Equilibrium Job Search A. Diamond’s critique 1. Partial equilibrium model takes the distribution of wages F(w) as given; but how does this distribution arise? 2. If workers are identical (have the same search strategies, arising from the same costs and benefits of search and the same expectations) and if employers know this, a. employers will simply set their wages at the reservation wage level; b. the wage distribution collapses to a single point, and c. the rationale for searching disappears 3. For a general equilibrium model with a non-degenerate wage distribution, we will need to modify the model of job search behavior; two useful and more realistic modifications are to account for job terminations and on-the-job search B. Job search model with job terminations 1. Assumptions a. Continue with most of the assumptions of basic job search model in a stationary environment with variation in the arrival rate of job offers b. However, instead of assuming that jobs last forever, assume that they can be dissolved at any given time with probability qc. If the job is dissolved, the person returns to unemployment d. Consideration of job terminations is important in an equilibrium framework because it provides a supply of unemployed people 2. The present discounted value of work, Ve, a. defined in terms of intervals of time, ∆, is ()[]ueeVqVqwrV ∆+∆−+∆∆+= 111 expression reflects the wage that will be received, the expected future value of employment or unemployment, Vu, (depending on whether the job continues) b. If we multiply both sides of the equation by (1+r∆), rearrange terms and divide through by ∆, we get the continuous time expression ())()( wVVqwwrVeue−+= c. Further rearranging terms, we get ueVqrqqrwwV+++=)( the value of employment reflects 1) the future value of wages, discounted by both the interest rate and the probability of job loss 2) the possible value of future unemployment also discounted by the interest rate and probability of job loss3. The present discounted value of job search from unemployment in continuous time is ()∫∞−λ+−=urVueuudwwfVwVcbrV )()( (student’s should verify that this is the appropriate expression even with job separations) 4. Substituting the term for the value of work and rearranging terms, we obtain the expression ()()∫∫∞∞−+λ+−=−+λ+−=rrwuwrurdwwFqrcbdwwfwwqrcbw)(1)( 5. Expression is nearly the same as in the standard model; the only difference is that the wage benefits of search are now discounted by both the interest rate and the probability of job loss C. Extension of model to include on-the-job search 1. Assumptions a. We will maintain the assumptions from the previous model; however, we will also assume that job offers continue to arrive after a person accepts a job b. The job offers during employment come from the same wage distribution as offers during unemployment; the only difference is that the arrival rate of offers during employment, λe, is lower thanthe rate during unemployment, reflecting different intensities of employed and unemployed search c. There are no costs to switching jobs; so, workers accept any job offer that pays more than they are currently making 2. With these assumptions, the value of employment is ()[]∫∞εε−ελ+−+=weeeeuedfwVVwVVqwwrV )()()()()( this is the same as our previous expression, except for the last term, which reflects the value of higher wage job offers 3. The value of unemployed search remains ()∫∞−λ+−=urVueuudwwfVwVcbrV )()( 4. Define the value of working at the reservation wage by substituting wr in for w in the value of employment equation; rearranging terms, we get ()[]∫∞−λ−−−=rwreeereurerdwwfwVwVwVVqwrVw )()()()()( 5. However, at the reservation wage, Vu = Ve(wr); thus, the expression simplifies to ()[]∫∞−λ−λ+−=rwueeurdwwfVwVcbw )()( 6. It can be shown that the expression further simplifies to ()()∫∞−λ++−λ−λ+−=rweeurdwwFqrwFcbw)(1)(1 7. Reservation wage for accepting a job from unemployed job search is lower when on-the-job search is alloweda. in the case where λu = λe, the reservation wage collapses to b – c 1) people accept the first job that is offered that exceeds the net benefits/costs of unemployment 2) if the net costs are zero or if b – c defines the floor of the wage distribution, spells of unemployment are only determined by the arrival rate of job offers b. in the case where λe = 0, the reservation wage returns to the simple case with job terminations D. Equilibrium model with an endogenous wage distribution 1. Assumptions a. We will assume that the number of workers and firms is fixed; for convenience, we will set the size of each group at one b. Continue to use the stationary search framework with job terminations and on-the-job search 1) terminations refresh the pool of unemployed people and help to anchor the wage distribution 2) separations from on-the-job search help to produce a non-degenerate wage distribution c. We will treat the separation and offer arrival rates, q, λu, and λe, as fixed (which is admittedly unrealistic); equilibrium will be achieved through the wage distribution (i.e., through the wages that firms offer) 2. Some additional notationa. let ℓ(w) represent the number (proportion) of workers working at a firm that pays w b. let ∫εεε=wdfwL0)()()( l represent the proportion of all workers who work in firms paying less than w c. let u denote the proportion of workers who are unemployed (unemployment rate) 3. Flows into jobs a. consider the flows of workers into jobs that pay wages of w or more b. the proportion of these jobs is 1–F(w) c. rate of flows from unemployment is λuu[1–F(w)] d. rate of flows from employment is λeL(w)[1–F(w)] 4. Flows out of jobs that pay w or more are q[1 – u – L(w)] 5. In equilibrium, flows into and out of jobs at each wage are equal [λuu + λeL(w)][1–F(w)] = q[1 – u – L(w)] a. because this relationship holds at all w and because u is constant at all w, [][])()()()(1)( wLqwfwLuwFwLeue′−=λ+λ−−′λ which is obtained by differentiating both sides of the previous expression by w b. noting that L′(w) = ℓ(w)f(w), we can rewrite this as ()[][]()[])()()(1)()()()()(1wLuwwFqwfwLuwfwwFqeueeueλ+λ=−λ+λ+λ=−λ+llc. since this relationship also holds at all w and u is constant at all w, we can differentiate again