Estimating the effects of weather and climate change on agricultural productivity

Abstract

1 Introduction

Most economists consider measures of productivity change to be measures of output quantity change divided by measures of input quantity change (e.g. Jorgenson and Griliches, 1967, p.250; Schreyer, 2001, p.11; O’Donnell, 2018, p.11). Measuring changes in output and input quantities (and therefore productivity) is an index number problem. One of the more worrying features of applied research on productivity is that most authors use output and input quantity indexes that do not satisfy a set of basic axioms from index theory. Indexes that do not satisfy at least one axiom include the well-known Fisher, Törnqvist, Malmquist, Elteto-Koves-Szulc (EKS), and Caves-Christensen-Diewert (CCD) indexes. The axioms that are violated most frequently are a transitivity axiom and a proportionality axiom. This has serious implications: indexes that violate these two axioms will say that outputs and inputs have increased and/or decreased when they may in fact have done the opposite, or perhaps not changed at all. This paper measures changes in output and input quantities using what O’Donnell (2016, 2018) calls proper indexes. Proper quantity indexes satisfy six basic axioms from index theory, including transitivity and proportionality.

If measures of productivity change are considered to be measures of output quantity change divided by measures of input quantity change, then explaining changes in productivity necessarily involves explaining changes in output and input quantities. Economists have many models that can be used for this purpose. For example, a widely-used model is one in which firms are assumed to be price takers in both output and input markets, and where managers are assumed to choose output and input quantities to maximize profits. Economists who use these types of models typically assume that market prices and characteristics of production environments are known at the time production decisions are made. These are unrealistic assumptions, at least in industries like agriculture. This paper considers an economic model that accounts for both weather and output price uncertainty. Estimating the model involves estimating a stochastic production frontier and a system of input demand equations. The paper shows how the estimated parameters of the model can be used to assess the effects of weather and climate change on total factor productivity (TFP).

The basic economic idea behind the paper is that changes in weather and climate potentially affect agricultural inputs and outputs (and therefore productivity) through two channels: first, realizations of weather variables potentially affect the outputs that can be produced using predetermined inputs; and, second, expectations about weather and climate potentially affect the input and planned output choices of managers.

Most previous studies that have examined the first channel have focused on measures of partial factor productivity (PFP): Anand and Khetarpal (2015), for example, use a biophysical simulation model to examine the effects of changes in surface air temperatures on wheat yield per hectare; and Nastis et al. (2012) use a production function model to estimate the effects of changes in temperature and precipitation on land productivity. And while several studies have looked at the effects of weather variables on TFP, most have used TFP indexes that have poor axiomatic properties: Salim and Islam (2010), for example, use a vector error correction (VEC) model to estimate the effects of changes in precipitation on Törnqvist index numbers; and Hughes et al. (2011) use a stochastic production frontier model to construct a climate effects index, which they then use to deflate Fisher index numbers. Only a handful of studies appear to have used proper TFP indexes: Sabasi and Shumway (2018), for example, use a seemingly unrelated regression (SUR) model to estimate the effects of changes in temperature and precipitation on Lowe index numbers; and Njuki et al. (2018, 2020) use stochastic production frontier models to estimate the effects of changes in temperature and precipitation on multiplicative index numbers. This paper goes a step further by considering the second channel: it considers the way in which expectations about weather and climate potentially affect input and planned output choices. No previous studies appear to have examined this second channel.

The structure of the paper is as follows. Section 2 deals with the problem of measuring TFP change.¹ It starts by using artificial data to demonstrate the properties of various indexes. It then uses U.S. agricultural data to demonstrate that there are real-world situations where the choice of index matters. Section 3 considers the first channel by which weather and climate potentially affect productivity. It starts by writing the relationship between observed outputs, observed inputs and observed weather variables in the form of a stochastic production frontier model. It then uses the estimated parameters of the model to decompose a proper TFP index into measures of technical progress, environmental change, technical efficiency change, scale-and-mix efficiency change, and changes in statistical noise. Section 4 considers the second channel. It starts by writing the relationship between observed input quantities, observed input prices, expected output prices, and expected weather variables in the form of a system of input demand equations. It then uses the estimated parameters of the system to further decompose the measure of scale-and-mix efficiency change into a measure of technical progress, a measure of input price change, various measures of changes in expectations, and a measure of changes in allocative efficiency and statistical noise. Section 4 summarizes the paper and offers some concluding remarks.

2 Measuring TFP change

Measures of TFP change are measures of total output quantity change (i.e. output quantity indexes) divided by measures of total input quantity change (i.e. input quantity indexes). Computing output and input quantity index numbers is a matter of assigning numbers to baskets of outputs and inputs. Measurement theory says such numbers cannot be assigned in an arbitrary way. Instead, they must be assigned in such a way that the relationships between the numbers reflect the relationships between the baskets (Tal, 2016). This section uses artificial data on apples and oranges to illustrate this basic principle. It then uses U.S. agricultural data to demonstrate that there are situations where the choice of index matters.

2.1 Comparing apples and oranges

Consider the baskets of apples and oranges presented in Table 1. This table also presents quantity index numbers that have been computed using six different indexes: the MEW index is a type of multiplicative index that gives the products equal weight; the MOLS index is a multiplicative index that uses OLS parameter estimates as weights; the geometric Young (GY) index is a multiplicative index that uses average value shares as weights; the binary Törnqvist (BT) index is a ‘superlative’ index that uses observation-varying value shares as weights; the chained Törnqvist (CT) index is obtained by treating the observations as time-series observations and chaining pairs of BT indexes; and the CCD index is obtained by treating the observations as cross-section observations and taking geometric averages of sets of BT indexes. Technical details concerning each of these indexes can be found in O’Donnell (2018, Ch.3).²R code for computing the numbers in Table 1 can be found in Figure A1 in the Appendix.

Table 1.

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Quantity index numbers.

.	.	MEW .	MOLS .	GY .	BT .	CT .	CCD .
A =		1	1	1	1	1	1
B =		1.732	1.209	1.860	1.581*	1.581*	1.688*
C =		2.449	2.797	2.386	2.598	3.150*	2.700
D =		1.732	1.209	1.860	2.215*	2.508*	2.118*
E =		2	2	2	2	1.886*	1.884*

.	.	MEW .	MOLS .	GY .	BT .	CT .	CCD .
A =		1	1	1	1	1	1
B =		1.732	1.209	1.860	1.581*	1.581*	1.688*
C =		2.449	2.797	2.386	2.598	3.150*	2.700
D =		1.732	1.209	1.860	2.215*	2.508*	2.118*
E =		2	2	2	2	1.886*	1.884*

Notes: MEW = Multiplicative with Equal Weights; MOLS = Multiplicative with OLS weights; GY = Geometric Young; BT = binary Törnqvist; CT = chained Tornqvist; CCD = Caves-Christensen-Diewert.

*Incoherent (and not because of rounding errors).

Table 1.

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Quantity index numbers.

.	.	MEW .	MOLS .	GY .	BT .	CT .	CCD .
A =		1	1	1	1	1	1
B =		1.732	1.209	1.860	1.581*	1.581*	1.688*
C =		2.449	2.797	2.386	2.598	3.150*	2.700
D =		1.732	1.209	1.860	2.215*	2.508*	2.118*
E =		2	2	2	2	1.886*	1.884*

.	.	MEW .	MOLS .	GY .	BT .	CT .	CCD .
A =		1	1	1	1	1	1
B =		1.732	1.209	1.860	1.581*	1.581*	1.688*
C =		2.449	2.797	2.386	2.598	3.150*	2.700
D =		1.732	1.209	1.860	2.215*	2.508*	2.118*
E =		2	2	2	2	1.886*	1.884*

Notes: MEW = Multiplicative with Equal Weights; MOLS = Multiplicative with OLS weights; GY = Geometric Young; BT = binary Törnqvist; CT = chained Tornqvist; CCD = Caves-Christensen-Diewert.

*Incoherent (and not because of rounding errors).

The MEW, MOLS, and GY indexes are proper indexes. Among other things, this means they yield numbers that are consistent with measurement theory. Observe, for example, that the MEW index number in row B of Table 1 is the same as the number in row D, reflecting the fact that basket B contains the same number of apples and oranges as basket D; and the GY index number in row E is twice as big as the number in row A, reflecting the fact that basket E contains twice as many apples and oranges as basket A.

The BT, CT, and CCD indexes are not proper indexes. Consequently, they do not yield numbers that are consistent with measurement theory. Observe, for example, that the BT index number in row B differs from the number in row D, even though baskets B and D contain the same numbers of apples and oranges; the CT index number in row C is 3.15 times greater than the number in row A, even though basket C contains less than 3 times as many apples and oranges as basket A; and the CCD index number in row E is less than twice the number in row A, even though basket E contains exactly twice as many apples and oranges as basket A. The basic problem with BT, CT, and CCD quantity indexes is that they measure changes in value shares as well as quantities. Of course, if there is no variation in value shares, then they will only measure changes in quantities, as required; in that case, BT, CT, and CCD index numbers will be equal to GY index numbers.

2.2 TFP change in U.S. agriculture

This subsection uses different indexes to make time-series and cross-section comparisons of output, input, and TFP change in U.S. agriculture. The dataset comprises observations on the prices and quantities of four inputs (capital, land, labor, and materials) and three outputs (livestock, crops, and other outputs) in forty-eight states over the years from 1961 to 2004. The data were assembled by the Economic Research Service (ERS) of the U.S. Department of Agriculture (USDA). Details concerning the data can be found in Ball et al. (1997).

Several sets of index numbers are summarized in Fig. 1. The panels on the left-hand side of this figure present GY, BT, and CT indexes of output, input, and TFP change in Alabama from 1961 to 2004. The panels on the right-hand side present GY, BT, and CCD indexes of output, input, and TFP change across six states in 1961. Observe from the panels on the left-hand side that different indexes sometimes tell quite different stories about output, input, and/or TFP change over time: in panel (c), for example, the GY index tells us that input use in Alabama in 1983 was 9% lower in than it had been in 1961, while the CT index tells us that it was 5% higher. The panels on the right-hand side reveal that different indexes can also tell quite different stories about output, input, and/or TFP change across space: in panel (f), for example, the GY index tells us that Oklahoma was 3% less productive than Alabama in 1961, while the BT and CCD indexes tell us that Oklahoma was at least 11% more productive.

Figure 1.

Output, Input, and TFP change in selected years and states (Alabama in 1961 = 1).

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2.3 The choice of index

There are at least three reasons why economists should choose proper indexes. First, all proper indexes satisfy important axioms from index theory (e.g. transitivity and proportionality). Second, and relatedly, proper quantity indexes always yield numbers that are consistent with measurement theory (i.e. the patterns in the numbers always mirror the patterns in the quantities). Third, any given proper index can be used to make comparisons across both time and space; in contrast, the CT index, for example, can only be used to make (erroneous) comparisons across time, while the CCD index can only be used to make (erroneous) comparisons across space.

In practice, the choice of proper index is largely a matter of taste. In the case of output indexes, for example, O’Donnell (2018) suggests that ‘analysts who want to minimize the amount of variation in the index numbers’ should use a multiplicative index with weights that are computed within a regression framework (p.97); the GY index ‘should be used by analysts who regard revenue shares as [appropriate] measures of relative value (e.g. analysts who might otherwise use a Tornqvist, chained Tornqvist or CCD index)’ (p.96); a primal index should be used by ‘analysts who regard marginal rates of transformation as [appropriate] measures of relative value (e.g. analysts who might otherwise use a generalized Malmquist index)’ (p.97); and a benefit-of-the-doubt (BOD) ‘index should be used by analysts who believe measures of relative value should vary from one output comparison to the next. It can also be used by analysts who have no information about output prices, revenue shares, or production technologies’ (p.99).

3 Explaining changes in TFP

If TFP change is defined as a measure of total output quantity change divided by a measure of total input quantity change, then explaining changes in TFP necessarily involves explaining changes in outputs and inputs. This paper assumes that managers choose outputs and inputs in two stages: first, at the beginning of the production period, managers choose inputs and planned outputs to maximize expected profits in the face of uncertainty about output prices and one or more characteristics of the production environment (e.g. rainfall); and, second, after inputs have been chosen, managers seek to maximize the outputs that can be produced using their chosen inputs in whatever production environment is realized. This section focuses on the second stage. If this second-stage behavior is true, then the relationship between outputs, inputs, and environmental variables can be written in the form of a stochastic production frontier model in which the explanatory variables are exogenous. This section uses this type of model to explain variations in the GY TFP index numbers reported in Section 2.2. To do this, the USDA input and output data used in that section is supplemented with observations on three characteristics of the production environment: DD830 measures the number of degree days between 8^○C and 30^○C between March and August; DD30 measures the number of degree days above 30^○C between March and August; and PREC measures total precipitation in inches between March and August. These weather variables were the only environmental variables that were available; data on these variables were also assembled and supplied by the USDA.

3.1 The stochastic production frontier model

3.2 Estimation

The unknown parameters in stochastic frontier models are most often estimated using the method of maximum likelihood (ML). The are two problems with the ML approach. First, there is no satisfactory way of incorporating inequality information into the estimation process; this is because binding inequality constraints will lead to parameter estimates that lie on the constraint boundary, with standard errors of zero (implying we know the values of the parameters with certainty). Second, we cannot make valid finite-sample inferences concerning nonlinear functions of the parameters; this because all the theory underpinning ML estimation is asymptotic. For these reasons, this paper estimates the parameters in (8) using a Bayesian approach. Bayesian estimation involves summarizing the information we have about the unknown parameters in the form of a joint posterior probability density function (pdf). One source of information is the data; this so-called sample information is summarized in the form of the usual likelihood function (i.e. the function that is maximized in the ML approach). Other sources of information include economic theory and common sense; this so-called nonsample information is summarized in the form of a prior pdf. The likelihood function and the prior pdf are combined using Bayes’s theorem to form the joint posterior pdf.

Evaluating characteristics of marginal posterior pdfs for the individual parameters (e.g. means, variances) involves evaluating multiple integrals. In practice, this involves sampling from the joint posterior pdf. In this paper, samples of size one million were drawn using the Markov Chain Monte Carlo (MCMC) sampling package of Plummer (2019). The thinning interval for monitors was set to 100. The convergence of the chains was confirmed both visually and using the convergence diagnostic of Geweke (1992). Relevant R code is presented in Figure A2 in the Appendix.

Bayesian point estimates of the first five intercept parameters and all the slope parameters are presented in Table 2. Estimates of all forty-eight intercept parameters are presented in Table A1 in the Appendix. These tables also report associated estimated standard errors and lower and upper bounds of 95% highest posterior density (HPD) intervals. HPD intervals are the Bayesian counterparts to confidence intervals. By construction, the point and interval estimates in Table 2 are all consistent with prior expectations. Given the assumptions made earlier about the nature of statistical noise, they have straightforward interpretations: the point estimates of α₁ and α₄, for example, indicate that fixed characteristics of the production environment in California are |$46\%$| more favorable for agricultural production than fixed characteristics of the production environment in Alabama; the estimates of λ₁, …, λ₅ indicate there was technical progress in every decade, and that the rate of technical progress reached a maximum of 2.3% per annum in the 1980s; the estimates of β₁, …, β₅ indicate that, all other things being equal, increasing any input will lead to an increase in output, and that the production frontier exhibits decreasing returns to scale (the elasticity of scale is estimated to be 0.858); the estimate of δ₁ indicates that an increase in the number of so-called ‘good degree days’ will increase output; the estimate of δ₂ indicates that an increase in the number of ‘bad degree days’ will decrease output; and the estimate of δ₃ indicates that an increase in precipitation will increase output.

Associated with the parameter estimates reported in Table 2 are estimates of output-oriented technical efficiency (OTE). Point estimates for the first five states in both 1961 and 2004 are presented in Table 3. Estimates for all states in all years are presented in Table A2 in the Appendix. These tables also report associated estimated standard errors and lower and upper bounds of 95% HPD intervals. Among other things, these estimates indicate that Alabama farmers were slightly less technically efficient in 2004 than they had been in 1961. The average level of technical efficiency across all states in all years is estimated to be 0.959.

3.3 Decomposing GY TFP index numbers

Figure 2 presents a decomposition of TFP change in selected states. The TFPI numbers depicted in the top-left-hand panel are the GY numbers presented earlier in panel (e) of Fig. 1. The OTI, OEI, and OSMEI numbers were computed by using the estimated slope parameters in Table 2 and the estimated intercept parameters in Table A1 in the Appendix to evaluate the first three terms on the right-hand side of equation (11). The OTEI numbers were computed using the point estimates of OTE reported in Table 3 and in Table A2 in the Appendix. The SNI numbers were computed as residuals. Results for all forty-eight states are presented in Table A3 in the Appendix.

Figure 2.

TFP change in selected states from 1961 to 2004 (AL in 1961 = 1).

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The results reported in Fig. 2 and Table A3 indicate that the main drivers of TFP change over time were technical progress and scale-and-mix efficiency change. In the case of Alabama, for example, TFP in 2004 was 1.776 times higher than it had been in 1961. This increase can be broken down as follows: TFPI = OTI × OEI × OSMEI × OTEI × SNI = 1.517 × 0.996 × 1.328 × 0.977 × 0.906 = 1.776. This decomposition indicates that, in Alabama, (i) technical progress provided for a 51.7% increase in TFP; (ii) changes in the production environment had virtually no impact on TFP; (iii) improvements in scale-and-mix efficiency led to a 32.8% increase in TFP; (iv) lower technical efficiency led to a 2.7% fall in TFP; and (v) changes in omitted variables and other sources of statistical noise accounted for a 9.4% fall in TFP.

The results reported in Fig. 2 and Table A3 also indicate that the main drivers of TFP change across states were differences in production environments and levels of scale-and-mix efficiency. In 1961, for example, Texas was 6.7% less productive than Alabama. This difference can be broken down as follows: TFPI = OTI × OEI × OSMEI × OTEI × SNI = 1 × 1.230 × 0.780 × 0.997 × 0.976 = 0.933. The reason the OTI component takes the value one is that Equation (8) plausibly does not allow for differences in rates of technical progress across states. The remaining components indicate that, in 1961, (i) a better production environment allowed Texas farmers to be 23% more productive than farmers in Alabama; (iii) Texas farmers were 22% less scale-and-mix efficient than farmers in Alabama; (iv) Texas farmers were only 0.3% less technically efficient than farmers in Alabama; and (v) omitted variables and other sources of statistical noise accounted for only 2.4% of the difference in TFP.

Finally, it is possible to decompose the environmental change component of TFP change (i.e. the OEI) into the product of an output-oriented fixed effects index (OFEI) and an output-oriented temperature and precipitation index (OTPI). The OFEI measures the direct effects of time-invariant environmental variables on TFP; mathematically, the index that compares the effects in state i with the effects in state k is exp (α_i − α_k). The OTPI measures the direct effects of changes in temperature and precipitation on TFP; mathematically, the index that compares the effects in state i in period t with the effects in state k in period s is |$\prod _{j=1}^{J}(z_{jit}/z_{jks})^{\delta _j}$|⁠. To illustrate, let us reconsider the finding that, in 1961, a better production environment allowed Texas farmers to be 23% more productive than farmers in Alabama. This difference can be broken down as follows: OEI = OFEI × OTPI = 1.265 × 0.973 = 1.23. This indicates that time-invariant characteristics of the production environment (e.g. soil type and terrain) allowed Texas farmers to be 26.5% more productive than farmers in Alabama, and that, in 1961, differences in temperature and precipitation accounted for only 2.7% of the difference in TFP. Results for other states and other years also indicate that changes in weather had a relatively small effect on TFP.

4 Explaining changes in OSME

Explaining changes in OSME involves explaining changes in input quantities. This section uses a system of input demand equations to decompose the OSME index numbers reported in Section 3.3. To do this, the USDA data used in previous sections were supplemented with observations on expected aggregate output prices and weather variables. Expected aggregate output prices were measured using one-period lagged values of GY output price index numbers, and expected values of weather variables were measured using arithmetic averages of observed weather variables over the previous 10 years. To the extent that they measure average atmospheric conditions over a long period of time, changes in these expected weather variables can be viewed as measures of climate change.

4.1 The input demand functions

4.2 Estimation

In this paper, each of the input demand equations defined by (12) is estimated separately using a Bayesian approach. The likelihood function for the mth input demand equation is formed under the assumption that e_mit is distributed as an independent |$N(0,\sigma _m^2)$| random variable. The prior pdf is chosen to be a truncated multivariate normal distribution that once again contains almost no information about the parameters: all it says is that ϕ_m must be nonnegative, and that λ_m and ξ_mm must be nonpositive. Again, characteristics of marginal posterior pdfs are evaluated using MCMC sampling. Again, samples of size one million were drawn using the sampling package of Plummer (2019). The thinning interval for monitors was again set to 100. The convergence of the chains was again confirmed both visually and using the convergence diagnostic of Geweke (1992). R code for estimating the parameters of the input demand equation for capital is presented in Figure A3 in the Appendix.

Bayesian point estimates of the slope parameters in each input demand equation are presented in Table 4. Estimates of the intercept parameters are presented in Tables A4 to A7 in the Appendix. These tables also report associated standard errors and lower and upper bounds of 95% HPD intervals. All of the estimates reported in these tables are plausible: the estimates of λ₁, …, λ₄, for example, indicate that technical progress was capital-, land-, and labor-saving; the estimates of ϕ₁, …, ϕ₄ indicate that increases in expected output prices lead to increases in the demand for all inputs; and the estimates of ξ₁₁, ξ₂₂, ξ₃₃, and ξ₄₄ indicate that, all other things being equal, an increase in the price of any input leads to a fall in the demand for that input.

4.3 Decomposing OSME index numbers

Figure 3 presents a decomposition of OSME change in selected states. The OSME index numbers depicted in the six panels are the same numbers that were depicted earlier in Fig. 2, but using a different vertical scale. The ITI, EEI, EPI, and WI numbers were computed by using the parameter estimates reported in Table 4 (and Tables A4 to A7 in the Appendix) to evaluate the first four terms on the right-hand side of Equation (13). The AESNI numbers were computed as residuals. Results for all forty-eight states are presented in Table A8 in the Appendix.

Figure 3.

OSME change in selected states from 1961 to 2004 (AL in 1961 = 1).

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The results reported in Fig. 3 (and Table A8 in the Appendix) indicate that the main drivers of OSME change over time were technical progress and input price change. In the case of Alabama, for example, OSME in 2004 was 32.8% higher than it had been in 1961. This increase can be broken down as follows: OSMEI = ITI × EEI × EPI × WI × AESNI = 1.067 × 1.000 × 0.998 × 1.212 × 1.029 = 1.328. This decomposition indicates that, in Alabama, (i) technical progress provided for a 6.7% increase in OSME; (ii) changes in expected environmental conditions had no impact on OSME; (iii) changes in expected output prices had virtually no impact on OSME; (iv) changes in input prices led to a 21.2% increase in OSME; and (v) changes in allocative efficiency and statistical noise led to a 2.9% increase in OSME.

The results reported in Fig. 3 (and Table A8 in the Appendix) also indicate that the main drivers of OSME change across states were differences in input prices and expectations about environmental conditions. In 1961, for example, Texas farmers were 22% less scale-and-mix efficient than farmers in Alabama. This difference can be broken down as follows: OSMEI = ITI × EEI × EPI × WI × AESNI = 1 × 0.750 × 1.000 × 1.029 × 1.010 = 0.780. The reason the ITI component takes the value one is that, again, the model does not allow rates of technical progress to vary across states. The remaining components indicate that, in 1961, (i) differences in expected environmental conditions led Texas farmers to be 25% less scale-and-mix efficient than farmers in Alabama; (iii) differences in expected output prices made no difference to levels of scale-and-mix efficiency; (iv) differences in input prices incentivized Texas farmers to be 2.8% more scale-and-mix efficient than farmers in Alabama; and (v) levels of scale-and-mix efficiency in Texas were 4% higher than levels in Alabama due to differences in allocative efficiency and statistical noise.

Finally, it is possible to decompose the expected environmental change component of OSME change (i.e. the EEI) into the product of an input-oriented fixed effects index (IFEI) and an expected temperature and precipitation index (ETPI). The IFEI measures the effects of time-invariant environmental variables on OSME; mathematically, the IFEI that compares state i with state k is |$\prod _{m=1}^{M}\exp (\alpha _{mi}-\alpha _{mk})^{(\beta _m-\bar{s}_m)}$|⁠. The ETPI measures of the effects of expectations about temperature and precipitation on OSME; mathematically, the ETPI that compares expectations in state i in period t with expectations in state k in period s is |$\prod _{m=1}^{M}\prod _{j=1}^J (z^e_{jit}/z^e_{jks})^{\delta _{mj}(\beta _m-\bar{s}_m)}$|⁠. To illustrate, let us reconsider the finding that, in 1961, levels of OSME in Texas were 25% lower than levels in Alabama due to differences in expected environmental conditions. This difference can be broken down as follows: EEI = IFEI × ETPI =0.734 × 1.022 = 0.750. This indicates that time-invariant characteristics of the production environment (e.g. soil type and terrain) led Texas farmers to be 26.6% less scale-and-mix efficient than farmers in Alabama, and that, in 1961, differences in expectations about temperature and precipitation accounted for only 2.2% of the difference in OSME. Results for other states and other years also indicate that changes in long-term average temperature and precipitation (i.e. changes in climate) have a relatively small effect on OSME.

5 Summary and conclusion

Measuring productivity change involves measuring changes in output and input quantities. Economists tend to use chained Fisher (CF) and chained Törnqvist (CT) indexes to measure changes in quantities over time (see, for example, Schreyer, 2001, p.83). They also tend to use Elteto-Koves-Szulc (EKS) and Caves-Christensen-Diewert (CCD) indexes to measure changes in quantities across space (see, for example, Coelli et al. 2005, p.117). This paper measured changes in output and input quantities (and therefore productivity) in U.S. agriculture using geometric Young (GY) indexes. There are three reasons why GY indexes are preferred to CF, CT, EKS, and CCD indexes: first, they are proper indexes in the sense that they satisfy basic axioms from index theory; second, and relatedly, they yield numbers that are consistent with measurement theory; and, third, they can be used to make valid comparisons across both time and space.

Measuring productivity change is relatively straightforward. Explaining it is much more complicated because it involves explaining changes in output and input quantities. This paper explained changes in output and input quantities (and therefore productivity) using an economic model in which farmers choose outputs and inputs in two stages: first, at the beginning of the production period, farmers choose inputs and planned outputs to maximize expected profits in the face of uncertainty about output prices and one or more characteristics of the production environment (e.g. temperature); and, second, after inputs have been chosen, managers seek to maximize the outputs that can be produced using their chosen inputs in whatever production environment is realized. These behavioural assumptions gave rise to a stochastic production frontier model and a system of input demand equations.

This paper used the estimated parameters of the stochastic frontier model to decompose GY TFP index numbers into measures of technical progress, environmental change, technical efficiency change, scale-and-mix efficiency change, and changes in statistical noise. The main drivers of TFP change over time were found to be technical progress (i.e. the discovery of new techniques for transforming inputs into outputs) and scale-and-mix efficiency change (i.e. changes in how well farmers captured economies of scale and substitution). The main drivers of TFP change across states were found to be environmental change (i.e. changes in variables that are physically involved in the production process but never within the control of farmers) and, again, scale-and-mix efficiency change. The environmental change component was further decomposed into the product of time-invariant effects and weather effects. The results indicated that changes in weather had a relatively small effect on TFP. The policy implications are that there are two main ways in which governments can improve U.S. agricultural productivity: by improving rates of technical progress, and by increasing levels of scale and mix efficiency. Governments can improve rates of technical progress by conducting, or funding others to conduct, more R&D. To increase levels of scale-and-mix efficiency, they must first identify the drivers of scale-and-mix efficiency change.

This paper used the estimated parameters of the input demand equations to decompose the measure of scale-and-mix efficiency change into a measure of technical progress, a measure of input price change, various measures of changes in expectations, and a measure of changes in allocative efficiency and statistical noise. The main drivers of scale-and-mix efficiency change over time were found to be technical progress and input price change. The main drivers of TFP differences across states were found to be differences in input prices and expectations about environmental variables. The expected environmental change component was further decomposed into the product of time-invariant effects and expected weather effects. The results indicated that changes in expectations about temperature and precipitation (i.e. changes in climate) had a relatively small effect on scale and mix efficiency. The policy implications are that there are two main ways in which governments can improve scale and mix efficiency: by improving rates of technical progress, and by changing input prices. Many governments routinely change the prices of capital and labor by changing interest rates and the minimum wage, and they often change the prices of land and materials through taxes and subsidies.

Finally, this paper found that a small proportion of measured productivity change in U.S. agriculture could be attributed to changes in statistical noise. Statistical noise is a combination of functional form errors, omitted variable errors, measurement errors, and included variable errors. An avoidable source of statistical noise in this paper is errors in the data supplied by the USDA. Among other things, the USDA measures changes in output and input quantities by dividing revenue and cost indexes by output and input price indexes. So-called implicit quantity indexes are not proper indexes, and they do not yield numbers that are consistent with measurement theory. Economists at the USDA might want to investigate whether the use of proper quantity indexes is feasible, and, if so, whether it would lead them to change their views on spatial and temporal differences in agricultural output and input quantities (and TFP).

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