Explaining Growth in Dutch Agriculture: Prices, Public R&D, and Technological Change

Using relationships among the derivatives and Hessian matrices of the restricted and unrestricted
profit functions (discussed by Lau and others), long-run output and variable input demand function
and the associated elasticities can also be obtained. Using these elasticities, the long-run effects of
research as well as pricing and other government policies can be assessed. Empirical derivation of the
long-run results from an estimated short-run variable profit function is facilitated when specifying it to
be a normalized quadratic function. The advantage of this functional form is that closed-form
expressions can be derived for the optimal levels for the quasi-fixed factors. The quadratic
specification of the short-run profit function is:

∏ = во + Σα>i,P_l + ΣjμjZj+ Σ ад

+ 2 ΣΣ_ιrW_r + 2 ΣjΣ√jjZjZj∙ + 2 Σ k Σ kXkk WW (2)

+ Σ₁Σj_t4.P^,Z_i+ ΣΣ kΦ*PW + ΣjΣ MW

Taking the partial derivative with respect to output/ variable input prices (Hotelling’s Lemma) of
this profit function, the short-run output supply and variable input demand functions are then:

± Q =^∂∏ = ^αi⁺ Σi^λ^p' + Σ^ηHZj⁺ Σ_k^φikWk ⁱ =¹, . . . .^, m^,

(3)

Where the sign modifier of Qi is positive for an output supply equation and negative for a variable
input demand equation. Equations (1) are linear and can be solved readily for the Z^*j. When capital
(j=K) and family labor (j=L) are assumed to be quasi-fixed, their optimal levels are:

K^*

L^*

γ_LL

-γ_LK

-^γKL RK ^μK Σ i^ηiKPi Σ k^θKkWk

Y-K J|_ RL — μL - Z^P — Σ θ W

(4)

where ∆ = γ_KKγ_LL — (γ_KL ) ² . Defining the partial derivatives of the optimal capital and labor
equations as:

^δKP_I	_д K * ≡ ∂ P_i	=—	1 -TLL^fLLKl- — YKL^iL ) ∆	i =1,..	, m
^δKWk	≡ ∂ K *	—	∆ (Y∣L^θKk — ^γKL^θLk )	k=1,..	, S
^δKWk	≡∂ Wk		∆ (Y∣L^θKk — ^γKL^θLk )	k=1,..	, S
^δLPi	∂L^*= = д Pi	1 ∆	(Y_KKη_iL — Y_LKη_iK )	i =1,.	, m
^δLWk	_ д L		ɪ (^γKK^θLk — ^γLK^θKk ) ^,	k=1,	, S
^δLWk	= ∂ W *		ɪ (^γKK^θLk — ^γLK^θKk ) ^,	k=1,	, S

The long-run output supply and variable input demand functions can be expressed as:

±Q* = α+Σ λ+ηδ + η₇δ )p,+(μδ. + μδ )—δ R — δ. R
i i _i_' ii' i'k kp_i i'l lp_i i' k kp_i l lp_i kp_i k lp_i l

(5)

Σ (φ + θ_tδ + δδ,_t W

_k ik kk kw_k lk lk_k k

Note that equation (5) accounts for the additional impacts of prices on output and variable inputs
(price slopes are adjusted as compared to (3)) through capital and family labor adjustment. Moreover,

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Explaining Growth in Dutch Agriculture: Prices, Public R&D, and Technological Change

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