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Homework answers / question archive / Producer Theory Practice Set 3 V

Producer Theory Practice Set 3

V. Bardis

1. Consider a price taking firm with production function q = F(L, K) = L^{1/4} K^{1/4}

a) Find the firm's conditional input demand functions given the prices of labour and capital are w and r.

b) Find the firm's cost function, fixed cost and variable cost.

c) Find the firm's marginal cost and average cost.

d) Characterize the firm's technology in terms of returns to scale by referring to its cost curves.

2. Consider a price taking firm with production function q = F(L, K) = √L + √K

a) Find the firm's conditional input demand functions given the prices of labour and capital are w and r.

b) Find the firm's cost function, fixed cost and variable cost.

c) Find the firm's marginal cost and average cost.

d) Characterize the firm's technology in terms of returns to scale by referring to its cost curves.

3. Consider a price taking firm with production function q = F(L, K) = √min(4K, 2 L)

a) Find the firm's conditional input demand functions given the prices of labour and capital are w and r.

b) Find the firm's cost function, fixed cost and variable cost.

c) Find the firm's marginal cost and average cost.

d) Characterize the firm's technology in terms of returns to scale by referring to its cost curves.

4. Consider a price taking firm with production function q = F(L, K) = (L-1)^{1/4} (K-4)^{1/4} 1 £ L and 4 £ K

Otherwise (Hint: Use your answer to question I.)

b) Find the firm's cost function, fixed cost and variable cost.

c) Find the firm's marginal cost and average cost.

d) Find the "efficient scale" output amount.

e) Characterize the firm's technology in terms of returns to scale by referring to its cost curves.

5. Consider a price taking firm with production function q = F(L, K)= √4L + 2K

b) Find the firm's cost function, fixed cost and variable cost.

c) Find the firm's marginal cost and average cost.

d) Find the "efficient scale" output amount.

e) Characterize the firm's technology in terms of returns to scale by referring to its cost curves.

6. Identify for which of the two-input firms speficied above can the Method of Lagrange be used to find the solution to the firm's cost minimization problem. Where applicable, use the method in each case to solve the problem.

7. For each of the two-input firms specified previously, use the cost function you found to formulate the firm's profit maximization problem such that output (q) is the choice variable of the firm. Assuming the firm is also a price taker in the output market, where the price of output is p, find the solution to the firm's profit maximization problem, that is, find the firm's supply function and maximum value of profit.

8. Define each of the following and explain how you calculate it:

a) 1) Own-price substitution effect for an input.

2) Own-price output effect of for input.

3) Own-price total effect of for input.

Depict the above on a graph where the input whose price increases appears on the horizontal axis and the other input appears on the vertical axis.

Depict the above on a graph where the input whose price increases appears on the horizontal axis and the price of the input appears on the vertical axis.

b) 1) Cross-price substitution effect for an input.

2) Cross-price output effect for an input.

3) Cross-price total effect for an input.

Depict the above on a graph where the input whose price increases appears on the vertical axis and the other input appears on the horizontal axis.

9. For each of the two-input firms specified previously, use your results up to this point to verify that labour's own-price total effect is equal to the sum of the own-price substitution effect and own-price output effect. To simpify, you may use the following price values, p=100, w = r = 1.

10. using a well-defined diagram argue each of the following:

a) When comparing two firms which employ the same K/L ratio, the one with greater ease of substitution will experience a smaller increase in cost after an increase in the price of an input.

b) When comparing two firms with the same ease of substitution, the one which employs the larger (smaller) K/L ratio will experience a larger increase in cost after an increase in the price of an capital (labour).