4. Unemployment (July 15, 2013)
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1 Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS Unemployment (July 15, 2013) Introduction Neoclassical model Basic efficiency wage model Shapiro-Stiglitz model Search and matching
2 Introduction (1) Umemploment Rate (ILO-Definition) Canada France Germany Italy Japan
3 Introduction (2) In any economy at almost any time there are many people who are not working but who say that they want to work in jobs held by individuals similar to them, at the wages those individuals are earning. Two basic questions in the field of macroeconomics of unemployment What are the determinants of average unemployment over extended periods of time? How can the cyclical behavior of the labor market be explained? Shifts in labor demand lead to large movements in employment and only small movements in the real wage. 3
4 Relation between Emplyoment and Output Gap 4
5 Relation between Wages and Output Gap 5
6 Introduction (3) Walrasian labor market (wages are flexible such that supply equals demand) In the neoclassical textbook model of the labor market, there is no unemployment. Technological shocks may lead to fluctuations in employment in a dynamic setting. Observed unemployment fluctua ons may be interpreted as op mal responses ( RBC Theory). Non Walrasian labor market ( imperfections are important ) Nominal wage rigidities might lead to unemployment (e.g. minimum wages). Real wage rigidities prevent the clearing of the labor market. This is to say that, despite nominal wage flexibility, it might not be optimal for firms to set the market clearing wage. If heterogeneity among workers and jobs is important, the labor market should not be viewed as a single market. In this case, the process of matching up workers and jobs occurs not through markets but through a complex process of search. 6
7 Introduction (4) Assume that an unemployed worker offers to work for slightly less than the firm is currently paying. The firm might respond in several ways: The firm might refuse to cut wages since this might reduce the efficiency of employed workers ( efficiency-wage models). The firm might wish to cut wages but implicit or explicit agreements with its workers prevents it from doing so ( contrac ng models, insider-outsider models). The firm might refuse the offer since the ability of this worker is lower than the ability of its current workforce. Heterogeneity among workers implies that the labor market is not a large single market. Each worker and each job should be thought of as distinct. As a result, the process of matching up workers and jobs occurs not through markets but through a complex process of search ( search models, matching models). The firm can accept the worker s offer. In this case, the labor market works like a Walrasian market. 7
8 The neoclassical labor market labor demand labor supply (L*,w*) Labor supply curve is upward sloping: substitution effect dominates income effect. No unemployment: equilibrium amount of labor is L* and the equilibrium wage rate is w*. Shifts in labor demand (due to technological shocks or shifts goods demand) lead to procyclical fluctuations in employment and wage rate. Extending this basic model by a minimum wage gives a first reasonable explanation for unemployment. 8
9 Basic efficiency wage model: introduction Basic idea: paying a higher wage represents a cost to the firm but there might also be a benefit of paying higher wages. A higher wage may represent a benefit to the firm due to increased food consumption, better nutrition and higher productivity of workers. This argument seems to be important in DC but not important in IC. increased worker effort in situations where monitoring is imperfect. Paying a wage rate above the market clearing level, the firm creates an incentive workers not to shirk. build up of loyalty among workers and hence induce high effort. Conversely, a low wage can anger and desire for revenge and thereby lead to shirking or sabotage. an increase average ability of the workers resulting from a higher reservation wage of high quality workers. 9
10 Basic efficiency wage model: model setup There is a large number N of competitive firms. The representative firm seeks to maximize profits The output technology is described by Effort e depends on the wage the firm pays ( crucial assumption ) There are L identical workers. Each worker supplies one unit of labor inelastically. 10
11 Basic efficiency wage model: analyzing the model (1) The problem of the representative firm is If there are unemployed workers, the firm can choose the wage freely. If unemployment is zero, the firm must pay at least the wage paid by other firms. When the firm is unconstrained, the first order condition for L and w are These two equations implicitly determine L * and w *. The condition F (.)e=w states that the firm hires workers until the marginal product of el equals its cost. 11
12 Basic efficiency wage model: analyzing the model (2) Plugging F (.)=w/e(w) into (**) and dividing by L gives This condition is known as the Solow condition. Profit maximizing firms want to hire effective labor, el, as cheaply as possible. The Solow condition is the first-order condition for minimizing the cost per unit of effective labor wl/(e(w)l)=w/e(w) w.r.t. w. When the elasticity of e(w) w.r.t. w is one, a marginal change in w has no effect on w/e(w). (When we /e=1 the two opposing effects of increasing w cancel.) 12
13 Basic efficiency wage model: illustration A parameterized effort function A w* is the wage rate which minimizes the cost per effective unit of labor w/e(w). Point A obviously satisfies e (w)=e(w)/w or we (w)/e(w)=1. α W* 13
14 Basic efficiency wage model: implications Since firms are identical each firm chooses the same values of w* and L*. Total labor demand is NL*. Two cases need to be distinguished If L>NL* firms are unconstrained in their choice of w. The wage is w*, employment is NL*, and there is unemployment of L NL*. If L<NL* firms are constrained. The wage is bid up to the point where L=NL*. In this case, the efficiency wage the firm wishes to pay is below the market clearing wage and hence efficiency wages do not contribute to unemployment. The efficiency wage w* is entirely determined by the effort function! Shifts in labor demand (due to shifts in aggregate demand) have no effect on the real wage but large effects on employment (assuming that L>NL*). As economic growth shifts the demand for labor outward, the real wage remains unchanged and unemployment trends downwards. Eventually, unemployment reaches zero, at which point further increases in demand lead to increases in real wage. 14
15 Shapiro-Stiglitz model (1) Assumptions (1) There is mass L of identical workers and mass N of identical firms. The representative worker maximizes lifetime utility There are only two possible effort levels, either e=0 or e=e>0. Hence, at any moment of time a worker is in one of three states employed unemployed exerting effort State E: employed and exerting effort not exerting effort State S: employed and not exerting effort State U: unemployed 15
16 Shapiro-Stiglitz model (2) Assumptions (2) The transitions between the three states follow Poisson processes State E State E exp[ b(t t 0 )] is the probability that a worker who is employed and exerting effort in t 0 is still employed in the job at t (notice: for t=t 0 exp[ b(t t 0 )]=1). That is, the probability of loosing the job when exerting effort per period dt is b>0. State S State U State S Employment exp[ (b+q)(t t 0 )] is the probability that a worker who is employed and shirking in t 0 is still employed in the job at t. The probability of loosing the job when shirking per period dt is (b+q)>0. That is, the differential effect of shirking, compared to exerting effort, is q>0. exp[ a(t t 0 )] is the probability that a worker who is unemployed in t 0 is employed at t. That is, the probability of finding a job per period dt is a>0. The respective probabilities depend only on t t 0 and are independent of the specific point in time. This implies that it is not necessary to keep track of how long workers have already been employed. 16
17 Shapiro-Stiglitz model (3) Assumptions (3) The instantaneous flow of profits reads (F (.)>0 and F (.)<0) The problem of the representative firm comprises two stages set w sufficiently high that its workers do not shirk then chose the amount of labor The firm s decision at any date does not affect future profits. Hence there is no need to analyze the present value of firms. It is also assumed that (see below) L(t) is the number of workers who are exerting effort. S(t) is the number of workers who are shirking. 17
18 Shapiro-Stiglitz model (4) Values of E, S, and U Let V i for i {E,S,U} denote the value of being in state E, S or U (i.e. expected lifetime utility). The determination of the V i is simplified by two assumptions Transitions among states follow Poisson processes transition probabilities are constant V i do not depend on how long the worker has been in his state. The focus is on steady states implying that the V i are constant over time. The dynamic programming approach is employed. That is, we look only at a brief interval of time and use the V i to summarize what occurs after the end of the interval. V E, V S and V U are determined by the following set of equations (see below) See the next slide for an explicit derivation of equation (1). For a heuristic reasoning see slide Search and Matching (4a). These 3 equations can be viewed as 3 no arbitrage conditions between assets V E, V S and V U. In what follows, we try to understand the economic implications of these conditions together with V E =V S. 18
19 Shapiro-Stiglitz model (4a) The expected PDV of being employed in time t=0 may be expressed as Noting that e at dt= (1/a)e at one can write the 1 st expression as Hence, one may write Let Δt 0 such that V E (0)=V E (Δt)=:V E one gets Solving for V E gives Applying L Hopital s rule gives (recall e 0 =1) Therefore, one finally obtains 19
20 Shapiro-Stiglitz model (5) The No Shirking condition (1) Firms must pay enough to prevent workers from shirking, i.e. V E V S. At the same time, profit maximizing firms do not want to pay more since the maximum effort is e. Hence, the firm chooses w such that V E =V S. Equ. (1), (2) and V E =V S (in fact, in equilibrium V E V S =ε with ε>0 arbitrarily small) imply Firms set wages high enough such that workers strictly prefer employment to unemployment, i.e. workers obtain a premium. What is the implied wage which gives rise to this premium e/q? Subtracting (3) from (1) gives Wage needed to induce effort is increasing in the cost of effort (e), the ease of finding jobs (a), the rate of job breakup (b), and the discount rate (ρ). It is decreasing in the probability that shirkers are detected (q). 20
21 Shapiro-Stiglitz model (6) The No Shirking condition (2) Next, we express the wage needed to prevent shirking in terms of NL. Since the economy is in a steady state (i.e. unemployment remains constant), the following relation must hold This equation implies a+b=bl/(l NL). Plugging the RHS into w=e+(a+b+ρ) e/q gives This is the No Shirking Condition (NSC). It shows the wage that deters shirking as a function of employment. The wage needed to deter shirking is increasing in NL. The higher is employment, the lower is unemployment. It is comparatively easy to find a new job and the costs of being unemployed are low. At full employment, unemployed workers find work instantly. Hence, the cost of being fired is zero and no wage can deter shirking. 21
22 Shapiro-Stiglitz model (7) The labor market Labor demand No-shirking condition Profit maximizing firms do two things Offer wages that induces effort, i.e. pay wages that satisfy the NSC. Firms then hire workers up the point where the MPL equals the wage, i.e. ef (el)=w. Labor supply is horizontal at e up to the number of workers (L) and then vertical. Equilibrium under imperfect monitoring is given by E. There is equilibrium unemployment! E E W The Walrasian equilibrium is given by point E W. ef (el/n)>e guarantees that labor demand intersects labor supply in the vertical segment. L 22
23 Shapiro-Stiglitz model (8) Implications The Shapiro-Stiglitz model explains equilibrium unemployment. Unemployed workers prefer to be employed at the prevailing wage and would exert effort to remain employed. Nonetheless, they cannot bid down the wage since firms know that if they hire additional workers at a lower wage, workers will prefer shirking. Implications for short run fluctuations: suppose that labor demand falls. Employment and the wage move down along the NSC. Since labor supply is perfectly inelastic, employment responds more than it would without imperfect monitoring. The model provides a candidate explanation for the observation that wages may respond less to demand-driven output fluctuations (see the wage-elastic branch of the NSC). The decentralized equilibrium is inefficient. Since the MPL at full employment is, by assumption, larger than the marginal disutility of exerting effort, i.e. ef (el/n)>e, the first-best allocation requires that everyone is employed. Shapiro and Stiglitz suggest that wage subsidies financed by lump sum taxes or profit taxes improve welfare. Such a policy shifts the labor demand curve up and thus increases w and NL along the NSC. Since the value of additional output exceeds the opportunity costs of production, overall welfare would increase. The distribution of the gain depends on how the wage subsidies are financed. 23
24 Search and Matching: introduction In a frictionless labor market, firms are indifferent about loosing their workers since identical workers are costlessly available at the same wage. Likewise, workers are indifferent about loosing their jobs. When workers and jobs are highly heterogeneous, however, the labor market has little resemblance to a Walrasian labor market. Rather than meeting in centralized markets where employment and wages are determined by the intersection of supply and demand, workers and firms meet in a decentralized one-to-one fashion, and engage in a costly process of trying to match up heterogeneous preferences, skills, and needs. Since this process is not instantaneous, it results in some unemployment. 24
25 Search and Matching: the model (1) Workers and jobs can be in two different states employed (number: E) filled (number: F) Workers Jobs unemployed (number: U) vacant (number: V) Each job can have at most one worker ( F=E). The labor force is fixed at L ( E+U=L). The model is dynamic, time is continuous, only steady states are considered. Vacancies can be created or eliminated freely. 25
26 Search and Matching: the model (2) Firms: Profits per period of time from a job An employed worker produces output A per period of time (A>C). Maintaining a job (filled or vacant) requires a fixed cost C per period of time. Firms maximize the expected PDV of lifetime profits (discount rate is r). Workers: utility per period of time is There is free entry into the firm sector. Setting up a firm is equivalent to setting up a position, which may be filled or vacant. Parsimonious modeling! ( But the real world is much more complex?!!). Costs of job search and effort are ignored. Workers maximize the expected lifetime utility (discount rate is r). 26
27 Search and Matching: the model (3) Unemployment and vacancies yield a flow of new jobs per unit of time This matching function proxies for the complicated process of employer recruitment, worker search, and mutual evaluation. If β+γ>1 there are thick market effects. Increasing the level of search makes the matching process operate more effectively; it yields more matches per period of time. If β+γ<1 there are crowding out effects. The dynamics of employment (E) is described by As in the Shapiro-Stiglitz model jobs end at an exogenous rate b>0 per period of time. Since we are focusing on steady states M(U,V)=bE (implying E=0) must hold. 27
28 Search and Matching: the model (4) Consider different states (workers: E and U; firms: F and V) as assets and apply the reasoning explained below to describe equilibrium (compare to equ. (*)). Then in equilibrium the following relations must hold a:=m/u denotes the rate per unit time that unemployed workers find jobs. α:=m/v is the rate per unit time that vacant jobs are filled. 28
29 Search and Matching: the model (4a) Consider a world with two assets A and B. Let V A denote the nominal value of A and V B the nominal value of B. A offers a payoff of π A and B offers a payoff of π B. Asset B may devaluate completely with probability p. Assuming that agents are risk neutral, the following equilibrium condition must hold Let a=π A /V A one may write 29
30 Search and Matching: the model (5) When a vacant position is getting filled, workers and firms must agree on the wage. Wage must be high enough that the worker wants to work in the job and low enough that the firm wants to hire the worker. There is a range of wages that makes both parties better off. Assumption: Workers and firms set the wage so that each gets the same gain, i.e. When a=α, the firm and the worker divide the output from the job equally. When a>α, workers can find new jobs more rapidly than firms can find new employees, and so more than half of output goes to workers (and vice versa). Since there is free entry into the firm sector, the value of vacancies must be zero in equilibrium, i.e. V V =0 ( free entry condition ). This condition is employed to determine E in equilibrium. 30
31 Search and Matching: solving the model Recall equ. (4) rv V = C+α(V F V V ) and substitute V F V V according to V F V V =(A w)/(a+b+r) to get a and α are, however, endogenous and do depend on the level of employment E. Next we express a and α as functions of E, i.e. a=a(e) and α=α(e). The requirement V V =0 then determines the level of employment E. Probability that unemployed find job. From M=bE and E+U=L one gets Recall a:=m/u denotes the rate per unit time that unemployed workers find jobs. Probability that vacant jobs get filled. From M=bE and M=KU β V γ one obtains α:=m/v is the rate per unit time that vacant jobs are filled. 31
32 Search and Matching: main result - illustration The equilibrium level of employment E is determined by V V =0. A-C E* L=1 A cyclical upswing can be described by an increase in A. The rv V locus shifts upward and E rises. In contrast, in a Walrasian labor market E remains constant (E=L). The frictions under study imply that workers are not costlessly available at the prevailing wage. The increase in A, with C fixed, raises profits. Hence, the number of firms (hence E) rises. -C The model does not imply substantial wage rigidity, however. The rise in E causes a to rise and α to fall. This implies that the wage rises more than proportionately with A. 32
33 Notation and abbreviations (1) Neoclassical model and efficiency wages 0<ß<1 constant parameter (effort function) b>0 constant parameter (effort function) e worker s effort el effective Labor L amount of labor L number of identical workers L* optimal labor chosen by firms N number of competitive firms u unemployment rate, utility of a worker w wage rate w* equilibrium wage rate w/e(w) costs per unit of effective labor w a Y η ew π wage rate paid by other firms final output elasticity of effort w.r.t. the wage profit of the firm Shapiro Stiglitz model a>0 constant parameter b>0 constant parameter (effort function) e worker s effort e fixed value of e L number of identical workers N large number of competitive firms q>0 constant parameter S number of worker who are shirking t time-index u(.) utility from consumption V i w value of being in state i wage rate Ρ>0 time preference rate 33
34 Notation and abbreviations (2) Search and Matching 0 β 1 constant parameter 0 γ 1 constant parameter A technology parameter b>0 constant parameter C consumption E number of employed workers E variation of E F number of filled jobs K>0 constant parameter M matching function p probability r interest rate U number of unemployed workers V i V w π i nominal value of asset i number of vacant jobs wage rate payoff of asset i Abbreviations DC developing countries FOC first-order conditions IC industrialized countries RBC real business cycle w.r.t. with respect to 34
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