please solve Q 1,2,7, and 13 with details and formulas and how it is solved
1. A firm can manufacture a product according to the production function (LO1, LO2, L05, LO6) Q = F(K, L) = K3/4 L 1/4 a. Calculate the average product of labor, APL, when the level of capital is fixed at 81 units and the firm uses 16 units of labor. How does the average product of labor change when the firm uses 256 units of labor? b. Find an expression for the marginal product of labor, MPL, when the amount of capi- tal is fixed at 81 units. Then, illustrate that the marginal product of labor depends on the amount of labor hired by calculating the marginal product of labor for 16 and 81 units of labor. 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 labor at $50 per unit of labor, how many units of labor 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 pro- duces output using capital (which it rents at $25 per hour) and labor (which is paid a wage of $30 per hour under a contract for 20 hours of labor services). Complete the following table and use that information to answer these questions. (LO], LO2, L05, LO6) a. Identify the fixed and variable inputs. b. What are the firm's fixed costs? c. What is the variable cost of producing 475 units of output? d. How many units of the variable input should be used to maximize profits? e. What are the maximum profits this firm can earn? f. Over what range of the variable input usage do increasing marginal returns exist? g. Over what range of the variable input usage do decreasing marginal returns exist? h. Over what range of input usage do negative marginal returns exist?7. A multiproduct firm's cost function was recently estimated as (L07) C(@1, 22) = 90 - 0.5 2102 + 0.4 07 + 0.3 02 a. Are there economies of scope in producing 10 units of product 1 and 10 units of product 2? b. Are there cost complementarities in producing products 1 and 2? c. Suppose the division selling product 2 is floundering and another company has made an offer to buy the exclusive rights to produce product 2. How would the sale of the rights to produce product 2 change the firm's marginal cost of producing product 1?13. You are a manager for Herman Miller, a major manufacturer of office furniture. You recently hired an economist to work with engineering and operations experts to esti- mate the production function for a particular line of office chairs. The report from these experts indicates that the relevant production function is Q = 2(K) 1/2 (1) 1/2 where K represents capital equipment and L is labor. Your company has already spent a total of $8,000 on the 9 units of capital equipment it owns. Due to current economic conditions, the company does not have the flexibility needed to acquire additional equipment. If workers at the firm are paid a competitive wage of $120 per day and chairs can be sold for $400 each, what is your profit-maximizing level of output and labor usage? What is your maximum profit? (LO1, LO2)
