Modelling an aggregate agricultural panel with application to US farm input demands

1. Introduction

Cost function and input demand measurement are among the most useful tools for economic analysis. Analyses of factor substitution, returns to scale and scope, technical change, industry performance and productivity all have relied on cost functions (Chambers, 1988). The usual econometric approach is to begin by specifying a flexible functional form of the cost function in prices, fixed inputs and output, followed by an application of Shephard's Lemma to derive choice functions that have additive errors with input quantities, expenditures or cost shares as dependent variables (McElroy, 1987).

Data at individual and at various levels of aggregation are increasingly available in panels, creating both opportunities and challenges. Beyond the standard questions of functional form and identification, panel data sets require attention to the specification of the cross section and time series elements of the error structure and the nature of any heterogeneity across cross-sectional units (Baltagi, 2008). Appropriate estimation methods depend on whether the data are aggregate (e.g. at the regional or country level, Ball, Hallahan and Nehring, 2004; Ball et al., 1997; Behrens and De Haen, 1980) or at the individual firm level (e.g. Thijssen, 1992, 1994; Lansink, 2000; Gardebroek, 2004; Rasmussen, 2010).¹ This paper presents a model for and analysis of a panel data set of state-level annual observations for US agricultural input use.² This requires consideration of several potential problems, including endogenous explanatory variables, aggregation across decision-makers, technological heterogeneity, production risk and the spatial and temporal properties of the error terms.

Opportunities associated with using panel data sets are well-known. In particular, common and heterogeneous effects can be separately identified. Cross-sectional heterogeneity in single equation econometric models can be accounted for using the well-known fixed- and random-effects estimators (Wooldridge, 2010: chapter 10). It is less clear how to appropriately account for cross-sectional heterogeneity in non-linear models and/or multiple equations frameworks. We focus on this issue by allowing a subset of the model parameters to vary cross-sectionally, and also provide a method for addressing statistical inference in a multiple equation framework where both spatial and temporal correlations among the error terms are present. While our empirical application is to state-level data, the empirical framework can be generally applied to more disaggregate panel data as well.

This paper presents a model of and an empirical application to a balanced panel of US state-level annual agricultural input use for the time period 1960–1999. The paper applies a relatively new approach to production modelling (LaFrance and Pope, 2010). This is expanded and the empirical model addresses: (i) measurement errors in outputs; (ii) model generality and flexibility; (iii) technical change; (iv) endogeneity; (v) heterogeneous technologies; (vi) aggregation across firms; (vii) structural breaks; (viii) heteroskedasticity; (ix) serial correlation and (x) spatial correlation. Following LaFrance and Pope (2010), production decisions are based on ex ante (i.e. planned or expected) outputs rather than ex post (i.e. realised) outputs. The reason is that most inputs are committed to production before outputs are realised. This approach also allows for flexibility to be treated consistently with a large literature in consumer theory (inter alia, Gorman, 1981; Lewbel, 1990, 1991).

The data set analysed in the empirical application has 40 annual observations on 48 states, so it is impractical to implement non-parametric generalised method of moments (GMM) estimation methods such as the Newey–West estimator (Newey and West, 1987). In addition, the structural model includes a subset of state-specific parameters that play the role of non-linear fixed effects. As a result, standard simultaneous equations with error components methods for balanced panels (e.g. Baltagi, 2008: chapter 7: 147–180) also cannot be used. Therefore, we use a flexible, parsimonious and parametric GMM estimation method to address individual-specific covariance matrices for the system of equations, serial correlation and spatial correlation.

The relatively short-time series requires a judicious choice of structural parameters to capture heterogeneous technologies between cross-sectional units of observation. Therefore, the model includes state-specific parameters that influence the levels, slopes and curvatures of the input demands with respect to prices, cost, quasi-fixed inputs, technical change and structural breaks.

Our main findings can be summarised as follows. We reject homothetic models as well as the price independent generalised logarithmic (PIGLOG) and price independent generalised linear (PIGL) models (Muellbauer, 1975, 1976), which have been ubiquitous in empirical work since Berndt and Christensen (1971) and Binswanger (1974a, 1974b), in favour of a substantially more general model of input demand choices. Second, the functional form of this model is explored through a pair of Box–Cox transformations over cost and input prices to nest the generalised PIGL, generalised PIGLOG and generalised quadratic functional forms. We strongly reject the generalised PIGLOG and generalised quadratic functional forms in favour of a flexible generalised PIGL model. Economic regularity, consistent with cost minimising behaviour, is satisfied at almost all data points. Substantial heterogeneity in economic response across states is found. Intertemporal and spatial correlations both are stable and important to valid statistical inferences. There is strong empirical evidence in support of structural changes due to the OPEC oil embargo and ensuing commodity price inflation that began in 1973 and the dramatic changes in agricultural policy that began with the 1986 omnibus farm bill.

The next section specifies the cost model. Section 3 describes estimation. Section 4 discusses the data set for the empirical application. Results are presented in Section 5. Section 6 discusses two robustness checks on the results. The final section presents our conclusions.

2. The cost model

The approach taken here stems from the conceptual model of LaFrance and Pope (2010). It is briefly reviewed focusing on issues of empirical implementation. Two common issues in economic models of production are: (i) most agricultural data have been aggregated across production units – across fields on a farm, farms in a county, state, country, or region; and (ii) in agricultural production, most inputs are committed to production before outputs are realised, leading to latent variables in input demand equations.

Thus, input demands are functions of input prices, quasi-fixed inputs and cost.

One motivation for this specification is the choice functions in equation (3) are based entirely on observable measures. Production risk is one reason this is a reasonable approach to modelling ex ante cost functions – output levels that are planned or expected at planting time are unknown and unobservable (Pope and Chavas, 1994; Pope and Just, 1996, 1998; Moschini, 2001). It has been argued that expected/planned output alone does not adequately characterise cost minimising behaviour under risk aversion (e.g. Pope and Chavas, 1994), in which case the state-contingent approach of Chambers and Quiggin (1998, 2000) is likely more appropriate. Re-defining $Y¯$ as a vector of state-contingent outputs in equation (1) accommodates this alternative. Thus, while our approach is largely motivated from an ex ante cost function perspective, it is sufficiently general to accommodate the state-contingent effort-cost function of Chambers and Quiggin (2000). Furthermore, one can also apply conventional errors-in-variables language, where $Y¯$ is the true measure of outputs and $Y$ is outputs measured with error.

If joint production also is subject to constant returns to scale – a well-accepted stylised fact in agriculture – then the variable cost function is homogeneous of degree one in $(Y¯,Z).$³ Dividing through by the last element of the quasi-fixed input vector – acres of farmland in the application below – gives the variable cost function in the normalised form $c~(W,z,θ(Y¯,z)),$ where $c~=C/ZL,Y¯=Y¯/ZL,$ and $z=[Z1/ZL…ZL−1/ZL]′,$ are variable cost, planned outputs and the first L − 1 quasi-fixed inputs, respectively, measured per unit of the Lth quasi-fixed input, and $θ(Y¯,z)=θ~(Y¯,[z′1]′).$ The cost function also is homogeneous of degree one in variable input prices. Dividing through by the Nth price – the farm wage rate in the empirical application – gives the cost function in real terms as $c(w,z,θ(Y¯,z)),$ where $c=c~/WN$ and $w=[W1/WN…WN−1/WN]′$ are real cost per unit of the Lth quasi-fixed input and real-input prices, excluding the Nth, respectively.

Note that the effects of expected outputs on cost enter directly through $θ(k,t,Y¯).$ Though separability is a common assumption in the applied production literature (Chavas, 2008), it would be ideal to assume as little as possible. While this assumption does restrict the underlying production function in some ways, it remains quite flexible in others.⁴ As discussed in section A1 of the Supplementary data at ERAE online, θ enters the cost specification non-linearly so that the marginal cost of increasing $y¯m$⁠, $m=1,…,M,$ is a function of cost, quasi-fixed inputs (land and capital), variable input prices, technology and the output levels themselves. In addition, the marginal rates of technical substitution can vary with variable input levels, quasi-fixed input levels, technology and expected outputs through θ. Furthermore, the marginal rates of production transformation can vary with quasi-fixed input levels, technology and expected output levels; however, they will be independent of variable input levels.

where $eijt=wijtxijt$ is real expenditure per acre and $uijt$ is a random error term. See section A2 in Supplementary data at ERAE online for the derivation of (6) from (5). Parameters of the structural model include $α0i=[α0i1…α0iN]′,i=1,…,I,$ reflecting possible state-specific land, climate or other effects and characteristics; $α1=[α11…α1N]′,$ reflecting the impacts of capital per acre; $α2=[α21…α2N]′,$ reflecting technological change;⁶ the n × n matrix B and n-vector $γ,$ to capture some of the input price effects; the N-vector $[δ1…δN]′,$ to capture additional input price effects and identify departures from a quasi-homothetic input use expansion path and the scalars κ and λ, to nest the functional forms for variable input prices and cost in the system of input demand equations.

We introduce cross-sectional heterogeneity through the state-specific $α0i$ parameters, which is similar in spirit to commonly used linear unobserved effects panel data models (Wooldridge, 2010: chapter 10). Given the non-linearity of the model, these parameters permit heterogeneous levels, slopes and curvatures of the input demands with respect to prices, cost, quasi-fixed inputs, technical change and structural breaks. In addition, this parameterisation permits heterogeneous expansion paths, as well as heterogeneous elasticities of substitution between factors and factor demand price elasticities.⁷ Thus, this framework provides a rich environment for incorporating production heterogeneity within a panel data setting.⁸

This model is non-linear in variable cost, creating aggregation properties similar to those in the theory of consumer choice (see LaFrance, Beatty and Pope, 2006; LaFrance, 2008 and references therein). In the present case, after some algebra, three functions of cost appear on the right-hand side of (6), $cit1−κ,$$cit$ and $cit1+κ.$ Thus, the number and functional form of the terms involving real-input prices and variable costs are both nested in this specification. Section A3 of the Supplementary data at ERAE online demonstrates how more simplified versions of this model are nested within (6). Since the non-linearity and large number of parameters in the general model are potentially difficult to work with in practice, these more simplified alternatives might prove attractive in other settings and can be easily tested for given the nested structure.

3. Estimation

The estimation method is a parametric form of GMM. As noted in the introduction, due to the appearance of the non-linear fixed effects parameters, equation (6) cannot identify state-specific random effects. It remains possible that $uijt$ are serially correlated over time, have state-level covariance matrices and are correlated over space. Given this complexity of the stochastic structure of the model, consistent estimation of the standard errors can be problematic and can have adverse effects on the validity of statistical inference. Thus, considerable attention must be paid to estimation of the standard errors. The approach is described in this section with section A4 of Supplementary data at ERAE online providing more detail.

To incorporate all three of these possibilities, let $ωijt$ denote an independent and identically distributed random variable with zero mean, unit variance and finite fourth moment, $∀i,j,t.$ Define $vijt=∑i∗=1Isii∗ωi∗jt,∀i,j,t,$ where $∑i∗=1Isii∗2=1$ and $∑ℓ=1Isiℓsi∗ℓ=ρii∗,$$i≠i∗.$ The spatial correlation matrix, $E(v⋅jtv′⋅jt)=SE(ω⋅jtω′⋅jt)S′=SS′=R,$ is finite, symmetric, positive definite, and has ones on the main diagonal, $∀j,t.$ Let $S−1=Q,$ so that $ωijt=∑i∗=1Iqii∗vi∗jt,$$∀i,j,t$ recovers the iid random variables. The spatially correlated random variables $vijt$ have zero means, unit variances, and are independent across j and t. Next, define $ϵijt=∑j∗=1naijj∗vij∗t,$$∀i,j,t.$ The matrix $E(ϵi⋅tϵ′i⋅t)=AiE(vi⋅tv′i⋅t)A′i=AiA′i=Σi$ is finite, symmetric and positive definite $∀i,t,$ and can differ across individuals. Let $Ai−1=Li,$ so that $vijt=∑j∗=1nℓijj∗ϵij∗t,$$∀i,j,t$ recovers the spatially correlated (0, 1) random variables. Finally, define $uijt=∑j∗=1nϕjj∗uij∗t−1+ϵijt,$$∀i,j,t,$ so that $uijt$ are correlated over time, inputs and space. The Eigen values of $Φ=[ϕjj∗]$ must be inside the unit circle for $uijt$ to be generated by a stationary time series process. Otherwise $Φ$ is unrestricted. The error covariance matrix, $E(ui⋅tu′i⋅t)=Ωi=ΦΩiΦ′+Σi,$ is finite, symmetric and positive definite $∀i,$ and reflects state-specific covariances, serial correlation and spatial correlation.

The estimation method, which is a variant of non-linear three stage least squares (NL3SLS) (see Ball et al., 2010), involves the following five steps.⁹ Steps (2), (3) and (4) are used to obtain consistent estimates of the covariance matrix of the moment conditions. Steps (1) and (5) are the first and last stage of the typical NL3SLS estimator. Conditional on the instrument set, this procedure produces consistent, asymptotically normal parameter estimates (Rothenberg and Leenders, 1964). One can check for additional heteroskedasticity of an unknown form with robust standard errors (White, 1980; MacKinnon and White, 1985). Results of doing so for this data set are reported in the empirical section.

1. Estimate the structural parameters through non-linear instrumental variables (NL2SLS, Amemiya, 1985);

2. Use the NL2SLS errors from step (1) to estimate the time series component by linear seemingly unrelated regressions (Zellner, 1962) methods;

3. Use the errors from step (2) to estimate state-specific covariance matrices for the demand system (Malinvaud, 1980);

4. Use the standardised errors from step (3) to estimate the spatial correlation process (Stohs and LaFrance, 2004) and

5. Holding the estimates of the time series, systems and spatial correlation components fixed, complete a second non-linear instrumental variables step.

4. Data

The main data set analysed in the empirical application is a state-level annual time series panel on US agricultural production. This data set is compiled and maintained by the Economic Research Service (ERS) of the United States Department of Agriculture (USDA), is described in detail in Ball, Hallahan and Nehring (2004), and is commonly known as the Ball data. These data contain measures of variable input quantities, prices, and expenditures, farm capital, land in farms and variables relating to realised (ex post) farm production and revenues. For this particular application, we require data on variable input prices and expenditures, farm capital and land in farms. The additional explanatory variable in equation (6) is variable cost, which essentially replaces farm outputs in the demand equations and is constructed as the sum of input expenditures.

To match the data used in the empirical application as closely as possible to the above theory, we modify the Ball data in three important ways. The measure of own labour cost in the Ball data is calculated from off-farm employment surveys and non-farm wage rates. We use the farm wage for own labour cost so that management skill is assigned to the owner/operator's net return to farming. A recent ERS survey found that 98 per cent of US farms are family farms (Hoppe and Banker, 2006). El-Osta (2011) and references therein provide an excellent overview of the difficulties associated with the valuation of human capital in a family farm setting. In general, one can either incorporate a ‘management’ premium for the owner into the farm labour wage, or value the owner's wage using the hired-labour wage rate and allow the premium to be reflected in the returns-to-operation. The former likely introduces significant measurement error into the wage variable as correctly measuring this premium is difficult in practice. This error would be especially problematic for our empirical application as we use wage as the numéraire to normalise input prices (Pope, LaFrance, and Just, 2007). Thus, we follow the latter approach.

The Ball data also contain an imputed value of capital that relies on a host of assumptions and secondary estimates. We use direct estimates of the value of farm capital from annual surveys reported by the ERS. The accounting of on-farm capital stocks and flows has received much attention in the literature. An excellent overview is provided in Andersen, Alston and Pardey (2011). The most widely used measures are those of the Ball and InSTePP series.¹⁰ The primary difference between the Ball and InSTePP series is that the former uses the perpetual inventory method while the latter uses a physical inventory method (Andersen, Alston and Pardey, 2011). In addition, estimates also differ in their treatment of depreciation and the retirement of capital assets, as well as the sources and categories of data used (Andersen, Alston and Pardey, 2011). Thus, the selection of an appropriate capital series is not straightforward due to the different inventory methodologies and the various underlying assumptions required in using them. We focus here instead on direct estimates of capital stock from ERS balance sheet data. The  slope estimate from a regression of our capital series on the Ball series is 0.77 with an R² of 0.89; thus, our measure is closely aligned to the Ball series.

Finally, we relate farmland to the Census of Agriculture, published in 1954, 1959, 1964, 1969, 1974, 1978, 1982, 1987, 1992, 1997, 2002 and 2007. In each year in our sample that coincides with a Census year, we use the Census data for land in farms. In other years, we use ERS estimates of the acres harvested of major crops in each state to adjust the Ball data. The difference in each state between farmland in the Ball data and the ERS measure of harvested acres is calculated for years that did not have a Census report. For each 3- to 4-year period between adjacent Census years, the average of this difference is added to harvested acres to connect annual changes in farmland to annual changes in crop acres.

Our proposed estimation method relies on instrumental variables methods, so identification requires that the instruments are correlated with the endogenous regressors and uncorrelated with the error terms. The first group of instrumental variables includes national averages of the real-input prices, variable cost per acre and the real value of capital per acre, all lagged two periods to ensure they are predetermined and thus uncorrelated with the error terms. These instruments are potentially weak given the two-period lag construction. To address this concern, our second group of instrumental variables includes contemporaneous variables for the general US economy: per capita disposable income, unemployment, the rate of return on AAA corporate 30-year bonds, the manufacturing wage, the producer's price index (PPI) for materials and components, and the PPI for fuels, energy and power.¹¹

Although contemporaneous, the variables in the second group are unlikely to be correlated with the error term as US agriculture is a small sector of the US economy with ∼2 per cent of employment and <1.5 per cent of GDP. California is the largest farm economy, accounting for <0.25 per cent of US GDP. As a result, the effects of changes in any state's farm prices, production costs or value of capital will be far smaller on the general economy than the impact of changes in the general economy on that state's farm economy.

Each general economy variable is considered a valid instrument for a specific explanatory variable. The manufacturing wage is a measure of the opportunity cost of farm work and serves as an instrument for the farm wage. Income affects the demand for agricultural products and serves as an instrument for the cost of production. The rate of return on corporate bonds is a measure of the opportunity cost of investing in agriculture. The PPI for fuels, energy and power is an instrument for the farm costs of fuels, energy and agricultural chemicals, which are largely hydrocarbon based. The PPI for materials and components is an instrument for the farm cost of materials. Our final set of instruments includes both groups mentioned above as well as a time trend.

The adding up condition requires that one input demand equation must be omitted from the estimated model. Farm labour is excluded from the estimated equations in this study. Input prices and variable cost are deflated by the farm wage to satisfy homogeneity. Input demands estimated are energy, agricultural chemicals and materials. Lagging the aggregate farm-level instruments shortens the sample period 2 years. Thus, 1,824 observations on three variable inputs, 48 states and 38 years (1962–1999) are used to estimate the input demand model. The final (normalised) input expenditure, variable cost, capital and input price series were tested for unit roots using the Im, Pesaran and Shin (2003) procedure for heterogeneous panels. We reject the null hypothesis that all panels have unit roots at standard significance levels for each of the series (all p-values are <0.01).

There are 144 input demand equations and 12 instruments, thus generating 1,728 moments for estimation. There are 599 model parameters, so the model is significantly over-identified and testing the model was important. We test this using the omnibus test, Hansen's (1982)J-statistic and fail to reject the null hypothesis (that the orthogonal conditions hold) at the 1 per cent significance level. This implies that the demand model is reasonably specified.

5. Results

This section reports our empirical findings as they relate to structural breaks, the stochastic process for the error terms and the functional form of the input demand equations. The final subsection analyses the economic regularity of the estimated cost model and reports own-price elasticity estimates for the factor demands.

5.1. Structural breaks

Gutierrez, Westerlund and Erickson (2007) investigates the role that structural breaks play in the agricultural economy with respect to finding a stable relationship between farmland prices and land rents. They use a panel data set of 31 US states over the time period 1960–2000. They find that all states have at some point been subject to structural breaks.

The results of Gutierrez, Westerlund and Erickson (2007) show considerable evidence of structural breaks in 1973 and 1986. The former is the beginning of the period of rapid commodity price inflation that followed the OPEC oil embargo. The latter is the start of major reforms in agricultural policy during the Reagan administration, including a movement to decouple farm subsidy payments from agricultural production, and lower price and income support levels overall. During the mid- to late-1970s, US agriculture experienced oil price shocks, a high rate of growth in farm income, growth in net agricultural exports with a falling exchange rate and poor weather conditions in competing production regions. On the other hand, increased uncertainty in the expected returns to agricultural investment, high real interest rates and lower commodity prices and support levels have characterised the farm economy since the second half of the 1980s.

Tests for structural breaks also are conducted in this paper. There are many diagnostic procedures for testing the adequacy of a model's specification and parameter stability (Brown, Durbin and Evans, 1975; Ploberger and Krämer, 1992). The methods in LaFrance (2008) work well in situations like the present, where the empirical model includes a relatively large non-linear simultaneous equation system and a relatively short-time series.

The $τs$ are unrestricted across states and inputs, but enter the model as structural parameters to accommodate economic theory both before and after the structural breaks.

strongly support including the structural change variables (all p-values are <0.001). The results that follow are therefore reported with these two structural breaks included in the model.

5.2. Intertemporal and spatial correlation estimates

The estimated serial correlation matrix with White/Huber robust asymptotic standard errors in parentheses is reported in Table 1. The parameter estimates imply stable dynamics, with the largest Eigen value of $Φˆ$ equal to 0.48. The F-test for all serial correlation parameters equal to zero is rejected at the 1 per cent significance level. Durbin–Watson statistics do not suggest a higher-order serial correlation, with the Durbin–Watson statistics for each input averaged across states equal to 1.911, 1.757 and 1.775, respectively.¹²

where $i≠i∗,i,i∗=1,…,48,$ index states and $sii∗=(dii∗−d¯)/σˆd$ is the standardised distance between each pair of states, to improve the computational precision during estimation.¹³ The estimated correlation function and asymptotic 95 per cent confidence band are presented in Figure 2. It is noteworthy that the estimated spatial correlation function is statistically well above the horizontal axis across the entire continental United States.

5.3. Functional form

We turn next to a subset of the parameter estimates for the structural model. Table 2 presents the estimates of B, $γ,$$[δ1δ2…δN]′$and the Box–Cox parameters $(κ,λ).$ Two interesting questions relating to functional form are the transformations of input prices and variable costs. Industry standards are the logarithmic and linear transformations, which are tested with the following null hypotheses:

1. linear–linear, $H0:κ=λ=1,$$χ2(2)=686.2,$p-value = 0.000;

2. log–log, $H0:κ=λ=0,$$χ2(2)=59.35,$p-value = 0.000;

3. log–linear, $H0:κ=0,λ=1,$$χ2(2)=709.2,$p-value = 0.000 and

4. linear–log, $H0:κ=1,λ=0,$$χ2(2)=1736.3,$p-value = 0.000.

Thus, all four hypotheses are rejected at the 1 per cent significance level. From Table 2, the null hypothesis $H0:κ=0$ has a p-value equal to 0.231. However, neither the log–log nor the log–linear specifications are supported by these data. It is clear that the additional flexibility of the power functions in prices and cost is useful in explaining these data, consistent with recent empirical results in consumer behaviour (LaFrance, 2008).

A Wald test for the null hypothesis that the parameters in $β(w)$ are jointly zero is rejected at the 1 per cent significance level. The conclusion from a similar test for the parameters in $δ(w)$ is rejected at the 10 per cent level, with a p-value of 0.0886. We conclude that there is evidence in support of including higher-order price effects and extending traditional models to include three separate functions of cost.

Testing whether per acre variable input demand use depends on capital per acre is equivalent to testing the hypothesis $H0:α11=α12=α13=α14=0.$ The resulting test statistic has a p-value of 0.005, which supports including capital per acre in the demand system. We also find evidence in support of the time trend; a Wald test for the null hypothesis $H0:α21=α22=α23=α24=0$ has a p-value of 0.059.

The remaining parameter estimates are the state-specific terms $α0ij,$$i=1,…,48,$$j=1,…,4.$ There are too many estimates to report in tables here. As an alternative, Figure 3 presents kernel density functions across states of the parameter estimates for each input. The figure also contains similar plots for the structural break parameters. These plots imply substantial heterogeneity across states in agricultural technologies and the post-1973 and post-1986 structural breaks. Wald tests for the null hypotheses $H0:α0i1=α0i2=α0i3=α0i4=0$ reject the null for 46 out of 48 states at the 1 per cent significance level and for all 48 states at the 5 per cent level.

Figure 4 reports Box plots for the test statistics for all inputs, which fail to reject zero means at the 10 per cent significance level; indeed, the largest z-statistic in absolute value is 1.19.

5.4. Economic regularity and elasticity estimates

$H33$ negative semi-definite is necessary and sufficient for $H$ to be negative semi-definite. We calculate $Hˆ33$ at each state-year observation and find that 96 per cent of the associated eigenvalues are negative.

where $wi$ and $qi$ are nominal input prices and input quantities, and $Δ(xi)$ is an N × N diagonal matrix with $xi$ as the ith main diagonal element. We have included a subscript i for the Hessian matrix to emphasise that it varies across states.¹⁴ These elasticities are calculated for each state-year pair in the data. Figure 5 reports histograms and kernel density estimates of time-averaged state-level own-price elasticities. This figure clearly demonstrates the model's ability to capture a wide range of price response, which would not be apparent from aggregate US data.¹⁵ The estimated density functions show that own-price elasticities are all negative and vary substantially across states. Two inputs that are often of policy interest are energy and agricultural chemicals, which tend to have the most elastic input demand functions.

Table 3 reports summary statistics for the state-level own-price elasticities for labour, energy and chemicals by the NASS production region. Several interesting patterns emerge. First, input use is price inelastic in all regions, except chemicals in New Mexico (−1.22) and Arizona (−1.02), and labour in Arizona (−1.71). Second, chemicals are the most price elastic input across all regions. Farm expenditures on chemicals are a large portion of total production expenses relative to labour and energy (USDA, 2012). From a policy perspective, it is interesting that this also is the most price sensitive variable input. The highest own-price elasticities for agricultural chemicals occur in the West, and the lowest occur in the South. For individual states, New Mexico (−1.22) is the most price sensitive, while Florida is the least (−0.57), which is not surprising given the adverse (favourable) production climate in New Mexico (Florida). Finally, labour has the lowest own-price response in all regions except the West, where some of the largest absolute own-price elasticities are observed (Arizona at −1.71, New Mexico at −0.97 and Colorado at −0.92). Interestingly, labour accounts for roughly 20 per cent of total production expenses in the West (USDA, 2012). This implies that wage volatility can lead to considerable volatility in agricultural production costs in this region.

6. Further robustness of model estimates

This section evaluates the robustness of the empirical model in two additional dimensions. First, we re-estimate the model under the assumption of zero spatial correlation across states to evaluate the importance of this aspect of the modelling process. Given the estimation steps summarised in Section 3, this is accomplished by replacing the spatial correlation matrix R with the identity matrix, so that $ωijt=∑i∗=1Iqii∗vi∗jt$ becomes $ωijt=vijt.$ Figure 6 presents kernel density functions of the p-values for the structural parameters with and without the spatial correlation correction. Excluding this correction shifts the distribution of p-values substantially to the right, implying that inconsistent statistical inferences are the likely result.

The second issue addressed is whether equation (6) can be applied to state-level data. In particular, suppose that this equation adequately describes farm-level behaviour, and that per acre production costs vary across farms. Then a similar equation also holds at the state-level, but with $E[citj]$ instead of $(E[cit])j$ for $j∈{1−κ,1+κ},$ with the expectation taken with respect to the distribution of costs across farms in a given state and in a given year. Jensen's inequality implies that these two quantities will not be the same when $κ≠0.$

We use information theory to investigate this question. The standard exponential probability density function on the positive real line is $fX(x)=e−x,x∈R+.$ One definition of the gamma function, $Γ(α),$ is the mean of $xα−1$ when x has a standard exponential distribution, $Γ(α)=∫0∞xα−1e−xdx.$ A change of variables to $y=μx,$$μ>0,$ implies that Y has an exponential distribution with mean $μ,$ hence, $E(Yα)=Γ(α+1)μα$ for all $α≥0.$

Setting $α=1−κand 1+κ,$ respectively, we find $E(Cit)=c¯it,$$E(Cit1−κ)=Γ(2−κ)c¯it1−κ$ and $E(Cit1+κ)=Γ(2+κ)c¯it1+κ,$$i=1,…,48,$$t=1,…,40.$

Estimates of the same parameters reported in Table 2 using this model of the farm-level cost distribution are presented in Table 4. Comparing these results to those reported previously, we find that the empirical model is robust to this concern. This is not surprising since the point estimate of $κ$ is small in both cases, and $Γ(2−κ)$ and $Γ(2+κ)$ both are close to $Γ(2)=1$ when κ is close to zero. In addition, the same qualitative results and conclusions for the hypothesis tests reported above also emerge with this specification for the distribution of year-to-year farm-level costs per acre.