A firm has a production function for output Y given by Y = F(K, AL, Q) = Kα(AL) βQγ , K is capital input, L is labor input, and Q is use of land. Capital can be hired at the rental rate R per unit, labor can be hired at the wage W per unit, and land may be rented at price P per unit.
Assume that F(K, AL, Q) = Kα(AL) βQγ , as above, but also set γ = 1 − α − β. The firm’s profit maximization problem, assuming perfect competition in input and output markets, is given by max K,L,Q Y − W L − RK − P Q.
i. Write the first-order conditions for profit maximization characterizing the optimal use of capital, labor, and land inputs by the firm. ii. What is the elasticity of the optimal choice of capital per unit of land K Q to the price ratio P R ? What is the elasticity of the optimal choice of labor per unit of land L Q to the price ratio P W ? iii. Given optimal input choices, does the land share P Q Y vary with the land price P? If so, in which direction does it vary when P increases? iv. Demonstrate whether, given optimal input choices, the firm earns positive profits, negative profits, or zero profits. You may wish to use the following result. Proposition (Euler’s Theorem): For any differentiable constant returns to scale production function with N inputs F(X1, ..., XN ), the following relationship holds F(X1, ..., XN ) = X N i=1 ∂F ∂Xi Xi.
Suppose that an economist observes data generated by the true production function F(K, AL, Q) = Kα(AL) βQγ with γ = 1 − α − β. This function has capital K, labor L, and land Q inputs, and inputs are chosen in a profit maximizing fashion. However, the economist is lazy and does not like working with functions of three variables. Instead, the lazy economist assumes that the production function is given by a function of capital and labor only Y = Kα˜ (AL) 1−α˜ . In the data, the economist observes that the labor share is equal to 0.6. i. What value of the capital elasticity α˜ would the lazy economist infer? ii. Demonstrate whether the inferred value of α˜ from part (3)(c)(i) is greater than, less than, or equal to the true capital elasticity α. Assume γ > 0 and recall γ = 1−α−β. (d) Assume that the firm production function F is given by F(K, AL, Q) = (αKρ + β(AL) ρ + (1 − α − β)Q ρ ) 1 ρ with α > 0, β > 0, α + β < 1, and ρ < 1.
i. Write the first-order conditions for profit maximization characterizing the optimal use of capital, labor, and land inputs by the firm.
ii. What is the elasticity of the optimal choice of capital per unit of land K Q to the price ratio P R ? What is the elasticity of the optimal choice of labor per unit of land L Q to the price ratio P W ?
iii. The production function in this part is an example of a constant elasticity of substitution (CES) production function, where the value σ = 1 1−ρ is commonly referred to as the elasticity of substitution. Assume σ > 1. Given optimal input choices, does the land share P Q Y vary with the land price P? If so, in which direction does it vary when P increases?
iv. Assume σ < 1. Given optimal input choices, does the land share P Q Y vary with the land price P? If so, in which direction does it vary when P increases? v. Compare your answers from (3)(d)(iii) and (3)(d)(iv) with your answer to (3)(b)(iii) above. Note that a Cobb-Douglas production function is a special case of the CES production function with σ = 1.