Columbia University Spring 2023 Intermediate Micro-UN321 1.004 Problem Set 5 (Consumer Choice: Expenditure Minimization; Theory of the Firm: Technology and Cost) Instructions: Complete the problem set on paper. Take pictures and save in PDI' format. Upload the l'l')l-' le to gradescope. ()ne 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 (Expenditure minimization) Consider a single parent with one child who has preferences for housing (X) and other items (Y) represented by the utility function U(x,y) = X't'zym. Currently, the parent works 40 hours a week and earns a montth income of M = 32,000 and lives in a city where the unit price of housingis P}: = $4 per square foot per month. The unit price of other items is Py = $1. 1. Find the parent's optima] bundle and label it (X1130)- 2. Compute the parent's indirect utility at the optimal bundle V0 2 U (M5311) The parent would like to reduce her number of hours to have more time to supervise her son when he is out from school. She is wondering whether by moving to another town where housing costs P's = $2 per square foot per month she could afford to work part time. We are interested in the minimum budget the parent would need in the new town to enjoy the same welfare (standard of living) she currently enjoys. 3. Write the parent's expenditure minimization problem. 4. Write the |.agmngian for this problem. 5. Compute the rst order conditions for the Lagrangian. (i. Solve the rst order conditions and flnd the cheapest bundle ofhousing and other items that in the new town would give the parent the same utility she currently enjoys (the results here might not be round numbers. Round up to the closest integer). T. livaluatc the cost ofthis bundle. 3. If she moved to the new town could the parent afford to cut her hours to 31! a week and earn only M' = 1,200 per month without lowering her standard of living? Problem 2 Technology and Cost The local cleaner launders white clothes using the production function q 2G + 25 , where G is the number of cups of a generic bleach and S is the number of cups of a super bleach that is twice as potent. 1. W'hat is the marginal product ofG and S? 2. In a graph where you measure cups of generic bleach along the horizontal axis draw a couple of isoquants. Add an arrow to indicate the direction ofincreasing quantity. 3. If the price of generic bleach is pt; 3 $1 while the price of super bleach is ps 3 31.5 what bleach should the cleaner use if it' wanted to keep the cost of production as low as possible? Problem 3 -- Technology and Cost Consider a rm that runs a medical clinic where it uses the services of labor (L) and equipment (K) to produce healthcare services. The clinic technology is summarized by the production function q = 0.5].l-i-'Kl'3' where q is the number of patients treated each day, I. is labor measured in daily number of full working shifts, and K is equipment measured in number of pieces used each day. Currently, the clinic is renting and/'or owns 512 pieces of equipment. 1. What is the clinic's short run production function? 2. Compute the expression for the clinic's marginal product of labor. Does the clinic's short run production function exhibit diminishing marginal returns to labor? 3. Starting from the short run production function, write the expression for the clinic's short run conditional demand for labor L" = 1.0.0. 4. \\'la'hat is the firm's short run cost function? Now, consider the long run production function. 5. Does the long run production function exhibit increasing, decreasing, or constant returns to scale? Suppose that the firm wished to double the number of patients its doctors visited each day. (1. Should the firm increase the size ofthe current clinic or should it open a second clinic ofthe same size? lirielly discuss. Problem Set 5 1