McCall Model of Job Search with an On-the-job Search Consider the continuous time McCall Model with an on-the-job search. An unemployed worker with an innite-time horizon wishes to maximise the present discounted value of wages. At a moment in time, the unemployed worker receives a wage offer w, where w is drawn from a known exogenous distribution Fa (w). with an upper bound r. Let 1' denote the discount rate and m] denote the job offer arrival rate while unemployed. If the unemployed worker rejects a wage offer, she continues searching for employment and receives a ow value of b where W 2} b. If the unemployed worker accepts a wage offer, she receives the offered wage and continues searching on the job while working. However, the job o'er arrival rate and the distribution of wage offer while employed are different from those while unemployed. Let 0:1 denote the job oer arrival rate for employed workers and assume that the wage offer for employed workers is always 15' (i.e. the PDF of wage offer for the employed worker f] (w) equals one if w = if and zero otherwise). Since the wage offer for the employed worker is always higher or equal to the current wage, the employed worker always accepts the on-the-job offer. Further, let's assume that the employed worker who already receives r also accepts the on-the-joh offer. (a) Express the Bellman equation associated with the value of employment N(w) at a wage of w for the worker and the Bellman equation associated with the value of being unemployed U. (Hint: N (15') = iii/r.) (b) Assume the worker follows a cut-off strategy of accepting wages above or equal to R and rejecting wages below R. Derive the reservation wage R. (c) Derive the comparative statics of the reservation wage with respect to the job oer arrival rate while em- ployed, i.e. nd dR/dal. Explain the intuition ofthe sign odefda] . Hint: lEmrLr [w ~ 176,01: ff - F{w]}dw. (:1) Now suppose we have exogenous job destruction. Let 3. denote the exogenous job destruction rate which is common for all jobs. Provide an analytical expression of the steady state unemployment rate R. Compare the result with the case in which there is no on-the-job search and explain the difference intuitively