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Determinants of long-run unemployment.

1. Introduction

Although a high rate of economic growth and a low rate of unemployment are two major goals of most governments, the relationship between these two goals is not well understood. For example, Pissarides (1990) shows that the long-run unemployment rate and growth rate are always negatively correlated, whereas Aghion and Howitt (1994) conclude that the former can be an inverted U-shaped function of the latter. More recent studies by Eriksson (1997) and Falkinger and Zweimuller (2000) suggest that growth can either increase or decrease unemployment depending on the sources of economic growth. The empirical evidence on this issue is equally ambiguous. Bean and Pissarides (1993) find that there does not exist any significant relationship between unemployment and growth across OECD countries. Caballero (1993) finds that these two series are weakly positively correlated in the UK and United States. However, Muscatelli and Tirelli (2001) find that though unemployment has a significant negative effect on growth in Canada, France, Germany, Italy, Norway, Japan, and Sweden, its impact is not significant in Australia, Austria, the UK, and the United States.

Aghion and Howitt (1994) and Pissarides (1990) are two influential theoretical studies that have investigated the long-run effect of growth on employment. (1) In Pissarides (1990), the long-run growth rate is exogenous. Because higher productivity growth raises the rate of return from job creation, and hence increases the exit rate from unemployment, the unemployment rate and growth rate are always negatively correlated in his model. To reconcile the conflict between the model prediction and the empirical evidence, Mortensen and Pissarides (1998) incorporate renovation costs into a model similar to that of Pissarides (1990). (2) They show that the relationship between the unemployment rate and growth rate depends on the renovation costs. That is, they are negatively correlated if the renovation costs are low and positively correlated if the renovation costs are high.

Aghion and Howitt (1994) identify two competing effects of growth on unemployment. On the one hand, as in Pissarides (1990), an increase in growth increases the returns from job creation, which reduces the unemployment rate (the capitalization effect). On the other hand, an increase in growth shortens the duration of job matches. Because shorter duration of job matches directly raises the job separation rate and indirectly discourages job creation (the creative destruction effect), a higher growth rate could increase the unemployment rate. The results (Propositions 1 and 2) in Aghion and Howitt (1994) suggest that the unique equilibrium unemployment rate can be represented as an inverted U-shaped function of the growth rate whenever the entry cost is positive but sufficiently small.

Because productivity growth is exogenous in Pissarides (1990) and Mortensen and Pissarides (1998), the cross-country variations in economic growth cannot be explained. In contrast, the long-run growth rate is endogenously determined in Aghion and Howitt (1994). However, they do not explicitly examine the impact of labor market parameters such as unemployment benefits and hiring costs on the unemployment rate and growth rate. (3) Consequently, Aghion and Howitt (1994) cannot answer some important questions such as whether differences in institutional settings between Europe and the United States are accountable for the differences in their unemployment rates. Because, as shown by Mortensen and Pissarides (1998), the relationship between the unemployment rate and growth rate can turn from negative to positive as the renovation costs rise, an explicit examination of the impact of labor market parameters is important to understanding the cross-country differences in the long-run growth rate and unemployment rate.

To examine the determinants of long-run unemployment and economic growth simultaneously, we extend the endogenous growth framework of Howitt and Aghion (1998) to allow for a more general treatment of the labor market in the spirit of Pissarides (1990). The major distinction between Pissarides (1990) and Mortensen and Pissarides (1998) and our model is whether growth is endogenously determined. (4) Endogenizing economic growth enables us to explicitly analyze the impact on unemployment of factors that are commonly considered as determinants of growth, but are largely overlooked by the unemployment literature, such as the productivity of research and development (R&D) and the speed of technological spillovers. Our model differs from Aghion and Howitt (1994) in that the impact of several important institutional factors, such as unemployment benefits and workers' bargaining power, on growth and unemployment is explicitly examined. Our model generates several interesting findings that are absent from Aghion and Howitt (1994), Mortensen and Pissarides (1998), and Pissarides (1990). (5)

First, we find that the long-run growth rate depends not only on the regular preference and technology parameters, as in the literature on endogenous growth with full employment, but also on certain labor market parameters; symmetrically, we find that the unemployment rate depends not only on the labor market parameters, but also on other factors that affect growth. Second, consistent with the empirical evidence, our model predicts that a rise in the growth rate can either increase or decrease the unemployment rate, depending on the model's parameters. Third, different types of government policies that directly or indirectly promote long-run growth can have opposite effects on the unemployment rate.

The remainder of the paper is organized as follows. The next section describes the environment and sets up the model. Section 3 derives the steady-state equilibrium conditions and the major results and discusses the policy implication of these results. Some concluding remarks are given in the last section.

2. The Model

This section develops the basic model. Our model extends the Schumpeterian endogenous growth model of Howitt and Aghion (1998) to allow for a more general treatment of the labor market in the spirit of Pissarides (1990).

Technologies

The economy is populated with a continuum of identical households with measure one. Each household consists of many infinitely lived members whose time endowment is normalized to unity. There are five types of production activities in this economy: final good production, intermediate good production, search in the labor market, physical capital accumulation, and R&D. It is assumed that in the intermediate sectors, producers are assumed to have temporary monopoly power, and in the labor market, wage rates are determined through Nash bargaining.

Final Good Production

Following Pissarides (1990), we make the following two assumptions: (i) there is a continuum of identical final-good producing firms with measure one and (ii) each firm employs many workers and is large enough to eliminate all uncertainty about the flow of labor. An individual firm uses a continuum of intermediate goods i [member of] [0, 1] and labor as its inputs subject to the following production technology:

Y = [N.sup.1 - [alpha]] [[integral].sup.1.sub.0] [A.sub.1][x.sup.[alpha].sub.i] di, 0 < [alpha] < 1, (1)

where Y is the output; N is the number of workers employed; (6) [x.sub.i] is the flow of intermediate good i used: [alpha] is a parameter that measures the contribution of the intermediate good to the final-good production, and its inverse measures the intermediate-good producer's market power; and [A.sub.i] s the productivity coefficient of intermediate good i that is determined by the technology from R&D.

Final output is allocated among consumption C, investment in R&D Q, expenditures on hiring in the labor market [GAMMA], and investment in physical capital K:

Y = C + [??] + Q + [GAMMA]. (2)

We implicitly assume that each unit of consumption good foregone can be used to produce one unit of capital and that there is no capital depreciation. Throughout this paper, the final good is used as a numeraire.

Search in the Labor Market

To produce final output, the final-good producers have to search for workers. Because N is the number of workers that are matched with jobs in the final good sector, 1 - N is the number of unemployed workers. Job-worker pairs are assumed to separate at a constant rate s, with 0 < s < 1. (7) To find a suitable employee, a firm has to incur a hiring cost [GAMMA]. We assume that the hiring cost is proportional to the wage rate W, that is, [GAMMA] = [gamma]W, where [gamma] > 0. The rate at which new jobs and workers match is governed by the constant-returns-to-scale aggregate matching technology

M (v, 1 - N) = M (v, u), (3)

where v is the number of vacancies, and u [equivalent to] 1 - N is the number of unemployed workers. Because the labor force is normalized to unity, v and u are also respectively the vacancy rate and the unemployment rate. With the matching technology (3), the instantaneous probability of a vacancy being filled is m([theta]) [equivalent to] M(v, u)/v = M(1, 1/[theta]) with m'(*) < 0, where [theta] [equivalent to] v/u is the vacancy-unemployment ratio that is outside the control of firms. As a result, the employment of an individual firm evolves according to (8)

N = m([theta])v - sN, (4)

where a dot over a variable represents the time change rate of that variable. Equation 4 states that the net change of employment N is the difference between the inflow of workers m([theta])v and the outflow of workers sN. Given the matching technology (3), the wage rate determined through Nash bargaining W, and the prices of intermediate goods [p.sub.i], the final-good producer chooses the number of vacancies v and quantities of intermediate inputs [x.sub.i] to maximize its discounted expected profit (9)

[[integral].sup.[infinity].sub.0] [e.sup.rt][[pi].sub.Y]dt, [[pi].sub.Y] = [N.sup.1 - [alpha]] [A.sub.i][x.sup.[alpha].sub.i] di - WN - [GAMMA]v - [[integral].sup.1.sub.0][p.sub.i] [x.sub.i] di, (5)

subject to the dynamic employment of Equation 4. In Equation 5, [p.sub.i] is the price of intermediate good i in terms of final good. The current-value Hamiltonian function for the final-good producer's maximization problem is

[H.sup.f] = [N.sup.1 - [alpha]] [[integral].sup.1.sub.0] [A.sub.i] [x.sup.[alpha].sub.i] di - WN - [GAMMA]v - [[integral].sup.1.sub.0] [p.sub.i][x.sub.i] di + [xi][m([theta])v - sN],

where [xi] is the co-state variable associated with this maximization problem. The first-order conditions are

[partial derivative][H.sub.f]/[partial derivative][x.sub.i] = [alpha][A.sub.i][x.sup.[alpha] - 1.sub.i][N.sub.1 - [alpha]] - [p.sub.i] = 0, (6)

[partial derivative][H.sub.f]/[partial derivative]v = -[GAMMA] + [xi]m([theta]) = 0, (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Note that in Equation 8, as discussed in Shi and Wen (1997), that the difference between the marginal product of labor (1 - [alpha])[[integral].sup.1.sub.0] [A.sub.i][([x.sub.i]/N)sup.[alpha]] di and the wage rate W is the final-good producer's surplus from hiring an additional worker. Solving the first-order conditions gives the conditions that determine the final-good sector's demand for labor and intermediate good i:

(1 - [alpha]) [[integral].sup.1.sub.0] [A.sub.i] [([x.sub.i]/N)sup.[alpha]] di = W + (s + r - [g.sub.w])[GAMMA]/m([theta]), (10)

[alpha][A.sub.i][x.sup.[alpha] - 1.sub.i] [N.sup.1 - [alpha]] = [p.sub.i] [for all]i [member of] [0,1], (11)

where [g.sub.w] [equivalent to] W/W. Equation 10 equalizes the marginal benefit (the left-hand side) and the marginal cost (the right-hand side) of employing an additional worker. The marginal cost of using an additional worker consists of two parts: the wage cost W and the expected hiring cost (s + r - [g.sub.w])[gamma]/m([theta]). Similarly, Equation 11 states that the marginal benefit (the left-hand side) and the marginal cost (the right-hand side) of using an additional unit of intermediate good i must be equalized. In the steady-state balanced growth equilibrium, [??] = 0. Then, the unemployment rate is determined by

s + s/[theta]m([theta]). (12)

Equation 12 implies that the unemployment rate depends positively on the job separation rate s and negatively on the vacancy-unemployment ratio [theta] and the matching efficiency m([theta]). (10)

Intermediate Good Production

Following Howitt and Aghion (1998), we assume that only capital is needed to produce intermediate goods. The production technology for intermediate good i is assumed to take the following form:

[x.sub.i] = [K.sub.i]/[A.sub.i], (13)

where the capital input [K.sub.i] is deflated by the productivity parameter [A.sub.i] to reflect the fact that higher-quality intermediate goods are more difficult to produce. Given the rental rate r and the final-good sector's demand for intermediate goods (Equation 11), intermediate good producer i chooses output level [x.sub.i] to maximize its monopoly profit flow

[[pi].sub.i] = [p.sub.i][x.sub.i] - r[K.sub.i] = [alpha][A.sub.i] [x.sup.[alpha].sub.i]][N.sup.1 -[alpha]] - r[A.sub.i][x.sub.i], (14)

where [[pi].sub.i] is the monopoly profit flow for intermediate good producer i. Then the first-order condition for this maximization problem is

r = [[alpha].sup.2] [([x.sub.i]/N).sup.[alpha]-1], (15)

which yields intermediate-good sector i's optimal output

[x.sub.i] = x [equivalent to] N [([[alpha].sup.2/r)sup.1/(1 - [alpha]) (16)

Because both the marginal revenue and marginal cost of each intermediate monopolist are proportional to the quality of its product, every intermediate monopolist produces the same amount of output regardless of the quality of its product. From Equation 16 and the capital market equilibrium condition [[integral].sup.1.sub.0] [K.sub.i] di = [[integral].sup.1.sub.0] [A.sub.i][x.sub.i] di = Ax = K, where A [equivalent to] [[integral].sup.1.sub.0] [A.sub.bi] di is the average productivity of intermediate goods, we have

x = K/A = kN, and r = [[alpha].sup.2][k.sup.[alpha] -1], (17)

where k [equivalent to] K/(AN) is the productivity-adjusted capital-labor ratio. Correspondingly, the intermediate monopolist's maximum profit is

[[pi].sub.i] = [A.sub.i][alpha](1 - [alpha])[k.sup.[alpha]]U. (18)

Substituting the quantity of each intermediate input given in Equation 17 into Equation 1 yields the output of final good

Y = A[k.sup.[alpha]] N. (19)

Note that Equation 19 is the standard neoclassical production function.

R&D

R&D is targeted at specific intermediate products. It is assumed that each successful innovation creates an improved version of the existing product, which replaces the existing product. As a result, the innovator becomes a temporary monopolist until the next innovation in that sector. Assume that innovations follow a Poisson process and that the arrival rate in any sector is

[phi] = [lambda]q, [lambda] > 0, (20)

where [lambda] is the R&D productivity parameter, and q = [Q/A.sup.max] is the (leading-edge) productivity-adjusted expenditure on R&D in each sector, where [A.sup.max] = max{[A.sub.i], i [member of] [0, 1]} is the productivity of the leading-edge technology. The reason for deflating R&D expenditure by the leading-edge productivity parameter is that the complexity of innovation increases proportionally as technology advances. An R&D firm chooses its R&D expenditure Q to maximize its expected return {[phi][PI] - Q}, where [PI] is the expected value of an innovation. The expected value of an innovation is given by [PI] = [[integral].sup.[infinity].sub.t] [[A.sup.max][alpha](1 - [alpha])[k.sup.[alpha].sub.[eta]][N.sub.eta] exp [- [[integral].sup.[eta].sub.t] ([r.sub.[tau]] + [[phi].sub.[tau]]) [d.sub.[tau]]] d[eta] (where [eta] is a time subscript), which yields

[PI] = [A.sup.max] [alpha](1 - [alpha])[k.sup.[alpha]]N/r + [lambda]q. (21)

Equation 21 shows that the discount rate is the sum of the interest rate r and the arrival rate [lambda]q of innovation. The arrival rate of innovation in the discount rate implies that the higher the arrival rate of innovation, the shorter the period during which the intermediate monopolist can enjoy its monopoly profits. The first-order conditions for an R&D firm's maximization problem are

[lambda][PI]/[A.sup.max] [less than or equal to] 1, Q [greater than or equal to] 0, and Q ([lambda][PI]/[A.sup.max] -1) = 0. (22)

Because we will consider only interior solutions Q > 0, the first-order conditions (Equation 22) reduce to

[lambda][PI]/[A.sup.max] = 1. (23)

Equation 23 states that the expected marginal benefit of R&D (the left-hand side) equals the marginal cost of R&D (the right-hand side).

Knowledge Spillovers

Following Caballero and Jaffe (1993) and Howitt and Aghion (1998), we assume that growth of the leading-edge productivity [A.sup.max] comes from knowledge spillovers produced by innovations. Specifically,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where [sigma] is a parameter that measures the marginal impact of each innovation on the stock of public knowledge. Howitt and Aghion (1998) have shown that the ratio of the leading-edge productivity [A.sup.max] to the average productivity A converges monotonically to the constant 1 + [sigma]. Since we are interested only in the steady-state balanced growth path, we assume that [A.sup.max]/A = 1 + [sigma] for all t, so the average productivity A also grows at the same rate as the leading-edge productivity, i.e., [??]/A = [g.sub.a].

Preferences

As mentioned above, the economy is populated with a continuum of identical households, and each household consists of many infinitely lived members. Each household member spends his time working if he is employed or searching for a job if he is unemployed. Unemployed household members are randomly matched with job vacancies. To simplify our analysis, we follow Shi and Wen (1997) to assume that each household consists of a continuum of members who care only about the household's welfare. As a result, there is no uncertainty in the household's income and consumption. The representative household's preferences are given by

U(C) = [[integral].sup.[infinity].sub.0] [e.sup.[rho]t] ([C.sup.1 - [epsilon]]/1 - [epsilon]) dt, (25)

where C is consumption; [rho] is the constant rate of time preference; [member of] is the elasticity of marginal utility; and t represents time. Note that the time subscript is omitted whenever no confusion can arise. The household's budget constraint is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

where r is the rate of interest, W the wage rate, K the household's capital assets, N the fraction of employed household members, [[pi].sub.Y] the profit from the final-good production, Z the unemployment benefits, and T the lump-sum tax. (1l) We assume that the employment benefits are proportional to the wage rate, i.e., Z = zW, where z > 0. The household chooses consumption C to maximize its utility (Equation 25) subject to its budget constraint (Equation 26) and the following law of motion for N:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

Equation 27 states that the net change of the number of employed members in each household is the difference between the flow of job matches [theta]m([theta])(1 - N) and the flow of job separations sN. The current-value Hamiltonian function for the household's maximization problem is

[H.sup.c] = [C.sup.1 -[epsilon]/1 - [epsilon] + [eta] [rK + [[pi].sub.Y] + WN + Z (1 - N) - C - T] + [zeta][[theta]m([theta]) (1 - N) (1 - N) - sN], (28)

where [eta] and [zeta] are the co-state variables associated with this maximization problem. The first-order

[partial derivative][H.sup.c]/[partial derivative]C = [C.sup.[epsilon]] - [eta] = 0, (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Solving the above first-order conditions gives conditions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Equation 33 states that the growth rate of consumption depends positively on the interest rate r and negatively on the subjective discount rate 9 and the elasticity of marginal utility [epsilon]. In Equation 34, the difference between the wage rate W and the unemployment benefit Z is the worker's surplus from employment.

Wage Determination

Now we consider the determination of wage rates. We assume that the wage rates are determined by Nash bargaining. As in Shi and Wen (1997), a matched worker (who cares only about the household's welfare) negotiates with the firm. (12) The wage rates derived from the Nash bargaining solution maximize the weighted surpluses of the household and the firm, (13)

[(W - Z)sup.[beta] [[(1 - [alpha])A[k.sup.[alpha]] - W].sup.1 - [beta]], 0 < [alpha] < 1, (35)

where [beta] is the workers' bargaining power. The first-order condition for this maximization problem is

[beta]/W - Z = 1 - [beta]/(1 - [alpha])A[k.sup.[alpha]] - W, (36)

which gives the following solution:

W = A[beta](1 - [alpha])[k.sup.[alpha]]/1 - (1 - [beta])z. (37)

From Equation 37, we can see that the equilibrium wage rate depends positively on the average productivity A of intermediate goods, the productivity-adjusted capital-labor ratio k, the workers' bargaining power [beta], and the unemployment benefit z.

3. Steady-State Equilibrium and Results

We focus our discussion on steady-state balanced growth equilibria. In a steady-state balanced growth equilibrium, the values of the interest rate r, the productivity-adjusted capital intensity k, the vacancy rate v, the unemployment rate u, and the vacancy-unemployment ratio [theta] are all constant, and output Y, consumption C, capital stock K, investment in R&D Q, the wage rate W, the leading-edge productivity [A.sup.max], and the average productivity A all grow at the same constant rate g, i.e., [g.sub.y] = [g.sub.c] = [g.sub.k] = [g.sub.q] = [g.sub.w] = [g.sub.a] = g, where [g.sub.y] = [??]/Y, [g.sub.k] = [??]/K, and [q.sub.q] = [??]/Q. (14) Now we derive equilibrium conditions. First, we obtain the usual positive relationship between the interest rate and the growth rate from Equation 33:

r = [epsilon]g + [rho]. (38)

Note that this relationship is exactly the same as in the literature, with an exogenous growth rate and an endogenous interest rate. Given the growth rate g, a higher elasticity of marginal utility (i.e., a lower elasticity of intertemporal substitution) gives rise to a higher interest rate. Second, Equation 17 is rewritten as

k = [([[alpha].sup.2]/r).sup.1/(1 - [alpha]), (39)

where the productivity-adjusted capital intensity k depends negatively on the interest rate. The intuition behind this is that a higher interest rate increases the cost of capital, thus reducing the demand for capital. Third, combining Equations 21, 23, 24, and 38, we rewrite the optimal R&D condition as

[lambda][alpha](1 - [alpha])[k.sup.[alpha]](1 - u)/g([epsilon] + 1/[sigma]) + [rho] = 1. (40)

Fourth, Equations 10, 16, and 37 lead to

s + r - g/m([theta]) = (1 - [beta])(1 - z)/[beta][gamma]. (41)

Using the unemployment rate given by Equation 12, i.e.,

u = 1/s + [theta]m([theta]), (42)

we know that the vacancy-unemployment ratio 0 depends positively on the separation rate s and negatively on the unemployment rate u, that is,

[theta] = f(u, s) with [partial derivative]f/[partial derivative]u < 0 and [partial derivative]f/[partial derivative]s > 0. (43)

Combining Equations 41 and 43 gives

m[f(u, s)] = [beta][gamma](s + r - g)/(1 -[beta])(1 - z). (44)

Since m'(x) < 0 and [partial derivative]f/[partial derivative]u < 0, the unemployment rate u depends negatively on the difference between the growth rate and interest rate (g - r). If the interest rate r is fixed or the elasticity of marginal utility [epsilon] < 1, then an increase in the growth rate g reduces the unemployment rate u; if the interest rate is endogenous and [epsilon] > 1, then the effect reverses. Finally, substituting Equations 38 and 39 into Equations 40 and 44, we obtain the following two equations that determine the steady-state values of the unemployment rate u and growth rate g:

u = 1 - [([epsilon] + 1/[sigma])g + [rho][([epsilon]g + [rho]).sup.[alpha]/(1 - [alpha])]/ [lambda] (1 - [alpha])[[alpha].sup.(1 + [alpha])/(1 - [alpha])], (R)

m[f(u, s)] = [beta][gamma][s + ([epsilon] - 1)g + [rho]]/(1 - [beta])(1 - z). W

Equation R is essentially the optimal R&D condition. We can easily see that this equation indicates a negative relationship between the unemployment rate and the growth rate. The reason is that an increase in the unemployment rate decreases the profit flow of a successful innovator, which reduces the value of innovation. As a result, investment in R&D drops, and the growth rate falls.

Equation W is the wage determination condition. This equation implies that the relationship between the unemployment rate and the growth rate depends on the magnitude of the elasticity of marginal utility (see the proof of Proposition 1). The unemployment rate increases with (is independent of, decreases with) the growth rate if the elasticity of marginal utility [epsilon] is greater than (equal to, smaller than) unity. The reasons are as follows. As discussed in Pissarides (1990), the unemployment effects of changes in the growth rate are related to technological progress and the intertemporal nature of the final-good producers' employment decisions. On the one hand, the final-good producers have to incur hiring costs now to hire workers who bring profits to the final-good producers in the future. Since the hiring cost increases proportionally with profits as technology advances, the final-good producers have the incentive to economize on future hiring costs by bringing forward some hiring provided that the interest rate remains the same. As a result, given the interest rate, the final-good producers increase (decrease) the number of vacancies as the growth rate rises (falls) in order to economize on the hiring costs (the growth effect).

On the other hand, the final-good producers discount the future hiring costs and profits at the equilibrium interest rate. As the equilibrium interest rate rises, the final-good producers have less (more) incentive to bring forward (postpone) hiring because future hiring becomes less expensive relative to current hiring (the interest rate effect). These two effects offset each other: the growth effect decreases unemployment while the interest rate effect does the opposite. The net effect depends on which effect dominates.

From the equilibrium relationship between the interest rate and the growth rate, r = [epsilon]g + [rho], we have [partial derivative]r/[partial derivative]g = [epsilon] . As discussed in Eriksson (1997), the strength of the interest rate effect relative to the growth effect depends on the elasticity of marginal utility: If the elasticity of marginal utility [epsilon] > (=, <) 1, then the interest rate effect is stronger than (as strong as, weaker than) the growth effect. Accordingly, the unemployment effect of an increase in the growth rate depends on the elasticity of marginal utility. With an increase in the growth rate, the equilibrium unemployment rate increases (remains unchanged, decreases) if the elasticity of marginal utility is greater than (equal to, less than) unity.

[FIGURE 1 OMITTED]

Figure 1 shows the two curves that illustrate the relationships between the unemployment rate and the growth rate determined by Equations R and W. The R&D curve (R) is always downward-sloping, whereas the wage determination curve (W) is upward-sloping (horizontal, downward-sloping) if the elasticity of marginal utility is greater than (equal to, less than) unity.

Now we investigate the properties of steady-state equilibria by examining the above two equilibrium conditions, R and W. First, we look at the conditions for the existence of steady-state equilibria. Let [u.sub.1] be the solution to Equation W with g = 0 (thus [u.sub.1] > 0) and B be the expected marginal benefit of R&D when g = 0, that is, B = (1 - [u.sub.1])[lambda](1 - [alpha])[[alpha].sup.(+ [alpha]/(1 - [alpha])][[rho].sup.1/[alpha] - 1)] (see Equation 40), then the condition for the existence of equilibria is given by:

PROPOSITION 1. If B [less than or equal to] 1, then there exists a unique steady-state equilibrium with [u.sup.*] = [u.sub.1] and [g.sup.*] = 0 (a development trap equilibrium); if B > 1, then there exists a unique steady-state equilibrium with [u.sup.*] > 0 and [g.sup.*] > 0.

PROOF: From Equation R, we have

[partial derivative]u/[partial derivative]g < 0, [u|.sub.g = 0] = [u.sub.2] [equivalent to] 1 - [[rho].sup.1]/(1 - [alpha])]/[lambda](1 - [alpha])[[alpha].sup.(1 + [alpha])/(1 - [alpha])], (46)

and u = 0 if g is sufficiently large. From Equation W, we obtain

[partial derivative]u/[partial derivative]g = [beta][gamma]([epsilon] - 1)/[psi], (47)

where [psi] [equivalent to] (1 - [beta])(1 - z)m'(x) [partial derivative]f/[partial derivative]u) > 0 because [partial derivative]f/[partial derivative]u < 0 and m'(x) < 0. Because the denominator [psi] is always positive, the sign of [partial derivative]u/[partial derivative]g depends on the sign of ([epsilon] - 1). As a result, we have [partial derivative]u/[partial derivative]g > (=, <) 0 if [epsilon] > (=, <) 1. Furthermore, [u|.sub.g = 0] = [u.sub.1] > 0, and u > 0, [mu] g > 0. The properties of these two equations imply that if [u.sub.2] [less than or equal to] [u.sub.1], then the two curves R and W given by these two equations do not intersect each other; if [u.sub.2] > [u.sub.1], then these two curves have a unique intersection point. Note that the condition [u.sub.2] [less than or equal to] (>) [u.sub.1] is equivalent to the condition B [less than or equal to] (>) 1. QED.

This proposition is intuitive. The condition B [less than or equal to] 1 implies that the expected marginal benefit of R&D is less than or equal to the marginal cost (the marginal cost = 1). In this case, R&D firms do not have incentives to invest in R&D. As a result, there is no growth ([g.sup.*] = 0). This is a development trap equilibrium. In this equilibrium, unemployment still exists because job-worker separation always occurs, and it is costly to match workers with jobs. From the definition of B, we can see that the expected marginal benefit B of R&D (when g = 0) depends positively on the productivity of R&D [lambda], and negatively on the subjective discount rate [rho] and the unemployment rate [u.sub.1]. The unemployment [u.sub.1] in turn depends positively on the separation rate s, the subjective discount rate [rho], the hiring cost [gamma], the unemployment benefits z, and the workers' bargaining power [beta]. As a result, the development trap equilibrium occurs if one or more of the following happen: (i) the productivity of R&D) [lambda] is too low, (ii) the subjective discount rate [rho] is too large, (iii) the separation rate s is too high, (iv) the hiring cost 7 is too high, (v) the unemployment benefits z are too generous, and (vi) the workers' bargaining power [beta] is too strong.

The condition B > 1 indicates that the expected marginal benefit of R&D is greater than the marginal cost if there is no investment in R&D. Obviously, it is optimal for the R&D firms to invest in R&D to the extent that the expected marginal benefit and the marginal cost of R&D are equalized. The rest of the paper will focus on the unique steady-state equilibrium with positive unemployment and growth (i.e., B > 1). The steady-state equilibrium growth rate exhibits the following comparative statics: (15)

PROPOSITION 2. The equilibrium growth rate [g.sup.*] depends positively on the productivity of R&D [lambda] and the effectiveness of knowledge spillovers [sigma], and negatively on the separation rate s, the hiring cost [gamma], the unemployment benefits z, the workers' bargaining power [beta], the subjective discount rate [rho], and the elasticity of marginal utility [epsilon].

PROOF: From Equation R, we can easily see that

[partial derivative]u/[partial derivative][lambda] > 0, [partial derivative]u/[partial derivative][sigma] > 0, [partial derivative]u/[partial derivative][epsilon] < 0, and [partial derivative]u/[partial derivative][rho] < 0 . (48)

Similarly, from Equation W, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (49)

where [partial derivative]/[partial derivative]u < 0 and [partial derivative]f/[partial derivative]s > 0. The conditions in Equations 48 and 49 imply [partial derivative][g.sup.*]/[partial derivative][lambda] > 0, [partial derivative][g.sup.*]/[partial derivative][sigma] > 0, [partial derivative][g.sup.*]/[partial derivative]s < 0, [partial derivative][g.sup.*]/ [partial derivative][gamma] < 0, [partial derivative] [g.sup.*]/[partial derivative]z < 0, [partial derivative] [g.sup.*]/[partial derivative][beta] < 0, [partial derivative][g.sup.*]/[partial derivative][epsilon] < 0, and [partial derivative][g.sup.*]/[partial derivative][rho] < 0. QED.

[FIGURE 2a OMITTED]

[FIGURE 2b OMITTED]

[FIGURE 2c OMITTED]

The growth effects of changes in the model's parameters can be easily shown graphically. Changes in the parameters shift either the curve R or the curve W or both. For example, though an increase in the productivity of R&D [lambda] does not affect the curve W, it shifts the curve R upward to the right. Hence, the equilibrium growth rate increases (see Figure 2a). A rise in the separation rate s shifts the curve W upward to the left while leaving the curve R unchanged. As a result, the equilibrium growth rate decreases (see Figure 2b). An increase in the subjective discount rate [rho] decreases the equilibrium growth rate by shifting the curve R downward and the curve W upward (see Figure 2c). The effects on the equilibrium growth rate of changes in other parameters can also be seen graphically.

The comparative-static results concerning those regular parameters ([alpha], [lambda], [sigma], [rho], [epsilon]) remain the same as those in the literature on endogenous growth with full employment (e.g., Aghion and Howitt 1998). Changes in these parameters affect the equilibrium growth rate by directly influencing the expected marginal benefit of R&D. (16) For example, an increase in the productivity of R&D [lambda] directly increases the expected marginal benefit of R&D by increasing the probability of success, leading to an increase in the monopolists' expected profits.

However, different from the endogenous growth literature with full employment, the labor market parameters ([beta], z, [gamma], s) also affect the equilibrium growth rate by indirectly influencing the expected marginal benefit of R&D. The growth effects of the labor market parameters can be easily seen from the two equilibrium conditions R and W in the benchmark case with [epsilon] = 1 (i.e., log utility). (17) With [epsilon] = 1, Equation W becomes m[f(u, s)] = [beta][gamma](s + [rho])/[(1 [beta])(1 - z)]. Since m'(x] < 0, [partial derivative]f/[partial derivative]u < 0, and [partial derivative]f/[partial derivative]s > 0, an increase in any of the four parameters ([beta], z, [gamma], s) raises the unemployment rate u. This is because an increase in any of the four parameters reduces the final-good producers' profits (an increase in the hiring cost [gamma] or the separation rate s directly decreases the final good producers' profits while an increase in the workers' bargaining power [beta] or the unemployment benefits z reduces the final-good producers' profits indirectly by raising the wage rate) and thus lowers the final-good sector's employment, leading to a higher unemployment rate. From Equation R, we can see that a higher unemployment rate u lowers the growth rate g. The reasons are as follows: A lower level of employment in the final good sector reduces the final good sector's demand for intermediate goods, which reduces the profits of intermediate monopolists and hence reduces the expected marginal benefit of R&D.

The result that the labor market parameters also indirectly affect the long-run growth rate has important policy implications. Although those policies that promote R&D activities (direct growth-promoting policies) play a critical role in increasing long-run growth, those policies that ensure a well-functioning labor market (indirect growth-promoting policies) are equally important. Moreover, as we will see in Proposition 3, the indirect growth-promoting policies are "better" than the direct growth-promoting policies in the sense that, although both of them promote growth, the former always reduce unemployment, whereas the latter may do the opposite.

Now we turn to the determinants of the steady-state equilibrium unemployment rate. The determinants of the equilibrium unemployment rate are summarized by: (18)

PROPOSITION 3. (i) The equilibrium unemployment rate [u.sup.*] increases with the separation rate s, the hiring cost [gamma], the unemployment benefits z, the workers' bargaining power [beta], and the elasticity of marginal utility [epsilon], regardless of the value of the elasticity of marginal utility [epsilon]; (ii) the equilibrium unemployment rate [u.sup.*] increases with (decreases with, is independent of) the productivity of R&D [lambda] and the effectiveness of knowledge spillovers [sigma] if the elasticity of marginal utility [epsilon] > 1 ([epsilon] < 1, [epsilon] = 1); and (iii) the equilibrium unemployment rate increases (decreases) with the subjective discount rate 9 if the elasticity of marginal utility [epsilon] = 1 ([epsilon] < 1).

PROOF: The results in Proposition 3 come from the conditions in Equations 48 and 49. QED

As mentioned above, changes in the model's parameters shift either the curve R or the curve W or both (see Figures 2a-c). For example, an increase in the hiring cost [gamma] shifts the curve W upward while leaving the curve R unchanged, thus raising the equilibrium unemployment rate. The responses of the equilibrium unemployment rate to changes in other parameters can also be seen easily by examining the curves R and W.

The effects of the labor market parameters, such as the separation rate, the hiring cost, and the unemployment benefits, on the equilibrium unemployment rate are very intuitive. For

example, an increase in the workers' bargaining power [beta] or the unemployment benefits z increases the equilibrium unemployment rate by raising the equilibrium wage rate. Similarly, a rise in the job separation rate s increases the equilibrium unemployment rate by reducing the labor market tightness and by increasing the number of unemployed workers during a given period.

However, the unemployment effects of those parameters that directly affect the equilibrium growth rate (such as the productivity of R&D and the effectiveness of knowledge spillovers) are less straightforward. We start from the case with a fixed interest rate (e.g., for a small open economy). As discussed earlier (after Equation 44), a change in any of the growth-related parameters (e.g., an increase in the productivity of R&D) that raises the growth rate always reduces the unemployment rate. This is the growth effect on unemployment discussed in Pissarides (1990).

Now consider the case with an endogenous interest rate (i.e., r = [epsilon]g + [rho]). In this case, in addition to the growth effect, changes in growth also affect unemployment through the interest rate channel: an increase in the growth rate increases the unemployment rate by raising the interest rate. This is the interest rate effect on unemployment discussed in Eriksson (1997), Pissarides (2000), and Falkinger and Zweimuller (2000). As a result, a change in any of the growth-related parameters has two offsetting effects on unemployment: the growth effect and the interest rate effect. The growth effect decreases the unemployment rate while the interest rate effect does the opposite. (19) The net effect depends on the relative strength of the two effects: (20) (i) if the elasticity of marginal utility c < 1 and thus the interest rate effect is relatively weak, then the unemployment effects of the growth-related parameters remain quantitatively the same as in the case with a fixed interest rate; (ii) if the elasticity of marginal utility [epsilon] = 1 and thus the interest rate effect exactly offsets the growth effect, then changes in the growth-related parameters do not affect unemployment; and (iii) if the elasticity of marginal utility [epsilon] > 1 and thus the interest rate effect is relatively strong, then a change in any of the growth-related parameters that raises the growth rate always increases the unemployment rate. As a result, for example, an increase in the productivity of R&D increases (does not affect, decreases) the unemployment rate if the elasticity of marginal utility is greater than (equal to, less than) unity.

Propositions 2 and 3 show that the relationship between the unemployment and growth rates depends on the model's parameters. For example, an increase in the productivity of R&D [lambda] always increases the equilibrium growth rate, but the effect on the equilibrium unemployment rate depends on the elasticity of the marginal utility: the equilibrium unemployment rises (remains unchanged, falls) if the elasticity of marginal utility [epsilon] > (=, <) 1. As a result, it is possible to have various combinations of unemployment and growth: (i) high growth and high unemployment; (ii) low growth and low unemployment; (iii) high growth and low unemployment; and (iv) low growth and high unemployment. More formally, we have:

COROLLARY. In response to exogenous changes in the model's parameters, the unemployment rate may or may not change. If the unemployment rate does change, it may rise or fall with the growth rate.

Although we do not explicitly investigate the impact of various government policies on the unemployment rate and growth rate, it can be verified that government policies can affect the equilibrium unemployment and growth rate in the same way as changes in the model's parameters. (21) For example, the unemployment and growth effects of a subsidy to R&D are similar to those of an increase in the productivity of R&D. Similarly, the effects of government-funded employment services on unemployment and growth are also similar to those of a decrease in the hiring costs. Because of this, we can derive some policy implications from the comparative-statics results in Propositions 2 and 3.

The first policy implication comes directly from the corollary. That is, high growth does not necessarily come at the expense of high unemployment. In other words, it is possible for a country to achieve both high growth and low unemployment, even in the long run, as long as the right government policies are used. For example, any policies that can shift the curve W downward and keep the curve R unchanged would be able to raise the growth rate and lower the unemployment rate.

Related to the first implication, the second policy implication is that government policies aimed at promoting growth may cause the unemployment rate to rise or fall depending on the nature of the policies and the nature of the economy. The determinants (including government policies) of the equilibrium unemployment and growth rate summarized in Propositions 2 and 3 can be divided into three groups: (i) parameters that appear in growth models with full employment (e.g., the productivity of R&D [lambda] and the effectiveness of knowledge spillovers [sigma] (ii) preferences parameters (e.g., the subjective discount rate [rho] and the elasticity of marginal utility [epsilon]); and (iii) labor market parameters (e.g., the separation rate s, the hiring cost [lambda], and the unemployment benefit z). There is an important difference between those determinants in the first group and those in the last two groups: changes in the determinants in the last two groups have in general definite effects on the equilibrium unemployment and growth rate; changes in those determinants in the first group also have definite effects on the equilibrium growth rate, but they have indefinite effects on the equilibrium unemployment rate. For example, an R&D subsidy will increase growth, but it may increase or decrease the unemployment rate depending on whether individuals' elasticity of marginal utility is greater or smaller than unity. In contrast to the R&D subsidy, government-funded employment services will raise growth and lower unemployment by reducing the hiring costs.

In terms of promoting growth, the direct policies such as R&D subsidies and the indirect policies such as government-funded employment services are both helpful. (22) However, in terms of reducing unemployment, the indirect policies can serve the purpose but the direct policies may do the opposite. In this sense, the indirect policies that affect the labor market efficiency are "better" than the direct growth-promoting policies.

4. Conclusions

This paper develops a model of endogenous growth with unemployment to investigate the determinants of and the relationship between long-run unemployment and growth. Our framework incorporates the spirit of Pissarides' (1990) search model into Howitt and Aghion's (1998) endogenous growth model with both innovation and capital accumulation. In our model, both long-run growth and unemployment are endogenously determined.

We find the following results. First, both the long-run growth rate and unemployment rate depend not only on factors, such as the productivity of R&D and the elasticity of marginal utility, that affect long-run growth as in those endogenous growth models with full employment, but also on the labor market parameters, such as the unemployment benefits and the hiring costs.

Second, the relationship between long-run growth and unemployment depends on the model's parameters. That is, various combinations of growth and unemployment are possible. This may explain why previous empirical studies cannot detect any deterministic link between these two variables.

Third, both policies that directly provide incentives for investment in R&D and policies that indirectly encourage investment in R&D through the labor market promote long-run growth. However, these two types of policies may affect the unemployment rate differently. Indirect policies can reduce the unemployment rate while direct policies may raise it. As a result, in terms of reducing unemployment, indirect policies that improve the labor market efficiency are better than direct growth-promoting policies. We believe that these results can help us to further understand the determinants of long-run unemployment.

Received May 2005; Accepted November 2006.

References

Aghion, P., and P. Howitt. 1994. Growth and unemployment. Review of Economic Studies 61:477 94.

Aghion, P., and P. Howitt. 1998. Endogenous growth theory. Cambridge, MA: MIT Press.

Bean, C. R., and C. A. Pissarides. 1993. Unemployment, consumption and growth. European Economic Review 37:837 59.

Boone, J. 2000. Technological progress, downsizing and unemployment. Economic Journal 110:581 600.

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Caballero, R. J. 1993. Unemployment, consumption and growth: by Charles Bean and Chris Pissarides. European Economic Review 37:855-9.

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Mortensen, D., and C. A. Pissarides. 1998. Technological progress, job creation, and job destruction. Review of Economic Dynamics 1:733 53.

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(1) Some other studies on this issue include Bean and Pissarides (1993), Boone (2000), Brecher, Chen, and Choudhri (2002), Daveri and Tabellini (2000), Gordon (1997), Hoon and Phelps (1997), Manning (1992), and Mortensen and Pissarides (1998).

(2) The renovation costs consist of the expenditure on updating machinery and the cost to train workers to operate this new equipment.

(3) More recently, Aghion and Howitt (1998) explicitly consider how the labor market parameters affect the long-run growth rate and unemployment rate in a modified version of the model in Aghion and Howitt (1994) with a simplifying formulation of the labor market. We follow the spirit of Pissarides (1990) to model the labor market; in particular, wages paid to workers are determined through Nash bargaining between workers and firms. Our results confirm the findings in Aghion and Howitt (1998).

(4) Another important difference between Pissarides (1990) and Mortensen and Pissarides (1998) and our model is the assumption about firms and workers' attitudes toward risk. In their models, both firms and workers are risk-neutral. However, in our model, workers are risk-averse while firms are still risk-neutral. Because of this difference, those asset-pricing type of equations concerning wage determination in Pissarides (1990) and Mortensen and Pissarides (1998) cannot be used in our model. To avoid analytical complexity arising from the fact that individuals face risk in the labor market, we follow Merz (1995) and Shi and Wen (1997) to adopt the "large household" assumption that each household consists of a continuum of members who care only about the household's welfare; as a result, individual risks in the labor market are completely smoothed within each household.

(5) Job destruction is exogenous in our model. The models in Aghion and Howitt (1994) and Mortensen and Pissarides (1998) can be used to study the impact of growth on unemployment, but the analysis will be more complicated because the influence goes through both job creation and job destruction.

(6) An alternative interpretation is that, in the production function (1), the variable N is actually rain[N, K], where K is the capital stock and v [equivalent to] K - N [greater than or equal to] 0 is the number of vacancies.

(7) The assumption that the separation rate is constant may not be realistic. The separation rate may depend on the rate of technological progress. An interesting extension of this paper is to examine whether the main results in the paper will still hold true if the separation rate depends on the rate of technological progress.

(8) Because we assume that there are a continuum of identical firms with measure one, this is also the law of motion for the aggregate employment.

(9) Because we will focus on the steady state, steady-state conditions (e.g., [r.sub.t] = r) will be used when we derive relevant expressions and equations.

(10) The unemployment rate depends negatively on the vacancy-unemployment ratio [theta] because [theta]m([theta]) is increasing in [theta].

(11) We assume that the unemployment benefits Z are financed by the lump-sum tax T and that the government's budget is balanced at each point in time, that is, Z(1 - N) = T.

(12) Alternatively, the Nash bargaining game can be reinterpreted as one between the matched worker's household (on behalf of the matched worker) and the firm.

(13) Note that the firm and the worker in this bargaining game are assumed to be myopic because only the current surpluses are shared (rather than the present-discounted values as in more general search models). As discussed earlier, the worker's surplus is (W - Z), whereas the firm's surplus is [(1 - [alpha])[[integral].sup.1.sub.0] [A.sub.i][([x.sub.i]/N).sup.[alpha]] di]. Using the definition A = [[integral].sup.1.sub.0] [A.sub.i] di and [x.sub.i] = x = kN from Equations 16 and 17, we rewrite the firm's surplus as [(1 - [alpha])A[k.sup.[alpha]] - W].

(14) In equilibrium, the hiring cost [GAMMA], the unemployment benefits Z, and the lump-sum tax T also grow at the constant rate g.

(15) Note that the effect of [alpha] on the expected marginal benefit of R&D is ambiguous. This can be seen from the determinants of the intermediate-good producer's profit. The intermediate-good producer's profit (Equation 18) can be rewritten as [[pi].sub.i] = [[A.sub.i](1/[alpha] - l)r]x, where [A.sub.i](1/[alpha] - 1)r is the profit per unit of output and x = N[([[alpha].sup.2]/r).sup.1/(1 - [alpha])] is the optimal output. An increase in a has two offsetting effects on the profit hi. On the one hand, the increase in [alpha] reduces the profit by decreasing the price [A.sub.i]r/[alpha] and thus lowering the profit per unit of output. On the other hand, the increase in [alpha] raises the optimal output x by increasing the capital input. The net effect on the profit depends on which of the above two effects dominates. As a result, the growth and unemployment effects of changes in [alpha] are ambiguous.

(16) Note that the marginal cost of R&D is normalized to 1.

(17) In the case with [epsilon] [not equal to] 1, the growth effects of the labor market parameters remain qualitatively the same. However, the magnitudes of these growth effects will depend on the feedback effects of growth on unemployment.

(18) If the elasticity of marginal utility [epsilon] > 1, then the effect of the subjective discount rate [rho] on the equilibrium unemployment is ambiguous. As will be discussed in Proposition 3, this is due to the fact that changes in [rho] have two offsetting effects: the growth effect and the interest rate effect.

(19) As a result, a rise in the elasticity of marginal utility increases the unemployment rate by strengthening the interest rate effect.

(20) The relative strength of the two effects is determined by the elasticity of marginal utility [epsilon]. The larger the elasticity of marginal utility, the stronger the interest rate effect. Empirical studies find that the elasticity of marginal utility is in general greater than unity (see, e.g., Evans 2005; Hall 1988).

(21) Actually, some of the model's parameters are directly or indirectly related to government policies. For example, the unemployment benefits are directly affected by the government's unemployment insurance policies, and the hiring costs are indirectly related to the government's labor market regulations.

(22) Here, direct (indirect) policies refer to those policies that directly (indirectly) affect long-run growth.

Haoming Liu, Department of Economics, National University of Singapore, Singapore 117570; E-mail ecsliuhm@nus.edu.sg.

Jinli Zeng, Department of Economics, National University of Singapore, Singapore 117570; E-mail ecszjl@nus.edu.sg; corresponding author.

We would like to thank two anonymous referees for valuable comments and suggestions and the National University of Singapore for financial support (under Academic Research Grant R-122-000-037-112). All remaining omissions and errors are our own.
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Title Annotation:trends in labor market and whole economy due to unemployment
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Author:Liu, Haoming; Zeng, Jinli
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