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solpr12ec70204

# solpr12ec70204 - Suggested Solutions to Problem Set 12 Econ...

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Suggested Solutions to Problem Set 12 Econ 702, Spring 2004 Prepared by Ahu Gemici and Omer Kagan Parmaksiz May 3, 2004 Solution 1 Suppose we have a world described by the following Γ matrix, in which probability of finding a job and loosing a job is .1. Find the stationary distrubution of this economy and calculate the average duration of unemploy- ment. Γ = 0 . 9 0 . 1 0 . 1 0 . 9 Stationary distribution x * is the solution to the following equation, Γ T x * = x * (1) 0 . 90 e + 0 . 10 u = e (2) 0 . 10 e + 0 . 90 u = u (3) both of which implie e = u and with normalization e + u = 1 (4) the stationary distribution for employment states is x * = 1 2 , 1 2 (5) also the duration of unemployment is given by the inverse of probability of transiting from unemployment to employment which is given by, D u = 1 . 1 = 10 . Solution 2 Consider the optimal unemployment insurance problem we covered in class, c ( V ) = min c,a,V u { c + β [1 - p ( a )] c ( V u ) } (6) s.t V = u ( c ) - a + β p ( a ) V E + (1 - p ( a )) V u (7) Derive the envelope condition explicitly and show that c ( V ) is convex. 1

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To solve the problem, construct Lagragian function L = c + [1 - p ( a )] βc ( V u ) + θ V - u ( c ) + a - β p ( a ) V E + (1 - p ( a )) V u FOC: (c) θ = 1 u c (8) (a) c ( V u ) = θ 1 βp ( a ) - ( V E - V u ) (9) (V u ) c ( V u ) = θ (10) To derive EC, we write c ( V ) = min c,a,V u c + [1 - p ( a )] βc ( V u ) - θ V - u ( c ) + a - β p ( a ) V E + (1 - p ( a )) V u Take first derivative of c ( V ) with respect to V when c, a, V u take optimal value, we get c ( V ) = ∂c * ∂V - p ( a ) ∂a * ∂V βc ( V u * ) + [1 - p ( a * )] βc ( V u * ) ∂V u * ∂V + θ 1 - u ( c ) ∂c ∂V + ∂a ∂V - p ( a ) ∂a ∂V β ( V E - V u ) - [1 - p ( a )] β ∂V u ∂V Substitute (9) and (10), c ( V ) = ∂c * ∂V - p ( a ) ∂a * ∂V βθ 1 βp ( a ) - ( V E - V u ) + [1 - p ( a * )] βθ ∂V u * ∂V + θ 1 - u ( c ) ∂c ∂V + ∂a ∂V - p ( a ) ∂a ∂V β ( V E - V u ) - [1 - p ( a )] β ∂V u ∂V (11) = ∂c * ∂V - ∂a * ∂V θp ( a ) ∂a * ∂V βθ ( V E - V u ) + [1 - p ( a * )] βθ ∂V u * ∂V (12) + θ 1 - u ( c ) ∂c ∂V + ∂a ∂V - p ( a ) ∂a ∂V β ( V E - V u ) - [1 - p ( a )] β ∂V u ∂V (13) Combine (8), we can show envelope condition as c ( V ) = θ Showing the convexity or strict convexity of the cost function of the insurer is complicated primarly due to a possible nonconvexity of the promise keeping constraint. Basically, the cost function is strictly convex in promised utility since efficiency implies increasing V is possible through increased consumption.
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