Equilibrium Unemployment Theory

Transcription

1 Equilibrium Unemployment Theory The Baseline Model Matthias S. Hertweck University of Basel February 20, 2012 Matthias S. Hertweck Equilibrium Unemployment Theory 1/35

2 Lecture Outline The Job Flow Condition Workers Optimization Problem Wage Determination Matthias S. Hertweck Equilibrium Unemployment Theory 2/35

3 Literature Required Readings: Pissarides, C. A. (2000), Equilibrium Unemployment Theory, The MIT Press, Cambridge, Massachusetts, pp Optional Readings: Diamond, P. A, (1982), Wage Determination and Efficiency in Search Equilibrium, Review of Economic Studies, Vol. 49(2), Nash, J. (1953), Two-Person Cooperative Games, Econometrica, Vol. 21 (1953), Matthias S. Hertweck Equilibrium Unemployment Theory 3/35

4 The Matching Function captures labor market frictions implicitly ml = m(ul, vl) (1) output: the number of new job matches ml inputs: job searchers ul & open job vacancies vl monotonically rising and concave in ul and vl linear homogeneous in L (labor force) henceforth, L is assumed to be constant Matthias S. Hertweck Equilibrium Unemployment Theory 4/35

5 Matching Rates in the Labor Market during a small time interval t, a firm expects to meet a suitable worker with probability: q(θ) m(ul, vl) vl ( u ) = m v, 1 (2) the corresponding job finding probability is given as: p(θ) m(ul, vl) ul ( = m 1, v ) = θq(θ) (3) u by linear homogeneity of the matching function, the probabilities depend only on the value of labor market tightness θ Matthias S. Hertweck Equilibrium Unemployment Theory 5/35

6 Congestion Externalities the tighter the labor market, the longer the expected time to fill a vacancy: q (θ) 0 but the shorter the expected search for a job: p (θ) 0 both parties do not internalize the adverse effects of their search decisions on the aggregate return rates this gives rise to congestion externalities Matthias S. Hertweck Equilibrium Unemployment Theory 6/35

7 Law of Motion of Unemployment mass of existing matches: n = 1 u job destruction probability: λ law of motion of the unemployment share: u = λ(1 u) θq(θ)u } {{ } } {{ } inflow outflow (4) Matthias S. Hertweck Equilibrium Unemployment Theory 7/35

8 Equilibrium Condition 1: The Job Flow Condition in the steady state, the inflow equals the outflow: λ(1 u) = θq(θ)u (5) hence, the equilibrium unemployment rate reads as: u = λ/[λ + θq(θ)] (6) the job matching model is able to explain the co-existence of equilibrium vacancies and unemployment Matthias S. Hertweck Equilibrium Unemployment Theory 8/35

9 The Beveridge Curve the Beveridge Curve predicts a negative relationship between equilibrium vacancies and unemployment concavity of the matching function in the input factors v and u implies that the Beveridge curve is convex to the origin V BC 0 U Matthias S. Hertweck Equilibrium Unemployment Theory 9/35

10 Firms in the Labor Market Active Firms: firms are small: one worker per firm every active match produces output of value p measure of active matches: n = 1 u Inactive Firms: there is an infinite mass of firms with unfilled positions free entry: each firm with an unfilled position may decide whether or not to post a vacancy Matthias S. Hertweck Equilibrium Unemployment Theory 10/35

11 Asset Value Methodology firms in the labor market are either active or vacant workers in the labor market are either employed or unemployed the asset value of a labor market state is equal to the present discounted value of the expected income stream all agents are risk-neutral Matthias S. Hertweck Equilibrium Unemployment Theory 11/35

12 The Vacancy Posting Decision V = δpc + q(θ)δj + [1 q(θ)]δv (7) V: asset value of an unfilled vacancy J : asset value of an occupied job δ: discount rate pc : vacancy posting cost, proportional to output Matthias S. Hertweck Equilibrium Unemployment Theory 12/35

13 The Vacancy Posting Decision - contd Simplifying equation (7) yields: (1 δ)v = δpc + δq(θ)(j V) (8) rv = pc + q(θ)(j V) (9) where r = (1 δ)/δ denotes the interest rate. Matthias S. Hertweck Equilibrium Unemployment Theory 13/35

14 The Vacancy Posting Decision - contd free entry implies that there are no arbitrage opportunities inactive firms open vacancies until V = 0 holds remember that q (θ) < 0 pc q(θ) = J (10) 1/q(θ) expected time to fill a vacancy expected costs to fill a vacancy equals the asset value J Matthias S. Hertweck Equilibrium Unemployment Theory 14/35

15 The Asset Value of an Occupied Job J = δ(p w) + (1 λ)δj (11) rj = (p w) λj (12) p w: output minus labor costs λ: job destruction rate Matthias S. Hertweck Equilibrium Unemployment Theory 15/35

16 Equilibrium Condition 2 combining equation (10) with equation (12) yields: p w (r + λ)pc q(θ) = 0 (13) in equilibrium, output covers labor costs w and hiring costs hiring cost are sunk reservation wage of the firm: p but if p was equal to w, no firm would open a vacancy Matthias S. Hertweck Equilibrium Unemployment Theory 16/35

17 Labor Demand Curve the job creation condition (13) produces a negative relationship between θ = v/u and the wage w W JC/D 0 θ Matthias S. Hertweck Equilibrium Unemployment Theory 17/35

18 Workers in the Labor Market Workers Optimization Problem Wage Determination there are L workers in the labor market the share u is unemployed and searches for a job there is no on-the-job search, search intensity is fixed the share n = 1 u is employed (existing matches) all employed workers produce output of value p Matthias S. Hertweck Equilibrium Unemployment Theory 18/35

19 Workers Optimization Problem Wage Determination The Asset Value of Unemployment U = δz + θq(θ)δw + [1 θq(θ)]δu (14) ru = z + θq(θ)(w U) (15) U: asset value of unemployment W: asset value of a job z: per-period income during unemployment r U: the reservation wage of an unemployed worker ru > z reflects the option value of labor market search Matthias S. Hertweck Equilibrium Unemployment Theory 19/35

20 Workers Optimization Problem Wage Determination The Asset Value of Employment W = δw + λδu + (1 λ)δw (16) rw = w λ(w U) (17) the worker s valuation of a job is smaller than the wage w this reflects the risk of unemployment Matthias S. Hertweck Equilibrium Unemployment Theory 20/35

21 The Worker s Benefit from a Job Workers Optimization Problem Wage Determination workers prefer to work when (W U) > 0 we assume: r λ θq(θ) > 0 then w > z is the necessary and sufficient condition for this: W U = w z r + λ + θq(θ) (18) Matthias S. Hertweck Equilibrium Unemployment Theory 21/35

22 Workers Optimization Problem Wage Determination Characterization of the Permanent Income aim: characterization of the permanent income in terms of model parameters and variables we substitute equation (18) into equations (15) and (17), respectively: ru = rw = rz + λz + θq(θ)w r + λ + θq(θ) rw + λz + θq(θ)w r + λ + θq(θ) (19) (20) Matthias S. Hertweck Equilibrium Unemployment Theory 22/35

23 The Mutual Surplus The Job Flow Condition Workers Optimization Problem Wage Determination matching requires job search (costly, time consuming) matched agents are better off than searching agents there is a mutual surplus: the reservation wage of the firm w R,F = p is higher than the reservation wage of the worker w R,W = ru J V = 0 J = 0 w R,F = p W U = 0 W = U w R,W = rw = ru (21) Matthias S. Hertweck Equilibrium Unemployment Theory 23/35

24 Wage Bargaining The Job Flow Condition Workers Optimization Problem Wage Determination r U p bargaining set w the bargaining set contains infinitely many possible equilibrium wages we need additional assumptions to identify the wage that splits the mutual surplus standard assumption: Nash bargaining Matthias S. Hertweck Equilibrium Unemployment Theory 24/35

25 Nash Bargaining The Job Flow Condition Workers Optimization Problem Wage Determination static two-player one-shot bargaining game threat points: outside option (i.e. V and U, respectively) both parties divide the mutual surplus according to a constant sharing parameter Matthias S. Hertweck Equilibrium Unemployment Theory 25/35

26 Workers Optimization Problem Wage Determination The Nash Product identifies a wage w that maximizes the weighted product: w = arg max w (W(w) U)β (J (w) V) 1 β (22) W U: the worker s surplus share J V: the firm s surplus share β: worker s bargaining power symmetric Nash bargaining presumes β = 0.5 Matthias S. Hertweck Equilibrium Unemployment Theory 26/35

27 Workers Optimization Problem Wage Determination The Nash Wage solving equation (22) yields: β(w(w) U) β 1 (J (w) V) 1 β = (W(w) U) β (1 β)(j (w) V) β (1 β)(w(w) U) = β(j (w) V) (23) using (12), (17) and V = 0 we obtain: ( w + λu (1 β) r + λ ) U = β ( ) p w r + λ (24) Matthias S. Hertweck Equilibrium Unemployment Theory 27/35

28 The Nash Wage - contd Workers Optimization Problem Wage Determination since all matches produce output of value y, all workers earn the same wage w = βp + (1 β)ru (25) the wage is a weighted average of the reservation wages r U low β high β p bargaining set w Matthias S. Hertweck Equilibrium Unemployment Theory 28/35

29 Workers Optimization Problem Wage Determination Alternative Characterization of the Wage remember that J equals average hiring costs pc/q(θ) this feeds into the worker s surplus share (equation 23): (W U) = β pc 1 β q(θ) (26) hence, also ru (equation 15) is a function of hiring costs: β pc ru = z + θq(θ) 1 β q(θ) (27) ru = z + β pcθ (28) 1 β Matthias S. Hertweck Equilibrium Unemployment Theory 29/35

30 Workers Optimization Problem Wage Determination Equilibrium Condition 3 now we substitute ru in equation (25): w = (1 β)(z + (β/(1 β)pcθ)) + βp (29) w = (1 β)z + βp(1 + cθ) (30) upon matching, a firm stops searching employed workers are rewarded for the saving of hiring costs pcθ: average hiring costs per unemployed worker Matthias S. Hertweck Equilibrium Unemployment Theory 30/35

31 The Wage Curve The Job Flow Condition Workers Optimization Problem Wage Determination equation (30) produces an upward-sloping relationship between w and θ, representing labor supply W WC/S (1-β)z + βp 0 θ Matthias S. Hertweck Equilibrium Unemployment Theory 31/35

32 Equilibrium Conditions job flow condition (inflow equals outflow): u = λ λ + θq(θ) (31) job creation condition (no arbitrage): p w (r + λ)pc q(θ) = 0 (32) wage equation (Nash bargaining): w = (1 β)z + βp(1 + cθ) (33) Matthias S. Hertweck Equilibrium Unemployment Theory 32/35

33 Graphical Representation three variables (v, u, w), three equations W WC V JCL W* (1-β)z + βp 0 θ* JC θ V* 0 θ* U* BC U Matthias S. Hertweck Equilibrium Unemployment Theory 33/35

34 Equilibrium Description JC: a high wage reduces the incentives for job creation WC: a tight labor market increases worker s wage demands the unique equilibrium is at the intersection of JC and WC, corresponding to the Walrasian equilibrium key difference: equilibrium tightness θ, not labor input L Matthias S. Hertweck Equilibrium Unemployment Theory 34/35

35 Equilibrium Description contd. substitution of w in JC yields the so-called job creation line JCL: (1 β)(p z) r + λ + βθq(θ) pc = 0 (34) q(θ) BC: displays equilibrium v and u for a given JCL Matthias S. Hertweck Equilibrium Unemployment Theory 35/35