Important job creation - job destruction rate Account for labor supply and labor demand behavior to explain unemployment

Transcription

1 Labor Economics and Employment Policies Pierre Cahuc and Sébastien Roux, Lecture 5: Job creations, job destructions and unemployment: the matching model Important job creation - job destruction rate Account for labor supply and labor demand behavior to explain unemployment Matching model: Pisssarides (2000)

2 Contents The basic model The large firm model: Investment and unemployment Dynamics and impact of aggregate shocks 2

3 1. The basic model Thematchingfunction M(V,U): instantaneous flow of hires per unit of time, - V job vacancies - U stock of persons looking for work Number of hires over interval [t, t+dt] is equal to M(V t,u t )dt M(V,D) - strictly increasing with respect to each of its arguments - M(V,0) = M(0,D)=0 - constant returns to scale. 3

4 The probability of filling a vacant job per unit of time M(V,U) = M(1,U/V) m(θ); θ V/U (1) V θ labor market tightness Differentiating (1) with respect to U: m 0 (θ) = U2 V 2M0 U (1,U/V) < 0 Exit rate from unemployment: M(V,U) = V M(V,U) = θm(θ) (2) U U V Differentiating with respect to V, we find: [θm(θ)] 0 m(θ)+θm 0 (θ) =M V (V,U) > 0 4

5 Trading externalities:. - Between-group externalities: positive - Within-group externalities: negative (congestion effects) 5

6 Equilibrium of flows and the Beveridge curve stock of unemployed persons: U, employment: L labor force: N. Labor force grows by quantity N 0;rate of growth n = N/N The variation U in the stock of unemployed persons is U = N + ql θm(θ)u (3) u = U/N the rate of unemployment, as we have N = L + U u = q + n [q + n + θm(θ)] u (4) Steady state u = q + n q + n + θm(θ) (5) 6

7 In the plane (v, u), this relationship yields the Beveridge curve (decreasing, convex) The Beveridge curve reflects the efficiency of the matching technology 7

8 v (BC ) (BC) u The Beveridge curve 8

9 The behavior of firms Two goods in the economy: a good produced by the firms and consumed by all individuals; and labor, assumed to be homogeneous, which is the sole factor of production Thegoodproducedbythefirmsisthenumeraire Each firm has one job that can be either vacant or filled Filled job produces y per unit of time, pays wage w Vacant job costs h per unit of time 9

10 At every instant, a job can either be filled or vacant Filled job yields an expected profit Π e Vacant job yields an expected profit Π v rπ e = y w + q(π v Π e )+ Π e (6) rπ v = h + m(θ)(π e Π v )+ Π v (7) 10

11 Labor demand free entry condition: Π v =0: h m(θ) = y w r + q Analogous to labor demand in the neo-classical theory of the firm Equations (5) and (8) and θ = v/u define (v, u) when wages are exogeneous. (8) 11

12 v y, m y z, γ, h, r, q θ * u * m q, n (BC) u Labor market equilibrium 12

13 Impact of changes in labor supply on steady state unemployment: Increase in N and increase in n Counseling to unemployed workers - Share s (0, 1) of unemployed workers are counseled - Counseling increases the efficiency of search from 1 to 1+δ, δ>0 Does counseling creates jobs? Are non counseled unemployed workers crowded out by counseled unemployed workers? 13

14 Define Equilibrium value of θ : θ = Exit rate out of unemployment v s(1 + δ)u +(1 s)u h m(θ) = y w r + q θm(θ) for non counseled unemployed (1 + δ)θm(θ) for counseled unemployed No crowding out: counseling create jobs! 14

15 The behavior of workers Any worker can be either employed, with an expected utility V e, or unemployed, with an expected utility V u V e. Workers are risk neutral: rv e = w + q(v u V e )+ V e (9) rv u = z + θm(θ)(v e V u )+ V u (10) 15

16 Wage bargaining Wage bargaining outcome is a simple surplus sharing rule surplus S =sumoftherents that a filled job paying negotiated wage w procures Rent represents the difference between what individuals obtain through the contractual relationship and what the best opportunity outside the contract would bring them For the employee the rent amounts to (V e V u ) For the employer it is equal to (Π e Π v ). The surplus is thus defined by: S = V e V u + Π e Π v (11) 16

17 Bargaining: V e V u = γs and Π e Π v =(1 γ)s (12) γ [0, 1] be the relative power of the worker; the result of the negotiation Several ways of explaining such a division of the surplus Equal the maximum of the generalized Nash criterion: max w (V e V u ) γ (Π e Π v ) 1 γ 17

18 Non-cooperative bargaining game. Let us assume, for example, that the bargaining unfolds, at each instant, as a two-stage game with the following characteristics: Stage 1) The two players propose a contract that stipulates a wage to be paid in the future small interval of time dt. Stage 2) If one of the two players has refused to sign the contract proposed in stage 1, the worker makes a new, take-it-or-leave-it offer with probability γ, and the employer in turn makes an offer of the same kind, with the complementary probability (1 γ). If there is again no agreement, the job is destroyed. 18

19 If it is the worker who makes the offer in stage 2, the employer obtains Π v, the worker gets (S + V u ) If, it is the employer who makes the offer in stage 2, the worker obtains V u, the employer gets (S + Π v ) So in the first stage, the worker anticipates (1 γ)v u + γ(s + V u )=V u + γs. Symmetrically, the employer anticipates (1 γ)(s + Π v )+γπ v = Π v +(1 γ)s. If we assume that there is a cost attached to going to stage 2, the bargaining game possesses a single equilibrium, corresponding to the immediate agreement of a surplus sharing contract as described by condition (12). 19

20 The negotiated wage From (6) and (9) we have: and V e V u = w rv u r + q (12), (14) and (13) imply S = y r(v u + Π v ) r + q and Π e Π v = y w rπ v r + q (13) (14) w = rv u + γ(y rv u ) (15) The wage negotiated is a linear combination of the value y of the production and of the reservation wage, rv u, weighted by the respective power of the employee and the employer 20

21 Wage curve and labor supply Labor demand (8) relation between w and θ Need other relationship between w and θ Definition (10) of V u and surplus sharing rule (12) entail rv u = z + γθm(θ)s Form (13) of the value of surplus S at free entry equilibrium: rv u = z(r + q)+γyθm(θ) r + q + γθm(θ) 21

22 Substituting this expression of rv u in (15), we get the Wage Curve: w = z +(y z)γ(θ) with Γ(θ) = Γ(θ) increases with θ γ[r + q + θm(θ)] r + q + γθm(θ) (w, θ) defined by the wage curve and the Labor Demand h m(θ) = y w r + q (16) 22

23 w (LS) w m w * E (LD) L d L * L s L Wage Curve and Labor Demand 23

24 Empirical elements relating to bargaining power Empirical work has been devoted to estimating wage equations similar in form to the one given by relation (16) Blanchflower and Oswald (1995), TheWageCurve,MIT Press Abowd and Lemieux (1993), wages are higher in Canadian firms with little exposure to international competition, γ =.3 Van Reenen (1996), partition of rents created by innnovation, using British data for the period ; γ =.3 Results suggest that workers do in fact capture a portion of the rent of jobs 24

25 Labor market steady state equilibrium Equilibrium value of (u, v) From equations (8) and (16) we get (1 γ)(y z) r + q + γθm(θ) = h m(θ) defines (u, v) with θ = v/u, and the Beveridge Curve q + n u = q + n + θm(θ) 25

26 v y, m y z, γ, h, r, q θ * u * m q, n (BC) u Labor market equilibrium 26

27 Comparative statics z γ h m y q r n w θ u Comparative statics of stationary equilibrium 27

28 2. The large firm model: Investment and employment Include capital Large number of identical firms At every instant, representative firm utilizes quantities K of capital and L oflabortoproduceaquantityf (K, L) of the numeraire good. F strictly increasing with respect to each of its arguments, strictly concave, constant returns to scale. The behavior of workers is identical to the one described in the basic model 28

29 The problem of the firm In every firm, at every instant, decisions unfold in the following order: 1. The firm decides on its hires 2. The employer negotiates over wages with each worker, one to one, capital is chosen simultaneously with the wage bargaining The problem of this firm is written: subject to: Max V,I Π = Z + 0 [F (K, L) wl hv I] e rt dt (17) L = m(θ)v ql (18) K = I δk (19) 29

30 The optimal solutions The Hamiltonian of this problem is written: H =[F (K, L) wl hv I] e rt + µ [m(θ)v ql]+λ(i δk) (20) The first-order conditions read: H H =0 and I K = λ (21) H H =0 and = µ (22) V L To these equations must be added the transversality conditions: Lim µl =0 and Lim λk =0 (23) t t 30

31 Equalities (21) entail e rt = λ and λδ F K (K, L)e rt = λ. The first equation entails that λ = rλ. Substituting this expression of λ into the second equation, we arrive at: F K (K, L) =r + δ (24) Conditions (22) entail he rt = µm(θ) and qµ [F L (K, L) w]e rt = µ. At stationary equilibrium, θ =0,we get: h(r + q) F L (K, L) =w + m(θ) (25) 31

32 Wage bargaining In stage 2 each employee bargains over his or her wage individually with the employer Thus, bargaining concerns the marginal surplus created by each job The value π of a marginal job is written as follows: µ 1 π = {[F 1+rdt L (K, L) w] dt +(1 qdt) π} π = F L(K, L) w r + q Identical to that giving the value of a filled job in the basic model with Π v =0and y = F L (K, L). 32

33 Let us set k = K/L and f(k) =F (K, L)/L (24) and (25) entail: f 0 (k) =r + δ (26) f(k) kf 0 h(r + q) (k) =w + (27) m(θ) Equation (27) is identical to relation (8) defining labor demand in the basic model with y = f(k) kf 0 (k). 33

34 The impact of the interest rate on unemployment In the period from 1981 to 1993 interest rates rose sharply Country United States Japan Germany France United Kingdom Long-term interest rates. Source: Rowthorn (1995; table 3, p. 36) 34

35 Simulation of the large firm model, with the parameter values given in table Cobb-Douglas production function Y = AK 0.3 L 0.7. We observe that the interest rate differentials among the G5 countries, which rarely exceed 1%, can only explain small differences among the unemployment rates in these countries, since a 10 point increase in the interest rate induces a rise on the order of only 3 points in the unemployment rate. 35

36 u y r r 0.09 The impact of the interest rate on unemployment. 36

37 Investment in specific capital, holdup, and unemployment Other source of under-investment: holdup problems The investment decisions analyzed to this point in the present chapter concern general capital, which the firm can utilize in any job at all, and resell at will Certain investments go into specific capital, commiting the firm to irreversible expenditures that only have value in the context of the relationship between the employer and the employee, who share the benefits of this investment Williamson (1975), Grout (1984), Caballero (2007): the incentives to invest in specific capital may be drastically reduced when contracts are incomplete 37

38 Let us suppose that the individual productivity of a worker depends on a specific investment in training, the entire cost of which, denoted by i, is paid by the employer at the time of hiring Production per capita increasing and concave function y(i) of the initial investment in training 38

39 Investment with a complete contract Two-stage game. 1. bargaining determines a wage for the whole duration of the employer-employee relationship 2. The employer decides how much to invest Backward induction In the second stage max i y(i) w(i)+qπ v (Π e i) =max i r + q i = y 0 (i) w 0 (i) =r + q (28) In the first stage, the employer and the employee bargain over the wage The employer obtains (Π e i) 39

40 The surplus, net of the investment cost reads S n (i) =V e V u + Π e i Π v = S(i) i Where S(i) corresponds to the definition (13) of the surplus, so that here: S(i) = y(i) r(π v + V u ) r + q Bargaining gives a share (1 γ) of the net surplus to the employer: Π e i Π v =(1 γ)s n (i) Utilizing expressions (14) and (13), the negotiated wage is written, with Π v =0: w(i) =γ [y(i) (r + q)i]+(1 γ)rv u (29) 40

41 Entails w 0 (i) =γ y 0 (i) (r + q), condition (28) then yields y 0 (i )=r + q (30) The investment defined by this last relation maximizes the net surplus S n (i). In this sense, a complete, non-renegotiable contract ensures efficient investment in specific capital. 41

42 Investment with an incomplete contract Two-stage game 1. The employer decides on the investment 2. The wage is bargained over. Backward induction In the second stage w(i) =γy(i)+(1 γ)rv u In the first stage of the game max i y(i) w(i) (Π e i) =max i r + q i = y 0 (i) = r + q 1 γ As function y(i) is increasing and concave, i > i. 42

43 When the contract in incomplete, productivity is lower, unemployment is higher. Ways to solve the holdup problem: - Allocate property rights before investing (for example, giving the employer the right to determine wages unilaterally to ensure that the employer will be the residual claimant of his or her investment) - Make provision for transfers should the contract be broken which makes it possible to avoid renegotiation Only through close analysis of labor contracts is it possible to assess the extent of the holdup problem. 43

44 3. Out-of-stationary-state dynamics The study of out-of-stationary-state dynamics allows us to exhibit a significant contrast between the movement over time of vacant jobs, and that of unemployment. Dynamic analysis also sheds light on the propagation mechanisms of shocks affecting the economy 44

45 Bargaining and the dynamics of the surplus Value functions read rv e = w + q(v u V e )+ V e (31) rv u = z + θm(θ)(v e V u )+ V u (32) rπ e = y w + q(π v Π e )+ Π e (33) rπ v = h + m(θ)(π e Π v )+ Π v (34) The surplus reads S = V e V u + Π e Π v and Ṡ = V e V u + Π e Π v (35) Free entry condition Π v =0, implies Π v =0. With the definitions (35), adding up equations (31) and (33), entails: (r + q)s = Ṡ + y + V u rv u (36) 45

46 Wage bargaining: V e V u = γs and Π e Π v =(1 γ)s (37) 46

47 The dynamics of vacancies and unemployment Free entry condition (Π v = Π v =0)and (34) yield Π e = h/m(θ) The second of the sharing rules (37) then entails: h S = (1 γ)m(θ) = Ṡ = hm0 (θ) (1 γ)m 2 (θ) θ (38) with, relation (32) and the first of the sharing rules (37) we get: rv u V u = z + θm(θ)γs = z + γθh (39) 1 γ Bringing the values of S, Ṡ and (rv u V u ) given by relations (38) and (39) into differential equation (36) we get: hm 0 (θ) (1 γ)m 2 (θ) θ + h[r + q + γθm(θ)] (1 γ)m(θ) y + z =0 (40) 47

48 First-order, non-linear differential equation of the form ϕ( θ, θ) =0 Linearizing function ϕ around point ( θ =0,θ = θ ): θ + aθ = aθ with a = γ m2 (θ ) m 0 (θ (r + q) < 0 ) General solution of the form θ = Be at + θ, where B is a constant. Parameter a being negative,the unique stable path of θ corresponds to B =0. We then have, at every instant, θ = θ. Evolution of the unemployment rate defined by u +[q + n + θ m(θ )]u = q + n First-order linear differential equation in which the coefficient of u is positive. 48

49 The unemployment rate thus exhibits a monotonic convergence to its stationary value given by relation (5) 49

50 Aggregate shock and reallocation shock Aggregate shock refers to a change in aggregate demand or supply of goods, and would not shift the Beveridge curve (changes in y, r, z, γ) Reallocation shock, on the other hand, refers to a restructuring of production units, which would shift the Beveridge curve without noticeably affecting the components of aggregate supply or demand (changes in the matching function m(.) or in the job destruction rate q). 50

51 v r v* E* v1 * E1 E (BC) u* u Aggregate shocks 51

52 v q v* E* E (BC ) v1 * E1 q (BC) u* u Reallocation shocks 52

53 The propagation of shocks Impact of an increase of 2% in the rate of growth of the labor force on the dynamics of the unemployment rate in the basic model Adjustment takes place very swiftly, since the unemployment rate rises from 10.9% to 11.7% in one year 53

54 Unemployment rate Years The impact of a permanent 2 percentage points increase in n 54

55 Underestimates the adjustment lags of the rates of unemployment and employment Only mechanism through which shocks are propagated is the delay necessary to effect hires Merz (1995) and Andolfatto (1996) bear witness to this insufficiency Den Haan et al. (2000) have constructed a matching model that takes the adjustment costs of capital into account. Their model exhibits persistence effects that clearly fit better with reality. Hall (1995, 1999) fragility of newly created jobs constitutes a potentially important source of propagation of shocks to unemployment and employment (see also Cole and Rogerson, 1999) Shimer (2005), Hall (2005): wage rigidity 55

56 Further Readings Davis, S. and Haltiwanger, J. (1999) Gross Job Flows, in O. Ashenfelter andd.card(eds), Handbook of Labor Economics, North Holland, vol 3B. Hall, R., (2005), Employment Fluctuations with Equilibrium Wage Stickiness, American Economic Review, March 2005, 95(1), pp Mortensen, D. and Pissarides, C. (1999), Job Reallocation, Employment Fluctuations and Unemployment, in Woodford, M. and Taylor, J. (eds), Handbook of Macroeconomics, Elsevier Science Publisher, vol 1B, chap 18, pp Petrongolo, B. and Pissarides, C (2001), Looking into the blackbox: a survey of the matching function, Journal of Economic Literature, 39, pp Pissarides, C. (2000), Equilibrium unemployment theory, ed. 2, Cam- 56

57 bridge, MIT Press. Rogerson, R., Wright, R. and Shimer, R., (2005), Search Theoretic Models of the Labor Market, Journal of Economic Literature, 43(4), pp Shimer, R., (2005), The Cyclical Behavior of Equilibrium Unemployment and Vacancies, American Economic Review, 95(1), pp