prelims_Micro Prelim ANSWERS June 2007

prelims_Micro Prelim ANSWERS June 2007 - University of...

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University of California, Davis Date: June 25, 2007 Department of Agricultural and Resource Economics Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE ANSWER KEY Part 1 Insurance (a)-(b) Given q and α , the consumer’s consumption contingent to the bad state is α + ω = ] 1 [ 1 1 q x , and her consumption contingent to the good state is α ω = q x 2 2 . The consumer solves the following optimization problem. Consumer’s Problem . Given q , choose α in order to maximize ) ] 1 [ ( ) 1 ( ) ] 1 [ ( 1 1 α + ω π + α + ω π q u q u subject to ] , 0 [ 1 2 ω ω α . Because it is explicitly assumed that the consumer always chooses a positive α , we disregard the nonnegativity constraint on α and consider only the constraint 1 2 ω ω α . The Lagrangian now becomes L ( α , λ ) = )] ( [ ) ( ) 1 ( ) ] 1 [ ( 1 2 2 1 ω ω α λ α ω π + α + ω π q u q u , with Kuhn-Tucker conditions (KT-1) 0 ] )[ ( ' ) 1 ( ] 1 )[ ] 1 [ ( ' 2 1 = λ α ω π + α + ω π q q u q q u , (KT-2) 0 )] ( [ 1 2 = ω ω α λ , (KT-3) 0 λ . If α were equal to 1 2 ω ω , then one would have x q q α ω = α + ω 2 1 ) ] 1 [ , and (KT-1) would become 0 ) ( ' ) 1 ( ] 1 )[ ( ' = λ π π q x u q x u , i.e., 0 ) ( ' ) 1 ( ] 1 )[ ( ' π π q x u q x u (by KT- 3), or q q π π 1 1 , contradicting the assumption that q > π . Hence, 1 2 ω ω < α , and the consumer chooses not to fully insure, and hence to bear some risk, i. e., 2 2 1 1 ] 1 [ x q q x α ω < α + ω . ( 1 ) It follows from (2) that λ = 0, and therefore (KT-1) becomes the FOC equality F ( α , π ) 0 ) ( ' ) 1 ( ] 1 )[ ] 1 [ ( ' 2 1 = α ω π α + ω π q q u q q u , (2)
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2 which implies the tangency condition q q x u x u = π π 1 ) ( ' ) 1 ( ) ( ' 2 1 , ( 3 ) see Figure 1.1. Graphically, because q > π , the tangency condition requires x 1 < x 2 , i. e., the consumer chooses a point off the certainty line. The SOC, assumed to hold with strict inequality, is then 0 < α F . ( 4 ) (c) Using (4), we apply the Implicit Function Theorem to (2), and obtain α π = π α F F d d . Because of (2) the sign of π α d d is that of π F . We compute 0 ) ( ' ] 1 )[ ( ' 2 1 > + = π q x u q x u F . In words, an increase in the probability of the loss, while the premium rate remains fixed, leads the consumer to demand more insurance. Intuitively, the increase in the probability of the bad state increases the (relative) marginal ex ante utility of consumption in the bad state, leading the consumer to substitute consumption in the bad state for consumption in the good state. Graphically, the indifference curve through the previously chosen point A becomes steeper (dashed curve), leading the consumer to choose a point
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This note was uploaded on 02/09/2010 for the course ECON 200D taught by Professor Pontusrendahl during the Winter '06 term at UC Davis.

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prelims_Micro Prelim ANSWERS June 2007 - University of...

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