Chapter 5 Henderson and Quandt Microeconomics, Questions 5.1-5.3_ Solutions

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5-1 Each of the following production functions is homogeneous of degree one. In each case,

derive the marginal products for X, and X₂ and demonstrate that they are homogeneous of

degree zero:

(a) a = (ax₁x₂-- )/( + )

(b) q = A + bx₁ + cx₂

(a) To derive the marginal product of X₁, we take the partial derivative of the production function with respect to X₁ holding X₂ constant:

∂q/∂X₁ = aX₂ - 2bX₁/(aX₁ + βX₂)^2

To derive the marginal product of X₂, we take the partial derivative of the production function with respect to X₂ holding X₁ constant:

∂q/∂X₂ = aX₁ - 2cX₂/(aX₁ + βX₂)^2

To demonstrate that the marginal products are homogeneous of degree zero, we need to show that multiplying each input by a constant λ results in the marginal products being multiplied by the same constant λ. Let's consider the marginal product of X₁:

∂q/∂(λX₁) = aX₂ - 2b(λX₁)/(a(λX₁) + βX₂)^2

= λ(-2bX₁/(aX₁ + βX₂)^2) + aX₂/(aX₁ + βX₂)^2

= λ(∂q/∂X₁)

Similarly, for X₂:

∂q/∂(λX₂) = aX₁ - 2c(λX₂)/(aX₁ + β(λX₂))^2

= λ(-2cX₂/(aX₁ + βX₂)^2) + aX₁/(aX₁ + βX₂)^2

= λ(∂q/∂X₂)

Thus, both marginal products are homogeneous of degree zero.

(b) To derive the marginal product of X₁, we take the partial derivative of the production function with respect to X₁ holding X₂ constant:

∂q/∂X₁ = Aax₁^(a-1)x₂^(1-a) + b

To derive the marginal product of X₂, we take the partial derivative of the production function with respect to X₂ holding X₁ constant:

∂q/∂X₂ = A(1-a)x₁^a x₂^(-a) + c

To demonstrate that the marginal products are homogeneous of degree zero, we need to show that multiplying each input by a constant λ results in the marginal products being multiplied by the same constant λ. Let's consider the marginal product of X₁:

∂q/∂(λX₁) = Aa(λX₁)^(a-1)x₂^(1-a)

= λ(a/X₁)(AaX₁^(a-1)x₂^(1-a))

= λ(∂q/∂X₁)

Similarly, for X₂:

∂q/∂(λX₂) = A(1-a)x₁^a (λX₂)^(-a)

= λ((1-a)/X₂)(A(1-a)x₁^a x₂^(-a))

= λ(∂q/∂X₂)

Thus, both marginal products are homogeneous of degree zero.

5-2 An entrepreneur uses two distinct production processes to produce two distinct goods, Q₁

and . The production function for each good is CES, and the entrepreneur obeys the

equilibrium condition for each. Assume that has a higher elasticity of substitution and a lower

value for the parameter a than Q₂ [see (5-12)]. Determine the input price ratio at which the input use ratio would be the same for both goods. Which good would have the higher input use ratio if the input price ratio were lower? Which would have the higher use ratio if the price ratio were higher?

Assuming that the entrepreneur is maximizing profit subject to the production functions and input prices, the equilibrium condition for each good implies that the marginal rate of substitution between inputs is equal to the input price ratio for each good. Let p₁ and p₂ be the prices of inputs 1 and 2, respectively, and let x₁ and x₂ be the amounts of inputs used to produce each good. Then, we have:

For Q₁:

MRS₁₂ = (p₁/p₂) = a[αx₁^(-ρ) + (1-α)x₂^(-ρ)]^(-1/ρ) * (α/p₁)x₁^(-ρ-1)

For Q₂:

MRS₂₁ = (p₂/p₁) = a[αx₁^(-ρ) + (1-α)x₂^(-ρ)]^(-1/ρ) * [(1-α)/p₂]x₂^(-ρ-1)

where ρ is the elasticity of substitution and a is a constant parameter.

To find the input price ratio at which the input use ratio is the same for both goods, we set x₁/x₂ for each good equal to each other and solve for p₁/p₂:

(α/p₁)x₁^(-ρ-1) = [(1-α)/p₂]x₂^(-ρ-1)

Multiplying both sides by p₁/p₂ and rearranging terms, we get:

(p₁/p₂)^2 = (α(1-α)x₂^(-ρ-1))/(x₁^(-ρ-1))

Simplifying further, we have:

p₁/p₂ = [α(1-α)x₂/x₁]^(-1/2ρ)

This shows that the input price ratio at which the input use ratio is the same for both goods depends on the relative values of α and ρ, as well as the input use ratio x₁/x₂ and the parameters of the production functions.

If the input price ratio were lower than the value determined by the above equation, the good with the higher elasticity of substitution (Q₁) would have a higher input use ratio, since it is more responsive to changes in input prices. If the price ratio were higher than the value determined by the above equation, the good with the lower elasticity of substitution (Q₂) would have a higher input use ratio, since it is less responsive to changes in input prices.

5-3 An entrepreneur has the production function q = A. She buys inputs and sells the

output at fixed prices, but is subject to a quota that allows her to purchase no more than

units of X₁. She would have purchased more in the absence of the quota. Use the Kuhn-

Tucker analysis to determine the entrepreneur's conditions for profit maximization. What is the

optimal relation between the value of the marginal product of each input and its price? What is

the optimal relation between the RTS and the input price ratio?

The profit maximization problem subject to the quota can be formulated as follows:

Maximize π = pq - w₁x₁ - w₂x₂

subject to q = Ax_1^a x_2^(1-a)

and x₁ ≤ x_1^0

where p is the output price, w₁ and w₂ are the input prices, and x_1^0 is the quota on input 1.

The Lagrangian for this problem is:

L = pq - w₁x₁ - w₂x₂ + λ[Ax_1^a x_2^(1-a) - q] + μ(x₁ - x_1^0)

where λ and μ are the Lagrange multipliers for the production constraint and the quota constraint, respectively.

The first-order conditions are:

∂L/∂q = p - λ = 0

∂L/∂x₁ = -w₁ + λaAx_1^(a-1) x_2^(1-a) + μ = 0

∂L/∂x₂ = -w₂ + λ(1-a)Ax_1^a x_2^(-a) = 0

Ax_1^a x_2^(1-a) = q

x₁ - x_1^0 ≤ 0

μ(x₁ - x_1^0) = 0

The Kuhn-Tucker complementary slackness conditions are:

μ(x₁ - x_1^0) = 0

λ ≥ 0

μ ≥ 0

Using these conditions, we can solve for the optimal input and output levels:

x₁ = min{x_1^0, (w₁/λaA)^(1/(a-1))x_2^(a-1)}

x₂ = (w₂/λ(1-a)A)^(1/a)

q = A[x₁^a x₂^(1-a)]

The optimal values of λ and μ can be found by substituting these expressions for x₁, x₂, and q into the first-order conditions.

The optimal relation between the value of the marginal product of each input and its price is:

VMPL₁/w₁ = λaAx_1^(a-1) x_2^(1-a)

VMPL₂/w₂ = λ(1-a)Ax_1^a x_2^(-a)

The optimal relation between the RTS and the input price ratio is:

RTS = -(∂lnx₂/∂lnx₁) = -(1-a)/(a(∂lnx₁/∂lnx₂)) = w₁/(λaA) = w₂/(λ(1-a)A).

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