Theoretical Model

We consider a cost-minimizing firm that makes input choice decisions in period t and makes or influences R&D funding decisions in k earlier periods that augment technology available in period t. To reduce notational clutter, we ignore time subscripts in the equations in this section and refer to the time when input choice decisions are made as period 2 and the time when R&D funding decisions are made as period 1. We start with input choice decisions in period 2 without imposing any restrictions on factor augmentation. We then turn to the R&D resource allocation decision in period 1 and its factor augmentation implications. By combining both components, we show how relative price changes impact both factor augmentation and subsequent input choice decisions. In doing so, we document that the IIH's implications of augmenting and saving the more expensive input is limited to certain ranges of the firm's elasticities of substitution and the innovation function's curvature parameter.

To allow parsimonious representation of a multiple-input production function with the possibility that elasticities of substitution are not the same for all input pairs, we consider a two-level constant elasticity of substitution (CES) functional form as frequently used in IIH tests (e.g., Kawagoe, Otsuka, and Hayami Reference Kawagoe, Otsuka and Hayami1986; Thirtle Schimmelpfennig, and Townsend Reference Thirtle, Schimmelpfennig and Townsend2002; Piesse, Schimmelpfennig, and Thirtle Reference Piesse, Schimmelpfennig and Thirtle2011; Cowan, Lee, and Shumway Reference Cowan, Lee and Shumway2015):Footnote ²

(1)$$Y = F\lpar {X_1\comma \; X_2} \rpar = \left[{\delta X_1^{{{\rho -1} \over \rho}} + \lpar 1-\delta \rpar X_2^{{{\rho -1} \over \rho}}} \right]^{{\rho \over {\rho -1}}}\; {\rm for} \;\rho \in \lsqb 0\comma \; \infty \rpar \comma \; $$

where 0 < ρ < ∞ is the elasticity of substitution between input indices X _i > 0, i ∈ (1, 2), and 0 < δ < 1 is the share parameter.Footnote ³ The input indices are produced by pairs of inputs that also follow a CES form:

(2)$$X_i = F_i\lpar {x_{i1}\comma \; x_{i2}\semicolon \; a_{i1}\comma \; a_{i2}} \rpar = \left[{\delta_i{\lpar a_{i1}^{} x_{i1}\rpar }^{{{\rho_i-1} \over {\rho_i}}} + \lpar 1-\delta_i\rpar {\lpar a_{i2}^{} x_{i2}\rpar }^{{{\rho_i-1} \over {\rho_i}}}} \right]^{\textstyle{{\rho _i} \over {\rho _i-1}}}\comma \; {\rm} $$

where x _ij > 0 is the quantity of input j ∈ (1, 2) used in production of input index i, and a > 0 is a factor-augmenting parameter that captures technical progress, 0 < ρ _i < ∞ and 0 < δ _i <1, i  ∈ (1, 2). The restrictions on parameter magnitudes and input levels ensure the production function is quasi-concave and that a competitive cost-minimizing solution exists for positive input prices.

The firm selects optimal input quantities in period 2 that minimize the cost of producing a given output level with known input prices and technology. Since the two-level CES production function maintains homotheticity, minimizing cost provides the same optimal input ratios as maximizing profit with the same input prices and technology. Taking the first-order conditions and with a little reorganization documented in Appendix I, we obtain the optimal input demand relationship:

(3)$$\displaystyle{{x_{i1}^{^\ast}} \over {x_{i2}^{^\ast}}} = \left({\displaystyle{{\delta_i} \over {1-\delta_i}}} \right)^{\rho _i}\left({\displaystyle{{w_{i1}} \over {w_{i2}}}} \right)^{-\rho _i}\left({\displaystyle{{a_{i1t}^{}} \over {a_{i2t}^{}}}} \right)^{\rho _i-1}\comma \; $$

where w _ij > 0 is the price of input x _ij, and the asterisk on x denotes the cost-minimizing input level.

Equation (3) documents that the qualitative effect of technical change (represented by the ratio of factor augmentation parameters) on the input ratio is dependent on the magnitude of the elasticity of substitution, as shown by Acemoglu (Reference Acemoglu2002, Reference Acemoglu2007), Funk (Reference Funk2002), and Armanville and Funk (Reference Armanville and Funk2003). Specifically, for two inputs, say labor and capital, without relative price changes, labor-augmenting technical change (that augments labor relatively more than capital) results in a labor-saving production decision if and only if the elasticity of substitution is less than one (i.e., the two factors are gross complements). However, if the inputs are gross substitutes (i.e., elasticity of substitution is greater than one), labor-augmenting technical change results in relatively greater use of labor because it is more easily substituted for capital. When the elasticity of substitution is exactly one (as in the Cobb-Douglas production function), technical change does not lead to changes in the cost-minimizing input ratio.

We now turn to the cost-minimizing R&D resource allocation decision made in period 1. In doing so, we show that the relationship between relative price and relative factor augmentation is a non-monotonic function of the elasticity of substitution. More importantly, we document that competitive cost-minimizing behavior in both periods results in saving the more expensive input in period 2 even if the inputs are gross substitutes as long as the elasticity of substitution is not too large.

We consider a simple but very general homothetic innovation function that produces an innovation possibilities frontier that is strictly concave to the origin, where innovation is defined as augmentation of at least one factor.Footnote ⁴ It can accommodate both non-neutral and time-varying marginal R&D costs. For a given R&D budget $\bar{R}$, the innovation function is given by:Footnote ⁵

(4)$$R_i = \lpar {c_{i1}{\hat{a}}_{i1}} \rpar ^{\theta _i} + \lpar {c_{i2}{\hat{a}}_{i2}} \rpar ^{\theta _i}$$

where R _i is expenditure on R&D in period 1 to augment the ith input index, the total R&D budget is assumed to be exogenously given and fully expended, i.e., $\bar{R}\equiv R_1 + R_2$, $\hat{a}_{ij}$ is the expected factor-augmentation in period 2; Footnote ⁶ c _ij > 0 denotes marginal R&D costs (or degree of R&D difficulty considering all R&D costs and probability of success) in period 1 for technology that is expected to augment x _ij by one percent in period 2; and θ _i > 1 is a parameter representing the rate of trade-off between the two R&D outcomes $\hat{a}_{i1}$ and $\hat{a}_{i2}$; that is, the larger the parameter, the more concave is the frontier to the origin (hereafter referred to as the “concavity parameter”). The marginal R&D cost parameters represent both explicit cost (e.g., wages for scientists) and the likelihood of success in the R&D.

The innovation function is neutral if and only if c _i1 = c _i2. Figure 1 displays (a) a neutral innovation function and (b) a non-neutral innovation function. In panel (b) of the figure, the condition c _i1 > c _i2 implies that R&D on the x _i1 input-augmenting technology is more difficult (more costly) than that on input x _i2 to achieve the same level of augmentation. Increases in R&D funds expand the innovation function radially parallel to the origin.Footnote ⁷ The condition θ _i > 1 is sufficient to ensure that $\textstyle{{\partial {\hat{a}}_{i2}} \over {\partial {\hat{a}}_{i1}}} \lt 0$ and $\textstyle{{\partial ^2{\hat{a}}_{i2}} \over {\partial \hat{a}_{i1}^2}} \lt 0$ as demonstrated in Appendix II.

Figure 1.

Neutral and Non-Neutral Innovation Function

This simple formulation of the innovation function results in an inverse relationship between degree of R&D difficulty and R&D outcome. An increase in the marginal cost of R&D shifts the innovation frontier toward the origin. Because the innovation function is output-homothetic on the frontier, the share of R&D outputs measured by the ratio of the factor-augmenting parameters does not depend on the total R&D budget.

Considering the opportunity to invest (or influence investment) in R&D in period 1, perhaps many years before production inputs are selected, and assuming a two-level CES production function as in the input decision, the firm's R&D resource allocation problem in period 1 can be written as follows:

(5)$$\eqalign{\mathop {\min} \limits_{{\tilde{x}}_{ij}\comma \hat{a}_{ij}^{}} \; \sum\limits_i {\sum\limits_j E} \lpar w_{ij}\rpar {\tilde{x}}_{ij}\ s.t.\; {\bar{Y}}_{} = \left[{\delta \tilde{X}_1^{{{\rho -1} \over \rho}} + \lpar 1-\delta \rpar \tilde{X}_2^{{{\rho -1} \over \rho}}} \right]^{{\rho \over {\rho -1}}}{\rm and}\ \bar{R}\equiv R_1 + R_2\comma \; } $$

where $\tilde{X}_i = \left[{\delta_i{\lpar a_{i1}^{} {\tilde{x}}_{i1}\rpar }^{{{\rho_i-1} \over {\rho_i}}} + \lpar 1-\delta_i\rpar {\lpar a_{i2}^{} {\tilde{x}}_{i2}\rpar }^{{{\rho_i-1} \over {\rho_i}}}} \right]^{{{\rho _i} \over {\rho _i-1}}}\comma \; \, R_i = \lpar {c_{i1}{\hat{a}}_{i1}} \rpar ^{\theta _i} + \lpar {c_{i2}{\hat{a}}_{i2}} \rpar ^{\theta _i}\comma \;$ a tilde on input level denotes that it is a period 2 value “conceived” in period 1 and thus distinguished from the value that is actually chosen by the firm in period 2 when input prices are known, E is the expectations operator E[ · |Ω], and Ω is the firm's information set in period 1. Maintaining restrictions sufficient to ensure the production function is quasi-concave and the innovation possibilities frontier is strictly concave also ensures a competitive cost-minimizing solution for positive input prices.

Combining and rearranging the first-order conditions, the optimal innovation ratios in period 1 are obtained (see Appendix I for derivation of equations (6) and (7)):

(6)$$\displaystyle{{\hat{a}_{i1}^{^\ast}} \over {\hat{a}_{i2}^{^\ast}}} = \left({\displaystyle{{\delta_i} \over {1-\delta_i}}} \right)^{\displaystyle{{\rho _i} \over {\psi _i}}}\left({\displaystyle{{E\lpar w_{i1}\rpar } \over {E\lpar w_{i2}\rpar }}} \right)^{\displaystyle{{1-\rho _i} \over {\psi _i}}}\left({\displaystyle{{c_{i1}} \over {c_{i2}}}} \right)^{\displaystyle{{-\theta _i} \over {\psi _i}}}\comma \; $$

where ψ _i = 1 + θ _i − ρ _i and the asterisk denotes an optimal value. It is apparent that the effect of an expected price change on the ratio of expected R&D outcomes depends on the sign and magnitude of ${{1-\rho _i} \over {\psi _i}}$ and thus on the magnitudes of both ρ _i and θ _i (see the first row in Table 1). Since ρ _i  ≥ 0 and θ _i > 1, an increase in the (expected) relative price of x _i1 brings about factor-augmenting technical change such that the input that is expected to become more expensive is augmented more than the other whenever the two inputs are gross complements (0 ≤ ρ _i < 1). Conversely, when the inputs are gross substitutes (ρ _i > 1) and the elasticity of substitution is less than 1 plus the concavity parameter (ρ _i < θ _i + 1), the same increase will induce technical change that augments the input that is expected to become less expensive. Note that the sign change at ρ _i = 1 has been shown in previous literature, e.g., Acemoglu (Reference Acemoglu2002). However, a new result arises in the two-period model with endogenous technical change: if the inputs are gross substitutes (ρ _i > 1) and the elasticity of substitution is greater than 1 plus the concavity parameter (ρ _i > 1 + θ _i), then the same increase will again induce technical change that augments the input that is expected to become more expensive. Thus, it is clear that cost-minimizing firms respond not only to price incentives but also to the technological opportunity costs in making R&D resource allocation decisions. Further, the relationship between the elasticity of substitution and the impact of an input price change on factor augmentation is neither monotonic nor continuous.Footnote ⁸

Table 1.

Effect of a Change in Expected Input Price Ratio, ${\rm \Delta} \left({{{E\lpar w_{i1}\rpar } \over {E\lpar w_{i2}\rpar }}} \right)\gt 0$

The firm optimizes in period 2 by solving the input choice decision problem in which the R&D outcomes are taken as given. Substituting the expected augmentation parameters, ${{\hat{a}_{i1}^*} / {\hat{a}_{i2}^*}}$, into the equilibrium condition (3) yields the following input demand relationship:

(7)$$\displaystyle{{x_{i1}^{^\ast}} \over {x_{i2}^{^\ast}}} = \left({{{\delta_i} \over {1-\delta_i}}} \right)^{{{\theta _i\rho _i} \over {\psi _i}}}\left({{{w_{i1}} \over {w_{i2}}}} \right)^{-\rho _i}\left({{{E\lpar w_{i1}\rpar } \over {E\lpar w_{i2}\right) }}} \right)^{{{-{\lpar 1-\rho _i\rpar }^2} \over {\psi _i}}}\left({\displaystyle{{c_{i1}} \over {c_{i2}}}} \right)^{{{\theta _i\lpar 1-\rho _i\rpar } \over {\psi _i}}}\comma \; $$

which is homogeneous of degree zero in current prices, in expected prices, and in both current and expected prices.Footnote ⁹

With this combined condition, it is now easy to distinguish optimal input choice effects of technical change caused by changes in the expected price ratio from input substitution effects caused by changes in the current price ratio. As with factor augmentation, the effect of an expected relative price change on the optimal input ratio depends on the magnitudes of both ρ _i and θ _i (see the second row in Table 1). An increase in the expected input price ratio, ${{E\lpar w_{i1}\rpar } \over {E\lpar w_{i2}\rpar }}$, results in a production decision to save the input that becomes more expensive only if inputs are gross complements (0 ≤ ρ _i <1) or if they are gross substitutes (ρ _i > 1) and the elasticity of substitution is less than 1 plus the concavity parameter (ρ _i < θ _i + 1). If the inputs are gross substitutes (ρ _i > 1) and the elasticity of substitution is greater than 1 plus the concavity parameter (ρ _i > 1 + θ _i), the increase results in a production decision to save the input that becomes less expensive.Footnote ¹⁰ Our parametric relationships are qualitatively the same as Acemoglu's (Reference Acemoglu2007, Example I) static results when ρ _i <1 and when ρ _i > 1 + θ _i. When ρ _i > 1 + θ _i, both the static and two-period results show that the demand curve for an input can be positively sloped in the long run. However, our two-period result differs from his static result when 1 < ρ _i < 1 + θ _i.

Empirical Model – Two-Way Fixed-Effects Approach

In this section we document how a time-and-state two-way fixed-effects formulation can be used to control for unobserved non-neutral marginal R&D costs. We focus on the agricultural production sector and treat states as though they were price-taking cost-minimizing firms, a hypothesis previously not rejected by empirical testing (Lim and Shumway Reference Lim and Shumway1992). Although little formal R&D is conducted by agricultural firms, they may have influence through the political process on the allocation of public R&D funds. Thus, we treat public agricultural R&D in the state as an appendage to the representative state-level agricultural firm.

Including the state subscript s, time subscript t on period 2 variables, and time subscript t-k on period 1 variables and expectations, and taking the natural logarithm of both sides of equation (7), we obtain the regression model:

(8)$$\ln \left({\displaystyle{{x_{i1st}} \over {x_{i2st}}}} \right)= \beta _{i0} + \beta _{i1}\ln \left({\displaystyle{{w_{i1st}} \over {w_{i2st}}}} \right)+ \beta _{i2}\ln \left({\displaystyle{{E_{t-k}\lpar w_{i1st}\rpar } \over {E_{t-k}\lpar w_{i2st}\rpar }}} \right)+ \beta _{i3}\ln \left({\displaystyle{{c_{i1s\lpar t-k\rpar }} \over {c_{i2s\lpar t-k\rpar }}}} \right)+ \Gamma _i^{\prime} Z_{ist} + \varepsilon _{ist}$$

where Z is a vector of other control variables, ɛ_ist is an error term, and the β’s and the vector Γ are parameters. The parameters β _i1, β _i2, and β _i3 correspond to − ρ _i, ${{-{\lpar 1-\rho _i\rpar }^2} \over {\psi _i}}$, and ${{\theta _i{\lpar 1-\rho _i\rpar }^{}} \over {\psi _i}}$, respectively, in equation (7).Footnote ¹¹ Empirical support for the IIH (i.e., factor-saving behavior in response to the relative price increase) is provided by a significantly negative β _i2, the coefficient on the expected input price ratio.Footnote ¹²

If the marginal cost of R&D to augment one percent of an input is the same across inputs, i.e., c _i1t = c _i2t (which represents a neutral innovation function), the ratio of marginal R&D costs vanishes from the regression model and does not affect the optimal input ratio. With few exceptions (e.g., Deininger Reference Deininger1995, Fulginiti Reference Fulginiti1994), this has been implicitly assumed in the IIH testing literature until recently. Formal tests have been intrinsically limited to the demand side without accounting for possible non-neutrality in the supply of innovation. However, there is no reason to expect that the innovation function is neutral, so we must deal with the omitted variables problem due to the lack of explicit data on marginal R&D costs. To surmount this problem, we use a two-way fixed-effects approach in which we allow marginal costs to vary across inputs, across states, and over time.

The time-difference equation can be written as follows:

(9)$$\eqalign{& \Delta _t\ln \left({\displaystyle{{x_{i1st}} \over {x_{i2st}}}} \right)= \beta _{i1}\Delta _t\ln \left({\displaystyle{{w_{i1st}} \over {w_{12st}}}} \right)+ \beta _{i2}\Delta _t\ln \left({\displaystyle{{E_{t-k}\lpar w_{i1st}\rpar } \over {E_{t-k}\lpar w_{i2st}\rpar }}} \right)+ \beta _{i3}\Delta _t\ln \left({\displaystyle{{c_{i1s\lpar t-k\rpar }} \over {c_{i2s\lpar t-k\rpar }}}} \right)\cr & {\rm} + \Gamma _i^{\prime} \Delta _tZ_{ist} + \Delta _t\varepsilon _{ist}} $$

where Δ_t is the time-difference operator. If the ratio of marginal R&D costs varies across states but is invariant over time, then

$$\Delta _t\ln \left({\displaystyle{{c_{i1s\lpar t-k\rpar }} \over {c_{i2s\lpar t-k\rpar }}}} \right)= \ln \left({\displaystyle{{c_{i1s\lpar t-k\rpar }} \over {c_{i2s\lpar t-k\rpar }}}\displaystyle{{c_{i2s\lpar t-k-1\rpar }} \over {c_{i1s\lpar t-k-1\rpar }}}} \right)= \ln \left({\displaystyle{{c_{i1s}} \over {c_{i2s}}}\displaystyle{{c_{i2s}} \over {c_{i1s}}}} \right)= 0$$

and the equation would be estimable in its current form without data on marginal R&D costs. However, since it is unknown whether the marginal R&D cost is invariant over time, we estimate the two-way fixed-effects formulation:

(10)$$\eqalign{& \Delta _{t\comma s}\ln \left({\displaystyle{{x_{i1st}} \over {x_{12st}}}} \right)= \beta _{i1}\Delta _{t\comma s}\ln \left({\displaystyle{{w_{i1st}} \over {w_{i2st}}}} \right)+ \beta _{i2}\Delta _{t\comma s}\ln \left({\displaystyle{{E_{t-k}\lpar w_{i1st}\rpar } \over {E_{t-k}\lpar w_{i2st}\rpar }}} \right)+ \beta _{i3}\Delta _{t\comma s}\ln \left({\displaystyle{{c_{i1s\lpar t-k\rpar }} \over {c_{i2s\lpar t-k\rpar }}}} \right)\cr & {\rm} + \Gamma _i^{\prime} \Delta _{t\comma s}Z_{ist} + \Delta _{t\comma s}\varepsilon _{ist}} $$

where Δ_t,s is the time-and-state difference operator. Thus, defining the change rate term,

$$ \eqalign{& g_{i1\lpar t-k\rpar } = \displaystyle{{c_{i1\lpar t-k\rpar }} \over {c_{i1\lpar t-k-1\rpar }}}\comma \cr& \Delta _{t\comma s}\ln \left({\displaystyle{{c_{i1s\lpar t-k\rpar }} \over {c_{i2s\lpar t-k\rpar }}}} \right)= \ln \left({\displaystyle{{g_{i1s\lpar t-k\rpar }} \over {g_{i2s\lpar t-k\rpar }}}/\displaystyle{{g_{i1\bar{s}\lpar t-k\rpar }} \over {g_{i2\bar{s}\lpar t-k\rpar }}}} \right)\comma \;}$$

where $\bar{s}\ne s$ is a reference state.

We maintain the sufficient condition that the change rate in relative marginal costs of R&D over time is equal across states.Footnote ¹³ More restrictive assumptions such as state-invariant marginal R&D costs (as typically assumed) and/or time-invariance are sufficient but not necessary for the equation to be identified. Importantly, identification under our sufficient condition does not require identical marginal R&D costs across inputs, and thus the equation is estimable with a non-neutral innovation function even without data on marginal R&D costs. Further, the assumption still allows for differences in marginal R&D costs across states and for changes in the innovation function over time so long as the pattern of change is common across states.

If the sufficient condition noted above holds, the two-way fixed-effects formulation surmounts the omitted variable problem caused by the lack of marginal R&D cost data and makes estimation of β _i3 unnecessary. To the extent that condition is not a reasonable approximation, this approach is unable to fully capture the effects of a non-neutral innovation function. Equation (10) is estimated using a two-way fixed-effects (state and time) panel data estimator.Footnote ¹⁴

We include two additional control variables—total public agricultural R&D expenditures for each state s and year t-k and total agricultural output for each state s and year t. These variables are included to control for potential size effects of total R&D expenditures in the innovation creation industry and of total agricultural output in the innovation-implementation industry. Inclusion of these control variables renders homotheticity as a local property rather than a global constraint for both the production function and the innovation function.

Data and Variable Specification

The data used for our test of the IIH include annual agricultural input quantities and prices for four exhaustive input categories (land, non-land capital, labor, and intermediate inputs) as well as total agricultural output and total public expenditures for agricultural productivity R&D for each of the 48 contiguous U.S. states for the years 1960–2004. The panel data set is balanced over the entire period. Summary statistics are presented in Table 2.

Table 2.

Summary Statistics

^a Measured relative to a 1996 Alabama price of 1.00.

^b Measured as the value of the input, in thousands of dollars, used for agricultural production in the state divided by the price index.

The agricultural input prices and quantities and total output are from the U.S. Department of Agriculture (USDA ERS 2015). For details regarding original data sources and construction of this state-level aggregated series, see Ball, Hallahan, and Nehring (Reference Ball, Hallahan and Nehring2004) and Ball et al. (Reference Ball, Gollop, Kelly-Hawke and Swinand1999). Total annual public expenditures for agricultural productivity R&D by state are from Huffman (Reference Huffman2012). They were compiled following the procedures outlined in Huffman (Reference Huffman, Singh, Joseph and Johnson2015). Specifically, R&D expenditure data collected by the USDA in its Current Research Information System (CRIS) were used. CRIS includes funding from all sources for agricultural R&D undertaken by both federal R&D organizations (U.S. Department of Agriculture) and state public R&D entities (state agricultural experiment stations and veterinary colleges at land grant universities). Expenditures on post-harvest R&D and R&D on households, families, and communities were excluded from Huffman's series.

Test Results

The estimation model, equation (10), was estimated using a two-way fixed-effects panel data estimator. The statistical estimates based on rational price expectations are presented in Table 3. Estimation results are reported from two model specifications in which only price expectation differed: the first used a single price expectation lagged five years and the second used a weighted average of expected prices lagged two to five years. Each column contains estimates for one input pair.

Table 3.

Statistical Estimates with Rational Expectation Price Forecasts^a

^a Standard errors are in parentheses; p-value of estimated parameters: * p < 0.1, ** p < 0.05, *** p < 0.01. The t-tests conducted for the coefficients on current and expected price ratios are one-tailed tests (with the null hypothesis that the coefficient is greater than or equal to zero). T-tests for coefficients on other regressors are two-tailed tests (null hypothesis that the coefficient is equal to zero). Time fixed effects and state fixed effects parameter estimates are suppressed.

^b Standard errors of the R&D concavity parameter are obtained using the delta method.

For comprehensive coverage, we tested for consistency with the IIH by considering all six exhaustive pairs of the four inputs (i.e., labor/land, capital/land, intermediate inputs/land, capital/labor, intermediate inputs/labor, and intermediate inputs/capital). Due to potential heteroskedasticity and autocorrelation, robust standard errors were obtained using a clustered sandwich estimator with the state as the unit of cluster (Wooldridge Reference Wooldridge2002).Footnote ¹⁸ Time and state fixed effects were included in all equations, but these estimates are not reported in the table to conserve space. The number of observations included in each of the regressions varied across input ratios due to the different number of lags involved in the calculation of rational expectations of future price.

Estimates of the effect of the current price ratio on the input quantity ratio, β _i1, are the negative of the estimated elasticity of substitution, ρ _i, and estimates of the effect of the expected future price ratio on the input quantity ratio correspond to β _i2 in equation (10). For all 12 estimated equations in this table, the elasticity of substitution was estimated to be substantially less than 2.0 (0.03–0.54).Footnote ¹⁹ Thus, empirical support for the IIH and cost-minimizing behavior is provided by a significantly negative β _i2.

Using a single price expectation based on the information set available five years earlier, five of the six pairings of inputs (83%) supported the IIH: labor/land, capital/land, intermediate inputs/land, capital/labor, and intermediate inputs/capital. Using a weighted average of price expectations given the information set available two to five years earlier, three equations (50%) supported the IIH: capital/labor, intermediate inputs/labor, and intermediate inputs/capital. Apart from statistical significance, the estimated signs on the expected future price ratio as well as on the current price ratio are negative for all 12 equations, and thus the directional impact of technical change inferred from the point estimates are found to be consistent with the IIH in all the estimated equations.

Effects of total agricultural output were estimated to have a significantly positive effect in 10 of the 12 equations and an insignificantly negative effect in the other two. Based on the results, we conclude that increases in total output lead to more intense use of non-land inputs (i.e., labor, capital, and intermediate inputs) than land—see the first three columns of Table 3—and to more intense use of intermediate inputs (energy, fertilizer, etc.) than labor and capital—see the last two columns. These results that reject the null hypothesis imply that the agricultural production function is not globally homothetic.

Total public R&D funds were estimated to have a significantly positive effect in two equations and a significantly negative effect in two, while the rest were statistically insignificant. These results cause us to reject the hypothesis that the innovation function is globally homothetic. They imply that increases in total R&D funds lead to behavior that saves labor and intermediate inputs relative to capital. Consequently, the evidence implies that the marginal cost of R&D to save 1 percent of the capital input relative to that of labor or intermediate inputs increases as total R&D investment increases. Non-homotheticity of the innovation function could have biased several earlier studies away from consistency with the IIH as it renders neutrality only as a local property. For example, in both Binswanger (Reference Binswanger1974b) and Huffman and Evenson (Reference Huffman and Evenson1989), a locally non-neutral innovation function could have contributed to their rejection of changes in machinery use being consistent with the induced innovation hypothesis. The same could have occurred in Liu and Shumway's (Reference Liu and Shumway2009) direct econometric test results with variations in the labor/capital input ratio.

However, it does not help with Kawagoe, Otsuka, and Hayami's (Reference Kawagoe, Otsuka and Hayami1986) rejection of the induced hypothesis in the land/labor or land/fertilizer input ratios. Total public R&D funds were not found to be significant in any of the equations where land was included as an input (see the first three columns of Table 3). Unlike de Janvry, Sadoulet, and Fafchamps (Reference de Janvry, Sadoulet, Fafchamps and Bardhan1991), we fail to find evidence that total public R&D expenditures bias technical change toward land-saving technology in U.S. agriculture.

Derived estimates for the R&D concavity parameter θ _i are also reported in Table 3. Except for the intermediate inputs and capital ratio, all estimates are greater than 1.0, as required for a concave innovation possibility frontier. The estimated concavity parameter is highly sensitive both to input pair and to the information used to estimate price expectations, but none is significantly different from one. The variance of the input quantity ratio explained by the estimated models ranged from 44 to 89 percent of total variance.

Robustness Checks

We consider several alternative model specifications to examine the robustness of initial model results. We consider variants of our price expectations assumption, the type of estimator used, and the nature of fixed effects.

Adaptive and Naïve Price Expectations

We use the adaptive expectations hypothesis and the naïve expectations hypothesis as two alternative specifications in our robustness analysis. To create adaptive expectations, we first forecast prices one year ahead at time t-k using geometrically declining distributed lags on five years of realized prices:

$$E_{t-k}\lpar w_{is\lpar t-k + 1\rpar }\rpar = \varphi w_{is\lpar t-k\rpar } + \varphi \lpar 1-\varphi \rpar w_{is\lpar t-k-1\rpar } + \varphi \lpar 1-\varphi \rpar ^2w_{is\lpar t-k-2\rpar } + \cdots + \varphi \lpar 1-\varphi \rpar ^4w_{is\lpar t-k-4\rpar }$$

where φ denotes the weight. The subsequent, 2-, ..., k-year ahead forecasts are sequentially updated as

$$E_{t-h}\lpar w_{ist}\rpar = \sum\limits_{\,p = 1}^{h-1} {\varphi {\lpar 1-\varphi \rpar }^{\,p-1}E_{t-h}\lpar w_{is\lpar t-p\rpar }\rpar } + \sum\limits_{\,p = h}^5 {\varphi {\lpar 1-\varphi \rpar }^{\,p-1}w_{is\lpar t-p\rpar \rpar }}$$

for h = 2, …, k. We select the optimal weight φ based on the Akaike Information Criterion. To ensure that the cumulative weights are over 0.9 within the assumed R&D period (i.e., $\sum\nolimits_{t = 1}^5 {\varphi {\lpar 1-\varphi \rpar }^{t-1}} \gt 0.9$), we only consider φ = 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. When φ = 1, the expectation of future price is the realized lagged price, E _t−k(w _ist) = w _is(t−k). This is the naïve expectation. Forecasts of future prices from these two expectation hypotheses are created using a single price expectation lagged five years and a weighted average of expected prices lagged two to five years.

Robustness Results

Table 4 documents whether estimated signs and magnitudes of the parameters on current and expected price ratios are consistent with multistage cost minimization conditions and the IIH under each of the alternative specifications. Compared to the test results with the rational expectation hypothesis, somewhat less support for the IIH was found with both the adaptive and naïve expectations.Footnote ²⁰ For capital/labor, intermediate inputs/labor and intermediate inputs/capital equations, all four of these estimated equations were found to be supportive of the IIH. For labor/land and intermediate inputs/land, three of the four estimates had expected signs and magnitudes, but none was statistically significant.

Table 4.

Parameter Signs and Magnitudes Consistent with the IIH and Cost Minimization, Alternative Price Expectations, and Estimation Methods ^a

^a “Yes” indicates that the estimate is consistent (in terms of sign) with the IIH and cost minimization conditions, and the asterisk (*) indicates that the estimate is statistically significant at the 10 percent level; “No (β _i1)” and “No (β _i2)”, respectively, indicate that the elasticity of substitution is negatively estimated or that the expected input price ratio positively affects the input quantity ratio.

^b Based on a single five-year rational expectations price forecast.

The SUR results provided the same support for the IIH as previously (50 percent) when using the weighted average of price expectations and slightly lower support than previously (67 percent vs. 83 percent) when using a single price expectation lagged five years.

While all parameters were consistent with cost minimization and the IIH with two-way fixed effects, inconsistencies were found with each of the alternative fixed-effects specifications. Not accounting for time fixed effects results in four input pairs being inconsistent with the joint hypothesis. Failing to account either for state fixed effects or any fixed effects (i.e., pooled regression) results in one input pair being inconsistent with the joint hypothesis.

Considering only statistically significant estimates, five of the six input pairs were consistent with the joint hypothesis when estimated using two-way fixed effects, time fixed effects, or pooled regression. Only two were consistent when estimated using state fixed effects.

The null hypothesis that fixed effects are zero was soundly rejected at the 1 percent level of significance in all specifications. Thus, unobserved state and time effects are concluded to be important for explaining changes in relative input demands. Two-way fixed-effects estimation provides the greatest support for cost minimization and the IIH. Any omitted variables bias in the alternatives reduces support for this joint hypothesis.Footnote ²¹