Lecture search 4 最近の更新履歴 Keisuke Kawata's HP

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Keisuke Kawata

ISS, UTokyo

1 Search theory (2017)

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Review: Property of search model

Search theory (2017) 2

• We focusing on two side-matching model where two types of agents try to match each other type.

• Property of sequential search model with/without endogenous price distribution are

Without With _{This slide}

Search activity Someone Nobody Someone

Price dispersion Yes No No

Diamond paradox No Yes No

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DMP model

• Most popular search model is the Diamond-Mortencen-Pissarides model (DMP model, Pissardes 2000)

• The model has three components

Matching function: The number of new match is determined by the exogenous function of unemployed and vacancy.

Nash bargaining: The price is determined by a surplus sharing rule. Free entry condition.

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Matching function

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15

0.4 0.6 0.8 1 1.2 1.4

有効求人倍率

入職率充足率

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Matching function

• Matchi g fu ctio = Productio fu ctio of e e plo e t.

⇒ _,

where v is the number of unfilled job, and u is the number of unemployed.

• Consistently empirical findings (Petrongolo & Pissarides 2001 ), we assume is (i) increasing function of both arguments, (ii) constant return to scale.

• Additionally, assuming twice differentiable is assumed by technical reasons.

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Matching function

• By using constant-return-to scale, the matching probabilities of a worker and a job are

#

# ⁼

, = , ≡ ,

#

# ⁼

, = , ⁻¹ ≡ ,

where = (market tightness).

• is increased with , while is decreased.

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Nash bargaining

• Let introduce the individual surplus as

Worker’s surplus: life-time value as employed^－ life-time value as unemployed Fir ’s surplus: life-time profits as filled-job ^ー life-time profits as unfilled job Joi t surplus: su orker’s a d fir ’s surplus

• The wage is determined by the following rule,

′ _{= × �}

� ′ = − × �

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2. One-shot model

• Let consider the model with L workers (they are initially unemployed).

• The timing of game is

1. Jobs entry the labor market with a fixed costs c.

2. Matching is occurred the following matching function.

3. If a job and a worker are matched, they jointly produce y value, while an unmatched worker produces b, while unfilled jobs produce no value.

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Equilibrium conditions

• Matching function: �, .

• Free entry condition: q − − =

• Nash Bargaining: ∈ − − ¹⁻

⇒ ₋ ₋ ₌ ₋

⇒ ₌ ₊ ₋ _.

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Equilibirum conditions

• Matching function: �, .

• Free entry condition: ^{� �,} ₋ _{− =}

⇒ ₋ ₋ ₌ ₋

⇒ ₌ ₊ ₋ _.

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Equilibrium equations

⇒ ₋ ₋ ₌ ₋

⇒ ₌ ₊ ₋ _.

• Free entry condition is q − − =

• Market tightness is,

q − − − = .

Total differentiation yields

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Social optimal

• The social elfare is defi ed as a si ple su of orker a d fir ’s e pected payoff as

= �, − − + �, + − �,

= �, + � − �, −

rewriting as a function of tightness,

= + − − �.

The optimal condition is

= ^′ − −

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Efficiency of market equilibrium

• The equilibrium condition is

− − − = .

• The optimal condition is

= ^′ − − .

• The market equilibrium is efficient only if

− = ^′

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Efficiency of market equilibrium

• Because ₌ , the Hosios condition can be rewritten as

= −^��_{� �}^′ ^� ≡ (Hosios condition)

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Efficiency: Alternative intuition

• Because only jobs pay costs to make a new employee-employer match, all surplus should be for jobs due to avoid the hold-up problem.

• However, a new entry has a negative externality for other firms as ^′ ^.

• To i ter alize the egati e e ter alit , a e tr fir should pa the ta as T = − ^′ , and their expected payoff also should be _{− − .}

• Above tax can be implemented by the expected wage ₌ , and the wage should be then ₌ .

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3. Dynamic model

• The timing in each period is

1. Jobs entry the labor market with a fixed costs c, addition to unemployed workers.

2. Matching is occurred the following matching function.

3. If a job and a worker are matched, they jointly produce y value, while an unmatched worker produces b, while unfilled jobs produce no value.

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Value functions

• The value of an employed worker with wage w is

= + � � + − � ,

the value of an unemployed worker is

= + � + − ,

the value of a filled job is

� = − + � � + − � � ,

and the value of a vacancy is

= − + � � + − ,

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Nash bargaining

• Surplus of a worker and a job are _{− and �} _{− .}

• Joint surplus is _{+ �} _{− − .}

• The Nash bargaining solution is

− = + � − − ,

and

� − = − + � − − .

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Market equilibrium

• Market equilibrium is defined by _{, , �} , , , , which is characterized by value functions, the wage determination rule,

− = + � − − ,

and the free-entry condition,

= .

• Almost same as in the one-shot model

←Only key difference is the outside option of employed worker is endogenized as U.

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Intermediate step

• A intuitive way to solve the model is reducing the equilibrium conditions into three equations with wage, market tightness, and outside option.

• The free entry condition and values of jobs obtain

= � _{− � + �� .}⁻

→Negative relationship between market tightness and wages.

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Intermediate step

• With the free entry condition (V=0), the values of employed worker and filled job, and the wage determination rule, can be combined as

− − � = − − � ,

and

= + − − � .

← Positive relationship between wage and outside option(value of unemployed worker).

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Intermediate step

• The free entry condition (V=0), the values of unemployed and unfilled job, and the wage determination rule, can be combined as

− � _{= + −}

→ Positive relationship between the value of unemployed and market tightness.

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Intermediate step

• Reduced equilibrium conditions can be shows as;

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Final step

• For the comparative statics, it is useful to additionally reduce into two equations with wage and tightness as

= � _{− � + �� ,}⁻ and

= + + − ,

or

= � − � + �� + �⁻ ⁻ ^{− .}

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Dynamics

• While wage and tightness have no dynamics, the dynamics of number of unemployment is

+1 ⁼ ⁻ ^{+ � � −} ^.

• Above has a stable steady-state as

= ^�_{+ � �}

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Social planner problem

• Social planner maximizes the sum of life-time payoff given a motion of unemployed worker.

← The value function of social planner is

= � − + − + � ₊₁

subject to

+1 ⁼ ⁻ ^{+ � � −} ^.

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Social planner problem

• The optimal conditions are

�

� ^{= = −} ^{− �} ^{′ ′} ^,

and

′ _{= − + −} _{+ � −} _{− �} ′_.

Combining them yields,

= _{− � + �[}^� ^′ ₋ ⁻_′ _{] + ��}

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Social planner problem

• By using elasticity, the optimal condition can be rewritten as

= ^{� −}_{− � + �} ⁻_{+ �� .}

The market equilibrium was

= ^{� −}− � + �� + �⁻ ^, which is efficient if and only if _{= .}

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DMP model

Hosios, A. J. (1990). On the efficiency of matching and related models of search and unemployment. The Review of Economic Studies, 57(2), 279-298.

Petrongolo, B., & Pissarides, C. A. (2001). Looking into the black box: A survey of the matching function. Journal of Economic literature, 39(2), 390-431.

Pissarides, C. A. (2000). Equilibrium unemployment theory. MIT press.