of each unit of blue ice is $2000. The capital costs $10 per unit per hour total a. Find the profit function. b. How many hours will Walter employ Jesse, if he is maximizing profits. c. Now consider the long run in which Walter can choose how much capital i according to the production function: Q(E,K)=(K4)(E!2). Find the optim capital and labor to use in the long run. Find the optimal amount of employment and capital in the long run using th mix of capital and labor from part c. Hint: Write out the new profit function

of each unit of blue ice is $2000. The capital costs $10 per unit per hour total a. Find the profit function. b. How many hours will Walter employ Jesse, if he is maximizing profits. c. Now consider the long run in which Walter can choose how much capital i according to the production function: Q(E,K)=(K4)(E!2). Find the optim capital and labor to use in the long run. Find the optimal amount of employment and capital in the long run using th mix of capital and labor from part c. Hint: Write out the new profit function

Microeconomics: Principles & Policy

14th Edition

ISBN:9781337794992

Author:William J. Baumol, Alan S. Blinder, John L. Solow

Publisher:William J. Baumol, Alan S. Blinder, John L. Solow

Chapter7: Production, Inputs, And Cost: Building Blocks For Supply Analysis

Section: Chapter Questions

Problem 1DQ

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2. A competitive firm uses two variable factors to produce its output, with a production function q min{x₁, x₂}. The price of factor 1 is $8 and the price of factor 2 is $5. Due to a lack of warehouse space, the company cannot use more than 10 units of x₁. The firm must pay a fixed cost of $80 if it produces any positive amount but doesn't have to pay this cost if it produces no butput. What is the smallest integer price that would make a firm willing to produce a positive amount? b. d. $44 $41 $29 $13 $21
PRICE (Dollars per tracker) 30 80 70 60 ATC 20 AVC MC 10 0 0 10 20 30 40 50 60 70 80 90 100 QUANTITY (Thousands of trackers per day) In the short run, given a market price equal to $45 per tracker, the firm should produce a daily quantity of trackers. On the preceding graph, use the blue rectangle (circle symbols) to fill in the area that represents profit or loss of the firm given the market price of $45 and the quantity of production from your previous answer. Note: In the following question, enter a positive number regardless of whether the firm earns a profit or incurs a loss. The rectangular area represents a short-run of $ thousand per day for the firm.
2. Suppose the hourly wage is $10 and the price of each unit of capital is $25. The price of output is constant at $50 per unit. The production function is f(E, K) = E'/²K'/2. (a) Write a firm's short-run profit maximization problem. (b) Find a firm's marginal product of labor MP(E). (c) Find the second order condition and derive the labor demand function (i.e. An increase in wage decreases the number of workers hired). (d) If the current capital is fixed at 1,600 units, how much labor should the firm employ in the short run? (e) How much profit will the firm earn?
12. If the demand function for a product is p = 100 – 2x where is price in dollars and x is the number of units sold, and the average cost function for a product is C(x) = 100 +4 where x is the number of units sold and C is dollars per unit, what is the maximum profit? Answer and Work Explanation of Work
You are given the profit function: Profit = 100Q – Q2 – 100 – 0.5Q2a. In words, how would you find the quantity you should produce to maximize profits?b. What is the slope of this line at that optimal profit point, and how does its value tell you that you areat the optimal quantity?c. In words, what would the slope of that line be if you produced a quantity smaller than the optimalquantity? In words, what would the slope of that line be if you produced a quantity greater than theoptimal quantity?
Suppose the marginal product of the last worker employed by a firm is 40 units of output per day and the daily wage the firm must pay is K20, while the marginal product of the last machine rented by the firm is 120 units of output per day and the daily rental price of the machine is K30. Is the firm maximizing output in the long run? If not what should be done? b) Mr. Phiri has estimated that he can sell 1000 additional hamburgers per day by renting more automated equipment at a cost of K100 per day. Alternatively, he could sell an extra 1200 hamburgers per day by keeping the restaurant open for two more hours at a cost of K50 per hour. Which of the two alternatives should Mr. Phiri purseu?
K A cereal factory has weekly fixed costs of $22,000. It costs $1.26 to produce each box of cereal. A box of cereal sells for $4.06. Find the rule of the cost function c(x) that gives the total weekly cost of producing x boxes of cereal. OA. c(x)=1.26x OB. c(x)=22,000+ 1.26x OC. c(x)=22,000 +4.06x OD. c(x)=22,000+ 2.8x
Roland Umbrellas has a production function given by Q = L0.5K0.5. The wage (W) is $80 per day and the rental per unit of capital (R) is $5 per day. In the long run, how many units of capital will Roland want to buy for each unit of labor?
2. A firm producing hockey sticks has a production function given by: q = 2 vkl In the short run, the firm's amount of capital equipment is fixed at k = 100. The rental rate for k is v=$1, and the wage rate for / is w=$4. (a) Calculate the firm's short-run total cost curve. Calculate the short-run average cost curve. (b) What is the firm's short-run marginal cost function? What are the SC, SAC, and SMC for the firm if it produces 25, 50, 100 and 200 hockey sticks? (c) Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point.
6. Each of the 10 firms has the production function q = √KL. The wage is 5 and the rent on capital is 10. Assume we are in the long run, so firms can vary both factors. Write an expression for one firm's cost minimization problem. Use whichever method you prefer to minimize cost and derive an expression for one firm's total cost: TC (9₁) c. Compute one firm's marginal cost and derive the inverse supply curve for one firm and the inverse supply curve for the market. What is the elasticity of supply? a. b.
Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.
8. Consider a Firm producing a given output in the long run; the inputs employed in production are labour L, which costs w > 0 per unit, and capital K, which costs r > 0 per unit. The Firm produces a quantity Q according to the following production function: Q(K, L) = 4K/3 L²/3. Suppose the Firm's target is to produce 1,000 (Q = 1,000). (a) Find the cost-minimising combination of inputs K* and L when w = 160 and r = 80, then compute the corresponding long-run cost of production. (b) Suppose that the rental price of capital decreases to w' 20. Find the new cost-minimising input combination. (c) Use your answers to points a) and b) to calculate the firm's price elasticity of demand for capital over this range of prices (rounded to two digits).