The Model

The economy is populated by the representative large household and the representative large firm. As in Andolfatto (1996) and Fang and Rogerson (2009), we adopt the assumption of the large household setup. Family members in a larger household pool all resources regardless of their labor market status which assures perfect consumption insurance. The large household comprises a continuum of members (of measure one), who are either employed or non-employed. Like Fang and Rogerson (2009), the employed engage in work or leisure and obtain a wage when working. Yet, unlike Fang and Rogerson (2009), the non-employed take on job search or leisure and

the cost of job search is foregone leisure. Also, unlike these authors, there is a large firm.

The large firm creates and maintains multiple vacancies and rents capital and hires labor to produce goods using a technology that is concave in employment. The job finding and recruitment rates are endogenous, depending on the masses of both matching parties.

Unfilled vacancies and job seekers are met bilaterally through the matching technology.

Filled vacancies and employed workers are separated at an exogenous rate. Finally, there is a fiscal authority that levies taxes and offers non-employment benefits.

2.3.1 Households

The representative household has a unified preference and pools all resources for its members. In a period t, a fraction et of the members is employed and the remaining fraction (1−et) is non-employed. Given a fixed time endowment normalized at unity, each employed member allocates a fraction lt of the total time to work and the remaining fraction (1−lt) to leisure. Non-employed members devote a fraction st of their time to job search and the remaining fraction (1−st) to leisure. From the household’s perspective, the employment changes according to

( )

1 μ (1 ) ψ ,

+ − = − −

t t t t t t

e e s e e (1) where st(1-et) is an aggregate search made by the non-employed, μt is the effective job finding rate and ψ is the (exogenous) job separation rate. Thus, the change in employment (et+1−et) is equal to the inflow of non-employed workers into the employment pool (μtst(1−et)) net of the outflow as a result of separation (ψet).

Denote c_t as consumption and k_t as capital with δ the depreciation rate. Further, denote by wt and rt the wage rate and the interest rate, respectively. Let the profit be πt, non-employment benefits be b, the labor income tax rate be τ and the lump-sum tax per household be Tt. The household’s budget constraint is

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The large household has four sources of income: capital rental, after-tax wage earned by employed members, the compensation received by non-employed members, and profits remitted from firms. It allocates income to consumption, investment and lump-sum taxes. The household obtains utility from consumption and leisure. Following Andolfatto (1996), the utility of an employed member is u c( )_t +χ₁V(1−l_t) and the utility of a non-employed member is u c( )_t +χ₂V(1−s_t), where χ1 and χ2 are the degree of leisure utilities for an employed and a non-employed member, respectively. We assume that u and V exhibit the standard concavity property of positive and decreasing marginal utilities.⁵ If χ₂V(1−s_t)≤ χ₁V(1−l_t), a non-employed member is not better-off than an employed member, as opposed to the standard Pissarides matching model adopted in Fang and Rogerson (2009). The utility of the large household is the sum across all household members and thus u c( )_t +e V_tχ₁ (1−l_t) (1+ −e_t)χ₂V(1−s_t).

The household’s optimal control problem is written as the following Bellman equation,

subject to the constraints (1) and (2), where ρ>0 is the time preference rate.

The first-order conditions with respect to kt+1 and st and the Benveniste-Scheinkman conditions for k_t and e_t are, respectively,

(

)

5 For simplicity, we use the same form of the leisure utility for employed and non-employed members in the household. Results are the same if different forms are used.

(

)

(

)

While (4a) is standard, (4b) equates the marginal cost of search effort in terms of foregone leisure to the expected marginal gain of employment from a successful match in the next period. The last two conditions are the representative household’s marginal gain of capital and employment, respectively, in the beginning of the period.

Forwarding (4c) by one period and substituting it into (4a) gives the following standard Euler equation

The representative large firm rents capital and hires labor in order to produce a single final good y_t. The production technology is neoclassical, given by the following function.

( ) ,1

α −α

t = t t t

y Ak e l (6) where A>0 is a productivity parameter and α∈(0, 1) is the share of capital. The

production function is concave in employment, as opposed to that in the standard Pissarides matching model adopted by Fang and Rogerson (2009) wherein aggregate production is linear in employment.

From the firm’s perspective, employment is increased by the inflow of employees and decreased by the outflow due to separation.

1 η ψ ,

+ − = −

t t t t t

e e v e (7) where ηt is the (endogenous) recruitment rate and v_t is (endogenously) created

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vacancies.

To create and maintain vacancies, firms need to pay a cost to adjust the vacancy numbers. We assume the following quadratic cost function: Λ( )v_t =λ₀v_t +λ₁v_t², where The representative large firm chooses capital and vacancies in order to maximize the discounted sum of flow profits. The Bellman equation associated with the firm is

, 1

The first-order conditions with respect to k_t and v_t and the Benveniste-Scheinkman condition for et are, respectively,

( )^α 1 ,

Capital is determined by the marginal product of capital equal the rental rate in (9a).

(9b) is the vacancy-employment condition which equates a firm’s marginal cost of vacancies in this period to the expected marginal benefit of employment/recruitment from a successful match in the next period. A firm’s marginal benefit of employment in (9c) is the sum of the marginal product of labor net of the wage rate multiplied by hours worked per worker and the discounted future marginal benefit.

It is straightforward to rewrite (9a) as

1

Thus, the market effective capital-labor ratio, denoted by q, is decreasing in the rental rate.

2.3.3 Labor Matching and Bargaining

The labor market exhibits search frictions with aggregate flow matches depending on the masses of job seekers and vacancies. Following Diamond (1982), we assume pair-wise random matching. The matching technology takes the constant-returns form:⁶

(

) ( )

= −

t t t t

M m s e v where m>0 measures the degree of matching efficacy and γ∈(0, 1) the contribution of job seekers in matching. Aggregate search and recruitment behave like two inputs in the matching function and the output is the aggregate matched pair M_t. The matching function facilitates the endogenous determination of job finding rates and recruitment rates. As in Andolfatto (1996), since st is search effort per job seeker, aggregate search effort by job seekers is s_t(1- e_t).

A job seeker’s surplus acquired from a successful match is evaluated by its augmenting value from employment Ue in (4d), whereas a vacant job’s surplus of a successful match is gauged by its incremental value from recruit Πe in (9c). In a frictionless Walrasian world, taking the wage as given, the household maximizes Ue and the firm maximizes Πe in order to decide their supply of and demand for labor. There is implicitly an auctioneer in the labor market which sets an equilibrium wage so as to equate labor supply to labor demand. In a frictional labor market, however, there is no

6 In a survey of micro foundations underlying the matching function and its empirical success, Petrongolo and Pissarides (2001) referred to the matching function as a useful modeling device for building labor market frictions into equilibrium macroeconomic models of wages, employment, and unemployment that occupies the same place in the macroeconomist’s tool kit as other aggregate functions such as the production function.

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auctioneer and a job seeker would meet at most one unfilled job one time and similarly, an unfilled job would meet at most one job seeker one time. This creates a bilateral monopoly.

Following conventional wisdom, the wage rate and working hours are determined simultaneously by a matched worker-job pair through a cooperative bargaining game.

Like Fang and Rogerson (2009), an employed worker does not devote all the time endowment to work and thus the pair of a successful match also bargains over working hours. In the game, the following joint surplus is maximized:

[ ( , )] [ ( )] ,U k e_{e t t} ^β Π_e e_t 1^−β where β∈(0, 1) measures a labor’s bargaining power. In solving the bargaining problem, the worker-job pair treats matching rates (μt and ηt), the beginning-of-period level of employment (et), and the market interest rate (rt) as given.

The worker also takes as given the wage and working hours of all others. The first-order conditions are

The government’s behavior is passive; it levies labor income and lump-sum taxes and offers non-employment benefits. The government budget constraint is

(

)

+ = −

t t t t t

T w e l b e (11) In order to isolate the effects of policy changes carried out later, we include lump-sum taxes Tt. When the labor tax rate τ is changed, with non-employment benefits b being held unchanged, lump-sum taxes/subsidies T will change accordingly in order to

balance the budget. Similarly, when non-employment benefits are increased, with the labor tax rate being held constant, lump-sum taxes will adjust to balance the budget.