Lecture Notes # 4 Tito Boeri

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1 Lecture Notes # 4 Tito Boeri 1 Labour market "rigidities" in the MP model Before discussing the implications of having employment and wage rigidities in the MP model, it is important to evaluate the welfare properties of the laissezfaire equilibrium. 1.1 Is Nash-bargaining efficient? There is an optimal or efficient level of unemployment in this model, which maximises social output (market output net of recruiting costs plus unemployment income). Alternative bargaining rules (as well as other policies) can be evaluated in terms of their capacity to attain this level. The economics behind this normative discussion is that bargaining rules should internalise the congestion and trading externalities involved by job matching. These externalities are positive in the other side of the market because they make it thicker (trading externalities) and negative on the same side because of the competition with other job-worker seekers (congestion-search externalities). In Nash bargaining wages are set after the match of the firm and the worker, hence they are unlikely to take into account the effects on workers and firms that are still searching. Let us consider the model with exogenous job destruction (the separation rate being λ) for simplicity. This is also the reason for the hold-up problem characterised by the literature. As we are interested in wage setting and search externalities, we can evaluate the efficiency properties in the simplest version of the MP, namely with exogenous job destruction and fixed match productivity equal to one unit (hence total output is equal to total employment). Formally, efficiency of Nash bargaining means that wage determination is consistent with the maximisation of social output of an infinitely lived economy: Ω = Z 0 e rt [p(1 u)+bu cθu]dt subject to the law governing the evolution of unemployment:. u = λ(1 u) θq(θ)u (1) The planner chooses unemployment and market tightness in such a way to maximise the objective. The first-order Euler conditions (necessary conditions) for unemployment and market tightness of this problem read respectively: e rt [p b + cθ]+µ [λ + θq(θ)] µ. =0 (2) where µ is the Lagrange multiplier associated with the law of motion of unemployment (1) and 1

2 e rt [cu + µq(θ)u + µq0(θ)θu] = e rt c + µq(θ)u[1 η(θ) ] =0 (3) where η is the elasticity of the vacancy filling rate (q(θ)) wrt market tightness, that is, a measure of negative search externalities associated with having more firms issuing vacancies. The second order conditions are satisfied by the concavity of the matching function. Now solving (3) for µ and substituting into??, weget (p b + cθ)+ λ + θq(θ) q(θ)[1 η(θ) ] ] µ. =0 We then evaluate this at the steady state (where µ. = r ) getting, after a few manipulations [1 η(θ) ](p b) =c[ λ + r + η(θ) )θq(θ) ] q(θ) It is now possible to compare this optimality condition to the (private) job creation condition under Nash bargaining. emember that the equilibrium wage equation (see lecture notes #3) w =(1 β)b + β(p + cθ) (4) and use the free entry condition J = c q(θ) to obtain the following alternative expression for competitive wages c(r + λ) w =1 q (θ) We can then equate this to the right-hand-side of (4) and rearranging 1 c(r + λ) q (θ) =(1 β)b + β(p + cθ) (1 β)(p b) =c[ λ + r + βθq(θ) ] q(θ) emind now the efficient unemployment condition [1 η(θ) ](1 b) =c[ λ + r η(θ) )θq(θ) ] q(θ) it is easy to see that Nash bargainining statisfies this optimality condition when 2

3 β = η(θ) (5) The interpretation of this result (Hosios condition for optimality) is that the labour share should equal the elasticity of vacancy filling with respect to market tightness (a number between 0 and 1). The latter captures the percentage change in the average duration of vacancies associated with a one percentage change in the unemployment stock. The lower the responsiveness of vacancy duration to unemployment, the higher the labour share consistent with the attainment of social optimum because more unemployment is needed to generate significant trading externalities. Under Nash-bargaining, β is a parameter (reflecting the degree of time impatience of bargaining partners) while the elasticity η(θ) is not constant under general specifications of the matching function. Even when the elasticity is constant (under iso-elastic matching technologies), there is no reason to expect it to equal the labour share. When the wage share is larger than the elasticity, equilibrium unemployment is higher than at the optimum. The opposite happens when β is lower than the elasticity of job finding wrt θ. Put another way, when congestion costs are high (when adding a new jobseeker significantly reduces job finding rates of the other unemployed), the wage share ought to be small in order to allow the equilibrium to maximise social output. The above result can be extended (Pissarides chapter 8) to the case where job destruction is endogenous. It can also be shown that the Hosios condition is satisfied when markets are completed in the sense that search and trade externalities (matching delays) are properly priced by third party market makers offering to unemployed and employers a tradeoff between future income and matching delay (Mortensen and Pissarides HLE, 1999). 1.2 Mimimum Wages and Excess Coverage Governments can affect wage setting in two ways i) by setting minimum wages, and ii) by artificially extending the coverage of collective wage agreements negotiated within the unionised segment of the workforce (which is often minoritarian). Option i) is isomorphic to an increase in the (flat-rate) unemployment benefit, b. The practicability of option ii) is related to the degree of acceptance by workers and firms of wage compressing institutions, which is explored below. 1.3 Extensions of the MP model Let us extend the Mortensen-Pissarides in three directions: 1. assume that there are a fixed number of risk-neutral workers of skill types s drawn from the interval s [0, 1]. Skill can be thought of as a nonrandom, identifiable component of productivity, which defines a submarket for labor across which mobility is ruled out. Firms can work with all types of workers; when matched, a firm and a worker generate a flow productivity sx, where x [0, 1] is a match-specific component referred to 3

4 as a shock. All new matches (i.e. filled jobs) will once more start at the highest possible x (x =1). Immediately thereafter, the idiosyncratic productivity term jumps at Poisson frequency λ and is a random draw from a cumulative distribution F (x). 2. assume that whenever a productivity shock occurs, workers have to pay a one-off renegotiation cost ρ. It is assumed that the worker pays this cost nominally, but since wages are bargained over continuously, both parties will in general share the costs of renegotiation in equilibrium. 3. introduce an employment protection tax T (deadweight) to be paid by employers whenever a job is terminated 1.4 Asset value conditions 1. The valuation by workers of unemployment (U), and employment (W (x)), and by firms of an open vacancy (V )versusajob(j (x)) arenowgiven by the following four functional equations given x: rw(x) =w(x)+λ rj(x) = sx w(x)+λ ru = b + θq(θ)[w (1) U] (6) rv = sc + q (θ)[j(1) V ] (7) (W (z) W (x) ρ)df (z)+λf ()(U W (x)) (8) ((J(z) J(x))dF (z)+λf ()(V st J(x)). (9) Equations (6) through (9) set normal returns on capitalized valuations of labor market states to their periodic payouts. In equation (6), the flow yield from the valuation of the state of unemployment at interest rate r is equated to income in unemployment or leisure equivalent b, plus an expected capital gain stemming from finding new employment at x =1. The function W (x) in (8) returns the value of employment in a job-worker match with current productivity x. Givenx, the implicit rate of return on the asset W is equal to the current wage plus the expected capital gain on the employment relationship. Although the cost of renegotiation, ρ, is payed by the worker, since wages are bargained, both parties will ultimately share the costs of renegotiation in equilibrium. One interpretation of ρ is an unavoidable investment necessary to maintain the existing employment relationship, given that the shock has occurred. Employers surveys suggest that renegotiation, bargaining, information and organisational costs associated with wage adjustments to plant-level productivity changes can be substantial and discourage the adoption of productivity-related pay structures, even in non-unionised firms. (see slides) 4

5 1.4.1 Wage Determination For an existing match, the Nash-bargained wage is given by w(x) = arg max [W (x) w U]β [J(x)+sT V ] (1 β) yielding the first order condition W (x) U = β [J(x)+W(x)+sT V U]. (10) It is convenient to solve first for the steady-state valuation of unemployment U. Using the free entry condition V =0with (10) evaluated at x =1and inserting the result into (6) yields ru = b + βscθ 1 β. (11) The equilibrium state valuation of unemployment is linear in θ, whichisa sufficient statistic for tightness in labor markets. As in lecture 3, we use (11) to obtain the equilibrium wage rule: w(x) =(1 β)[b + λ(1 F ())ρ]+βs(cθ + x + rt). (12) Notice that the equilibrium wage depends not only on familiar parameters such as b (the monetary value of unemployment or leisure), θ (labor market tightness), x (match productivity), but also on T (severance cost), and λ, theshock probability, and the renegotiation cost ρ. These latter two factors are more important, the more likely a job is to survive (1 F ()). Idiosyncratic productivity shocks which do not lead to match dissolution make the worker partially liable for renegotiation costs. By abandoning the match and passing into unemployment, renegotiation costs can be avoided; consequently, a higher wage is needed to indemnify for this contingency. Effectively, the fallback of the worker is increased by the savings on future renegotiation costs that is implied by a breakdown of negotiations and spell of unemployment. The more power the employer has, the more likely will the wage reflect this compensating differential as opposed to insider rents. In contrast, bargaining power of workers links wages more tightly with idiosyncratic productivity, local market conditions, as well as the lock-in effect of the firing tax Job Creation The derivation of the job creation condition follows the same steps as in lecture 3: (1 β)(1 ) T = c (13) r + λ q (θ) This condition on and θ can be represented as a downward-sloping JCcurve (for job creation). Noticethatneithers nor ρ affects the position of the JC curve. The intuition for this result is that renegotiation costs do not affect the incentive to create a job at any given skill level, but rather influence the viability of the job via the surplus available to the match. Insofar as setup costs are proportional to skill in the particular labor market, there is no bias on the job creation margin in favour of a particular skill level. 5

6 1.4.3 Job Destruction Jobs are destroyed when productivity falls below its corresponding reservation or threshold level. is implicitly defined for each skill s by the condition J() = st. (14) At the same time, Nash bargaining also implies that satisfies the zero matchsurplus condition: J()+sT V + W () U =0 (15) and, given the free entry condition V =0, it follows that W () =U that is, separations are still privately, but not necessarily socially, efficient in the sense of Hosios (1990). The reservation productivity level for the competitive search market,, is determined implicitly by the job destruction condition: s + sλ r + λ (z )df (z)+rst = b + βscθ + λ [1 F ()] ρ (16) 1 β The left-hand side is the flow benefit of a continuing match with productivity ; this is the current flow product plus the option value deriving from possible future improvements over the following time interval. The right-hand side represents the (opportunity) costs of maintaining the match at the threshold value of idiosyncratic productivity, plus the expected value of renegotiation costs. This job destruction (JD) condition defines an upward-sloping curve in the (θ, ) space Equilibrium The intersection of (16) with the job creation condition (13) defines a labor market equilibrium for submarket with skill s. For each skill level there exists a unique equilibrium reservation productivity and labor tightness pair (,θ ) given by the implicit functions of deterministic productivity s, the Poisson arrival rate λ, worker bargaining power β, vacancy setup costs c, renegotiation costs ρ, firing tax T and income-equivalent in unemployment b: = (s, λ, β, c, ρ, b, T ) θ = θ (s, λ, β, c, ρ, b, T ). Given the equilibrium and θ, the unemployment rate in the labor submarket for skill level s follows from the familiar flow condition for constant unemployment: u u (s, λ, β, c, ρ, b, T )= λf ( ) λf ( )+θ q(θ ). (17) 6

7 1.5 A rigid wage regime Suppose that for some segments of the skill distribution wages depend positively on observable productivity s, but not on idiosyncratic shocks. In particular let w r = w r (s) with wr s > 0 and parametrize the wage schedule by wr = w + φs with 0 <φ<1, wherew b is a social minimum or minimum wage, while φ reflects skill-dependence of compensation independent of match productivity. Low values of φ suggest egalitarian wage structures, with higher values linking pay more tightly to systematic (deterministic) productivity. The job creation condition for a job in the rigid wage labor market is given by 1 r r + λ T = c q (θ r ). (18) In fact, use the equilibrium valuation equation to value a job under the rigid wage regime at x = r,imposingv r =0and the fact that at the destruction margin by definition J r ( r )+st =0,tosolveforλ 1 J r (z)df (z): r λ J r (z)df (z) =w r s r [r + λ (1 F ( r ))] st r Substitute this into the asset value condition with V r =0and obtain (r + λ) J r (x) =s (x r ) (r + λ) st. Now set x =1and use the zero profit condition in the rigid wage regime to obtain the JC-condition: sc (r + λ) J r (1) = s (1 r (r + λ)t )=(r + λ) q (θ r ) (1 r ) (r + λ) T = c q (θ r (19) ) Job Destruction The hallmark of the rigid wage regime is that the value of a job to the employee is independent of match productivity. Hence, the set of idiosyncratic productivities for which the job is destroyed will not necessarily coincide with those for which the job has zero value to the worker at the assumed rigid wage. ather, the participation constraint implies that for a given skill level, W r ( r )=W r >U r. In rigid-wage labor markets, the consensual dissolution of an employment relationship no longer applies, and there are always too many separations from the workers perspective. Separations are inefficient in the sense that for some range of productivities workers will be fired, but at the given wage, they would prefer to continue working. Except on a set of measure zero, there are only involuntary layoffs in the rigid wage regime. In contrast, quits and layoffs are indistinguishable in competitive search labor markets. 7

8 Because the rigid wage is not the outcome of individual level bargaining, surplus division obeys a rule of the residual claimant type. Let S r (x) be the total surplus resulting from a match for any s, so for any x [ r, 1] J r (x) =max( st, S r (x) (W r U r )). (20) The firm obtains all surplus greater than (W r U r ). The maximum operator applies since the firm can always close operation, here at cost st. Unlike the individual-wage labor market, the decision to destroy a job is taken by employers unilaterally and given by J r < st for any s; yet in general at this point W r >U r. The reservation productivity r for a match for skill level s, thatis, the reservation value for jobs under rigid wages is given by: s r + λs (x r )df (x) =w + φs rst (21) r + λ r Unlike the flexible, individually bargained wage case, the component related to renegotiation costs is absent. This expression represents the job destruction condition in the rigid search market, the JD-curve, in (θ r, r ) space, is horizontal, reflecting the independence of r of local labor market conditions. The firing tax T reduces the job destruction threshold and raises the average duration of ajob. In contrast to (16), neither labor market tightness (θ r ) nor individual worker bargaining strength (β) appear in the job destruction condition. The rigid wage influences the outcome via r, which is endogenously determined as the intersection of the JC and JD curves for every s. Asintheflexible wage labor market, an increase in λ ceteris paribus shifts back the job destruction curve towards the origin. The derivation of the job destruction condition (21) in the rigid wage regime is obtained as follows. ewrite the job valuation equation and impose V r =0 to obtain (r + λ) J r (x) =sx w r + λ J r (z)df (z) λf ( r )st. (22) r Substitute J r (z) = s(z r ) (r+λ) st in the integral on the right hand side: (r + λ) J r (x) = sx w r s (z r ) + λ df (z) r + λ r λ [1 F ( r )] st λf ( r )st = sx w r + λs (z r ) df (z) λst. (23) r + λ r Imposing x = r and J r ( r )= st yields the condition for job destruction w r = s r + λs (z r ) df (z)+rst (r + λ) r 8

9 Now substitute w r = w + φs and rearrange: s r Equilibrium λs r + λ r (z r )df (z) =w + φs rst The intersection of the JD and the JC curves gives unique equilibrium values of the reservation productivity and market tightness for the labor market under the rigid wage regime, which we call r = r (s, c, w, φ, b, T) and θ r = θ r (s, c, w, φ, b, T) respectively. Analogous to (17), the equilibrium unemployment rate u r in a rigid-wage labor market with skill level s is given by. u r λf ( r ) = λf ( r )+θ r q(θ r ) ur (s, c, w, φ, b, T). (24) Comparative Statics The dependence of the endogenous variables on the model parameters in the two regimes is described in the table below. Table 1. Comparative Statics esults Effect of = s λ ρ b β T w φ...on Flexible wage regime θ + + u ? igid wage r regime θ r + + u r + +? + + An increase in s shifts the JD curve downwards and the JC curve outwards from the origin, so an increase in skill unambiguously tightens the labor market and lowers the firingthresholdinbothregimes. Anincreaseinthefrequencyof productivity shocks, renegotiation costs and the value of leisure unambiguously increases unemployment in the flexible labor markets via their effects on wages. To the extent that a rigid wage does not depend on b, λ and ρ (and w>b), job creation and destruction margins (hence unemployment) are unaffected by changes in these parameters. As noted above, increases in the minimum wage and in the slope of the wage-skill profile in the rigid segment have unambiguous effects on job duration (negative), market tightness (negative) and unemployment (positive). Finally the firing tax reduces both job creation and destruction while its effect on unemployment is ambiguous. The comparative statics results in the rigid wage regime are obtained as follows. The two equations for the job destruction and creation conditions under rigid wages, rewritten slightly: r + λ r + λ r (z r )df (z) = w s 9 + φ rt (25)

10 k q (θ r ) = (1 r ) (r + λ) T d Differentiation and rearrangement leads to the form A r r dθ r = b r, with " 1 λ r+λ A= (1 # " r ) 0 dw s 1, and b = # w s ds + dφ rdt Tdr 2 r+λ kq0 1 r q 2 (r+λ) (dr + dλ) dt dk. 2 q h Inspection reveals that the determinant of A is now given by r 1 λ r+λ (1 r ) which is also unambiguously positive. The comparative statics results are i ih kq0 q 2 r s r w r φ w kq 0 = s 3 q 2 r = kq 0 sq 2 r = kq 0 q 2 r < 0; θr > 0; θr > 0; θr s = s 2 (r+λ) w r > 0 w = 1 s(r+λ) φ = 1 (r+λ) r < 0 r < 0 r b r T r r = r β = r ρ = r kq 0 q 2 < 0; θr r = T kq 0 q 2 r =0; θr b = θr β T = r+λ λ r r = (r+λ) T r < 0 = θr ρ =0 > 0; θr > 0 r The JC curve in the rigid labor market remains strictly downward sloping in (θ r, r )-space, since q 0 < 0, and lies everywhere above that of the competitive labor market. It is also independent of skill level s. 1.6 Closed Labor Markets A market for labor may not exist for every skill level. It is useful to define s as the minimal skill class above which the labor market is open (θ > 0); that is to say, in which positive vacancies are observed. If no vacancies are posted, the unemployment rate is 100% and the labor market is said to be closed. Alternatively, a labor market is closed if there is no value of x (0, 1] for which match surplus is positive. The value taken by s will generally depend on the wage setting regime. In the case of individualized, flexible wage-setting, as θ 0, the JC condition (13) implies that approaches 1 (r+λ)t (1 β) from below. h i Consequently there are no open labor markets for which 1 (r+λ)t (1 β), 1. Notice that when T =0, there are no closed labour markets. The JC condition for the rigid wage regime (21) is r [1 (r + λ) T,1] and contains the same implication. In the flexible wage case, the value for s, says is implicitly given by s : =1 (r + λ) T. Now consider the job destruction condition (16), substitute the inverse of q for θ, and then let the threshold approach 1 (r + λ) T from 10

11 below: lim 1 (r+λ)t s = = b + λ [1 F (1 (r + λ) T )] ρ 1 1 (r+λ)t 1 (r + λ) T + λ r+λ (z [1 (r + λ) T ])df (z)+rt r + λ [b + λ [1 F (1 (r + λ) T )] ρ] >b. r + λf (1 (r + λ) T ) Note that when ρ = T =0,thenb is the lower bound for match productivity, below which labor markets will be closed; match productivity at the outset must strictly exceed the flow benefit from leisure. This will not be the case however, when T>0, a fact that has been neglected by the literature; here labor markets can be closed, although workers in these skill classes have productivity strictly greater than b and would be willing to work if matched. When wages are rigid, the same analysis as above can be applied to (18), taking the limit as approaches 1 (r + λ) T from below: lim 1 (r+λ)t sr = = w 1 λt + λ 1 r+λ (x [1 (r + λ) T ])df (x) φ 1 (r+λ)t (r + λ) w>w (26) r (1 φ)+λ (1 φ F (1 (r + λ) T ) By inspection, labor markets are closed not only be the direct effect of the minimum wage w exceeding maximal match productivity, but also by φ and T, which can create a band of productivities exceeding w for which workers cannot be productively employed. 2 Participation in the igid egime A worker participating in the rigid wage segment earns w u independently of her match productivity; w u is a parameter from the perspective of the individual match. Membership is voluntary. The decision is made before the match is formed and holds forever in this job. Hence, in the rigid wage regime (in the set of skill levels S u ), the following condition must hold: W u W (1) s S u (27) where W u denotes the asset value of union employment to the worker. We clearly have also to impose that W u > U u where U u demotes the value of unemployment for a worker in the unionised segment. The participation decision of employers is embedded in the decision to issue vacancies. See the simulation results (slides). 2.1 A pure monopoly union If we neglect participation and focus just on the wage rule in the unionised segment, we can take as reference the monopoly union case. In this environment, 11

12 the union sets the wage while the employers choose the employment level. In terms of the matching model this amounts to assuming that the union sets a given β and employers accordingly establish the continuation region, that is,. Unionscanhaveanefficient role to play insofar as they internalise matching externalities. It can be shown that unions will set the β level that satisfies the Hosios condition (Lecture Notes 3) only when all its members are unemployed. When union members are employed, instead, β will be larger than the optimal level. In other words, there is an insider-outsider conflict arising with monopoly unions. When the union represents the unemployed, it will maximise the asset value of unemployment, that is the net worth of unemployed workers, ru. Weknow from (Lecture Notes 2) that the latter can be written as a linear function of market tightness ru = b + βc 1 β θ Alternatively, we can use the job destruction condition to express the reservation wage as a function of the reservation product 1 λ (1 β) (1 β)[ ru]+ (r + λ) (r + λ) (1 β)[ ru]+ ru + λ (1 β) (r + λ) λ (r + λ) J() = 0 (z ) df (z) = 0 (z ) df (z) = 0 (z ) df (z) = 0 ru = + λ (r + λ) This is, clearly, increasing in. Hence, the optimal β will maximise. The equilibrium is concave in β and is maximised when the (Hosios) condition β = η(θ) holds. Intuitively, when β is too high, the welfare of employees (insiders) is increased, while outsiders are worse off as they face lower job finding probabilities ru and we know from the above that Hosios condition does that. When unions instead represent employees (it is actually sufficient that more than 50 per cent of the union members are employed so that the median voter is working) β will exceed its optimal level. 2.2 The hold-up problem In presence of firing costs, the worker has the incentive to renegotiate the wage as soon as hired, in order to take advantage of the firing tax. In this context, (z ) df (z) 12

13 it may be appropriate to consider that there is a two-tier wage structure, where the initial or outside wage, w 0 is given by w 0 =(1 β)b + β [cθ +1] while the inside wage is w 1 =(1 β)b + β (cθ +1+rT). which shows that, insofar as T>0, w 1 >w 0. 13