{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern