7 Unemployment. 7.1 Introduction. JEM004 Macroeconomics IES, Fall 2017 Lecture Notes Eva Hromádková

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1 JEM004 Macroeconomics IES, Fall 2017 Lecture Notes Eva Hromádková 7 Unemployment 7.1 Introduction unemployment = existence of people who are not working but who say they would want to work in jobs like those held by individuals similar to them at the same wages 2 basic macroeconomic issues: 1. Is the existence of nonzero average unemployment over extended periods a market failure? What are its causes and consequences? - 2 positions: unemployment as natural implication of frictions in the process of matching workers and jobs - no problem unemployment as result of non-walrasian features of the economy - waste of resources 2. Why do shifts in a labor demand lead to large movements in employment and only small changes in the real wage? Walrasian labor market + high elasticity of supply X non Walrasian labor market + low elasticity of supply in competitive (Walrasian) markets, higher supply of labor (unemployed workers) would drive the wages down until balance is restored possible departure mechanisms: 1. rms do not want to reduce wages (e.g. due to motivation) => eciency wages 2. agreement with workers (unions) prevents rm from cutting wages => contracting models 3. heterogeneity among jobs and workers who have to create optimal matches => search and matching models no departure => unemployed = people moving between jobs or looking for higher wages than they can obtain 1

2 7.2 Eciency Wages Basic assumption: there are not only costs but also benets to a rm that pays a higher wage, which might create incentives not to adjust (lower) wages when facing a higher labor supply it enhances workers' productivity it motivates workers' eort which a rm cannot monitor perfectly it works as a signalling device - attracts pool of higher ability workers it builds loyalty towards the rm it reduces turnover, lower the likelihood of emergence of an union 7.3 Shapiro-Stiglitz Model Main idea: In a Walrasian labor market with market clearing wage, workers are indierent about losing their jobs, since identical jobs are immediately available. Thus, if the only way rms can punish shirking workers is by ring them, workers in such a labor market have no incentive to exert eort. But if a rm pays more than the market clearing wage, its jobs are valuable. Its workers may then choose to exert eort even if the monitoring technology is not perfect Assumptions There is a large number of identical workers L There is a large number of identical rms N Time is continuous, both workers and rms have innite decision horizon Representative worker maximizes his expected lifetime utility U = E [ e ρt u(t)dt ], ρ > 0 (1) 0 where u(t) is state-dependent instantaneous utility u(t) = { w(t) e(t) if employed 0 if unemployed where w denotes wage and e eort exerted by the worker. assume that e can only take two values For simplicity, we { ē > 0 i.e. exerting eort e(t) = 0 shirking 2

3 At any moment t a worker is one of three states: employed and exerting eort (E), employed and shirking (S) or unemployed (U). Jobs end at an exogenous rate given by Poisson process: if a worker begins to work at some job at time t 0 and he exerts eort, than the probability that worker is still employed in the same job at time t is P (t) = e b(t t 0), b (0, 1) This implies that if a worker is employed at some time t, the probability that he is still employed some time τ later is P (t + τ) P (t) = e bτ i.e. this probability is independent on how long the worker has been already employed. An equivalent way to describe this process is to say that job breakup occurs with probability b per unit of time (b is hazard rate for job breakup). Previous expression can be also rewritten (assuming τ 0) as P (t) = bp (t) Firms detection of workers that are shirking is also a Poisson process: detection occurs with probability q per unit of time, where q is exogenous and independent of job breakup rate. If worker is caught shirking, he is red. Therefore, if a worker is employed but shirking, the probability that he is still employed at time τ later is e (b+q)τ. Unemployed workers nd employment at rate a per unit of time. Each worker takes a as given, but in the economy as a whole a is determined endogenously. When rms want to hire workers, they choose randomly out of the pool of unemployed workers. Since workers are identical, the probability of nding a job does not depend on how workers became unemployed or on how long they are unemployed (i.e. no reputation). Representative rm's prots at time t are π(t) = F [ēl(t)] w(t)[l(t) + S(t)] where L is the number of employees exerting eort and S is the number of employees shirking. Firms face the problem to set the wage w suciently high that its workers do not shirk (S = 0) and to choose L. 3

4 The production function has the following properties: F ( ) > 0, F ( ) < 0 and F [ē L/N] > 1 i.e. marg. product of labor at full employment (i.e. when each rm employs 1 N of workers) is bigger than marginal disutility from exerting eort We focus our analysis on steady states Dynamic programming Value functions V i, i = E, S, U - expected present value of discounted lifetime, conditional on being in state i today: look at the small time interval t -> summarize what happens during and in the end of the interval -> let t approach to 0 steady state => if it is optimal for a worker to exert eort (or shirk) in the beginning of the period it continues to be optimal V E ( t ) = t e bt e ρt (w ē)dt + e ρ t [e b t V E ( t) + (1 e b t )V U ( t)] 0 }{{}}{{} during interval after interval after computing the integral, 1 we obtain V E ( t ) = (1 e (b+ρ) t ) (w ē) + e ρ t [e b t V E ( t) + (1 e b t )V U ( t)] b + ρ we solve for V E ( t) V E ( t) = 1 b + ρ (w ē) e (b+ρ) t e ρ t (1 e b t )V U ( t) Finally, letting t > 0 (using L'Hospital rule) 2 V E = 1 ρ + b [(w ē) + bv U] 1 t 0 e bt e ρt (w ē)dt = (w ē) 2 lim t >0 e ρ t (1 e b t ) 1 e (b+ρ) t [ ] t [ ] e (b+ρ)t b+ρ = (w ē) e (b+ρ) t b+ρ + 1 b+ρ 0 = lim t >0 ρe ρ t +(b+ρ)e (b+ρ) t (b+ρ)e (b+ρ) t = b b+ρ 4

5 How to derive this "intuitively": Think of an asset that pays dividends at rate w ē per unit of time when the worker is employed and no dividends when the worker is unemployed. Assume also that the asset is being priced by risk-neutral investors with required rate of return ρ. Since the expected present value of lifetime dividends of this asset is the same as the worker's expected present value of lifetime utility, the asset's price must be V E when the worker is employed and V U when the worker is unemployed. For the asset to be held, it must provide an expected rate of return of ρ. That is, its dividends per unit of time plus any expected capital gains or losses per unit of time must equal ρv E. When the worker is employed, dividends per unit of time are w ē, and there is a probability b per unit of time of a capital loss of V E V U. Thus ρv E = (w ē) b(v E V U ) (2) For shirking worker, his dividend while working is equal to w and the expected capital loss every unit of time is (b + q)(v S V U ) ρv S = w (b + q)(v S V U ) (3) For unemployed worker, his dividend is 0 and the expected capital gain (under S.S. condition that rms pay high enough wage for employees not to shirk) it is a(v E V U ) ρv U = a(v E V U ) (4) No-Shirking Condition rm has to pay enough that an employed worker is indierent between exerting eort and shirking, 3 so that V E = V S from equations (2) and (3) we get V E V U = ē q rm sets wages high enough for worker to strictly prefer employment to unemployment - they obtain rent (increasing in eort and decreasing in probability of detection). nally, from equations (2) and (4) we can compute what the wage must be for the rent to equal ē/q w = ē + (a + b + ρ)ē q (5) 3 There is no incentive for paying more than minimum, as worker cannot exert higher eort than ē. 5

6 7.3.4 Equilibrium The economy is in steady-state, thus movements out and into unemployment must balance NLb = ( L NL)a This allows us to compute the endogenous rate of "job creation" a (= rate of nding jobs by workers), which is increasing in employment level (N L) a = NLb L NL After substitution into wage function equation (13) we get new expression for wage, also called no-shirking condition. w = ē + ( ρ + L L NL b) ē q The wage that a rm must pay to induce its workers to exert eort is an increasing function of level of employment (this implies higher rate of nding a job a and lower loss from being caught shirking). Figure 1: Shapiro-Stiglitz model. When employed workers exert eort, the prot of representative rm equals π = F (ēl) wl Labor demand is therefore determined by equalizing marg. product of labor with its cost ēf (ēl) = w 6

7 Labor supply is horizontal at ē up to L and then vertical. In the absence of imperfect monitoring, equilibrium occurs at intersection of labor demand and supply with respective wage higher than ē (because by assumption,f [ē L/N] > 1). Under imperfect monitoring, equilibrium occurs at the intersection of no-shirking condition and labor demand curve. This equilibrium implies unemployment Short-term uctuations: Fall in Labor Demand Consider a fall in labor demand like in Figure 4 Since labor supply is inelastic, unemployment necessarily responds more than in would without imperfect monitoring. The model thus suggests one possible reason why wages may respond less to demanddriven output uctuations than they would if workers were always on their labor supply curves. Figure 2: Fall in labor demand in the Shapiro-Stiglitz model. 7

8 7.4 Search and Matching Models In a Walrasian labor market, rms are indierent about losing their workers, since identical workers are immediately and without cost available at the same wage. Likewise, workers are indierent about losing their jobs. However, when workers and jobs are heterogeneous then rather than meeting in centralized markets where the employment level and the wage are determined by the intersection of labor demand and supply curves; workers and rms meet in a decentralized, one-to-one encounters; they engage in a costly process of match up idiosyncratic preferences, skills and needs. This process is not instantaneous, and therefore generates unemployment. 7.5 Mortensen-Pissarides Model Assumptions 1. The economy consist of worker who are either employed (E) or unemployed (U), and jobs that are either lled (F ) or vacant (V ). 2. Each job is lled by one worker, i.e. F = E. 3. The size of the labor force is xed E + U = L 4. The number of jobs is endogenous - vacancies can be created and eliminated costlessly. 5. Fixed cost of maintaining a job (per unit of time) is C regardless whether the job is lled or vacant. 6. An employed worker produces output at rate Y > C per unit of time, i.e. the surplus produced by match is Y C, and is paid an endogenous wage w per unit of time. 7. No costs of eort or search in this specication (but usually part of extension). 8. Workers maximize the expected present discounted value of lifetime utility U = E [ e rt u(t)dt ] 0 where r > 0 is the discount rate, which is exogenous and constant. Instantaneous utility u(t) can take two values u(t) = { w(t) if employed 0 if unemployed 8

9 9. Firms maximize the expected present discounted value of lifetime prots, where { Y w C if job is lled π(t) = C if job is vacant Also, rms face the same discount value r as workers 10. Number of people that nd a job in the given unit of time (or number of lled vacancies) is determined by matching function M(U, V ) - proxy for recruitment, search and evaluation procedures. M(U, V ) = KU β V γ, β, γ [0, 1]; K > Filled jobs end at an exogenous rate b per unit of time. Therefore we can express the change in the number of employed people as Ė = M(U, V ) be 12. We focus on the steady-state, i.e. outow and inow into employment is in balance 13. Dene Ė = 0 M(U, V ) = be a M(U, V ), α M(U, V ) U V where a is the probability with which an unemployed worker becomes employed, and α is the probability with which vacancy is lled Dynamic programming Value functions for worker: rv E = w b(v E V U ) (6) rv U = a(v E V U ) (7) Value functions for rms: rv F = (Y w C) b(v F V V ) (8) rv V = C + α(v F V V ) (9) Bargaining When unemployed worker and rm with vacancy meet, they bargain over wage. Our assumption - each of the sides will get the same gain, i.e. V E V U = V F V V 9

10 7.5.4 Equilibrium As new vacancies can be created and eliminated costlessly, the value of vacancy must be zero. V V = 0 Combining equations for workers and rms we will get expressions for the gains from employment V E V U = w a + b + r (11) V F V V = Y w α + b + r (12) Considering the outcome of bargaining, we get the expression for equilibrium wage a + b + r w = a + α + 2b + 2r Y (13) If a = α (i.e. market is equally tight from the both sides), rm and worker divide the output from the job equally. The ratio of α and a determines the bargaining power, e.g. if a > α workers can nd job faster than rms can ll their vacancies - more than half of the output goes to worker. Combining (9), (12) and (13) brings rv V = C + α(e) a(e) + α(e) + 2b + 2r Y wherea and α as rewritten as functions of employment E. From the fact that U = L E and M(U, V ) = be we get a(e) = By the same token be L E, a ( ) > 0 α(e) = K 1 γ 1 β γ (be) γ ( L E) γ, α ( ) < 0 This implies ( + picture 3): rv V is decreasing in E As lim E > L a = and lim E > L α = 0, then lim E > L rv V = C As lim E >0 a = 0 and lim E >0 α =, then lim E >0 rv V = Y C In the equilibrium, employment is determined by the free-entry condition V V = 0 and it obviously holds E < L - existence of average unemployment. In the absence of frictions (M, as well as C), labor supply would be perfectly inelastic at L (workers are always better o by working with positive wage) and labor demand is perfectly elastic at Y C > 0 - implies full employment at wage w = Y C - workers are in position of "power" because rms want to employ everybody, i.e. workers can extract all the rent from the match. 10 (10)

11 Figure 3: Equilibrium unemployment in the search and matching model Productivity drop Consider a permanent fall in (productivity) which implies that rms are willing to pay lower wage. Employment falls because in the absence of a frictionless market, workers are not costlessly available at the prevailing wage. The reduction in Y raises rms' costs of searching for a worker relative to the prots they obtain when they nd one (remember C is constant). Thus, the number of jobs,and hence employment, falls. The model does not imply substantial wage rigidity: a fall in Y leads to a fall in E, which in turn implies a fall in a and a rise in α. When unemployment rises, workers cannot nd jobs as easily as before, and rms can ll vacancies more rapidly. From equation (18), this implies that the wage falls more than proportionally with Y. Figure 4: Eects of a rise in labor demand in the search and matching model. If you are interested in these kind of models, start with Mortensen, D., and Pissarides, D.C., (1994). Job Creation and Job Destruction in the Theory of Unemployment. Review of Economic Studies 61,