# Answer to Question #24091 in Economics of Enterprise for Annette

Question #24091

Pat operates a board game manufacturing company at which he produces and sells board games. Heproduces board games according to the following production function:

q=3K2/3 +3L2/3

.

where K is the number of machines to produce the games (his capital) and L is the number of

workers (his labor). The rental price of machines is 0 and the wage rate of workers is 1. Pat wishes

to minimize the cost necessary to produce his output.

a. Write down the Pat’s optimization problem.

b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

c. Write down the marginal product for each input.

d. Are there diminishing marginal returns to capital? Labor?

e. Find the marginal rate of transformation (MRTS).

In the short run, Pat’s capital is fixed at -, = 1 (i.e. he only has one machine).

f. Find the short-run input demand function(s).

g. Find the short-run total cost function.

In the long run, both capital and labor are variable (i.e. Pat can adjust the number of machines a

q=3K2/3 +3L2/3

.

where K is the number of machines to produce the games (his capital) and L is the number of

workers (his labor). The rental price of machines is 0 and the wage rate of workers is 1. Pat wishes

to minimize the cost necessary to produce his output.

a. Write down the Pat’s optimization problem.

b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

c. Write down the marginal product for each input.

d. Are there diminishing marginal returns to capital? Labor?

e. Find the marginal rate of transformation (MRTS).

In the short run, Pat’s capital is fixed at -, = 1 (i.e. he only has one machine).

f. Find the short-run input demand function(s).

g. Find the short-run total cost function.

In the long run, both capital and labor are variable (i.e. Pat can adjust the number of machines a

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