Public inputs and dynamic producer behavior: endogenous growth in U.S. agriculture

Abstract

This paper is an attempt to understand the impact of public R&D and public infrastructure on the performance of the U.S. agricultural sector during the last part of the twentieth century. A neoclassical Solow growth model is not sufficient for this understanding given the sustained growth performance of the sector. We base our analysis on a well-known endogenous growth model, the ‘AK model’ where non-convexities are introduced through non-rival inputs. Based on these models and within the dynamic models that rationalize private and public decision making, we have identified three testable hypotheses regarding the aggregate agricultural production technology. They are: (1) increasing returns to scale over all inputs; (2) positive effect of additional units of public inputs on the long-run demand for private capital; and (3) negative impact of public inputs on cost. They are tested using two estimation procedures on two data sets for U.S. agriculture. One, covering the period 1948–1994, developed by USDA, the other, covering the period 1926–1990, from Thirtle et al. Maximum likelihood estimates do not conform to the regularity and behavioral properties of the economic model rendering them unusable for testing these hypotheses. Bayesian estimates, although not totally satisfactory, do not reject the hypotheses after prior imposition of some of the regularity conditions. This supports the notion of an important role for public inputs on the rapid and sustained growth of the sector. We calculate that, on average, one additional dollar spent on public R&D stock reduces private cost by $6.5, implying a return on these public expenses of 190%.

Appendices

Appendix 1

This Appendix presents conditions (A) and (B) that guarantee duality between cost and value functions of the firms.

1.1 Conditions (A)

It is assumed that C(y, Z, I; G) satisfies the following set of regularity conditions:

• (A.1) C(y, Z, I; G) ≥ 0.

• (A.2) C(y, Z, I; G) is increasing in y and decreasing in Z. Additionally, C_I > 0 when I > 0 and vice versa, which follows from the assumption of adjustment costs.

• (A.3) C(y, Z, I; G) is convex in I.

• (A.4) For each (Z₀, y, p; G) a unique solution exists for (1). This means that there are well-defined factor demand functions associated with (1).

• (A.5) For each (Z₀, y, p; G), problem (1) has a unique steady state (SS) stock $\overline{\hbox{Z}} (\hbox{y, p; G})$ that is globally stable. This condition establishes the uniqueness and stability of the steady state.

• (A.6) For any (Z₀, y, p; G), there exists $\hat{\hbox{p}}$ such that $\hat{\hbox{I}}$ is the optimal gross investment vector at t = 0 in (1) given (Z₀, y, p; G).

1.2 Conditions (B)

It is assumed that the value function J(Z, y, p; G) satisfies the following properties:

• (B.1) J(Z, y, p; G) ≥ 0.

• (B.2)

  • (i) $(\hbox{ru}+\delta)\hbox{J}_{\rm z} (\hbox{Z, y, p; G})-\hbox{p}-\hbox{J}_{\rm zz} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G}) < 0,$ where u is an identity matrix. This expression is dual to ${\hbox{C}_{\rm z}\, < \,0}.$

  • (ii) J_z(Z, y, p; G) < 0 when $\hbox{I}^{\ast}(\hbox{Z, y, p; G})\equiv \dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G})+\delta \hbox{Z} > 0$ and vice versa. This condition is dual to C_I > 0 when I > 0 and vice versa.

  • (iii) $\rho \hbox{J}_{\rm y} (\hbox{Z, y, p; G})-\hbox{J}_{\rm yz}^{\prime} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G})\, > \,0,$ where $\dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G})=\hbox{J}_{\rm pz}^{-1} (\hbox{Z, y, p; G})[\rho \hbox{J}_{\rm p} (\hbox{Z, y, p; G})-\hbox{Z}].$ This condition is dual to C_y > 0.

• (B.3) The following expression is concave in p:

  $$ \rho \hbox{J(Z, y, p; G)}-\hbox{p}^{\prime}\hbox{Z}-\hbox{J}_{\rm z}^{\prime} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G}) $$

  Under some specific functional forms (like the normalized quadratic presented above), J_z(Z, y, p; G) is linear in p and the curvature requirement reduces to concavity of J(Z, y, p; G) in p. This condition is dual to (A.3).

• (B.4) The demand for the variable input, X^*(Z, y, p; G), is positive.

• (B.5) The stock Z that solves $\dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G})=\hbox{J}_{\rm pz}^{-1} (\hbox{Z, y, p; G})[\rho \hbox{J}_{\rm p} (\hbox{Z, y, p; G})-\hbox{Z}],$ with Z(0) > 0, has a unique globally stable steady state $\overline{\hbox{Z}} (\hbox{y, p; G}).$

Then, under conditions (A) and (B), duality between C(y, Z, I; G.) and J(Z, y, p; G) can be established as in Eqs. 2 and 3. The following derivative properties then hold:

1.3 Derivative properties

• 1. With respect to I:

  C_I(y, Z, I; G) = −J_z(Z, y, p; G). From (A.2) or (B.2.ii), this expression must be positive when I > 0 and vice versa. Testing for J_z(Z, y, p; G) = 0 is equivalent to testing for adjustment costs in inputs Z.

• 2. With respect to Z:

  $$ \hbox{C}_{\rm z} (\hbox{y, Z, I; G}) = (\rho \hbox{u}+\delta)\hbox{J}_{\rm z} (\hbox{Z, y, p; G})-\hbox{p}-\hbox{J}_{\rm zz} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^\ast(\hbox{Z, y, p; G}) < 0 $$

  from (A.2).

  This expression gives the shadow price of quasi-fixed inputs.

• 3. With respect to y:

  $$ \hbox{C}_{\rm y} (\hbox{y, Z, I; G})=\rho \hbox{J}_{\rm y} (\hbox{Z, y, p; G})-\hbox{J}_{\rm zy}^{\prime} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^{\ast}(\hbox{Z, y, p; G}) > 0 $$

  from (A.2).

  This expression represents the output supply of the firms.

• 4. With respect to p:

  $$ 0=\rho \hbox{J}_{\rm p} (\hbox{Z, y, p; G})-\hbox{Z}-\hbox{J}_{\rm zp} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^{\ast}(\hbox{Z, y, p; G}) $$

  Then, $\dot{\hbox{Z}}^{\ast}(\hbox{Z, y, p; G})=\hbox{J}_{\rm pz}^{-1} (\hbox{Z, y, p; G})[\rho \hbox{J}_{\rm p} (\hbox{Z, y, p; G})-\hbox{Z}],$ which is the dynamic demand for Z.

• 5. With respect to G:

  $$ \hbox{C}_{\rm G} (\hbox{y, Z, I; G})=\rho \hbox{J}_{\rm G} (\hbox{Z, y, p; G})-\hbox{J}_{\rm ZG} (\hbox{Z, y, p; G})\dot{\hbox{Z}}^{\ast}(\hbox{Z, y, p; G}) $$

  This expression represents the shadow price of G when the firms are out of the SS. At the SS, the shadow price is

  $$ \hbox{C}_{\rm G} (\hbox{y, Z, I; G})=\rho \hbox{J}_{\rm G} (\hbox{Z, y, p; G}) $$

  If this expression is negative, the shadow price of G is positive, meaning that public inputs reduce cost of production.

Appendix 2

This Appendix presents conditions (C) to (D) that guarantee duality between the value function of the firms and the value function of the government.

2.1 Conditions (C)

It is assumed that J(y, Z, p; G)  +  AC(I_g) satisfies the following conditions:

• (C.1) J(y, Z, p; G) + AC(I_g) ≥ 0

• (C.2)

  • (i) J(y, Z, p; G) + AC(I_g) is increasing in I_g. Given that J(y, Z, p; G) is independent of I_g, AC(I_g) must be increasing in I_g.

  • (ii) J(y, Z, p; G) + AC(I_g) is decreasing in G. Given that AC(I_g) is independent of G, J(y, Z, p; G) must be decreasing in G.

• (C.3) J(y, Z, p; G) + AC(I_g) is convex in I_g. Then, AC(I_g) must be convex in I_g.

• (C.4) For each (Z, p, y, r, G₀), there exists a unique solution for (8). This means that there are well-defined supplies of public inputs.

• (C.5) For each (Z, p, y, r, G₀), (8) has a unique steady state stock $\overline{\hbox{G}} (\hbox{Z, p, y, r})$ that is globally stable.

• (C.6) For any (Z, p, y, r, G₀), there exists $\hat{\hbox{r}}$ such that $\hat{\hbox{I}}_{\rm g}$ is the optimal public gross investment vector at t = 0 in (8), given (Z, p, y, r, G₀).

2.2 Conditions (D)

It is assumed that J^g(y, Z, p; r, G) satisfies the following conditions:

• (D.1) J^g(y, Z, p; r, G) ≥ 0

• (D.2)

  • (i) $\hbox{J}_{\rm G}^{\rm g} (\hbox{y, Z, p; r, G}) < 0.$ This condition is dual to (C.2)(i) and means that there are adjustment costs in the provision of public inputs.

  • (ii) $(\theta \hbox{u}+\delta_{\rm g})\hbox{J}_{\rm G}^{\rm g} (\hbox{y, Z, p; r, G})-\hbox{J}_{\rm GG}^{\rm g} (\hbox{y, Z, p; r, G})\dot{\hbox{G}}^{\ast} < 0.$ This expression is dual to (C.2)(ii): J_G(y, Z, p; r, G) < 0 (positive shadow prices of public inputs). Given $\hbox{J}_{\rm G}^{\rm g} (\hbox{y, Z, p; r, G}) < 0,$ it is sufficient for (D.2)(ii) to hold that $-\hbox{J}_{\rm GG}^{\rm g} (\hbox{y, Z, p; r, G}) < 0$ (that is, increases of the public good decrease the shadow price of it).

• (D.3) ${}_\theta\hbox{J}^{\rm g}(\hbox{Z, y, p, r, G})-\hbox{r}^{\prime}\hbox{G}-\hbox{J}_{\rm G}^{{\rm g}{\prime}}(\hbox{Z, y, p, r, G})\dot{\hbox{G}}^\ast$ must be concave in r. This is dual to condition (C.3).

• (D.4) $\hbox{I}_{\rm g}^{\ast} (\hbox{y, Z, p; r, G})\equiv \dot{\hbox{G}}^\ast(\hbox{y, Z, p; r, G})+\delta_{\rm g} \hbox{G}$ is positive.

• (D.5) The stocks G that solve $\dot{\hbox{G}}^\ast(\hbox{y, Z, p; r, G})=\hbox{J}_{\rm Gr}^{{\rm g}-1} (\hbox{y, Z, p; r, G})[{}_\theta\hbox{J}_{\rm r}^{\rm g} (\hbox{y, Z, p; r, G})-\hbox{G}],$ with G(0) > 0, has a unique globally stable steady state $\overline{\hbox{G}} (\hbox{Z, p, y; r}).$

Then, under conditions (C) and (D), duality between J^g(y, Z, p; r, G) and J(y, Z, p; r, G)  +  AC(I_g) can be established as in Eqs. 9 and 10. The derivative properties presented below then hold.

2.3 Derivative properties

• 1. With respect to I_g:

  $$ 0=\hbox{AC}_{{\rm I}_{\rm g}} +\hbox{J}_{\rm G}^{\rm g} (\hbox{p, Z, y; r, G}) $$

  or

  $$ -\hbox{J}_{\rm G}^{\rm g} (\hbox{p, Z, y; r, G})=\hbox{AC}_{{\rm I}_{\rm g} } > 0, $$

  This is positive given $\hbox{AC}_{{\rm I}_g} > 0.$

• 2. With respect to G:

  $$ {}_\theta\hbox{J}_{\rm G}^{\rm g} (\hbox{p, Z, y; r, G})=\hbox{J}_{\rm G} (\hbox{Z, y, p; G})+\hbox{r}+\hbox{J}_{\rm GG}^{\rm g} (\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast}-\delta_{\rm g} \hbox{J}_{\rm G}^{\rm g} (\hbox{p, Z, y; r, G}) $$

  or

  $$ \hbox{J}_{\rm G} (\hbox{Z, y, p; G})=(\theta \hbox{u}+\delta_{\rm g} )\hbox{J}_{\rm G}^{\rm g} (\hbox{p, Z, y; r, G})-\hbox{r}-\hbox{J}_{\rm GG}^{\rm g} (\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast} $$

  This expression is the firms’ willingness to pay for G (shadow price) when the firms are at the steady state. If the expression is negative (condition (D.2)(ii)), then public inputs reduce cost of production. When the government is also at the SS, that expression can be rewritten as

  $$ -\hbox{J}_{\rm G}^{\rm g} (\hbox{p, Z, y; r, G})=(\theta \hbox{u}+\delta_{\rm g})^{-1}(-\hbox{J}_{\rm G} (\hbox{Z,y,p;G})-\hbox{r}) $$

  which could be interpreted as a ‘social’ shadow price: the net social benefit (the firms’ shadow price of G minus the government’s cost of providing G) adjusted by the ‘social’ discount rate plus the depreciation rate of public inputs.

• 3. With respect to r:

  $$ {}_\theta\hbox{J}_{\rm r}^{\rm g} (\hbox{p, Z, y; r, G})=\hbox{G}+\hbox{J}_{\rm Gr}^{\rm g} (\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast} $$

  or

  $$ \dot{\hbox{G}}^{\ast} =\hbox{J}_{\rm Gr}^{\rm g-1} (\hbox{p, Z, y; r, G})[{}_\theta\hbox{J}_{\rm r}^{\rm g} (\hbox{p, Z, y; r, G})-\hbox{G}] $$

  which gives the optimal path of G.

• 4. With respect to Z:

  $$ {}_\theta\hbox{J}_{\rm z}^{\rm g} (\hbox{p, Z, y; r, G})=\hbox{J}_{\rm z} (\hbox{Z, y, p; G})+\hbox{J}_{\rm Gz}^{\rm g} (\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast} $$

  or

  $$ \hbox{J}_{\rm z} (\hbox{Z, y, p; G})={}_\theta\hbox{J}_{\rm z}^{\rm g} (\hbox{p, Z, y; r, G})-\hbox{J}_{\rm Gz}^{\rm g} (\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast} < 0 $$

  where the sign is given by condition B.2(ii): the value function of the firm is decreasing in Z.

• 5. With respect to y:

  $$ {}_\theta\hbox{J}_{\rm y}^{\rm g} (\hbox{p, Z, y; r, G})=\hbox{J}_{\rm y} (\hbox{Z,y,p;G})+\hbox{J}_{\rm Gy}^{{\rm g}{\prime}}(\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast} $$

  or

  $$ \hbox{J}_{\rm y} (\hbox{Z,y,p;G})={}_\theta\hbox{J}_{\rm y}^{\rm g} (\hbox{p, Z, y; r, G})-\hbox{J}_{\rm Gy}^{{\rm g}{\prime}}(\hbox{p, Z, y; r, G})\dot{\hbox{G}}^{\ast} > 0 $$

  where the sign is given by condition B.2(iii): the value function of the firm is increasing in y. Finally, at the SS level of G (or with no adjustment cost of G),

  $$ \hbox{J}_{\rm y} (\hbox{Z, y, p; G})={}_\theta\hbox{J}_{\rm y}^{\rm g} (\hbox{p, Z, y; r, G}) > 0 $$