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Problem Set 6 1 Columbia University Spring 2023 Intermediate Micro-UN3211.004 Problem Set 6 – (Cost) Instructions: Complete the problem set on paper. Take pictures and save in PDF format. Upload the PDF file to gradescope. One upload per group. Match each problem to the page where you wrote its solution. Last, don’t forget to add all other group members to the upload. Problem 1 (Short Run cost functions) Ms. Fox can produce artificial fur coats using the services of labor and equipment according to the production function q(L,K)= 100L1/2K1/2 Last week, Ms. Fox has signed and pre-paid a one-year lease for twenty five machines. 1. Find the expression for Ms. Fox’s Short Run production function ? = ?SR(L). 2. Does this production function exhibit diminishing returns to labor? 3. Do you expect Ms. Fox’s marginal cost to be increasing, constant or decreasing in quantity? Why? 4. Find Ms. Fox’s short run conditional demand for labor. Mrs. Fox pays each full-time worker a daily wage rate of $400. The daily rental cost of equipment is $100. 5. Find Ms. Fox’s Variable Cost function and Total Cost function. 6. What is Ms. Fox’s Marginal Cost function MC(q)? Does this function confirm your conjecture from part 3? 7. Compute Ms. Fox’s daily fixed cost. 8. Find Ms. Fox’s Average Total Cost ATC(q) and Average Variable Cost (AVC(q)) 9. In a diagram where you measure quantity along the horizontal axis and monetary amounts along the vertical axis, plot Ms. Fox’s AVC, ATC, and MC curves. Suppose the rental cost of equipment increased to $250. 10. How would Ms. Fox’s AVC, ATC and MC curves change? Problem 2 (Producing a target output at the lowest cost possible) Mr. Lee runs an orange grove, SweetOrange. It’s harvest season and each week Mr. Lee hires workers (L) and rents equipment (K) to bring q(L,K) = 250L1/3 K1/3 pounds of oranges to market. 1. Does technology at SweetOrange exhibit diminishing returns to labor? Does it exhibit diminishing returns to equipment? Explain. 2. Does technology at SweetOrange exhibit increasing, constant, or decreasing returns to scale? 3. Suppose that SweetOrange employs 2 full time workers and rents 3 pieces of equipment. What is the firm’s Marginal Rate of Technical Substitution? How do you interpret this number? This week, Mr. Lee received an order for 1,000 pounds of oranges. Hiring a full-time worker costs $100 per day and renting a piece of equipment also costs $100 per day. 4. How many workers and how many pieces of equipment should the firm employ to keep cost lowest? Use the Lagrangian method to provide your answer. a) Write SweetOrange’s cost minimization problem. b) Write the Lagrangian function c) Write the first order conditions for the Lagrangian problem. d) Solve the first order conditions and find L^ and K^ and ^. 5. At the cost minimizing technique, what is the cost of raising output by one unit by hiring more workers (w/MPL)? What is the cost of raising output by one unit by renting more equipment (r/MPK)? In words, explain why if these two amounts were different the firm would not be minimizing cost. We are interested in the SweetOrange’s cost of operations. 6. This week, at SweetOrange what is the cost of bringing 1,000 pounds of oranges to market? What is the unit cost of a pound of oranges? 7. In a graph where you measure the labor input along the horizontal axis and the equipment input along the vertical axis, represent SweetOrange’s q=1,000 isoquant. Then add some iso-cost lines indicating the direction of declining cost. 8. In the same graph illustrate the cost minimizing technique. We are also interested in how Mr. Lee would change its operations if a crack-down on illegal immigrant labor raised the cost of hiring a full-time worker to $150 per day. 9. Draw a new isoquant/isocost diagram. In this new diagram illustrate how the higher cost of labor would affect the cost minimizing technique Mr. Lee would choose to employ. Problem 3 (from technology to the cost function – long run) Drin uses labor time (L) and equipment time (K) to produce vintage alarm clocks. Drin’s daily production function is q = LK. 1. Does the production function displays increasing, constant, or decreasing returns to scale? Do you expect Drin’s long run average cost of production to be increasing (diseconomies of scale), constant, or decreasing (economies of scale) in quantity? Why? Problem Set 6 2 Columbia University Spring 2023 Intermediate Micro-UN3211.004 At Drin, the daily wage rate is $200 while the daily rental cost of capital is $400. 2. Leaving Drin’s level of output indicated as a parameter q, solve the firm’s cost minimization problem and find Drin’s conditional labor demand L^(q) and conditional equipment demand K^(q). 3. Substitute these functions into the expression for Total Cost = wL^ + rK^ and find Drin’s long run total cost function. 4. What share of total cost comes from the cost of labor? What share comes from the cost of equipment? 5. Find Drin’s Average Total Cost function. Problem 4 (from the pre-class activity – long run cost) Consider a firm that manufactures pieces of furniture according to the production function q = {L0.5 + K0.5}2. The firm can manufacture the rigs in North America where labor costs w = $1,000 and equipment costs $1,000 or in East Asia where labor costs $300 and equipment costs $600. Suppose the firm manufactures the pieces of furniture in North America. 1. Find the firm’s conditional demand for labor and conditional demand for equipment. 2. Find the firm’s Total cost function. 3. What is the average cost of manufacturing a piece of furniture in North America? Suppose the firm manufactures the pieces of furniture in East Asia 4. Find the firm’s conditional demand for labor and conditional demand for equipment. 5. Find the firm’s Total cost function. 6. What is the average cost of manufacturing a piece of furniture in East Asia? Suppose that shipping furniture from East Asia to North America costs $50 per piece of furniture. 7. Should the firm manufacture the furniture in North America or in East Asia? What would happen to the firm’s profit margins if the cost of shipping increased three folds to $150 per piece of furniture? Suppose last that the firm could relocate its operations to Central America. There, the cost of labor is $400 while the cost of equipment is $800. 8. What is the highest cost of transportation from Central America to North America that would make it worthwhile for the firm to relocate its operations? Problem 5 (Policy application) In the US, the Social Security Administration (SSA) runs a government paid insurance program for retired workers and their spouses called Medicare. In 1983, Medicare changed the way it reimbursed hospitals for treating Medicare patients. Originally, Medicare paid hospitals fees that for each patient covered a portion of the hospital’s total cost including both capital (equipment and space) and labor expenses. Under the new Prospective Payment System (PPS), the government reimbursement was made up of a payment computed proportionally to the hospital’s capital expenses (equipment and space) and a flat rate based on a patient’s diagnosis that did not take into account the hospital’s actual use of labor inputs. Effectively, the shift to PPS left hospital’s opportunity cost from using equipment and structures the same while significantly increasing the opportunity cost of using labor services. With the aid of an iso-cost-isoquant diagram, explain how the shift to PPS might have affected hospitals’ relative use of capital and labor when treating incoming Medicare patients. Problem Set 6 – (Cost) Instructions: Complete the problem set on paper. Take pictures and save in PDF format. Upload the PDF file to gradescope. One upload per group. Match each problem to the page where you wrote its solution. Last, don’t forget to add all other grou...