University of California, Davis Date: June 25, 2007 Department of Agricultural and Resource Economics Department of Economics Time: 5 hours MicroeconomicsReading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE ANSWER KEY Part 1 Insurance (a)-(b) Given qand α, the consumer’s consumption contingent to the bad state is α−+ω=]1[11qx, and her consumption contingent to the good state is α−ω=qx22. The consumer solves the following optimization problem.Consumer’s Problem. Given q, choose αin order to maximize )]1[()1()]1[(11α−+ωπ−+α−+ωπququsubject to],0[12ω−ω∈α. Because it is explicitly assumed that the consumer always chooses a positive α, we disregard the nonnegativity constraint on αand consider only the constraint 12ω−ω≤α. The Lagrangian now becomes L(α, λ) = )]([)()1()]1[(1221ω−ω−αλ−α−ωπ−+α−+ωπququ, with Kuhn-Tucker conditions(KT-1) 0])[(')1(]1)1[('21=λ−−α−ωπ−+−α−+ωπqquqqu, (KT-2) 0)]([12=ω−ω−αλ, (KT-3) 0≥λ. If αwere equal to12ω−ω, then one would have xqq≡α−ω=α−+ω21)]1[, and (KT-1) would become 0)(')1(]1)[('=λ−π−−−πqxuqxu, i.e., 0)(')1(]1)[('≥π−−−πqxuqxu(by KT-3), or qq−≥π−π11, contradicting the assumption that q> π. Hence,12ω−ω<α, and the consumer chooses not to fully insure, and hence to bear some risk, i. e., 2211]1[xqqx≡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[('21=α−ωπ−−−α−+ωπqquqqu, (2)
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