Diamond-Mortensen-Pissarides Model

Transcription

1 Diamond-Mortensen-Pissarides Model Dongpeng Liu Nanjing University March 2016 D. Liu (NJU) DMP 03/16 1 / 35

2 Introduction Motivation In the previous lecture, McCall s model was introduced McCall s model focuses on the optimal decision rule of unemployed workers However, it does not shed light on employers optimal behavior existence and/or creation of job vacancies matching of unemployed workers and job vacancies determination of wage Diamond-Mortensen-Pissarides (DMP) model fix these problems DMP model is the currently prevailing framework to analyze unemployment The model is named after three Nobel Prize laureates D. Liu (NJU) DMP 03/16 2 / 35

3 Structure of the Lecture Introduction DMP model Workers Firms/Employers Matching technology and wage negotiation Determination of unemployment rate Steady state equilibrium Dynamic equilibrium Extension: effi ciency of search equilibium Concluding remarks D. Liu (NJU) DMP 03/16 3 / 35

4 Assumptions about Workers Workers In each period, a worker is either employed or unemployed An unemployed worker searches for a job Flow utility function { wt η if employed b if unemployed where w t is negotiated wage, b is unemployment benefit, and η is disutility from working D. Liu (NJU) DMP 03/16 4 / 35

5 Workers Assumptions about Workers In period t, an unemployed worker will successfully find a job with probability µ t Workers take µ t as given If an unemployed worker get a job offer in period t, she will start to work from period t + 1 If an unemployed worker does not get a job offer in period t, she will continue to search in period t + 1 If a worker is currently employed, there is a probability x that she will lose the job at the end of the period D. Liu (NJU) DMP 03/16 5 / 35

6 Workers Life-time Utility of a Worker Denote the life-time utility of an unemployed and employed worker measured in period t by U t and W t Hence, we have U t = b + β[µ t W t+1 + (1 µ t )U t+1 ] and W t = w t η + β[(1 x)w t+1 + xu t+1 ] D. Liu (NJU) DMP 03/16 6 / 35

7 Life-time Utility of a Worker Workers Economic intuitions: The economic intuitions of the two equations are very similar. Take the life-time utility of an unemployed worker as an example. Since the worker is currently unemployed, her utility in the current period would be b. In this period, the worker may get a job offer with a probability µ t. Given that, her life-time utlity measured in the next period would be W t+1. The worker may also fail to secure a job position in the current period with a probability of 1 µ t. If that happens, her life-time utility would be U t+1. Hence, the second term on the RHS of U t is the expected life time utility of the currently unemployed worker in period t + 1 discounted back to period t (often called the continuation value). The sum of the two terms is thus the life-time utility of the unemployed worker. D. Liu (NJU) DMP 03/16 7 / 35

8 Net Benefit of Employment Workers Define the net benefit of employment by H t = W t U t It follows that H t = w t b η + β(1 x µ t )H t+1 D. Liu (NJU) DMP 03/16 8 / 35

9 Net Benefit of Employment Workers Economic intuitions: Note that w t b η is the flow utility gain for a worker to be employed rather than unemployed. In the next period, there is a probability of 1 x that the worker will be still employed and can continue to enjoy the net benefit of employment. Meanwhile, if the worker were currently unemployed, there would have been probability µ t that the worker would be employed from the next period and start to receive the benefit of employment. Given that the worker is actully employed in the current period, she will not be able to receive this part of benefit and thus it must be deducted from the continuation value. Thus, the sum of flow utility gain and the continuation value measures the life-time utility gain of being currently employed rather than unemployed D. Liu (NJU) DMP 03/16 9 / 35

10 Firms/Employers Assumptions about Firms/Employers Small firm: each firm only contains one job position (a job = a firm) There are many employers. In each period, employers determine how many job vacancies to post (v t ) The cost of posting a vacancy for one period is q For each vacancies posted in period t, there is a probability g t that the vacancy is filled by an currently unemployed worker If a newly created job vacancy is matched with an unemployed worker, the firm will produce an output of z starting from next period, till job separation happens Employers take g t as given D. Liu (NJU) DMP 03/16 10 / 35

11 Values of a Firm and a Vacancy Firms/Employers Denote the value of a firm and the value of a vacancy in period t as J t and V t, respectively and J t = z w t + β(1 x)j t+1 V t = q + βg t J t+1 The economic intuitions of the value functions above are very similar to those of the recursively defined life-time utility functions for workers. Note that once job separation happens, the firm ceases to exist and the value of the firm becomes zero. If a vacancy is not filled in the current period, it will no longer exist in the next period. If the employer still wants to open an vacancy, she has to post a new one next period. D. Liu (NJU) DMP 03/16 11 / 35

12 Firms/Employers Free Entry Condition We assume there is no barrier to the market of vacancy posting In the equilibrium, employers should get zero economic profits by simply posting vacacies q = βg t J t+1 This condition states that the cost of posting a vacancy equates the discounted future benefits of the vacancy D. Liu (NJU) DMP 03/16 12 / 35

13 Matching Technology and Wage Negotiation Matching Technology In each period, the total number of matches depends on the number of unemployed workers (u t ) and the number of vacancies (v t ) The matching technology is homogeneous of degree 1 m t = av γ t u 1 γ t The job finding rate and vacancy filling rate are thus µ t = m t u t = a( v t u t ) γ = aθ γ t g t = m t v t where θ t = v t /u t is market tightness = a( v t ) γ 1 = aθ γ 1 t u t D. Liu (NJU) DMP 03/16 13 / 35

14 Matching Technology and Wage Negotiation Wage Negotiation We assume that the equilibrium wage is determined through Nash Bargaining The workers and employers negotiate the wage as if they maximize where λ (0, 1) H λ t J 1 λ t D. Liu (NJU) DMP 03/16 14 / 35

15 Wage Negotiation DMP Model Matching Technology and Wage Negotiation w t = arg max H λ t J 1 λ t FOC wrt w t or λh λ 1 t Jt 1 λ H t + (1 λ)ht λ J λ J t t = 0 w t w t λj t = (1 λ)h t Hence, H t = λ(h t + J t ). Note that H t + J t is the joint surplus from the matching and H t is the worker s gain. This is why λ is often referred to worker s bargaining power. D. Liu (NJU) DMP 03/16 15 / 35

16 Wage Negotiation DMP Model Matching Technology and Wage Negotiation It follows that Hence λ[z w t + β(1 x)j t+1 ] = λ[z w t + (1 x)q/g t ] = (1 λ)[w t b η + β(1 x µ t )H t+1 ] λ = (1 λ)[w t b η + (1 x µ t ) 1 λ q/g t] w t = λ(z + q µ t ) + (1 λ)(b + η) g t = λ(z + qθ t ) + (1 λ)(b + η) D. Liu (NJU) DMP 03/16 16 / 35

17 Wage Negotiation DMP Model Matching Technology and Wage Negotiation Economic intuitions: The equilibrium wage is the weighted average of (1) the output gains of the match and (2) the worker s opportunity cost of employment. For each worker employed, the extra output produced is z and there is one fewer unemployed worker. Therefore, θ t fewer vacancies are posted and the vacancy posting costs saved are qθ t. Hence, z + qθ t is the total output gains from the match. In addition to the disutility from working, an employed worker is also giving up the unemployment benefit b. Hence, b + η is the opportunity cost of employment for the worker. D. Liu (NJU) DMP 03/16 17 / 35

18 Dynamics of Unemployment Rate Determination of Unemployment Rate The pool model introduced in the last lecture also applies here (with a few modifications) u t+1 = (1 µ t )u t + x(1 u t ) The size of unemployment in the next period depends on current size of unemployment u t, current inflow of unemployment x(1 u t ) and current outflow of µ t u t. D. Liu (NJU) DMP 03/16 18 / 35

19 Steady State Unemployment Rate Steady State Equilibrium Any endogenous variable without time subscript represent the steady state value of the variable u = (1 µ)u + x(1 u) or u = x x + µ D. Liu (NJU) DMP 03/16 19 / 35

20 Steady State Equilibrium DMP Model Steady State Equilibrium According to the wage equation w = λ(z + qθ) + (1 λ)(b + η) Hence, the steady state wage is an increasing function of the steady state market tightness. It characterizes the relationship between w and θ from the worker s perspective. As there are more vacancies per unemployed worker, workers are at a better position and they will earn higher wages. D. Liu (NJU) DMP 03/16 20 / 35

21 Steady State Equilibrium DMP Model Steady State Equilibrium In the steady state J = q βg = q βaθ γ 1 = z w 1 β(1 x) The second line of the equation is often referred to Job Creation Equation. According to this equation, w is a decreasing function of θ. It characterizes the relationship between w and θ from the employer s perspective. As the wage becomes higher, employers will get lower profits, which discourage them from posting many vacancies and lead to smaller θ. D. Liu (NJU) DMP 03/16 21 / 35

22 Steady State Equilibrium DMP Model Steady State Equilibrium By the wage equation and job creation equation, there exists a unique steady state equilibrium D. Liu (NJU) DMP 03/16 22 / 35

23 Comparative Steady State Analysis Steady State Equilibrium Consider what happens when unemployed workers receive higher unemployment benefits b increases Wage equation shifts to the left Higher wages and lower profits for firms θ decreases (for each unemployed workers, employers will post fewer vacancies) µ decreases and g increases It is harder for an unemployed worker to find a job but easier for an employer to fill a vacancy u increases Higher unemployment rate in EU (compared to US unemployment rate) is often attributed to the significantly more generous unemployment benefits D. Liu (NJU) DMP 03/16 23 / 35

24 Dynamic Equilibrium Dynamic Equilibrium Note that there are many endogenous variables in the model (u t, v t, θ t, w t, µ t, g t, J t, V t, W t, U t, H t ) It can be easily shown that all endogenous variables are functions of u t and θ t, e.g. v t = θ t u t The choice of the key variables of the dynamic system is not unique (try other combinations if you want!) Note that u t is a state variable (based on the pool model, current unemployment is predetermined) However, θ t is a choice variable (θ t = v t /u t and the number of vacancies are chosen by employers concurrently) D. Liu (NJU) DMP 03/16 24 / 35

25 Dynamic Equilibrium Dynamic Equilibrium Since u is a state variable, u 0 is given Two first order difference equations are needed to pin down the equilibrium path of the model economy The first difference equation is given by the dynamics of unemployment u t+1 = (1 µ t )u t + x(1 u t ) Note that µ t = aθ γ t The second difference equation is given by the dynamics of J t and the free entry condition D. Liu (NJU) DMP 03/16 25 / 35

26 Dynamic Equilibrium DMP Model Dynamic Equilibrium It follows that J t+1 = q βg t = z w t+1 + (1 x)q/g t+1 Note that g t is a function of θ t. Furthermore, w t+1 and g t+1 are functions of θ t+1 D. Liu (NJU) DMP 03/16 26 / 35

27 Dynamic Equilibrium DMP Model Dynamic Equilibrium Thus, the two difference equations that characterize the equilibrium path are (1 aθ γ t x)u t + x = u t+1 and q βaθ γ 1 t = z λ(z + qθ t+1 ) (1 λ)(b + η) + (1 x) q aθ γ 1 t+1 The equation above is the law of motion for θ t under the search equilibrium D. Liu (NJU) DMP 03/16 27 / 35

28 Effi ciency of Equilibrium Effi ciency of Market Equilibrium A natural question to ask is whether the search equilibrium is socially optimal Assume that all profits (z w t )(1 u t ) qv t are consumptions of employers. Furthermore, assume that the flow utility of employers equals to their level of consumption. Let b = 0, such that there is no government intervention at all With the assumptions above, we can construct the goods market equilibrium without changing the wage equation and the the law of motion for market equilibrium θ t What we need to compare is social planner s solution vs. the search equilibrium. D. Liu (NJU) DMP 03/16 28 / 35

29 Effi ciency of Equilibrium Social Planner s Problem A social planner is a person who only cares about the welfare of the agents in the economy A social planner try to maximize the life-time utility of the whole economy by allocating scarce resources In the case of this specific economy, the social planner needs to determine How much resources should be used to recruit workers How to allocate consumptions between all agents Since flow utility with respect to consumption is assumed to be linear, the allocation of consumptions between agents does not really matter under this assumption Generally speaking, there is no price in any planner s economy. Note that there is no wage in a planner s economy as wage is the price for labor. D. Liu (NJU) DMP 03/16 29 / 35

30 Social Planner s Problem DMP Model Effi ciency of Equilibrium A social planner try to maximize the life-time utility of the whole economy subject to the employment constraint s.t. max t=0 β t [(z η)(1 u t ) qu t θ t ] u t+1 = (1 aθ γ t )u t + x(1 u t ) D. Liu (NJU) DMP 03/16 30 / 35

31 Social Planner s Problem DMP Model Effi ciency of Equilibrium Lagrangian L = β t {[(z η)(1 u t ) qu t θ t ] t=0 +φ t [u t+1 (1 aθ γ t )u t x(1 u t )]} FOCs L θ t = β t [ qu t + φ t aγθ γ 1 t u t ] = 0 L u t+1 = β t φ t β t+1 [(z + qθ t+1 η) + φ t+1 (1 aθ γ t+1 x)] = 0 D. Liu (NJU) DMP 03/16 31 / 35

32 Social Planner s Problem DMP Model Effi ciency of Equilibrium It follows that Hence or q aγθ γ 1 t φ t = q aγθ γ 1 t φ t = β[(z + qθ t+1 η) + φ t+1 (1 aθ γ t+1 x)] q βaθ γ 1 t = β(z + qθ t+1 η) + β = γ(z + qθ t+1 η) + q q aγθ γ 1 t+1 aθ γ 1 t+1 (1 aθ γ t+1 x) (1 x) qθ t+1 The equations above characterize the law of motion for socially optimal θ t D. Liu (NJU) DMP 03/16 32 / 35

33 Effi ciency of Equilibrium Market Equilibrium vs Social Planner s Problem Note that the law of motion for θ t under the market equilibrium q βaθ γ 1 t = z λ(z + qθ t+1 ) (1 λ)η + (1 x) and that under the planner s economy are similar q βaθ γ 1 t = γ(z + qθ t+1 η) + q aθ γ 1 t+1 q aθ γ 1 t+1 (1 x) qθ t+1 Problem Show that when λ = 1 γ, the two equations are the same D. Liu (NJU) DMP 03/16 33 / 35

34 Effi ciency of Equilibrium Hosios Condition Theorem The market equilibrium is socially optimal If and only if λ = 1 γ When λ = 1 γ, the law of motion for θ t under the market equilibrium and the social planner s problem are the same This is called the Hosios condition Economic intuition: The equilibrium is socially optimal if the bargaining power of the workers equals to the relative "matching contributions" of unemployed workers Policy implication: If the worker s bargaining power equals to the matching elasticity with respect to unemployment, the government should not intervene the labor market. Otherwise, the analysis shows that government intervention might be desirable. D. Liu (NJU) DMP 03/16 34 / 35

35 Conclusions Concluding Remarks In this lecture, the basic DMP model is introduced The DMP model sheds light on job creation decisions, wage bargaining, matches between workers and vacancies and equilibrium unemployment The labor market is well characterized by the DMP model The DMP model does not provide us with a complete picture of the economy as a whole because goods market is not well characterized no consumption vs. saving decisions by the consumer no investment and capital This problem can be fixed by a DGE model with search and matching D. Liu (NJU) DMP 03/16 35 / 35