1. A firm's analysts estimate that the firm can manufacture a product according to the production function :

Q = F (K ,L) = K^3/4 L^1/4

a. Calculate the average product of labour, APL, when the level of capital is fixed at 81 units and the firm uses 16 units of labour. How does the average product of labour change when the firm uses 256 units of labour?

b. Find an expression for the marginal product of labour, MPL, when the amount of capital is fixed at 81 units. Then, illustrate that the marginal product of labour depends on the amount of labour hired by calculating the marginal product of labour for 16 and 81 units of labour.

c. Suppose capital is fixed at 81 units. If the firm can sell its output at a price of $200 per unit of output and can hire labour at $50 per unit of labour, how many units of labour should the firm hire in order to maximize profits?

2. A firm's product sells for $4 per unit in a highly competitive market. The firm produces output using capital (which it rents at $25 per hour) and labour (which is paid a wage of $30 per hour under a contract for 20 hours of labour services). Complete the following table and use that information to answer these questions.

a. Identify the fixed and variable inputs. b. What are the firm's fixed costs? b. What is the variable cost of producing 475 units of output? c. How many units of the variable input should be used to maximize profits? d. What are the maximum profits this firm can earn? e. Over what range of the variable input usage do increasing marginal returns exist? f. Over what range of the variable input usage do decreasing marginal returns exist? g. Over what range of input usage do negative marginal returns exist?

Table: K L Q. MP_k AP_k AP_L VMPK 0 20 0 1 20 50 2 20 150 3 20 300 4 20 400 5 20 450 6 20 475 7 20 475 8 20 450 9 20 400 10 20 300 11 20 150

3. Explain the difference between the law of diminishing marginal returns and the law of diminishing marginal rate of technical substitution.

4. An economist estimated that the cost function of a single-product firm is:

C (Q) = 90 + 35Q + 25Q² + 10Q³

Based on this information, determine: a. The fixed cost of producing 10 units of output. b. The variable cost of producing 10 units of output. c. The total cost of producing 10 units of output. d. The average fixed cost of producing 10 units of output. e. The average variable cost of producing 10 units of output. f. The average total cost of producing 10 units of output. g. The marginal cost when Q=10.

6. A manager hires labour and rents capital equipment in a very competitive market. Currently, the wage rate is $9 per hour and capital is rented at $10 per hour. If the marginal product of labour is 50 units of output per hour and the marginal product of capital are 60 units of output per hour is the firm using the cost-minimizing combination of labour and capital? If not, should the firm increase or decrease the amount of capital used in its production process?

7. A firm produces output according to a production function: Q= F(K,L) = min{4K,4L}

a. How much output is produced when K=2 and L=3? b. If the wage rate is $60 per hour and the rental rate on capital is $40 per hour, what is the cost-minimizing input mix for producing 8 units of output? c. How does your answer to part b change if the wage rate decreases to $40 per hour but the rental rate on capital remains at $40 per hour?