Consider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 C = wL + rK Using Lagrange Multiplier Method, calculate the cost-minimizing values of L, K, and C of Firm Y if it decides to produce 1,000 units of good Y. Graphically illustrate the expansion path of Firm Y if it decides to double its target output Derive the short-run cost function of Firm Z if 25 units of capital are employed. Suppose that good Z is sold at a perfectly competitive price of 10 per unit, calculate Firm Z’s profit and discuss if the Firm Z should continue to operate

Consider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 C = wL + rK Using Lagrange Multiplier Method, calculate the cost-minimizing values of L, K, and C of Firm Y if it decides to produce 1,000 units of good Y. Graphically illustrate the expansion path of Firm Y if it decides to double its target output Derive the short-run cost function of Firm Z if 25 units of capital are employed. Suppose that good Z is sold at a perfectly competitive price of 10 per unit, calculate Firm Z’s profit and discuss if the Firm Z should continue to operate

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter11: Profit Maximization

Section: Chapter Questions

Problem 11.3P

See similar textbooks

Similar questions

There is a firm making custom stuffed animals. The demand function for custom stuffed animals is: P = 1000 - 20Q The cost depends on how much of each of their inputs they use. It takes 3 inputs to make a stuffed animal: cotton, oil, and chorizo. The amount required, input prices, and restrictions are below: It takes 1 pound of cotton to make a stuffed animal. Cotton costs $20 per pound. They can buy up to 23 pounds of cotton. It takes 2 barrels of oil to make a stuffed animal. Oil costs $10 per barrel. They can buy up to 44 barrels of oil. It takes 3 chorizo burritos to make a stuffed animal. Burritos cost $5 per burrito. They can buy up to 100 burritos. Set this up in Solver to help them maximize profit by choosing the best quantity of stuffed animals to produce, subject to the constraints of the maximum amount of cotton/oil/burritos they can buy. In the space below, tell the profit that the firm will earn when they maximize its profit. PLEASE SHOW EXCEL WALKTHROUGH USING SOLVER.…
A firm is jointly owned by Juan and Roda. The firm’s production function requires two inputs: effort by Juan, denoted by x, and effort by Roda, denoted by y. Effort is only observable by the person who exerts it. The cost to Juan of a unit of his effort is c j = 2 and the cost to Roda for a unit of her effort is cr = 2. The price received for the goods is p = 2. The production of the firm is given by Q = 10(ln(x + 1) + ln(y + 1)). Assume that both Juan and Roda are risk-neutral rational agents.a) What are the socially optimal amounts of effort x* and y*? What is the total surplus in that case? (Hint: Solve the problem of a social planner that cares equally for Juan and Roda.)b) Suppose that Juan and Roda have a contract that specifies that Juan pays a fixed amount w = 15 to Roda and that Juan gets to keep and sell all the output. What is the total surplus now? How much of that surplus goes to Juan? To Roda?c) Now suppose that the contract between Juan and Roda specifies that the total…
This question deals with cost curves Keep in mind that the formula for a firm's cost function is: TC= FC + C(Q) TC → Total Costs: FC →Fixed Costs: C(Q) → Cost of production*Quantity produced → also known as Variable Costs Q1: Firms A and B are two firms supplying products in two separate differentiated goods markets. Equations (1) and (2) give the total cost functions of the two firms: - Firm A: TC = 2Q --- Equation (1) - Firm B: TC = 10 + 2Q --- Equation (2) Each firm has the ability to produce a maximum quantity of 80,000 units in ten batches of 8,000. The cost per unit in each firm is $2. Firm B has a fixed cost of $10. (a) Explain the relationship between the zero-profit curve and the marginal cost curve for the two firms using the quantity schedule of the two firms and the relevant plots of equations (1) and (2).
Suppose that widgets can be produced using two different production techniques, A and B. The following table provides the total input requirements for each of five different total output levels. Q=1 Tech. K L 4 1 1 3 A B Total Cost L K Assuming that the price of labor (PL) is $1 and the price of capital (P) is $3, calculate the total cost of production for each of the five levels of output using the optimal (least-cost) technology at each level. To do this, complete the table below by calculating the total cost of production, filling in the missing values using the optimal (least-cost) technology at each level. (Enter your responses as whole numbers.) Q=1 $ 6 Q=2 K L 6 2 2 5 Q=1 Total Cost L K 31 Q=2 $ 11 Q=3 K L 9 4 4 8 How many labor hours (units of labor) would be employed at each level of output? How many machine hours (units of capital)? To answer this, complete the table below for the units of labor and units of capital that would be used to produce each level of output. (Enter…
Problem 3: Long-Run Cost minimization Consider two cost-minimizing firms producing drones as their single output. Firm 1 produces traditionally, using capital K and labor L as inputs. Firm 2 has a close relationship with an artificial-intelligence research laboratory and may also use androids A as an input.The input prices are r > 0 for capital, w > 0 for labor, and a > 0 for androids.The firms’ production functions are as follows: F1 (K, L)=5KLF2 (K, L)=(L + 2A)4K Question: A customer orders exactly ¯q = 2000 drones from each firm. Describe each firm’s cost-minimization problem in formal terms
Assume that the firm’s operation is subject to the following production function and price data: Q=3X+5Y-XY Px= 3 $ Py= 6 $ where X and Y are two variable input factors employed in the production of Q. A. In the unconstrained case, what levels of X and Y will maximize Q? B. It is possible to express the cost function associated with the use of X and Y in the production of Q as TC = 3X + 6Y subject to the firm’s constraint of an operating budget of $25. Use the Lagrange multiplier technique to determine the optimal levels of X and Y. What is the firm’s total output at these levels of input usage? What is the value of Lagrangian multiplier? C. What will happen to the firm’s output from a marginal increase in the operating budget? (Hint: This is another way of asking you to interpret the Lagrangian multiplier you found in B above).
This question deals with cost curves and isoprofit curves Keep in mind that the formula for a firm's cost function is: TC = FC + C(Q) TC → Total Costs: FC → Fixed Costs: C(Q) → Cost of production*Quantity produced → also known as Variable Costs Q1: Firms A and B are two firms supplying products in two separate differentiated goods markets. Equations (1) and (2) give the total cost functions of the two firms: - Firm A: TC = 2Q --- Equation (1) - Firm B: TC = 10 + 2Q --- Equation (2) Each firm has the ability to produce a maximum quantity of 80,000 units in ten batches of 8,000. The cost per unit in each firm is $2. Firm B has a fixed cost of $10. (a) Make relevant plots of equations (1) and (2) (b) Use the plots in (a) and plots of isoprofit curves valuing $34,000 and $60,000 for the two firms to identify any differences in the shape of the two firms' isoprofit curves. Can you provide an explanation for any differences that may exist? (c) Use the information on both firms to assess…
Suppose the production is characterized by the function Q(K,L) = 50K05L05. The price of labor is $5 and the price of capital is $20. Find the cost minimizing combination of capital and labor if the manufacturer has an order for 1000 units of output. Put differently, Given: Q = 50KO.5L0.5 w = $5, r = $20 Goal: Produce Qo = 1000 units of output as cheaply as possible.
Suppose that widgets can be produced using two different production techniques, A and B. The following table provides the total input requirements for each of five different total output levels. Q = 1 Tech. K L A B Q=2 K L 2 4 1 6 1 3 2 5 Total Cost L K Assuming that the price of labor (P₁) is $1 and the price of capital (PK) is $3, calculate the total cost of production for each of the five levels of output using the optimal (least-cost) technology at each level. Q=3 K L 4 9 4 8 To do this, complete the table below by calculating the total cost of production, filling in the missing values using the optimal (least-cost) technology at each level. (Enter your responses as whole numbers.) Q=4 K L 12 5 Total Cost How many labor hours (units of labor) would be employed at each level of output? How many machine hours (units of capital)? To answer this, complete the table below for the units of labor and units of capital that would be used to produce each level of output. (Enter your…
Suppose that widgets can be produced using two different production techniques, A and B. The following table provides the total input requirements for each of five different total output levels. Q = 1 Tech. K L A 4 1 B 1 3 Total Cost Q = 2 K L 2 5 6 2 Q=1 $6 L K Q = 3 Assuming that the price of labor (PL) is $2 and the price of capital (PK) is $1, calculate the total cost of production for each of the five levels of output using the optimal (least-cost) technology at each level. K 9 4 Q = 4 L K L 4 12 7 11 8 8 5 12 7 15 To do this, complete the table below by calculating the total cost of production, filling in the missing values using the optimal (least-cost) technology at each level. (Enter your responses as whole numbers.) Q = 5 $27 Q=2 $ 10 Q = 5 K L Q = 3 $ 17 Q = 4 $ 26 How many labor hours (units of labor) would be employed at each level of output? How many machine hours (units of capital)? To answer this, complete the table below for the units of labor and units of capital that…
4) A firm faces a production function of twittle-twaps: Q(K,Lp,Ln) = 5*K(2/5)*LP(1/3)*LN(1/5) per hour, where capital (K), production labor (LP), and non-production labor (LN) are input factors used in production. The firm operates in a competitive market, where they are a price taker within the capital & labor markets and its own price (r = 40, wP = 25, wN = 50, P = 20). Answer the following.a. If capital and non-production labor are fixed at K = 32 and LN = 243, what is the general form MPLP and graph Q wrt to LP changing [you do not need to solve for LP yet].b. Is this production function decreasing, constant, or increasing returns to scale and why.c. Given the wage of production workers and the price of twittle-twaps, what is the optimal number of LP to employ to maximize profits and the quantity produced (VMPLP = wP).d. If the firm can control both K and LP, what does the Isoquant curve look like and its slope in relative terms if LN is fixed at 243 units [IQ slope =…
4) A firm faces a production function of twittle-twaps: Q(K,Lp,Ln) = 5*K(2/5)*LP(1/3)*LN(1/5) per hour, where capital (K), production labor (LP), and non-production labor (LN) are input factors used in production. The firm operates in a competitive market, where they are a price taker within the capital & labor markets and its own price (r = 40, wP = 25, wN = 50, P = 20). Answer the following.a. If capital and non-production labor are fixed at K = 32 and LN = 243, what is the general form MPLP and graph Q wrt to LP changing [you do not need to solve for LP yet].b. Is this production function decreasing, constant, or increasing returns to scale and why.c. Given the wage of production workers and the price of twittle-twaps, what is the optimal number of LP to employ to maximize profits and the quantity produced (VMPLP = wP).d. If the firm can control both K and LP, what does the Isoquant curve look like and its slope in relative terms if LN is fixed at 243 units [IQ slope =…