Chapter 4 Henderson and Quandt Microeconomics, Questions 4.4-4.7_ Solutions

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4-4 Assume that an entrepreneur's short-run total cost function is C=q³-10q²+17q+66.

Determine the output level at which he maximizes profit if p = 5. Compute the output elasticity of

cost at this output.

The profit function for this entrepreneur is given by π = pq - C, where p is the price and C is the total cost. Substituting the given values, we get:

π = 5q - (q³-10q²+17q+66) = -q³+10q²-12q-66

To maximize profit, we take the first derivative of the profit function with respect to q and set it equal to zero:

dπ/dq = -3q² + 20q - 12 = 0

Solving this equation, we get:

q = 2 or q = 2/3

To check which of these values maximizes profit, we take the second derivative of the profit function with respect to q:

d²π/dq² = -6q + 20

At q = 2, we have d²π/dq² = 8, which is positive, indicating that this is a local minimum. At q = 2/3, we have d²π/dq² = 19.33, which is also positive, indicating that this is a local maximum. Therefore, the entrepreneur should produce q = 2/3 to maximize profit.

The output elasticity of cost is given by:

ε = (∂C/∂q)(q/C) = (3q² - 20q + 17)(q/(q³-10q²+17q+66))

At q = 2/3, we have:

ε = (3(4/9) - 20(2/3) + 17)(2/3)/[(2/3)³-10(2/3)²+17(2/3)+66] = -0.446

Therefore, the output elasticity of cost at the profit-maximizing level of output is -0.446.

4-5 A family of short-run total cost curves is generated by C=0.04q³ -0.9q²+(10-In k)q+8k²

where k> 1 denotes plant size. Determine the firm's long-run total cost curve.

To determine the firm's long-run total cost curve, we need to find the minimum cost of producing each level of output for all possible plant sizes. This involves minimizing the short-run total cost function with respect to q for each value of k.

Taking the derivative of the short-run total cost function with respect to q, we get:

dC/dq = 0.12q² - 1.8q + (10 - In k)

Setting this expression equal to zero to find the minimum, we get:

0.12q² - 1.8q + (10 - In k) = 0

Using the quadratic formula, we can solve for q:

q = [1.8 ± sqrt(1.8² - 4(0.12)(10 - In k))]/(2(0.12)) = [1.8 ± sqrt(3.24 - 0.48 In k)]/0.24

To find the minimum, we take the positive root:

q = [1.8 + sqrt(3.24 - 0.48 In k)]/0.24

Substituting this expression for q back into the short-run total cost function, we get the long-run total cost function:

C_LR = 0.04[1.8 + sqrt(3.24 - 0.48 In k)]³ - 0.9[1.8 + sqrt(3.24 - 0.48 In k)]² + (10 - In k)[1.8 + sqrt(3.24 - 0.48 In k)] + 8k²

This is the firm's long-run total cost curve, which gives the minimum cost of producing each level of output for all possible plant sizes.

4-6 An entrepreneur uses one input to produce two outputs subject to the production relation

x= A(+) where , β>1. He buys the input and sells the outputs at fixed prices. Express

his profit-maximizing outputs as functions of the prices. Prove that his production relation is

strictly convex for >0.

The entrepreneur's profit function can be expressed as:

π = p_1 q_1 + p_2 q_2 - w x

where p_1 and p_2 are the prices of the two outputs, and w is the price of the input. Substituting the production relation x= A(q_1^α+q_2^β), we get:

π = p_1 q_1 + p_2 q_2 - w A(q_1^α+q_2^β)

To maximize profits, we take the partial derivatives of π with respect to q_1 and q_2, and set them equal to zero:

∂π/∂q_1 = p_1 - w A α q_1^(α-1) = 0

∂π/∂q_2 = p_2 - w A β q_2^(β-1) = 0

Solving for q_1 and q_2, we get:

q_1 = [p_1/(w A α)]^(1/(α-1))

q_2 = [p_2/(w A β)]^(1/(β-1))

These expressions give the profit-maximizing outputs as functions of the prices.

To show that the production relation is strictly convex, we need to show that the Hessian matrix of the production function is positive definite. The Hessian matrix is:

H(q) = [ A α (α-1) q_1^(α-2) 0 ]

[ 0 A β (β-1) q_2^(β-2) ]

This matrix is diagonal, and the diagonal elements are all positive for q_1, q_2 > 0 and α, β > 1. Therefore, the Hessian matrix is positive definite, and the production function is strictly convex for q_1, q_2 > 0.

4-7 An entrepreneur produces one output with two inputs using the production function q =

A. He buys the inputs and sells the outputs at fixed prices. He is subject to a quota which

allows him to purchase no more than units of X₁. He would have purchased more in the

absence of the quota. 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 entrepreneur's profit function is:

π = pq - w₁x₁ - w₂x₂

where p is the price of the output, w₁ and w₂ are the prices of the two inputs, and x₁^0 is the quota on the purchase of the first input.

The entrepreneur's production function is q = Ax_1^α x_2^(1-α). We can rewrite this as:

q/x_1^α = Ax_2^(1-α) (dividing both sides by x_1^α)

The left-hand side of this expression is the marginal product of x₁, and the right-hand side is the value of the marginal product of x₂. Similarly, we can rewrite the production function as:

q/x_2^(1-α) = Ax_1^α (dividing both sides by x_2^(1-α))

The left-hand side of this expression is the marginal product of x₂, and the right-hand side is the value of the marginal product of x₁.

To maximize profits subject to the quota, the entrepreneur needs to choose x₁ and x₂ to maximize π, subject to the constraint that x₁ ≤ x₁^0. Using the production function, we can express x₂ as a function of x₁:

x₂ = (q/(Ax_1^α))^(1/(1-α))

Substituting this expression into the profit function, we get:

π = pq - w₁x₁ - w₂(q/(Ax_1^α))^(1/(1-α))

Taking the derivative of π with respect to x₁ and setting it equal to zero, we get the first-order condition:

pαAx_1^(α-1) = w₂/(1-α)(q/(Ax_1^α))^(α/(1-α))

Solving for x₁, we get the optimal input choice:

x₁* = [w₂/(pA(1-α))]^(1/α)q^(1/(α-1))

To find the optimal value of x₂, we substitute x₁* into the expression for x₂:

x₂* = (q/(A[x₁*]^α))^(1/(1-α))

The optimal relation between the value of the marginal product of each input and its price is given by the first-order condition:

pαAx_1^(α-1) = w₂/(1-α)(q/(Ax_1^α))^(α/(1-α))

This equation shows that the ratio of the value of the marginal product of x₁ to its price is equal to the ratio of the value of the marginal product of x₂ to its price. In other words, the entrepreneur should allocate inputs such that the marginal value of each input is proportional to its price.

The optimal relation between the RTS and the input price ratio can be found by taking the derivative of the optimal input choice with respect to ln(w₂/w₁):

∂ln(x₁*)/∂ln(w₂/w₁) = 1 - α

This equation shows that the RTS is equal to 1 minus the elasticity of substitution between the two inputs. As the elasticity of substitution between the two inputs increases, the RTS decreases.

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