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Problem Set $-$ DMP$.$pdf

Optimal unemployment

ECON $711$

due on Monday April $15$ at $11$:$30$am

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Optimal unemployment with endogenous hours

We consider an extension of the Pissarides model of unemployment where the number of hours of work supplied by each employed worker, $h,$ is endogenous. Time is continuous and there is a measure one of workers. The measure of unemployed workers is $u(t)($it is also the unemployment rate$)$ and the measure of employed workers is $n(t),$ with $u(t)+n(t)=1.$ All agents are risk neutral and the rate of time preference is $r>0.$

There are two types of jobs: filled jobs, $n(t),$ and vacant jobs, $v(t).$ The creation of a vacant job requires the payment of a flow cost, $>0,$ per unit of time. Each filled job generates a flow of output, $y(h)\frac{?}{b}=ar(y)h^{},$ per unit of time, where $h$ is the number of hours supplied by the worker and $\frac{?}{\text{b}\text{a}\text{r}}(y)>0.$ The disutility of work for an employed worker is $(h)\frac{?}{b}=ar()\frac{h^2}{2},$ with $\frac{?}{\text{b}\text{a}\text{r}}()>0.$ Each job is destroyed with an instantaneous probability, $s>0.$ Unemployed workers enjoy no utility from leisure