Excercise_Solution_CH3

Excercise_Solution_CH3 - Solutions to Excercise in CH3...

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Unformatted text preview: Solutions to Excercise in CH3 Slides Bin Xie 2/10/2015 Example on Page 13: Quasi-Linear Utility Function U = u(q1 ) + q2 . du U1 du Solution: U1 = , U2 = 1. M RS = − =− . dq1 U2 dq1 ρ ρ Excercise (1) on Page 26: U = (q1 + q2 )1/ρ ∗ ∗ d(q2 /q1 ) d(p2 /p1 ) substitution is a ∗ ∗ / p /p q2 /q1 2 1 1 ρ ρ 1−ρ Solution: U1 = (q1 + q2 ) ρ ρ ρ−1 q2 . U1 q1 M RS = − = −( )ρ−1 . U2 q2 ρ ρ (q1 + q2 ) Show the M RT and show that the elasticity of constant. ρ−1 ρ ρ · ρq1 = (q1 + q2 ) 1−ρ ρ ρ−1 q1 ; U2 = 1 ρ ρ−1 ρ 1−ρ (q1 + q2 ) ρ · ρq2 = ρ 1−ρ ρ M RS = ∗ q1 ρ−1 p1 M RT , i.e. −( ∗ ) =− . q2 p2 ∗ p1 1/(ρ−1) q1 Rewrite the equation: . The elasticity of substitution could also be rewritten ∗ = (p ) q2 2 ∗ ∗ ∗ ∗ ∗ ∗ d(q2 /q1 ) q2 /q1 d(q2 /q1 ) d(p2 /p1 ) = / . as ∗ /q ∗ / p /p q2 1 d(p2 /p1 ) p2 /p1 2 1 ∗ ∗ (Think q2 /q1 and p2 /p1 as two single variables instead of four variables. You can As we know, solving the utility maximization/expenditure minimization problem yields: ∗ ∗ regard q2 /q1 as q and p2 /p1 as p. So the equation becomes q = p1/(ρ−1) and the elasticity becomes So the dq p . Now it becomes like a problem to derive an elasticity of demand.) dp q 1 dq q 1 p 1 elasticity of substitution is = p ρ−1 −1 1/(ρ−1) = which is a constant. dp p ρ−1 ρ−1 p The rest two excercise questions on page 26 are two examples of a Cobb-Douglas utility function. You can do them by yourself and check the answers based on the notes you got from the lecture. 1 ...
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