Keisuke Kawata
ISS, UTokyo
DMP model
• DMP is a simple search model beyond the Diamond paradox.
• The model is popular to incorporate the unemployment. e.g.,)
Business Cycle: Survey ⇒ Shimer (2010)
Urban/regional economics: Survey ⇒ Zenou (2009)
International trade: (Classical trade theory) Davidson, Martin, & (New New Trade theory) Helpman, E., & Itskhoki, O. (2010)
Housing market: Wheaton (1990)
Alternative model: Competitive search model
• There are some alternative search models
⇒ A important alternative model is the competitive search model (Moen 1997).
• Competitive search model still uses the matching function, while has following two unique futures
Price posting: Firms first post and committee wage before meetings. Directed search: Workers can search focusing on a specific wage level.
• Similar properties but more tractable in some cases.
• just change the search model from DMP to competitive search
Plan
1. One shot model with homogeneous agents 2. One shot model with heterogeneous firms
1. Settings: Sub-markets
• We use su - a kets a alog ← game-theoretic foundation (Burdett, Shi, Wright 2001.
• A labor market is divided into sub-markets. Each sub-market is characterized by wage-level, in which all firms offer same wage.
⇒Firms and workers can choose searching sub-market.
• Matching process within sub-markets is still descried by the matching function.
1. Setting: Other assumption
• There are a unit mass workers and many firms.
• We start from a simple case where workers and firms are risk-nuetral and homogeneous.
⇒ Expected utility of a worker applying sub-market with w is
+ − ,
while the expected profit of a firm in the sub-market is
� − − �
• Workers can only one sub-market (without multiple applications).
1. Setting: Timing
• We summary the model setting as timing of game 1. Wage-posting (selecting sub-market).
2. Workers observe the number of vacancies in each sub-market and then select a sub-market.
3. Matching process.
• We consider two cases
Exogeneous firm entry: The number of vacancies is exogenously given.
Endogenous firm entry: The number is determined by the zero profits condition in the first stage.
1. Equilibrium: third stage
• Because the constant-return-to-scale matching function is assumed, the job- finding and filling probabilities in a sub-market w are
= ,
= ,
where
= ,
and are numbers of vacancies and seekers in the sub-market.
1. Definition: equilibrium
• The market equilibrium is defined over { , } and expected utility and profits, which is satisfied
1. A worker chooses sub-market to maximize her utility. 2. A firm chooses sub-market to maximize her profits. 3. Feasibility condition;
∫ � = , � ∫ � =
where v is the total number of firms.
4. Zero profit condition in the endogenous firm entry case.
1. Equilibrium: second stage
• I e uili iu , et ee a ti e su -markets ( , > ), the arbitrage condition must be hold;
+ − = ′ ′ + − ′ .
⇒ Let defi e the a ket-le el utilit = max∈� + − .
Because each worker cannot be manipulate the market tightness, the arbitrage condition can be summarized as
+ − =
1. Equilibrium: first stage
• Ea h fi a ot a ipulate the a ket-le el utilit , the posted wage of a firm is given by
max � −
subject to
= + − .
Above problem is mathematically equivalent to max,� � − subject to
= + − .
1. Equilibrium: wage and tightness
• The first order conditions can be summarized are
= � + − ,
Where = − ′/ .
←same as in the DMP with the Hosios condition. Substituting into the constrain obtains
= � − + .
⇒ The equilibrium wage and tightness as a function of the market level utility.
1. Exogeneous firm entry
• Let suppose the number of vacancies is given.
⇒Feasibility condition determines the market tightness as = .
⇒The equilibrium utility is endogenously determined.
= � − + .
⇒ If the number of vacancies are relatively larger than the number of seeker, the equilibrium utility is then high.
1. Endogenous firm entry
• Let suppose the number of vacancies is determined by the zero-profits condition:
= � − − �.
Incorporating = � + − � yields
= − � − − �
← determining same equilibrium as in the DMP with the Hosios condition.
1. Efficiency properties
• The social welfare still can be defined as
�� = , � + − , − �
Exogeneous firm entry: Trivially efficient. Endogenous firm entry: Efficient.
← Why?
1. Efficiency properties: Intuition
• The equilibrium condition with the endogenous entry can be summarized by Fi ’s opti izatio p o le
max,� � − , . . , = + − ,
and the zero profit condition = � − − �.
⇒an equivalent condition is
max,� + − , . . , = � − − �.
Wo ke ’s utilit is a i ized ith the ze o p ofit o ditio
1. Efficiency properties: Intuition
• The optimization problem can be more modified as
max� � + − − �
⇒ Social welfare is maximized in the equilibrium.
• The efficiency properties of the competitive search model is from same intuition as in the Warlasian market ←
Warlasian market: Competition to attract (employed) workers. Competitive search: Competition to attract applicants.
2. Heterogeneous firms: Setting
• Supposing firms with heterogeneous productivities.
• We modify the timing of game is as follow
0. D a i g fi ’s p odu ti it f o � � and firm entry after observing it. 1. Wage posting
2. Workers observe the number of vacancies in each sub-market and then select a sub-market.
3. Matching process.
← The equilibrium properties are similar even if firm productivity is determined
2. Definition: equilibrium
• Fi ’s e t de isio -making can be characterized by threshold strategy: a firm entries the market if and only if � ≥ ത�.
• The market equilibrium is defined over { � , � , ത�} and expected utility and profits, which is satisfied
1. A worker chooses sub-market to maximize her utility. 2. A firm chooses sub-market to maximize her profits. 3. Feasibility condition;
∫ � = , � ∫ � =
where v is the total number of firms. 4. Zero profit condition
2. Equilibrium wage and tightness
• We can obtain the similar equilibrium conditions after stage 1. Wage condition: � = � + −
Market tightness given market utility: = � � − + .
⇒With productivity, wage has positive relationship, while tightness has negative relationship.
Why?: More productive firms have stronger incentive to fill their vacancies because they have higher opportunity cost
→ paying higher wage and attracting more seeker ⇒lower tightness
2. Equilibrium threshold and utility
• The zero profit condition leads the equilibrium threshold ത� as
= ത� ത� − ത� − �
= ത� − ത� − − �.
⇒ Feasibility condition leads
නത� �
−1� � �� =
← determining the equilibrium utility.
2. DMP VS competitive search
• Both DMP and competitive search can explain the wage variation as a result of productivity difference. But intuitions are different;
DMP: As a result of the bargaining, more productive firms pay higher wage.
Co petiti e sea h: As a fi ’s opti izatio sea h eha io , o e p odu ti it firms pay higher wage to attract more workers.
3. Conclusion
• Competitive search model suppose price-posting and directed search with the matching function.
← have similar property as DMP with the Hosios condition.
Related Literature
Job search
Business cycle: Shimer, R. (2010). Labor Markets and Business Cycles. Princeton University Press.
Urban/region: Zenou, Y. (2009). Urban labor economics. Cambridge University Press.
International Trade: (Classical) Davidson, C., Martin, L., & Matusz, S. (1999). Trade and search generated unemployment. Journal of International Economics, 48(2), 271-299.
(New New Trade theory) Helpman, E., & Itskhoki, O. (2010). Labour market rigidities, trade and unemployment. The Review of Economic Studies, 77(3), 1100-1137.
Related Literature
Housing search
Wheaton, W. C. (1990). Vacancy, search, and prices in a housing market matching model. Journal of Political Economy, 98(6), 1270-1292.
Intermediator
Antras, P., & Costinot, A. (2011). Intermediated trade. The Quarterly Journal of Economics, 126(3), 1319-1374.
Fernández-Blanco, J. (2012). A directed search model of intermediated trade. European Economic Review, 56(8), 1481-1494.
