Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 8 Please help improve the course by sending me an email about typos or suggestions for improvements Question 1 Consider a company using K and L as imperfect substitute: Q = L K. (a) Suppose = 1/3 = 2/3. What are the returns to scale for this technology? (b) For what values of and will this technology exhibit increasing returns to scale? Decreasing returns to scale? Question 2 Consider a company using K and L as perfect substitute: Q = L + K. (a) Suppose = 1/3 = 2/3. What are the returns to scale for this technology? (b) Now suppose Q = L + K + . For what values of , and will this technology exhibit increasing, constant and decreasing returns to scale? Question 3 When does the complements production function Q = min(L, K) have increasing, constant, and decreasing RTS? Question 4 Jenn uses a complements technology with long run production function q = min(L, K) . Currently, = 1/2, =1/2, PL = $10 and PK = $10. (a) Suppose demand for Jenn's product is given the equation P = 100 10q. What is Jenn's 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain target output if she wants to maximize profits? (b) Given the target output q in part (a), how many workers should Jenn hire? How much capital should Jenn lease? (c) What is the long run cost of producing Jenn's target output? (d) Suppose Jenn's workers clamor for higher wages. The labor union negotiates a higher wage of PL = $20. With the expenditure in part (c), will Jenn be able to produce the target output? (e) How will Jenn react to higher wages? In particular, what is the percentage change in labor, capital and long run cost due to the higher wages? Question 5 InHart uses a perfect substitutes technology with long run production function q = L + K. Currently, =1, =1, PL = $10 and PK = $20. (a) Suppose demand for InHart's product is given the equation P = 100 10q. What is InHart's target output if he wants to maximize profits? (b) Given the target output q in part (a), how many workers should InHart hire? How much capital should InHart lease? (c) What is the long run cost of producing InHart's target output? (d) Suppose the price of leasing capital is volatile. For what range of PK will InHart use the same number of workers and capital as in part (b)? Question 6 (2008 Test 2 Question) A company uses Labor (L), Capital (K) and Materials (M) as inputs to produce target output q with the production function q = L K M. For what values of , and will this company have constant returns to scale? Show your calculations below. Question 7 (2008 Test 2 Question) A company uses Labor (L) and Capital (K) as inputs to produce target output q with the production function q = min(L, K) + . For what values of , and will this company have increasing returns to scale? Show your calculations below. 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 8 (20082009 Final Exam Question) Fuelled by easy access to drugs from the USA (United States of Addicts) the citizens of the Canadian town Crackotoa are fast becoming drug addicts. The mayor of Crackotoa, Ms. Cindy Bong, has started a drug treatment program in which police officers (P) and drug counselors (D) work as complements in a 1:1 ratio to treat each drug addict (A). That is, the treatment program uses police officers and drug counselors as complementary "inputs" to treat drug addicts as the "output". (a) The program currently has 1,000 police officers and 1,000 drug counselors. If Crackotoa's government doubles the budget for the drug treatment program, how many more police officers and drug counselors should Ms. Cindy Bong hire? Assume the price of police officers and drug counselors is constant. (b) Repeat part (a) for the case where the price of police officers and drug counselors also doubles. Question 10 This question was motivated by the recent announcement by the Ontario province and the federal government to give Toronto close to $1 billion to (principally) extend the Sheppard subway line. In this question, you will investigate whether it's better to receive an unrestricted grant that be, for example, spent on any combination of mass transit and highways, versus a block of money which specifies a specific level of mass transit and highways. Suppose the "inputs" for a city's transportation level ("output") are "mass transit" (xaxis) and "highways" (yaxis). Assume mass transit and highways are imperfect substitutes for transportation in a city. (a) Draw the isoquants for transport as the output and mass transit and highways as inputs. (b) Suppose the Federal government gives cities block grant money for a specific amount of mass transit and highways to "produce" a target level of transport. Depict this on the isoquant figure. (c) Now suppose a unit of mass transit costs Pm and a unit of highways costs Ph. Instead of giving cities block grants as in part (b), now suppose the Federal government gives cities an unrestricted block grant (budget) which they are free to spend on any combination of mass transit, highways. Draw a picture for two cases: Case (i): a city which benefits from the unrestricted block grant; 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Case (ii): a city which does not benefit from an unrestricted block grant. Question 11 This question is a review of the Coke example discussed in the lectures. In the 1970s, Coca Colathe number 1 global brandused sugar in its secret formula. Rising sugar prices and changing consumer tastes away from high calorie drinks to low calorie drinks, forced CocaCola to tinker with its formula. In particular, CocaCola considered switching from sugar to corn fructose syrup. In all questions, put fructose on the xaxis and sugar on the yaxis (that's the opposite of what I did in the lectures). (a) CocaCola initially could not develop a new formula for Coke that used fructose as an input while maintaining the "taste" of Coke. What do Coke's isoquants look like? How much of sugar and fructose will Coke use in its products? (b) CocaCola figured out a way to mix sugar and fructose and maintain Coke's taste. What will Coke's isoquants look like? At that time, Coke used a 5050 combination of sugar and fructose in its productsshow this on your diagram. (c) Fructose prices continued to drop and CocaCola got better at tweaking its formula to incorporate fructose. By the later 70s, CocaCola was using only fructose in its products. Show this on an isoquant/isocost diagram. Question 12 In this question, you will explore the idea that if a company has an imperfect substitutes technology, it is able to alter the mix of inputs in response to input prices. For example, a company with a complements technology is "hostage" to input prices if input prices increase it is unable to switch to a relatively cheaper input. In contrast, a firm using imperfect substitutes technology can defray higher input prices by switching. Consider a firm with a CobbDouglas production function Q = L K. For your convenience, here are the optimal demands for labor and capital and the optimized cost function: L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) C(q) = q1/( + ) PL /( + ) PK/( + ) [(/)/( + ) + (/)/( + )] 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain From the demands for labor and capital, show that if wages increase, then to still produce target output q, this firm will use less labor and more capital. Question 13 Consider a firm with a CobbDouglas production function Q = L1/2 K1/2. Currently, PL = $5 and PK = $5 and the firm has target output q. (a) Solve for the optimal levels of labor and capital: (b) What is the cost function? (c) Calculate the Lagrange multiplier and interpret it. 5 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto.
 Fall '08
 HUSSEIN
 Economics, Microeconomics

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