy implications concerning what is worth doing and what is worth avoiding.

The analysis is related to the macroeconomic literature on the determinants of economic growth and productivity for overall economies. The models used vary in the parsimony of the parametric specification and in the choice of what parameters are to be estimated and those to be imposed. The heterogeneity of approaches reflects the inability to obtain empirically reliable and robust results that follow from basic economic reasoning. This feeds the search for better specifications and for the appropriate ways of handling the data. Most growth studies include a set of core Solow-Swan variables related to human capital investment, physical capital investment, initial income conditions and either population or labor growth.<sup>3</sup> Studies of differences in productivity levels use similar variables. There is less of a consensus on the broader set of state variables, though a growing interest has arisen in the role of certain state variables such as culture, geography, institutions, and market integration.<sup>4</sup> Defined very broadly, institutions include the humanly devised rules that shape economic incentives and that are particularly related to the protection of property, the enforcement of contracts, and the dissemination of information (North 1990; Acemoglu, Johnson, and Robinson 2005). There is no unity in

> 2 According to the World Bank (2007), this includes 2.1 billion people living on less than $2 a day.

> 3 See for instance the review by Durlauf, Johnson, and Temple (2005) and Durlauf and Quah (1999).

> 4 See, for example, Sachs and Warner (1997), Hall and Jones (1999), Rodrik, Subramanian, and Trebbi (2004), and Presbitero (2006).

2
how the state variables are to be used in the analysis. In some models they are assumed to be the sole causes of accumulation and productivity changes (Hall and Jones 1999), while in others they are added to inputs in the estimation of the production function. They are also used as instrumental variables to deal with endogeneity of inputs, institutions, and measurement errors.

Finding a set of variables that adequately describes prevailing economic incentives is challenging for several reasons. First, while the objective of most research is to get at the long-run determinants of economic growth, much of the variation in economic activity, and thus in the data, is associated with short-term fluctuations, which may be linked to quasi-fixed constraints or gaps caused by misplaced expectations. Second, economic growth theory is open-ended, and the set of potential determinants of economic incentives is large relative to the panel datasets available for studying macroeconomic growth.<sup>5</sup> As a practical matter, many empirical models deal with the prevalence of shortterm variability in the data by averaging across periods, and thereby affect the empirical results.<sup>6</sup> Even so, some authors argue that this type of transformation is not arbitrary but rather necessary to eliminate nuisance variations and potential biases.<sup>7</sup> As we show, the short-term variations are essential for identifying the production function, and thereby to obtain the appropriate weights for calculating total factor (TF) and total factor productivity (TFP). On the other hand, weights based on country averages provide distorted estimates of productivity.<sup>8</sup> In terms of end results, part of the debate involves the importance of total factor productivity relative to accumulations of human and physical capital in determining patterns of growth. For example, Mankiw, Romer, and Weil (1992) and Henderson and Russell (2005) find that accumulations largely account for growth, while Klenow and Rodriguez-Clare (1997) and Easterly and Levine (2001) place greater emphasis on productivity growth.

> 5 In their review of applied macroeconomic growth studies, Durlauf, Johnson, and Temple (2005) compile a set of 145 variables that have been used as growth determinants.

> 6 For a review of applied techniques based on growth rates, see Durlauf, Johnson, and Temple (2005).

> 7 See, for example, Pritchett (2000).

> 8 See, for example, Sala-i-Martin, Doppelhofer, and Miller (2004).

3
Similarly, a variety of conclusions are reached concerning the core set of state variables. Hall and Jones (1999) emphasize the role of social capital and Rodrik, Subramanian, and Trebbi (2004) assert the dominance of institutions over other determinants. In other papers, trade, monetary policy, and cultural factors related to religion, language, and colonial heritage are viewed as key determinants.<sup>9</sup> But in their review of the recent literature, Durlauf, Johnson, and Temple (2005, p. 558) conclude that “(e)ven when the study of growth is viewed in terms of a collective endeavor, the various papers cannot easily be distilled into a consensus that would meet standards of evidence routinely applied in other fields of economics.” In terms of this literature, we deal with a sectoral production function, representing a lower level of aggregation, but many of the issues still remain here as well. Agricultural output has grown as a result of changes in technology and in resource allocation where the role of labor declined and that of capital increased. Regarding the decomposition of the output growth to TF and TFP, we rely on the empirical estimates of the production function parameters, and as such the outcome depends on the quality of the estimates. When the parameters of the production function depend on the state variables, the relative contribution of the TF and the TFP is also endogenous; hence TFP can not be considered to be the trigger of growth but rather is a result of it. The dependence of the parameters on the state variables accounts for the wide spread in the empirical results reported in the literature.<sup>10</sup> The inputs used in the analysis are land, capital, fertilizer, and labor. The capital variables, constructed for this analysis, revise and update an earlier series that was used in Mundlak, Larson, and Butzer (1999). The state variables consist of variables representing technology, institutions, incentives, and physical environment. A substantial portion of the empirical section of the paper is devoted to finding a robust set of variables that adequately accounts for agricultural output differences not explained by factors of production.

> 9 For example, Frankel and Romer (1999) and Sachs and Warner (1997) discuss the role of trade. Examples related to exchange rates include Dollar (1992) and Barro and Lee (1994). Barro and McCleary (2003), among others, write about the role of religion.

> 10 Mundlak, Larson, and Butzer (1999) summarize the empirical agricultural production functions based on cross-country data that had appeared prior to the writing of the paper. More studies have appeared since, but they only confirm the existence of diversity of the results.

4
# **I. The Model**

The underlying premise is that producers at any time face more than one technique of production, and their economic problem is to choose the techniques to be employed together with the choice of inputs and outputs. The outline of the approach follows Mundlak (1988, 1993). Let _X_ be the vector of inputs and _Fj(X)_ be the production function associated with the jth technique, where _Fj_ is concave and twice differentiable, and define the available technology, _T_ , as the collection of all possible techniques, _T = {Fj(X); j=1,...,J}_ . Firms choose the implemented techniques subject to their constraints and the environment. We distinguish between constrained ( _K_ ) and unconstrained ( _V_ ) inputs, _X=(V,K)_ , and assume for simplicity, without a loss of generality, that the constrained inputs have no alternative cost. Prices for inputs ( _W_ ) and output ( _P_ ) are given by the markets. The optimization problem calls for a choice of the level of inputs to be assigned to technique j so as to maximize profits. To simplify the presentation, we deal with a comparative statics framework and therefore omit a time index for the variables. The extension to the intertemporal version is conceptually straightforward.

Ignoring the analytic details, we turn to characterize the solution and its implication. Let _s=(K,P,W,T)_ be the vector of state variables of this problem and write the solution as: _Vj*(s), Kj*(s)_ , to emphasize the dependence of the solution on the state variables. The optimal level of inputs _Vj*, Kj*_ determines the intensity of implementing the jth technique. To the extent that the implementation of a technique requires positive levels of some inputs, when the optimal levels of these inputs are zero, the technique is not implemented. The optimal output of technique j is _Yj* = Fj(Vj*, Kj*)_ , and the implemented technology ( _IT_ ) is defined by _IT(s) = {Fj(Vj,Kj); Fj(Vj*,Kj*)_ ≠ _0, Fj_ ∈ _T}_ .

The empirical analysis is based on observations generated by production functions that are implemented. The aggregate production function expresses the aggregate of outputs, produced by a set of micro production functions, as a function of aggregate inputs. This function is not uniquely defined because the set of micro functions actually implemented, and over which the aggregation is performed, depends on the state variables and as such is endogenous. The aggregate production function is written as: 5
This production function is defined conditional on _s_ , but changes in _s_ imply changes in _X*_ as well as in _F(X*,s)_ , and this is summarized by the reduced form _φ(s)_ . It is therefore meaningless in this framework to think of changes in _X_ , except by ‘error’, which are not instigated by changes in _s_ , or more precisely by a change in the implemented techniques. This means that whenever the implemented technology is affected by some state variables, it is impossible to reveal a stable production function from a sample of observations taken over points with different state variables. Thus, in general, the aggregate production function is not identifiable.

For (I.1) to be a production function in the usual sense, we need to introduce an allocation error to identify that portion of the applied inputs that is disjoint from s. With this in mind let _ε_ = _X –X*_ ; _E(εX*)_ = _E(εs)_ = 0; we elaborate further on the allocation error in the next section. With this modification we write the empirical the production function as: The function _F(X, s)_ can be approximated by a Cobb-Douglas-like function where the coefficients vary with the state variables and possibly with the inputs: where _y_ is the ln output, _x_ is ln _X_ , ε is redefined in terms of logs, ε = x – x*, _β(s, ε)_ and _γ(s)_ are the slope (vector) and intercept of the function respectively, and _u_ is a stochastic term.

Variations in the state variables affect _γ(s)_ and _β(s, ε)_ directly, as well as indirectly through their effect on inputs. The elasticity of output with respect to a given state variable is 6
The first term on the right hand side shows the output response to a change in inputs under constant technology. The remaining terms show the response of the implemented technology to a change in the state variables. This part is contained in the unexplained productivity residual in the standard productivity analysis under the assumption of constant technology. On the other hand, the elasticity with respect to the allocation error is The main message of this discussion is that to obtain a consistent estimate of the slope we need to estimate ∂ _y_ / ∂ε . Of course the allocation error is unobserved, but panel data can help us to deal with this problem.

# **II. The Statistical Model**<sup>11</sup>

To relate the discussion to the literature on empirical production functions, we start with the generic Cobb-Douglas model, and thus suppress the dependence of β on ε, which is the quadratic component of the function. The formulation is based on the micro model, but it is oriented toward macro data analysis by the introduction of additional state variables. The production function implemented under state s is: where _m0_ is an idiosyncratic term known to the firm but not to the econometrician, _u0_ is a random term whose value is unknown when the production decision is made.<sup>12</sup> _Eeu_ 0 ≡ _e_ η , and without a loss in generality, it is absorbed in Γ(s). The expectation of output, conditional on _X_ and _s_ , is _Y e_ = Γ ( _s_ ) _X_ β ( _s_ ) _em_ 0 , and the choice criterion is:

> 11 The discussion is based on Mundlak and Hoch (1965) and Mundlak (1996).

> 12 The formulation does not allow for delayed response to the transmitted error _m0_ as discussed in Chamberlain (1982).

7
where now W is the price of X measured in units of Y. The first order condition from the vantage of the econometrician, who is blind to idiosyncratic behavior, yields the optimal input _X*_ conditional on _s_ whereas actual input is given by _X_ = _X_ * _em_ 1 + _u_ 1 , where _m1_ summarizes the idiosyncratic behavior and _u1_ is a random term. The actual input, _X_ , differs from _X*_ , partly due to optimization error, and partly due to the econometrician’s failure to read the firms’ e decision correctly. Let _γ_ = ln _Г_ , _b_ = ln _β_ , _y_ = ln _Y_ , etc., substitute y = y + u 0 , and write the first order condition and the production function in log form: Solve for x: The system of equations (II.5) and (II.6) extend the standard analysis by the inclusion of _s_ . Both _s_ and _m0_ affect jointly _x*_ and _y_ , and thus cause a bias in the OLS estimation of the production function. The inclusion of the state variables in the model is likely to reduce the direct impact of _m0_ on output. We return to this subject in section VI below. To isolate the joint role of _s_ and _m0_ we make two initial assumptions.

= <u>Assumption 1: β(s) β</u> which also implies _b(s)=b._ To simplify the analysis, we linearize _γ(s)_ : 8
<u>Assumption 2: γ(s) = sγ.</u> A common approach to overcome the impact of _m0_ on the estimates is to utilize the panel structure of the sample. Let _zit_ be the i,t-th observation of a raw vector _z_ for country i and year t, i =1,...,N, and t=1,...,T. We rewrite the jth input demand, and without a loss in generality we normalize _x_ by - _c(s), b(s)_ is absorbed in _γ(s)_ , and note that xjit-xjit* is the composite ε _jit_ = _u jit_ + _m ji_ + _m jt_ : where u0it ~ IID (0, σ00). ujit ~ IID (0, σjj), E(u0uj) = 0; where _m0t_ and _m0i_ are the time effect and country effect on the production function (respectively), and the matrices are: _xit_ is 1xk, as is εit, _sit_ is 1xh, _γ_ is hx1.<sup>13</sup> Let _zi_ .. and _z_ . _t_ denote the averages of _zit_ over t and i respectively, and let _z_ .. be the overall mean. Let _W(it)_ be a projection matrix defined by its operation on a vector _z_ : _W_ ( _it_ ) _z_ = ( _zit_ − _zi_ . − _z_ . _t_ + _z_ ..) . Then the system reduces to: Observed state variables are to be included in the regression as exogenous variables and thus cause no identification problem. State variables that are not included, observed or unobserved, are part of the error term and as such lead to OLS biased estimates. When, however, _s_ consists only of country and time dummies, _W(it)s_ vanishes and OLS of (II.10) yield consistent and efficient estimates of _β_ . The precision of the estimate increases with the variance of _W(it)ε_ and decreases with the variance of _W(it)u0._ In contrast to the within-country-time transformation, the between transformations amplify the transmitted impact of _s_ and _m0_ on the estimates. This is the case for the

> 13 To simplify the presentation, we assume here that all inputs obey the first order condition in (II.3) and thus j=1,...,k. More accurately, some inputs are determined by longer term contracts (such as ‘fixed’ inputs) and could be thought of as exogenous, but even in this case, they may be affected by _s_ and _m0_ .

9
between-country, defined by the projection matrix _B(i)x=(xi_ **.** _. – x_ **_.._** _)_ , and the between-time, defined by the projection matrix _B(t)x=(x_ **.** _t – x_ **..** _)._ The transformed between-system is _B(.)_ is either _B(i)_ or _B(t)_ for the between-country or between-time transformation respectively. By construction, _W(it)_ , _B(i)_ , and _B(t)_ are orthogonal. Note that _B(.)s_ does not disappear, and as such it is part of the equation disturbance and leads to the bias in the OLS estimates. The impact of changes in _s_ on _y_ is summarized in equation (I.4).

The regression coefficients of interest can be written in a generic form for a − 1 projection matrix _P_ as: _b_ = [( _x_ , _s_ )` _P_ ( _x_ , _s_ )] ( _x_ , _s_ )` _Py ,_ where _P_ can be any one of the projection matrices of interest listed above with rank not smaller than the rank of the composite matrix ( _x,s_ ), where _x_ and _s_ are matrices built by stacking the i,t rows of the corresponding vectors for all i and t. It is to be noted that the three regressions mentioned here, within-country-time, between-country, and between-time, constitute a canonical set in the sense that regressions obtained from any other linear transformation of the data, such as pooled, within-time (country dummies), or within-country (time dummies), or by time differencing are matrix weighted combinations of the three canonical regressions.

Some authors use first differences to eliminate the i-effect and thus eliminate the bias in the _b(i)_ estimates.<sup>14</sup> This approach is less efficient than that of the within transformation. Both estimators are linear in the observations, and under the GaussMarkov condition, the within estimator is the efficient one. It should be noted in this connection that the transformation by the projection matrix _P_ changes the variance matrix of the disturbance. Thus, for the vector _u0_ , var _u0_ = _σ00I_ , and var _Pu_ = _σ00P_ , indicating heteroscedasticity. However, as indicated in the appendix, when _P_ is a projection matrix, then GLS of the transformed equation is equal to OLS, and hence it is the best linear unbiased estimate.

> 14 For instance, see Lau and Yotopoulos (1989), Mairesse (1990), and Griliches and Mairesse (1998).

10
As the data generated by _W(it)_ is cleaned from the time and country effects, they should best represent the more stable technology, referred to here as the core implemented technology.

# **III. Data**<sup>15</sup>

# **<u>Output and inputs</u>**

We estimate a cross-country agricultural production function where agricultural output depends on inputs, agricultural technology, and the state of the economy. In this analysis we use a measure of agricultural capital which revises and updates the previously constructed data set from Larson, Butzer, Mundlak, and Crego (2000). The inclusion of agricultural capital is one of several aspects which differentiate this study from most studies of agricultural production functions based on a panel of countries.

Agricultural output is measured as agricultural GDP in 1990 US dollars. We choose the GDP variable rather than the more often used agricultural production because it comes from the national accounts used for the construction of the fixed capital variable. Inputs to agricultural production include land, capital, labor, and fertilizers. Hectares of agricultural area are used for the measure of land. This includes arable land, land under permanent crops, and permanent pastures. Agricultural labor is defined as the economically active population in agriculture. Fertilizer consumption is often viewed as a proxy for the whole range of chemical inputs. The data on agricultural capital consists of two components: fixed capital, consisting primarily of structures and equipment, and capital of agricultural origin, consisting of livestock and trees. The two components differ in the method of construction, and also in terms of markets and pricing.

# **<u>Technology</u>**

As the available technology is unobserved, what we can do in empirical analysis is to identify variables associated with variations in the implemented technology. In the case of agriculture, there is a natural variable to measure the level of technology for a given

> 15 For a description of the construction of the data, see Mundlak, Larson, and Butzer (1997). While the methodology is the same, sources have been updated and in some cases revised. Details can be obtained from the authors.

11
crop; this is the yield or output per unit of land. The yield has been the main criterion for the introduction of the modern varieties of cereals and other crops, termed as the green revolution, beginning around the middle of the last century. The higher is the yield of the modern varieties, or the larger is the area devoted to these varieties, the larger is the average yield. Extending this concept to aggregate output, we construct an aggregate peak yield variable. For each country and each commodity, the maximum of the past yields is computed, thereby reflecting the potential output from the implemented technology in any given year. Country-specific Paasche indices (1990=1) are constructed of these peak commodity yields, weighted by land area. A Paasche index is used since changing the composition of output changes the relevance of existing technologies.

The most common variable used in empirical studies as a carrier or representative of technology is some measure of human capital, mostly schooling. The basic idea is that higher levels of education are conducive to technological progress.<sup>16</sup> We include the average schooling years of the total labor force, taken from Barro and Lee (2000).<sup>17</sup> Empirical studies show the relevance of various public goods that are associated with productivity, such as infrastructure in transportation and communication, measures of public health, and research and extension.<sup>18</sup> In this study we do not attempt to determine the contribution of these variables individually, but rather allow for the overall effect of the group on the estimation. We do this by selecting the per capita output in the country as a comprehensive measure of capital and technology (Mundlak and Hellinghausen 1982). We measure it as the ratio of the country per capita output to that of the United States and refer to it as a development indicator. This variable replaces the need for introducing a dummy variable to differentiate between developed and developing countries as some studies do.

> 16 However, the causality could go in either direction in that economic progress generates a demand for schooling. Therefore, the interpretation of a schooling variable in empirical analysis is somewhat ambiguous.

> 17 Education data are reported for every five years through the World Bank website

> <u>(http://devdata.worldbank.org/edstats/). Data for other years are obtained through linear interpolations.</u>

> 18 See for instance Evenson and Kislev (1975), Antle (1983), Binswanger et al. (1987), Lau and Yotopoulos (1989), and Craig, Pardey, and Roseboom (1997).

12
# **<u>Institutions</u>**

It is assumed that the physical, legal, and regulatory infrastructure and institutions support overall, including agricultural, development. We measure this influence with two variables obtained from the Freedom House – political rights and civil liberties. The measure of political rights reflects the electoral process, political pluralism and participation, and functioning of the government. The civil liberties measure includes aspects of freedom of expression and belief, associational and organizational rights, rule of law, and personal autonomy and individual rights. Both measures are on a scale of 1 to 7, where 1 represents the most free and 7 the least free. If these variables matter, they are expected to be correlated with development and reflected in the development indicator. They are nevertheless introduced here explicitly because of our interest in trying to isolate the effects of institutions on agricultural productivity. Hence, the expected contribution of these variables in the present analysis is over and above that of the development variable.

# **<u>Incentives</u>**

We introduce two measures of incentives to allow for the direct effect of incentives on productivity over and above their indirect effect through resource allocation and accumulation. The measures are the terms of trade between the agricultural sector and the overall economy, obtained as the relative price (agricultural GDP deflator to total GDP deflator, lagged one period), and its variability, calculated as a moving standard deviation from the three previous periods.<sup>19</sup> Note that this measure confounds in it the various taxes or subsidies, direct or indirect, applied to the sector. The variability in agricultural prices reflects the market risk faced by agricultural producers. In addition to the sector-specific risk, there is an economy-wide market risk, that of price volatility for the economy as a whole, measured by the rate of inflation. This is calculated as the rate of change in the total GDP deflator.

> 19 In an earlier paper, Mundlak, Larson, and Butzer (1999) used the price ratio of agriculture to that of manufacturing. The reason for the change is that the alternative to agricultural resources is not limited to manufacturing, but includes also services, which may, in fact, have become more important than manufacturing, particularly in developed countries.

13
# **<u>Physical environment</u>**

Agricultural production depends on the physical environment or natural conditions. We represent the environment by using two variables: potential dry matter (PDM) and a factor of water availability (FWA).<sup>20</sup> The first variable is intended to measure the theoretical potential production of dry matter. The production of dry matter requires moisture. Arid areas may have a large value for PDM, but actual production is small due to water deficit. The relative water availability is measured by the ratio of actual transpiration to potential transpiration. These two variables are country specific and do not vary with time.

# **IV. Sample Description**

The sample was determined by the data availability and the preference for a balanced data panel in order to simplify the analysis. It consists of annual data from 30 countries<sup>21</sup> for a 29-year period (1972-2000). The information conveyed by the sample is summarized in Table 1. The first column presents the average annual growth rate of the variables over the sample period. Agricultural output grew at a rate of 5.43 percent, whereas agricultural capital grew at a higher rate of 5.77 percent. Agricultural labor declined at the average rate of 0.6 percent. Thus, the average labor productivity increased at the average rate of 6.03 percent, and the ratio of capital to labor increased at the average rate of 6.37 percent. The growth rate of capital of agricultural origin (4.94 percent) is lower than that of fixed capital (5.80 percent). Fertilizer grew on average at the rate of 1.87 percent, whereas the agricultural area grew at the rate of 0.01 percent, implying a growth in the fertilizer-land ratio. The growth of agricultural output took place in spite of unfavorable prices as indicated by the decline in the terms of trade of agriculture at the average rate of 1.26 percent. These results signal an increase in productivity. The technology measures show a growth rate of schooling of 1.67 percent and 1.41 percent for peak yield.

> 20 The measures are based on Buringh, van Heemst, and Staring (1979) and were used in Mundlak and Hellinghausen (1982) and Binswanger et al. (1987).

> 21 Countries included in the study are: Australia, Austria, Canada, Cyprus, Denmark, Egypt, Finland, France, Greece, India, Indonesia, Italy, Kenya, Republic of Korea, Malawi, Mauritius, Morocco, Netherlands, Norway, Pakistan, Peru, Philippines, Sri Lanka, Sweden, Republic of Tanzania, Tunisia, Turkey, United Kingdom, United States, and Uruguay.

14
As mentioned above, the institutions measures are ordinal, and thus, growth rates would be meaningless. To give a picture of how institutions have evolved over the time period studied, we looked at the averages and medians of the indices for each year. For our sample of countries, the average measure of civil liberties is 3.27 in 1972 and 2.57 in 2000. For political rights, the averages are 3.17 and 2.37 respectively. Our sample covers countries on both ends of the spectrum, with a few countries advancing from 6 and 7 (“non-free”) to 1 and 2 (“free”),<sup>22</sup> while there are 7 countries which remain at 1 throughout the time period.<sup>23</sup> The qualitative nature of the above results is consistent with the common knowledge on agricultural development in the sample period. The results are highlighted here for two reasons: first to characterize the sample and second to show that the data are subject to a great deal of variability over time and across countries. This variability provides an insight into the relationships between the different variables of interest. To describe the variability we decompose the total sum of squares to the three orthogonal components (within-country-time, between-country, and between-time). Thus, SS total = SS(xit - x..) is decomposed to SSW(it) = SS(xit - xi. - x.t + x..), SSB(i) = SS(xi. - x..), SSB(t) 2 = SS(x.t - x..), where, for any variable z, we use the notation: SS(z) = ∑ zit . i<sup>∑</sup> t To standardize the results, we divide the components by the total sum of squares so that the numbers in Table 1 show the percentage of each component in the total sum of squares. The between-country differences account for most of the variability in output and more so in the inputs; about 89 percent of the output variability is due to the betweencountry differences. Thus, a regression which allows for a country effect, without any quantitative variables, would yield an R<sup>2</sup> of 0.89, so that the unexplained residual from country averages accounts for only 11 percent of the total sum of squares of output. If we add the time effect, the R<sup>2</sup> rises to 0.98. Similarly, the between-country variability accounts

> 22 The terms “free, partially free and non-free” are used by Freedom House as classifications corresponding to the indices of political rights and civil liberties.

> 23 If we restrict the sample to countries which have at least one value not equal to 1, the range of averages increases to 4.09 and 3.14 for civil liberties and 4.10 and 2.95 for political rights (in 1972 and 2000 respectively).

15
for 95 percent to almost 100 percent of the total variability in land, labor, livestock, and fertilizers. The situation is similar when the output and inputs are measured per worker.

The relative importance of the country and time components is different for the variability of the state variables. The between-country component is important in schooling, development, political rights, and civil liberties and less important in the other variables. In part, this difference among the state variables reflects the way the variables are measured. Schooling, development, political rights, and civil liberties are measured in units that allow cross-country comparisons, and interestingly, the relative importance of the between-country component in the total sum of squares is similar to that of output. We can relate this discussion to the determinants of the inputs as shown in equation (II.6). It seems that schooling, development, and the institutional variables can be identified with _s_ in that they are associated with the technology level and also affect the input level. They seem to have a strong correlation with the country effect. On the other hand, the price variables are indices, and as such, do not allow cross-country comparisons. They have a strong deviation component and perhaps are associated with the allocation error. A strong between-time effect is represented by the peak variable.

To sum up, the relative importance of the between-country component is dominant. This can lead to the erroneous conclusion that the within analysis has little to contribute. As a matter of principle, this conclusion is not well-founded because the precision of the estimated coefficients depends not only on the spread in the regressors but also on the variance of the equation disturbance which usually contains a component that is time invariant. Consequently, the variance of the within component is considerably smaller than the total variance. This is validated below where we show that the within estimates are meaningful empirically and informative substantively.

# **V. Empirical Results**

Our ultimate interest is the estimation of the role of the inputs in production, or simply the production elasticities. To do this we have to eliminate the jointness effect, or the transmitted effect consisting of the state variables and of the country and time idiosyncratic variables. To accomplish this we present here two models: the first is a pure 16
production function where inputs are the only regressors, and the second is an extended function which contains also the state variables described in section III.

We organize the empirical results of each model in three blocks. The first block presents the within-country-time estimates, _b(it)._ The working hypothesis is that these estimates are based on observations taken from the core technology. The second block presents the between-time estimates, _b(t)_ , representing the time-series component, common to all countries, and as such it captures the impact of changes over time in the available technology. The last block presents the between-country estimates, _b(i)_ , summarizing the between-country variability. The estimates are based on the locus of points that go across the different techniques implemented by the countries which, in principle, operate under the same available technology.

The general form of the estimated equation is: _yit_ = β 0 + _W_ ( _it_ )( _xit_ β+ _s_ it γ ) + _B_ ( _i_ )( _xit_ β _bc_ + _sit_ γ _bc_ ) + _B_ ( _t_ )( _xit_ β _bT_ + _sit_ γ _bT_ ) +ν 0it (V.1) where β0 is the intercept, and ν 0 _it_ = _m_ 0 _i_ + _m_ 0 _t_ + _u_ 0 _it_ .

Due to the orthogonal structure of the regressors, it is possible to estimate the three blocks separately. We can do it by estimating (II.10) and (II.12), or by estimating (V.1).<sup>24</sup> The difference will be in the dependent variable, and consequently in the value of R<sup>2</sup> and in the degrees of freedom used in the derivation of the t-score. We present the results from both approaches for reasons to be discussed below.

# **<u>Inputs only</u>**

The results are presented in Table 2. The dependent variable for columns termed ‘block’ is the transformed variable, _Py_ , where _P=W(it)_ , _B(t)_ , and _B(i)_ respectively, associated with (II.10) and (II.12). The R<sup>2</sup> and the t-score in the independent-block column are obtained from this regression. The R<sup>2</sup> appearing in the joint-block column is the proportion of the total variability of _y_ (not of _Py_ ) explained by this regression. The ratio of the two values of R<sup>2</sup> reflects the proportion of the block sum of squares in the total sum of

> 24 We refer to estimations of equations (II.10) and (II.12) as ‘independent-block’, while the estimation of equation (V.1) is termed ‘joint-block’.

17
squares of output, referred to as the weight. Thus, the R<sup>2</sup> of the joint-block equation (V.1), 0.9566, is equal to the weighted average of the independent-block values for R<sup>2</sup> . The t- score in the joint-block regression is obtained from the estimation of equation (V.1). It is clear that the contribution of the within variables to the explanation of total output ( _y_ ) is relatively small and that the t-score of the coefficients is lower than that obtained from the independent-block estimation where the dependent variable is _W(it)y_ . On the other hand, the difference between the two versions is smaller for the between-country estimates. This is a demonstration of the consequences of an implicit or explicit preference for using the between estimates, which in the case of panel data, would be the between-country estimates. The within estimates are avoided by working with country averages. However, the within variables provide information for identifying the production function and are less contaminated by variables leading to inconsistency in the estimates.

The key question of this analysis is whether the coefficients of the variables common to the three canonical regressions are the same, aside from sampling error. A casual inspection of the results indicates that they are quite different, confirming the basic initial hypothesis that the regressions summarize the combined effect of changes in inputs and technology, and therefore the within and between regressions summarize different processes.

To introduce uniformity in the results of the various models we impose constant returns to scale on the within estimates. This constraint is imposed only on the within estimates, because the between estimates are subject to the jointness effect and therefore do not present pure input elasticities. The sum of the within elasticities without this constraint is 1.25, and the difference between the input elasticities in the constrained and unconstrained models is absorbed mostly in the land elasticity.<sup>25</sup> The Wald test of constant returns to scale in the within regression is not rejected at the 4 percent level.

The sum elasticities of the two types of capital is 0.42, and the elasticity of land is 0.33. With sum elasticities of 0.75 for capital and land, there is little scope left for labor

> 25 To save space we do not present the unconstrained results.

18
and fertilizer.<sup>26</sup> This is the most important substantive result which indicates that agriculture is capital-cost-intensive. The elasticity of fertilizers, 0.13, is considered to be high for several reasons. First, we deal with the aggregate agricultural production function, whereas fertilizer is used only on plant products. Thus the corresponding elasticity related to the plant products should be higher than that obtained for the aggregate product. Second, note that the dependent variable is value added, and in a competitive economy the elasticity of fertilizer, whose cost is allowed for in the computation of value added, should be nearly zero (an outcome of the envelope theorem). A higher value for the fertilizer elasticity is likely to signal constraints on the supply of fertilizer causing the shadow price to exceed the official price used in the national accounts to compute the cost of fertilizer. The labor elasticity appears low, and we return to this subject later on. In Mundlak, Larson, and Butzer (1999), the early literature on cross-country studies was reviewed, and it stands out that our results differ from those reported in that literature. In part, it may be due to the fact that we use a complete capital series, which was absent from the other studies, and in part it is due to the fact that we use the within-time-country estimates, whereas the reported results are mostly cross-country in nature and resemble the between-country regression in the present study.

Turning to the between regressions, we note that the values of R<sup>2</sup> are by far higher than that of the within equation, and it is particularly high for the time-series component as given in the between-time regression. The sum of the capital elasticities is 0.85 in the between-time regression and 0.32 in the between-country regression. The between-time elasticity is particularly high, and this suggests that the pace of the implementation of changes in the available technology was strongly constrained by the level of the capital stock in agriculture. Similarly, the land coefficient in the between-time regression is high, but its t-score is low. A high value for the land coefficient suggests an increase in the shadow price of land associated with the increase in productivity, while at the same time there was little increase in the time series of agricultural area (Table 1).

> 26 These values are of the same order of magnitude obtained in an earlier study (Mundlak, Larson, and Butzer 1999) for a different sample of countries and different time period.

19
What is striking in the between-country regression is the low elasticity of land, 0.02, and the high elasticity of fertilizer, 0.41. This suggests that the techniques used by the more productive countries were land-saving and fertilizer-using. The subject is taken up in section VI.

# **<u>The reduced form</u>**

The country and time effects are estimated as residuals, and the question is to what extent they can be replaced by the state variables. The potential list of pertinent state variables is of unknown length, but we can only deal with observed variables. To get an idea on the relevance of our set of state variables, we estimate the reduced form of output, equation (I.1), which in view of equation (I.4) amounts to a quasi-supply function, in the sense that it allows for changes in the supply function. The state variables are decomposed to their orthogonal components, and their impact on the estimates of the various blocks is determined accordingly. The within estimates are determined by the interaction term, _W(it)s_ , and the between regressions are determined by _B(.)s._ The time behavior of the state variables is demonstrated in Figure 1 which presents plots of the annual averages of _B(t)s_ over time. It is seen that schooling and peak yield show a positive trend and are highly correlated (0.995). On the other hand, the relative prices and the price variability show a negative trend. Civil liberties and political rights are also subject to negative trend, which means overall improvements over time of these attributes. Inflation is fairly stable except for a big jump around 1990.

The OLS results organized by blocks appear in Table 3.<sup>27</sup> The values of the R<sup>2</sup> are not high for the within and the between-country regression in contrast to the time series. This means that our variables capture well the changes over time, and less so for the cross section. The R<sup>2</sup> for the model as a whole is 0.613. Comparing these values to those in Table 1, where the country effect by itself accounts for 89 percent of the output variance, indicates that our set of state variables is far from coming close to being a perfect substitute

> 27 To save space, we present here t-scores only from the independent-block estimates.

20
for the country effects. The D.W. test statistic (2.171) is reported only for the between-time regression, where it is relevant.<sup>28</sup> It is obvious that the coefficients in the three blocks are different and therefore represent different processes. Specifically, the relative price is positive and significant in the within and the between-country regressions and negative in the between-time regressions.<sup>29</sup> The price variability has a negative coefficient in the within and in the between-country regressions but not in the between-time regression. Schooling is positive in all the three regressions, as is the development indicator. Civil liberties has the expected negative sign in the within and between-time regression but not in the between-country regression, while political rights has a negative coefficient in the between regressions, but not in the within regression. Most of these results are carried over to the production function with the state variables, referred to as the extended model.

To sum up, it is important to note that even though the within regression has a low R<sup>2</sup> , it presents a supply function with expected signs. This is achieved by the interaction terms _W(it)s_ which are expected to cause input variations and to have a smaller impact on the technology choice. This is not the case for the between regressions.

# **<u>The extended production function</u>**

The OLS estimates of the extended model appear in Table 4. The R<sup>2</sup> of this model is 0.9694 as compared with 0.9566 in Table 2. This means that 29 percent of the unexplained error of the model in Table 2 was reduced by the introduction of the state variables. An F-test indicates that this addition is significantly different from zero. Constant returns to scale is imposed on the within inputs elasticities. The sum elasticities without this constraint is 1.22, and the difference between the constrained and unconstrained elasticities is mostly in the elasticity of land.

An examination of the input elasticities shows that the big picture presented in Table 2 has not changed in a dramatic way. Specifically, for the within estimates, the sum

> 28 We have also computed a principal components version. The results are not different in a substantive way from the OLS estimates and are therefore not reported here.

> 29 A similar result is reported in Binswanger et al. (1987).

21
elasticities of capital is 0.37 and the elasticity of land is 0.45, and this again leaves little scope for fertilizer and labor. The input coefficients in the between-regressions are also quite similar to those observed in Table 2. What then is the contribution of the state variables to the within regression? The answer is the rise of R<sup>2</sup> , or the reduction of the equation variance. To see where it comes from we first review the impact of the state variables in the within block. The main contribution comes from the price block and the development indicator. The price coefficient is positive and that of the price variability is negative. This is consistent with a positive response to price changes and a negative one to risk. In interpreting the role of price, we note that _w_ which appears in the theoretical model discussed above is the vector of real factor prices, which is unobserved. The relative price in the regression is the terms of trade of agriculture, which is the denominator of the components of the vector _w_ . As we deal with the log of the price variables, the denominator of the vector _w_ can be separated from the nominal factor prices, and it is introduced here explicitly into the equation. Thus the positive sign of the relative price coefficient in the within regression is interpreted as a positive supply response of the implemented technology. The price elasticity of productivity is 0.29, and that of the price variability is -0.31. These are quite sizable values. Using a somewhat different formulation, to which we return below, Fulginiti and Perrin (1993) reports a price elasticity of productivity of 0.13.<sup>30</sup> The development indicator, which reflects the overall infrastructure of the economy, as well as the institutional and technological environment, seems to be the most robust variable. This indicates that the more productive is the economy as a whole the higher is the productivity of agriculture. The civil liberties variable has the anticipated (negative) sign, whereas schooling has a weak negative impact. Note that the variables in the within regression are the interaction after the main effects were extracted, and therefore the sign indicates the correction to the influence of the variables as given by the main effects.

The productivity response observed in the between regressions is not always consistent with that of the within regression. In the between-country regression, the sign of

> 30 See also Hu and Antle (1993) and Binswanger et al. (1987) for price response in different formulations.

22
both the price and the price variability is the same as in the within regression. However, for the between-time estimates, the price coefficients have the opposite signs, as the productivity growth was associated with a decline in price and in a rise of the price variability. This reflects a downward trend in the relative price associated with productivity rise in world agriculture.

The price variability coefficient has a negative sign in the between-country regression, indicating negative response to risk. Inflation has a negative coefficient in the between-country regression.

The magnitude and the sign of the development indicator are robust across the three equations. Schooling has a weak positive impact in the between regressions.<sup>31</sup> The impact of political rights and civil liberties is ambiguous and weak. The two physical environment variables vary across countries but are time invariant. The sign of the water availability is positive, as expected. The sign of potential dry matter is negative. This is inconsistent with our earlier results and indicates that in this sample the high PDM countries were less productive.

# **<u>Stability of results</u>**

What happens when we remove the assumptions made in section II above? Assumption 2 of the linearity of γ(s) is not crucial and can be ignored here. Assumption 1 (constant β) however, is more crucial. One way to find out the validity of this assumption is to run the regression for subperiods. Table 5 presents results for two subperiods, 19721985 and 1985-2000.<sup>32</sup> A comparison of the two periods, and with the results in Table 4 for the whole period, indicates some changes but the qualitative nature of most of the main results is preserved. The strength of the capital elasticities is preserved even though there is some change in the composition of the two components of capital. The fertilizer elasticity is 0.14 and 0.13, and the interpretation of this value remains intact. The labor elasticity increased in the second period. The rise in some of the elasticities reduces the elasticity of

> 31 Because of the strong correlation between the time averages of schooling and peak yield, the latter variable is not included in the between-time regression.

> 32 We present here only the t-scores from the independent-block estimation. This is sufficient to establish the point of the dependence of the estimates on the period chosen.

23
land, but still the sum elasticities of capital and land exceed 0.5. The rise in the elasticity of labor in the second period may reflect the decline in the agricultural labor force.

Turning to the state variables, schooling has the wrong sign as before, but the peak yield is positive and significant in the first period, which experienced a stronger rise in yields. Finally, the role of price is positive and significant, whereas the price variability is negative. Thinking of the within estimators as representing the core technology, we see that the core technology is not detached from the economic environment and therefore is not invariant to the sample.

The between-time regressions also preserve the important result of high elasticity of capital. The sum elasticities of the two capital components are 0.94 for the first period, 0.70 for the second period as compared with 0.83 for the period as a whole. The betweencountry estimates of the input elasticities show little change. The main changes are in the coefficients of the state variables.

There are two possible approaches to incorporate the variability of _β(s,x)_ in the analysis.<sup>33</sup> The first requires knowledge of the factor shares and consists of estimating the elasticities from a system of factor shares and the production function.<sup>34</sup> This approach has problems of its own which are related to the interpretation we can give to the factor shares. The second approach is to write out _β(s,x)_ as a linear function of _s_ and _x_ which leads to a quadratic production function in _s_ and _x_ .<sup>35</sup> Such a function is blessed with many terms which are intercorrelated and thus create a problem for the extraction of reliable results. Note that the system reported in Table 4 already has a very high R<sup>2</sup> , and there is little scope for squeezing in many additional terms. One possibility is to be selective with the number of quadratic terms. For instance, Fulginiti and Perrin (1993) used the heterogeneous technology framework to estimate such an equation including quadratic terms of the inputs with some of the state variables (they refer to the state variables as technology-changing variables) for a sample of 16 developing countries for the period 1961-1985. A key issue in

> 33 See Mundlak (1988, 2001).

> 34 See Mundlak, Cavallo, and Domenech (1989).

> 35 The dependence on _x_ alone yields the translog function (Christensen, Jorgenson, and Lau 1973). We suppress here the dependence on _x_ and concentrate the discussion on the dependence on _s_ .

24
that study was to obtain a positive supply response to prices, and the outcome is an elasticity of 0.13 for output price and -0.09 for wages.

# **<u>VI. The Role of the Effects</u>**

Ordinarily, panel data analysis starts with the estimation of the coefficients of the quantitative variables, and this is followed up with the introduction of discrete qualitative variables, namely the effects. This natural course of action emphasizes the role of the quantitative variables and diverts attention from the information embedded in the effects. To clarify this point we can reverse the order and start the analysis by examining the role of the effects. In our case this calls for the decomposition of the output sum of squares and the computation of the R<sup>2</sup> of an equation consisting solely of country and time dummies. In our sample, such an equation explains about 98.5 percent of the total output sum of squares, as shown in Table 1. All this, to be sure, is without an inclusion of any input in the equation. But the inputs are there, because there is a strong correlation between the inputs and the effects, and this goes back to the optimization described in equations (II.1) and (II.2). As long as _m0_ affects the decision on input demand, the estimated effects reflect input variations. For a similar reason they also reflect variations in the state variables.

The introduction of inputs to the empirical equation yields significant coefficients but has little impact on the degree of explanation. The reason for this weak impact is that the inputs are subject to strong country and time effects. These effects, however, do not exhaust the input variability so that _W(it)x_ does not vanish, and it is this remaining variability that provides the information for the estimation of the coefficients.

To express the relationship between the effects and the regressors we rewrite equation (V.1): _yit_ = β 0 + _xit_ β+ _sit_ γ + _B_ ( _i_ )[ _xit_ π _xc_ + _sit_ π _sc_ ] + _B_ ( _t_ )[ _xit_ π _xT_ + _sit_ π _sT_ ] +ν 0it (VI.1) The within estimator provides an estimate of β and γ , whereas the betweencountry and between-time estimators provide estimates of βbc, γbc, βbT, and γbT respectively. Thus the π’s are the bias of the between estimators.

25
The relationship between the effects and the regressors is summarized by the following equations

> where ς 0i and ς 0t are the error terms.

An estimate of _m0i_ and _m0t_ is obtained from a regression of (II.8) with country and time dummies. The values of R<sup>2</sup> for the country regression (VI.2) are 0.554 with the state variables alone, 0.830 with the inputs alone, and 0.885 for both groups. Similar regressions for the time effect (VI.3) yield 0.982, 0.995, and 0.998 respectively. From this we learn that the state variables account for most of the time effect and less so for the country effect. The inputs account for a larger proportion of the country effect, but still less than of the time effect. Technical change is the main event which evolved over time, and the set of the state variables seems to be strongly correlated with it. The weaker relationship between the state variables and the country effect indicates that there is a scope for introducing additional state variables that are correlated with the country effect.

What are the implications of the estimates of equations (VI.2) and (VI.3)? The first one is that it provides a set of variables that account for the effect. The set is not unique, and we have already alluded to the long list of potential state variables used in the literature to account for growth and productivity. Thus one would have to provide a rational for preferring one set to an alternative one. The statistical analysis alone is insufficient to do the task. The second implication is related to the estimation itself. Suppose that we have a deterministic solution for the two equations, which means that we can replace the unobserved effects with observed variables. How would it affect the estimation? The answer is given by equation (VI.1). The estimate of β would be the within estimator, since the unobserved effects represented by _moi_ and _mot_ and reflected in the π would vanish. This means that explaining the effects can tell us something about how transformations of the data affect bias related to unobservables, but it should not change our choice of estimator.

26
# **VII. Evaluation**

As there are big differences between the three estimators, it is desirable to get a sense of reality and check how our estimates relate to the real world. We do it at a general level, starting with the calculation of the TFP. Using the growth rates in Table 1 and the within elasticities from Table 4 we obtain that TF increased at an average annual rate of 2.23 percent whereas TFP increased at an annual rate of 3.2 percent which accounts to 59 percent of output growth.

Using the elasticities, we compute the marginal value productivity, or shadow price, as the product of the average value productivity and the corresponding elasticity. Because the distribution is quite skewed, Table 6 presents the results of the median and of the mean. We note that the shadow rate of return on fixed capital is quite high, and this is consistent with the high growth rate of this input. Figure 2 presents the time path of the median shadow prices from which we learn that the capital deepening resulted in convergence to around 0.14. The shadow price of capital of agricultural origin is lower, and this may be related to the way the variable was constructed. The shadow wage of labor is relatively low which explains the migration of labor out of agriculture. The decline in the labor force and the rise of capital caused the shadow wage to grow at the annual rate of 5.4 percent, considerably higher than that of TFP. There is a problem in comparing the shadow wages to published wages. Published wages refer to payment for actual work, whereas the labor data refer to the available labor force which is not fully occupied due to the seasonality of farm work. This issue is discussed in some detail in a study on Asian agriculture (Mundlak, Larson, and Butzer 2004). The shadow price of fertilizer increased over time at a higher rate than that of TFP in spite of the fact that the fertilizer-land ratio has increased constantly over time. Recall that the output is value added, and thus it is net of fertilizer cost. It is likely that the price at the farm gate is higher than the price used in national accounts, but still there may be a gap reflecting the rise in demand due to the shift to fertilizer-intensive crops. Finally, the rent per hectare of land in 1990 dollar is 568 at the mean and 271 at the median. To get from this to the value of land, we assume a depreciation rate of 0.05 and subtract it from the shadow price of capital. We then capitalize it by dividing the rent on land by the net rate of return to capital to obtain 2185 and 2464 1990 dollars per hectare at the mean and median respectively. There is of course 27
considerable variability in the sample as in reality. To sum up this evaluation, it seems that our results have a realistic flavor which would not be the case if we repeated the calculations with the between estimates.

# **VIII. Perspective**

It is useful to relate the model briefly to the discussion in the literature on panel data.

1) Identification: The identification of the production function depends largely on the allocation error. The more the firms deviate from the first order conditions, the more accurate the estimates will be.

2) Consistency: In the absence of state variables, or under a weaker assumption where _W(it)s_ = 0, the OLS estimates of the within equation are consistent and those of the between equations are not. Some authors use first differences to eliminate the i-effect in order to eliminate the bias in the _b(i)_ estimates.<sup>36</sup> As indicated above, this approach is inefficient.

3) Sample size: Increasing the sample size does not eliminate the bias caused by the jointness effect; it only reduces the sampling error. This is true regardless of whether the sample is increased through N or T (the number of countries or years).

4) Input spread: The decomposition of the sum of squares of the inputs show that SSB(i) is dominating, and that SSW(it) is relatively small. It is, therefore, claimed that the within estimator does not utilize important information. This is true but not the whole truth, because SSW(it) also constitute a small fraction of the total SS of output. Thus, there is less information, but there is less to be explained by this information. We have demonstrated that the within estimator provides meaningful and statistically significant results.

5) Fixed or random effects: The foregoing discussion is invariant to the assumption about the nature of the idiosyncratic variables, or effects. Under the random effect model the GLS estimator is a matrix-weighted average of the within and between estimators

> 36 For instance, see Lau and Yotopoulos (1989) and Griliches and Mairesse (1998).

28
(Maddala 1971), and it is therefore inconsistent. The source of the bias is the jointness effect.

6) Measurement error: The within estimator is more sensitive to measurement errors.<sup>37</sup> This statement assumes implicitly that the measurement error is unaffected by the transformation, so that its relative contribution to the within SS is larger than to the between SS. This possibility is not ruled out, but it should be noted that there is good reason to believe that part of the measurement error is country (or firm) specific, and by the same token it is time specific, and is thus eliminated by the within transformation.<sup>38</sup> It is impossible to generalize on the relative importance of the measurement error in the universe of all panels. What we learn from this study is that the most sensible results come from the within transformation, and this transformation is consistent with the theory formulated above.

7) Diversity of results: Concern has been expressed from the fact that there is a great deal of diversity in the results obtained in production function estimates from panel data depending on how the data are pooled (Griliches and Mairesse 1998; Mairesse 1990). The diversity is a problem when the working hypothesis is that the estimates should be invariant to way the data are pooled. The general model presented here indicates that one should expect diversity, and in fact the diversity serves as a starting point for the construction of more meaningful models.

8) Instrumental variables: The use of instrumental variables was suggested as a way to overcome the bias in the estimates of panel data (Hausman and Taylor 1981). In the present framework the scope for the use of instrumental variables is rather limited because variables which are associated with the choice of inputs are assumed also to affect the choice of the function itself. In other words, the instrumental variables fall in the category of state variables in the present framework. The same argument also rules out the GMM estimator.

> 37  See Griliches and Mairesse (1998).

> 38  For a fuller discussion, see Mundlak (2001).

29
9) Input ratios: In the Cobb-Douglas a difference between the log of two inputs,<sup>39</sup> say j and g, xjit – xgit, eliminates the terms _m0_ and, in the absence of state variables, can serve as an instrumental variable. In this approach, when the production function is estimated in terms of average productivity, and constant returns to scale is imposed, the estimation is free of the jointness bias. Unfortunately, under heterogeneous technology, this is no longer the case.

# **IX. Summary and Conclusions**

The paper presents an estimate of the agricultural production function from a panel of countries.

1) Framework: In the world of heterogeneous technology, the implemented techniques and inputs are jointly determined conditional on the state variables that are assumed to specify the economic environment. Because of variability in the state variables, the production function of a sector is an aggregate of micro production functions. It is approximated by a Cobb-Douglas function with parameters that depend upon the state variables.

2) State variables enter as exogenous variables in the empirical equation, and the estimation is straightforward when they are observed. In contrast, unobserved state variables become part of the production function shock, thus creating a correlation between the inputs and the productivity shocks, similar in nature to the transmission of the idiosyncratic variables in panel data. Due to the structure of the problem, the only way to identify the production function is through allocation errors, namely, through input variations that are unaffected by the omitted state variables or the idiosyncratic productivity shock.

3) The sum of squares of the panel data is decomposed into the three orthogonal components. Most of the variability in output and inputs comes from between-country variations, whereas the within-country-time variations account only for a small proportion of the total sum of squares. Estimates obtained from between-country variations are

> 39 See Mundlak (1996).

30
popular because they are based on a wide spread in the regressors. They are, however, biased. On the other hand, the within-country-time variations of the inputs reflect largely allocation errors and thus produce consistent or low-bias estimates.

4) When not all state variables are observed, the choice of regression matters. We provide a practical example and present estimates obtained under the assumption of constant slopes, for the canonical set of regressions, between-country, between-time (timeseries component), and within-country-time variations. There are great differences in the estimates of the three canonical regressions. The elasticity of capital from the within regression is 0.37, as compared to 0.27 from the between-country regression, and 0.83 from the between-time regression. The latter suggests that capital was a constraint in the implementation of new capital-intensive techniques, in spite of the fact that capital grew faster than all other inputs and output. These numbers are indicative of the differences in the results obtained from the three regressions, and similar differences exist for the other variables. The elasticity of fertilizer from the within regression is 0.1, although it should be close to zero because output is measured by GDP, which is net of fertilizer costs. This indicates that the shadow price of fertilizer was higher than the market price. However, the value obtained for the fertilizer coefficient from the between-country regression is 0.44. If this were a true elasticity, it would mean that 44 percent of GDP should be attributed to fertilizer; clearly, this is absurd. In contrast, it is likely that land is a dominant factor of agricultural production. In this case, the elasticity of land in the between-country regression is 0.03 as compared to 0.45 in the within regression. These comparisons provide substantive evidence on the superiority of the within estimator. This is true even in comparison to linear estimates obtained from pooled data, since these are weighted matrixaverages of the three canonical regressions and reflect the bias of their components.

5) Agriculture: The new techniques were capital and fertilizer intensive. This is reflected in the growth rates of these inputs. On the other hand, the techniques were labor saving; this is consistent with the decline in the size of the labor force in agriculture. The land elasticity is high for the within and the time component, and low for the between countries. Thus, the more productive countries use land-saving and fertilizer-using techniques. The land elasticity reflects the terms of trade of agriculture for the period, which, on the whole, enjoyed important improvements in productivity. The decomposition 31
of output growth shows that TFP accounted for 59 percent of the output growth of 5.43 percent.

6) State variables: The relevance of the state variables was tested by estimating the reduced form, or a quasi-supply function. They account for most of the variability of the between-time output, and only slightly for the within output. Still the within regression provides a supply function with the right signs and significant coefficients. Turning to the production function, the relative price of agriculture has a positive impact, and its variability has a negative impact in the within regression. The development indicator indicates that the agricultural productivity is positively correlated with the strength of the economy as a whole. The indicator is assumed to represent total capital, physical and human, and the institutional infrastructure. Some of the variables which are confounded in the indicator were introduced explicitly into the regression, but their contribution was marginal.

7) Accounting for the effects: The country and time effects account for most of the variability in the data. It is shown that the effects are embedded in the country and time means of the inputs and the state variables. The state variables are particularly important in capturing the time effect, and less so for the country effect. This suggests that there is a scope for trying out additional state variables to account for the cross-country variations.

8) Stability of results: The estimates are sensitive to the economic environment. This is demonstrated by estimating the regressions for two sub-periods. Even though the estimates change, the main message is preserved.

10) The results are consistent with the changes that take place in the process of growth. Agricultural productivity rises, there is a shift to capital and fertilizer-intensive and labor-extensive techniques. The response in resource allocation leads to growth in the marginal productivities of labor, fertilizer, and land, and a slight decline in the marginal productivity of capital as a result of the fast growth of the capital-labor ratio. From the point of view of growth, policies should encourage the process to continue.

32