Question

Consider a firm which faces the following production function, Q = F(K,L) --- (1) where Q is the firm's output, K is the firm's capital input and L is the firm's labor input. This production function is assumed to have the following properties, FK > 0, FL > 0, FKK < 0, FLL < 0, FKL = FLK < 0, and FKK · FLL > FKL^2 --- (2) Given (1), the firm's total revenue R is given by R = PQ = PF(K, L) --- (3) where P is the price of the firm's output (assume that P > 0), and the firm's cost C is given by C = rK + wL --- (4) where r is the interest rate, w is the wage and C is the total cost. Given (3) and (4), the firm's profit function can be written as follows: ? = PF(K, L) - rK - wL --- (4) where ? is the firm's profit. If the firm operates under perfect competition, the prices of output and inputs (P, r and w) are exogenous, and this implies that the firm chooses K and L which maximize ?. Given this firm's profit-maximizing problem, answer the following questions: a) Obtain the FOC, ?K = 0, which is expressed in terms of r (i.e. r is a function of other variables). b) Obtain the FOC, ?L = 0, which is expressed in terms of w (i.e. w is a function of other variables). c) Obtain the SOC for maximum. From your answers in a) and b), there exist i) a value of K and ii) a value of L which optimize ?; let them be denoted by K* and L, respectively. Since the production function is stated in the general form, however, K and L* cannot be derived explicitly. To get around this problem, let's assume a specific, Cobb-Douglas production function as follows: Q = K^1/3L^2/3 --- (5) Then, the profit function becomes ? = PK^1/3L^2/3 - rK - wL --- (6)

          Consider a firm which faces the following production function,

Q = F(K,L) --- (1)

where Q is the firm's output, K is the firm's capital input and L is the firm's labor input.

This production function is assumed to have the following properties,

FK > 0, FL > 0, FKK < 0, FLL < 0, FKL = FLK < 0, and FKK · FLL > FKL^2 --- (2)

Given (1), the firm's total revenue R is given by

R = PQ = PF(K, L) --- (3)

where P is the price of the firm's output (assume that P > 0), and the firm's cost C is given by

C = rK + wL --- (4)

where r is the interest rate, w is the wage and C is the total cost.

Given (3) and (4), the firm's profit function can be written as follows:

? = PF(K, L) - rK - wL --- (4)

where ? is the firm's profit. If the firm operates under perfect competition, the prices of output and inputs (P, r and w) are exogenous, and this implies that the firm chooses K and L which maximize ?.

Given this firm's profit-maximizing problem, answer the following questions:

a) Obtain the FOC, ?K = 0, which is expressed in terms of r (i.e. r is a function of other variables).

b) Obtain the FOC, ?L = 0, which is expressed in terms of w (i.e. w is a function of other variables).

c) Obtain the SOC for maximum.

From your answers in a) and b), there exist i) a value of K and ii) a value of L which optimize ?; let them be denoted by K* and L*, respectively.

Since the production function is stated in the general form, however, K* and L* cannot be derived explicitly. To get around this problem, let's assume a specific, Cobb-Douglas production function as follows:

Q = K^1/3L^2/3 --- (5)

Then, the profit function becomes

? = PK^1/3L^2/3 - rK - wL --- (6)

Consider a firm which faces the following production function,

Q = F(K,L) — (1)

where Q is the firm's output, K is the firm's capital input and L is the firm's labor input.

This production function is assumed to have the following properties,

FK > 0, FL > 0, FKK < 0, FLL < 0, FKL = FLK < 0, and FKK · FLL > FKL^2 — (2)

Given (1), the firm's total revenue R is given by

R = PQ = PF(K, L) — (3)

where P is the price of the firm's output (assume that P > 0), and the firm's cost C is given by

C = rK + wL — (4)

where r is the interest rate, w is the wage and C is the total cost.

Given (3) and (4), the firm's profit function can be written as follows:

? = PF(K, L) - rK - wL — (4)

where ? is the firm's profit. If the firm operates under perfect competition, the prices of output and inputs (P, r and w) are exogenous, and this implies that the firm chooses K and L which maximize ?.

Given this firm's profit-maximizing problem, answer the following questions:

a) Obtain the FOC, ?K = 0, which is expressed in terms of r (i.e. r is a function of other variables).

b) Obtain the FOC, ?L = 0, which is expressed in terms of w (i.e. w is a function of other variables).

c) Obtain the SOC for maximum.

From your answers in a) and b), there exist i) a value of K and ii) a value of L which optimize ?; let them be denoted by K* and L*, respectively.

Since the production function is stated in the general form, however, K* and L* cannot be derived explicitly. To get around this problem, let's assume a specific, Cobb-Douglas production function as follows:

Q = K^1/3L^2/3 — (5)

Then, the profit function becomes

? = PK^1/3L^2/3 - rK - wL — (6)

Added by Stephen R.

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