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Macro1_PS7_solutions

Macro1_PS7_solutions - Macroeconomics 1 Master APE...

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Unformatted text preview: Macroeconomics 1. Master APE. 2010-2011. PS7 Prof. Gilles Saint-Paul / T.A : Eric Monnet 1 Equilibrium unemployment as a worker discipline device : the Shapiro-Stiglitz model Set up of the model : An economy 1 is populated with N identical, risk neutral workers. The utility function at each period is a function of wage and e ort as u ( w,e ) = w- e , where the e ort e is either null or positive. An unemployed worker received ¯ w with no e ort ( e = 0 ). There is a probability b that a worker will be separated from his job due to reallocation etc. Workers maximize the expected present discounted value of utility with a positive discount rate r . All the model is driven by a simple choice : the workers choice at any point in time is to exert e ort or not. Then, the odds of getting caught if shirking are q . If caught, the worker is red, enters unemployment and waits for a new job. 1) Consider a short, discrete time interval [0,t]. Then explain why in a world without e ort choice and shirking , the asset value of employment over this interval can be written as the following Bellman equation : V E = w.t + exp (- r.t )[ b.tV U + (1- b.t ) V E ] where V U is the value of unemployment. Then show how one can rewrite it as : rV E = w + b ( V U- V E ) 2) Given the decision to shirk or not and using the previous question, show that the funda- mental asset equation for a shirker is rV S E = w + ( b + q )( V U- V S E ) while for a nonshirker : rV N E = w- e + b ( V U- V N E ) Interpret these equations. 3) Solve for V S E and V N E 4) Using the previous expressions of V S E and V N E , state a simple condition for which the worker will choose not to shirk. Comment. What is the interpretation of this expression if q = 0 ? 5) Then introduce in the model M identical rms. The aggregate production function is 1 from Shapiro C. and Stiglitz J., 1984, 'Equilibrium unemployment as a worker discipline device', American Economic Review 1 Macroeconomics 1. Master APE. 2010-2011. PS7 Prof. Gilles Saint-Paul / T.A : Eric Monnet Q = F ( L ) , where L is the labor force. We assume that F ( N ) > e , that is, full employ- ment is e cient. We denote F ( L ) = ˆ w . What is the value of ˆ w given the condition you have found in question 4 ? 6) The key market variable which determines individual rms behavior is V U , the expected utility of an unemployed worker. Show that rV U = ¯ w + a. ( V E- V U ) . where a is the job acquisition rate (the probability of obtaining a job per unit of time). Then solve for rV U and rV E . Explain whether a is endogenous or exogenous in the model and show that a = bL/ ( N- L ) . 7) State a new non-shirking condition and then modify it to nd the following condition : w ≥ e + ¯ w + e q bN ( N- L ) + r we + ¯ w + ( e/q )( b/u + r ) ≡ ˆ w where u = ( N- L ) /N Explain why, from this equation, one can conclude that no shirking is inconsistent with full employment....
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