# PS8 - ECO 310 Fall 2008 Microeconomic Theory A Mathematical...

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ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Problem Set 8 - Answer Key Question 1: U ( W 0 ) = ( 1 2 + π ) U ( W 0 + h ) + ( 1 2 - π ) U ( W 0 - h ) U ( W 0 ) ( 1 2 + π )( U ( W 0 ) + hU 0 ( W 0 ) + 1 2 h 2 U 00 ( W 0 )) + ( 1 2 - π )( U ( W 0 ) - hU 0 ( W 0 ) + 1 2 h 2 U 00 ( W 0 ) 2 πU 0 ( W 0 ) + h 2 U 00 ( W 0 ) 0 π h 1 4 A ( W 0 ) where A ( W 0 ) = - U 00 ( W 0 ) U 0 ( W 0 ) is the absolute degree of risk aversion at wealth W 0 . Question 2: (a)The expected proﬁt for an insurance company on a contract for x dollars insurance coverage is π ( x ) = (1 - π ) px + π ( px - cx - x ) Competition drives the proﬁt to zero, so we have p = π (1 + c ). (b) EU ( W ) = πln ( x - px ) + (1 - π ) ln ( W 0 - px ) (c) max x πln ( x - px ) + (1 - π ) ln ( W 0 - px ) Take ﬁrst order diﬀerentiation with respect to x , we have π x = p (1 - π ) W 0 - px π = x p (1 - π ) W 0 - px = p (1 - π ) W 0 x - p x W 0 = π p Question 3 a) Farmer East consumes x er = y er + z er in scenario ”rain”, where x er and y er denote his levels of consump- tion and production when ”rain” occurs, respectively, and z er denote his residual claim to consumption in that state. Similarly, the constraint x ed = y ed

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PS8 - ECO 310 Fall 2008 Microeconomic Theory A Mathematical...

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