PS3sol - Economics 11: Microeconomics Theory 1 Professor...

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Economics 11: Microeconomics Theory 1 Professor Christian Hellwig Homework 3 Solutions 1. Labor Supply Bob’s utility function is U = CL ,wh e r e C is Consumption and L is Leisure. The commodity price is 1 and wage rate is $3 per hour. Assume the total amount of hours is 24 and Bob’s non wage income is $24 . (a) Find the labor supply. (b) If Bob’s employer wants him to work 10 hours per day, what would be the wage rate? (c) Now suppose the employer uses a overtime wage plan: w =3 for the f rst 8 hours of work and w =4 for each hour worked in a day in excess of eight hours. What is the labor supply? (d) Compare total wages and Bob’s utilities in part b and c. Answer: (a) Bob’s utility maximization problem can be written as Max CL s.t. C ( 2 4 L )+24 The corresponding Lagrangian problem is Max = + λ (24 + 3(24 L ) C ) FOCs are L + λ ( 1) = 0 C + λ ( 3) = 0 24 + 3(24 L ) C =0 >From the f rst two equations we have C L . Replacing it in the last equation, we get 24 + 3(24 L ) 3 L L =16 Labor supple equals to 24 L =8 . (b) We now solve the following problem: = + λ (24 + w (24 L ) C ) FOCs are L + λ ( 1) = 0 C + λ ( w )=0 24 + w (24 L ) C 1
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>From the f rst two equations we have C = wL . Replacing in the last equation, we get 24 + w (24 L ) wL =0 so L = 24 + 24 w 3+ w ,or w = 24 2 L 24 .
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This note was uploaded on 04/05/2009 for the course ECON 11 taught by Professor Cunningham during the Spring '08 term at UCLA.

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PS3sol - Economics 11: Microeconomics Theory 1 Professor...

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