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Unformatted text preview: ECON 610: Microeconomic Theory II Prof. Blume Problem Set 1 1 Problem Set 1 This is a solution key for problem set 1. Some intermediate steps are left to the reader. If you still have questions regarding any of the problems, please visit Prof. Blume or one of the TAs during office hours. Thanks, Liyan and Max Problem 1: In the coconut economy, Rob can produce at most al coconuts with l hours of work. His utility function for coconuts c and leisure time t is u ( c, t ) = b ln( c ) + t/ 12 . He has 24 hours in a day to allocate between work and leisure. Suppose a = 6 and b = 1 . (a) Find the Pareto optimal allocation. (b) Write this problem as a Kuhn-Tucker problem (if you did not do so in part (a) and find the multiplier for the production constraint. (c) How does the optimal allocation change with respect to the parameters a and b ? (d) What happens when b > 4 ? Solution: (a) The Pareto problem is to choose an allocation of leisure, work and coconuts to maximise Robs utility given the time and production constraints, as well as non- negativity constraints on each of these variables. That is: max ( c,l,t ) u ( c, t ) = b ln c + t/ 12 s.t. c al t + l 24 You can show that the solutions to this problem are fully characterised by the following KKT conditions (where is the multiplier on the production constraint): b c- = 0 (1) 1 12- a 2 p a (24- t ) (2) t 1 12- a 2 p a (24- t ) ! = 0 (3) p a (24- t )- c = 0 (4) ECON 610: Microeconomic Theory II Prof. Blume Problem Set 1 2 The solution to this set of equations is: t * = 24- 6 b if b 4 if b > 4 c * = 6 ab if b 4 24 a if b > 4 * = q b 6 a if b 4 b 24 a if b > 4 So, for ( a, b ) = (6 , 1) the Pareto optimal allocation is ( c, t, l ) = (6 , 18 , 6)....
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- Spring '05