A company is producing a product using capital with the production function f(K) satisfying f'(K) > 0 and ƒ"(K) < 0 for all K. The profit function is given by π = pf (K) - K where p is the price of the product. a) Find the expression for profit-maximizing K*. How does the K* change when p increases? Briefly, up to 50 words, discuss the economic intuition.

A company is producing a product using capital with the production function f(K) satisfying f'(K) > 0 and ƒ"(K) < 0 for all K. The profit function is given by π = pf (K) - K where p is the price of the product. a) Find the expression for profit-maximizing K. How does the K change when p increases? Briefly, up to 50 words, discuss the economic intuition.

Microeconomics

13th Edition

ISBN:9781337617406

Author:Roger A. Arnold

Publisher:Roger A. Arnold

Chapter8: Production And Costs

Section: Chapter Questions

Problem 11QP

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A company is producing a product using capital with the production function ƒ(K) satisfying f'(K) > 0 and ƒ"(K) < 0 for all K . The profit function is given by π = pf (K) - ) – K where p is the price of the product. a) Find the expression for profit-maximizing K*. How does the K* change when p increases? Briefly, up to 50 words, discuss the economic intuition. b) Assume that f(K) = √K. Suppose that there is an uncertainty over the price of the product. It is known that p is random with mean μ and variance o². A company seeks to maximize the expected profit. Set up the maximization problem. Find the expression for the optimal solution K*. How does the K* change when μ increases? Briefly, up to 50 words, discuss the economic intuition.
1. The production function for the personal computers of DISK, Inc., is given by q = 10K0.5L0.5,where q is the number of computers produced per day, K is hours of machine time, and L ishours of labor input. DISK’s competitor, FLOPPY, Inc., is using the production function q = 10K0.6L0.4. a. If both companies use the same amounts of capital and labor, which will generatemore output?b. Assume that capital is limited to 9 machine hours, but labor is unlimited in supply. Inwhich company is the marginal product of labor greater? Explain.
Question # 1: Majeda wrote the following answer on her microeconomics examination: “Virtually every production function exhibit diminishing returns to scale because my professor said that all inputs have diminishing marginal productivities. So, when all inputs are doubled, output must be less than double." How would you grade Majeda's answer? Question # 2: Suppose a firm had a production function with linear isoquants, implying that its two inputs were perfect substitutes for each other. What would determine the firm's expansion path in this case? For the opposite case of a fixed-portions production function, what would the firm's expansion path be? Question # 3: Suppose that a firm's production function is q = 30√L. In the short run, where there are fixed costs of $2,200, and labor is the variable input whose cost is $6,300 per units. What is the total cost of producing 20 units of output?
Explain why a profit function π(K,L) doesn't exist such that: The marginal benefit with respect to capital is 2KL
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Suppose the production function is given by Q = 5K+2L. What is the marginal product of capital when 5 units of capital and 10 units of labor are employed?
Answer each of the following questions as either true or false. For a statement to be “true,” it must always be true. If there is at least one case where the statement is not true (or if you need more information to be sure), answer “false.” You must justify each answer with an appropriate explanation or counterexample (which may include a relevant diagram). A firm can make widgets using capital and labor according to the production function f(K,L) = 100L + 0.5K. Denote the wage w and the rental rate on capital r. If r is sufficiently high, the firm will not hire any capital, no matter how many widgets it wants to produce.
A firm produces output according to the production function Q = F(K, L) = 4K + 8L.a. How much output is produced when K = 2 and L = 3?
For a firm to maximize profit, it must minimize the cost of producing whatever quantity it produces. Use the isocost and isoquant tools to present a firm that is choosing the optimal levels of labor and capital (i.e., tools) to produce a certain quantity and a certain cost. Then, show in your diagram how this firm would respond if it were to expand and spend more on its inputs, assuming it is best for the firm to become more “capital intensive” as it grows. Comment on WHY a firm might best become more capital intensive as it expands, even when the relative prices of labor and capital remain unchanged.
Suppose a firm has a Cobb Douglas production function and is initially producing 50 units of output using a long-run cost minimizing combination of inputs. The firm increases its output to 100 units. In the short run its level of capital remains fixed at the initial level (but the amount of labor used can be changed). In the long run, both inputs can change. On a graph, with amount of labor on one axis and amount of capital on the other, illustrate the expansion path in the long run and the short run. Draw an isocost line and an isoquant through each of the three solutions (i.e. before the increase in output, in the short run after the increase in output and in the long run after the increase in output. Note that both the second and third will be on the same isoquant). Your diagram does not need to be to scale, but should indicate the correct shapes and relative positions of the curves/lines.
Suppose that a production function of a firm is given by q = f(1) = 2√ī. It depends only on labor, whose price per unit is w. The firm is a price taker and the price for the good it produces is p. What is the quantity produced by the firm if the firm maximizes profits and what is the value of profit at that quantity?
Question 2 Show that the Cobb – Douglas production function f(x,,x,)= Ax“x , with 0 0, x, > 0. -