Lecture note 5 n 最近の更新履歴 Keisuke Kawata's HP Lecture note 5 n

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Keisuke Kawata

ISS, UTokyo

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Applications of DMP model

• An important extension of the DMP model is the endogenous productivity distribution ↔ given as y in the original model.

← can study the relationship between market structure and organizational decision- making.

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Plan

1. Solving strategy: Market variable 2. Human capital investment.

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1. Market variable

• In many cases, the equilibrium can be characterized as follow steps.

1. Given others strategy ( and pass of future strategy), optimal strategy of an agent is characterized.

2. Finding all agents strategies as consistent in step 1.

• Ma y odels ith zero ass orkers, there are arket aria les hi h are;

← ot affe ted y ea h age t’s strategy

← aggregator of other’s strategies

← able to be sufficient to characterize optimal strategy of each agent. E,.g.,) Price of Warlasian market model.

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1. DMP model

• Is Wage a market variable in the DMP model? ←

• DMP model has variables which is determined by interactions between a firm and a worker (match level variables) ← Wage is determined by the Nash

bargaining ← Each worker and firm can partially manipulate.

• Market tightness is determined by the wage distribution in a market ←market level variable.

• Outside option (value of unemployed) is determined by market tightness and the wage distribution ←market level variable and sufficient to characterize the match level variable (wage).

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2. Human capital investment

• Human capital investment is an important channel to determine the productivity.

• Becker (1962) shows classical argument with the perfect competitive market. General human capital: A firm has no incentives to invest the human capital if the capital can be used in other firm, while worker has an incentive.

Specific human capital: A firm has an incentive to invest human capital if the capital can be used only in the firm, while worker has an incentive.

• Empirical findings show many firms invest even if the general human capital.

←Theoretical challenge (Survey Acemoglu & Pischke 1999)

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2. DMP with human capital investment

• General capital model.

• To simplify, all workers are employed in period 0, and human capital investment is occurred in the period 0.

• The investment cost functions of a worker and a firm are � and � .

→ Total productivity after period 0 is given by � + � .

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2. Equilibrium conditions

• Market equilibrium is defined over value functions, wage, tightness, and investment levels _{� and � .}

←tightness is market variable, while outside option (U) is not.

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2. Value function in period 0

0 ^{= − �}� ^{+ �} ^�� ^{+ � ,}

and

�₀ = − � + �� _� + � . The first order conditions are

′ _�

� ^{= �} ^′

′ _{� = ��′}

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2. On-path after period 0

• 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

= −� + � � � � + � + − � ,

where _�₀ is an initial productivity (given).

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2. After period 0

• Let suppose a deviated worker and a firm hiring the worker.

→� is not affected, while � of the worker is affected by deviation. We need to characterize the value functions given _�^�.

Wage equation is obtained as

� � + � = � � + �_� + − � − � � + �_� , And the value of unemployed

− � � + � = − � + �� + � �^� � � + �_�

− � + �� + � �^� �

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2. After period 0

• The value of employed is

� + � = ^{� � + �}^� ^{+ [ − �} − � + ��] � + �_�

− � + �� ^,

� � + � = − � � + � − − � � + �_�

− � + �� ^,

where

− � � + � = − � + �� + � �^� � � + �_�

− � + �� + � �^� �

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2. After period 0

• The investment is

′ _�

� ^{= �} ^′^, ^′ ^{� = ��′}

where

′ ₌ ^{� +} ^{− �} ^{− � + ��} ^′

− � + �� ^{> ,}

�^′ = ^{− �} _{− � + ��}⁻ ^{− �} ^′ > ,

where

− � ′ = − � + �� + � �^{� �}^� ^� ^� �

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2. Main result

• The arket stri ture is ru ial for Be ker’s result.

• In the market with search friction, the wage is not elastic over the productivity, firms have then incentive to invest the human capital.

←What’s the effi ie y property?

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Reference

Acemoglu, D., & Pischke, J. S. (1999). Beyond Becker: training in imperfect labour markets. The Economic Journal, 109(453), 112-142.

Becker, G. S. (1962). Investment in human capital: A theoretical analysis. Journal of political economy, 70(5, Part 2), 9-49.