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# Lagrangian Economics Problems

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1a. Suppose a production function is given by F(K,L)=KL^2, the price of capital is \$10 and the price of labor is \$15. What combination of labor and capital minimizes the cost of producing given output? Explain.

1b. The production function for a product is given by q=100KL. If the price of capital is \$120 per day and the price of labor \$30 per day, what is the minimum cost of producing 1000 units of output?

2. Suppose that a semiconductor plant's production function is q=5KL, where q is its output rate, L is the amount of Labor it uses per period of time, and K is the amount of capital it uses per period of time. Suppose that the price of labor is \$1 a unit and the price of capital is \$2 per unit.

a. The firm's vice president for manufacturing hires you to figure out what combination of inputs the plant should use to produce 20 units of output per period. What advice would you give?

b. The same as in (a), but the plant needs to produce 40 units of output, explain.

c. compare your answers to part (a) and (b) and explain the scale effect.

d. The same as in (a) but the price of labor has risen to \$2 a unit, explain.

3. An econimist estimated that the cost function of a single-product firm is:
C(Q)= 50 + 25Q +30Q^2 +5Q^3

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 of producing 10 units of ouput

4. If q is the number of cars washed per hour amd L is the number of employers, A study of an auto laundry found the following short run relationship:
q= .8 +4.5L -.3L^2

a. Find the marginal product of L.
b. Do there appear to be diminishing marginal products?
c. If PL=\$4.5 (wage per hour), what is the marginal cost (MC) of this wash?
d. Find the optimal output of this firm given the price per car wash is \$5.

5. Suppose the production function for high quality brandy is give by:
q=K^1/2 L^1/2, where q is the output of brandy per week and L is labor hours per week. In the short-run, K is fixed at 100, so the short-run production function is q=10L^1/2

a. If capital rents for \$10 each and wages are \$5 per hour, show that short-run total costs are STC=1000+.05q^2
b. How much will the firm produce at a price of brandy of \$20 per bottle? How many labor hours will be hired per week? What profit will the firm make?