Search, unemployment, and growth

i

JOURNALOF

ELSEVIER Journal of Monetary Economics 35 (1995) 499-507

Monetary ECONOMICS

Search, unemployment, and growth

Ian King*, L inda Well ing

Department of Economics, University of Victoria, Victoria, BC V8W 3P5, Canada

(Received September 1993; final version received February 1995)

Abstract

We develop a model to study the effects of changes in the size of technological innovations upon search and 'waiting-time' unemployment when search is costly and shocks are positively autocorrelated. We find that increases in the size of innovations increase steady state search, decrease steady-state waiting-time unemployment, and decrease total unemployment.

Key words: Search; Sectoral reallocation; Unemployment; Growth

JEL classification: E24; J64; O41; R11

1. Introduction

Surprisingly, the theoretical effects of changes in the pace of technological change upon unemployment have not been widely examined. In recent studies, Pissarides (1990) and Aghion and Howitt (1992, 1994) considered these effects in models in which the creation of vacancies by firms is an increasing function of the size of technological innovations due to what Aghion and Howitt call the 'capitalization effect': the greater the size of innovations, the more important are

*Corresponding author.

Ian King's research was funded by grants from the Social Sciences and Humanities Research Council of Canada and the University of Victoria. Linda Welling's research was funded by a grant from the University of Victoria. We are grateful to David Andolfatto, Randy Gouge, Peter Kennedy, Richard Rogerson, Randy Wright, an anonymous referee, and participants at seminars at the Canadian Macroeconomics Study Group and at the Department of Economics at the University of Victoria for comments and helpful suggestions.

0304-3932/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 3 9 3 2 9 5 0 1 198 W

500 I. King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507

future returns, and since vacancies generate future returns for firms, then an increase in the size of innovations will increase the steady-state rate of vacancy creation. Through this channel, search unemployment is a decreasing function of the size of innovations in those models. ~

In this paper, we argue that this result relies heavily on the assumed asymmetry of the choices confronting firms and workers as they enter the search process. Vacancy creation is chosen by firms in these models and requires incurring a direct cost. Workers are passive, incurring an opportuni ty cost of search while they wait to be matched with a job. If, instead, workers bear a cost when they choose to actively search, then equilibrium search may be an increasing function of the size of innovations. To illustrate this point, we develop a variant of Lucas and Prescott 's (1974) search model in which the asymmetry is reversed: competitive firms in different locations create vacancies costlessly, but workers bear a cost when they choose to search. In this setting, the workers ' search decision is also subject to a capitalization effect: since the benefits of search depend on future expected payoffs and since these payoffs are greater when the size of innovations is greater, then more workers will choose to search when innovations are larger.

We also introduce the possibility of waiting-time unemployment into this model. When local productivity conditions are bad, workers can either search for new jobs, as in the Lucas-Presscot t model, or wait for local conditions to improve and be recalled to their old jobs. Workers only work if they receive a wage above some minimum value, the payoff they receive if they do not work. Equilibrium waiting-time unemployment is a decreasing function of the size of innovations; moreover, so is total unemployment even in the absence of vacancy costs for firms.

The paper is organized as follows. Section 2 introduces the model. Section 3 characterizes the properties of the stationary equilibrium. Section 4 presents a discussion of the results, and a conclusion.

2. The model

This model extends the model presented in King (1990) to allow technological progress. The economy consists of a large number of spatially distinct locations. In each location there is a continuum of firms of measure 1. Each firm produces

1The models presented in Aghion and Howitt (1992, 1994) are different, but in both cases the capitalization effect of increases in the size of innovations upon search unemployment is negative. Both of these papers also consider the effects of changes in the frequency of innovations on search, in the presence of creative destruction, with different results. In this paper, as in Pissarides (1990), attention is restricted to the changes in the size of innovations, in the absence of creative destruction.

I. King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507 501

the same type of output. A continuum of immortal worker-consumers choose their location in each time period.

In period t e {0, 1, 2 . . . . . } each firm in location i e [0, 1] has access to the production technology:

y i = g , s { f ( n { ) , (1)

where Yl denotes output, n{ is the amount of labour input, sl is the location- specific shock, and 9t is the (economy-wide) state of technology. The function f ( . ) is twice continuously differentiable, and

f ' > 0 , f " < 0 , l i m f ' ( n ) = ~ , lim i f (n) = 0. (2) n --~. 0 n ~ o c

The state of technology evolves exogenously in all locations according to

gt = }"go, (3)

where t' > 1 is a parameter and go > 0 is given. We restrict attention to stationary equilibria with constant growth rates. Accordingly, the steady-state growth rate of the economy is 7 - 1.

The production shocks are independent across locations and follow a first- order Markov process within each location. In each period, at any particular location, the shock can take on one of two possible values: s e {an, aL} where an > aL > 0. The transition matrix H for these shocks is assumed to be symme- tric with persistence parameter n. This implies that in the stationary equilibrium of this economy (defined below) one half of all locations will draw an, and the other half will draw aL in each period. We restrict attention to shocks that are positively autocorrelated, which implies that rc > ½.

Let x~ denote the number of workers in location i at the beginning of period t. The triple (g, s, x) therefore defines the state of a location at the beginning of a period. Since g is common to all locations in any time period, it is useful to use the pair (s, x) to index a particular 'type' of location. Let i~t(s, x) represent the equilibrium measure of locations of type (s, x) in period t. In stationary equilibria, l~,(s, x) = I~r-l(s, x) = I~(S, x), although g is growing at the same rate 7 - 1 in all locations.

Output is the numeraire. Firms act to maximize their expected present value by hiring labour in their local competitive labour market and selling the output in an economy-wide goods market. Since there are no intertemporal links for any firm, each firm's problem simplifies to single-period profit maximization, subject to the production function (1). Hence, wages in regions of type (s, x) are given by

wr(s, x) = 9,sf '(n,(s, x)),

where nt(s, x) denotes employment in regions of type (s, x) in period t. In stationary equilibria, n t ( s , x ) = n , l ( s , x ) = n ( s , x ) , so the above condition

502 L King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507

becomes

w,(s, x) = o t s f ' (n(s, x)). (4)

Let 2 denote the average number of workers across locations. Thus,

f x#(ds , dx) = 2. (5)

Workers observe all current information and have rational expectations. In any period, they can choose to stay in their current location or to move. The cost of moving in period t is given by kt. In the presence of growth in this model, to preserve stationarity, we must assume that kt grows at the same rate as productivity. Thus,

kt - - / k 0 , (6)

where ko is given. In any period, once workers have chosen their location, they can choose whether to work at the going wage, or to not work and collect co t . We assume that the sequence {~ot } is exogenously given, and grows at the same rate as productivity. Let x* be defined as the value of x that would drive the marginal product of labour equal to cot in low productivity locations: g : L f ' ( x * ) = ~Ot. Notice that x* will be constant in stationary equilibria, since tot grows at the same rate as gt, and so x* is determined by

g o a L f ' (x*) = O)o. (7)

Under these circumstances, it can be shown that the necessary and sufficient condition for some workers to choose leisure in the stationary equilibrium is

OOo > goOLf(.~), (8)

where ~ is defined by

goaLf(:~) = g o a u f ( 2 2 -- 2) -- ko[1 + fiT(1 -- 2r0].

'Sectoral reallocation' or 'search' in this model occurs when workers move from one location to another. Waiting-time unemployment in this model is incurred by those workers who do not move in the current period, and do not work, but simply collect mr, waiting for local conditions to improve.

The within-period sequence of events is as follows. The state (s, x) of each location is given, and observed by everyone. Workers then make their movement decision. Workers then choose whether to work or to collect ~ot. Finally, production takes place, workers are paid, and output is consumed.

Let mr(s, x) denote the number of net migrants in each location of type (s, x) in period t. In stationary equilibria: mr(s, x) = rnt+ l(S, x) = re(s, x). Hence, in locations of type (s, x), employment is bounded by n(s, x) <~ x + re(s, x). Moreover, x* represents the number of workers who actually work in locations where the

L King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507 503

equilibrium wage is driven down to cot. Hence, in stationary equilibria, employment in locations of type (s, x) is given by

f xx* + m(s,x) ifif g, sf'[Xnot. + m(s,x)) > o9, (9) n(s, x) k

Population dynamics in locations in the stationary equilibria are given by

x,+ 1 = x, + m(s,, x,). (10)

Workers are risk-neutral, and in each period choose their location to maximize the expected present discounted value of their consumption. Let/3 e (0, 1) denote workers' discount factor. We assume that fly < 1, to ensure that returns are bounded. We also assume that the {k,} sequence is not too large to rule out movement in the stationary equilibrium. A sufficient condition is

9oanf ' (P,) -- 090 ko < (11)

1 + 3y(1 - 27t) "

Let v,(s, x) denote the equilibrium expected present discounted value of the income stream available to workers in locations of type (s, x) in period t:

v,(s, x) = max {co,, g, s f ' (x + re(s, x))} +/3 E{v, + 1 (s', x') I (s,, x,) = (s, x)}. (12)

Let 2, denote the equilibrium expected present value a worker has access to if he or she moves in period t. Notice that we assume that this is independent of the worker's current location. In the stationary equilibrium, 2, grows at the rate 7 - 1. Since 2t represents the expected benefit from moving, and kt represents the cost, then we have the following relationship between migration and value functions in each location:

(i) wherever re(s, x) > O,

(ii) wherever re(s, x) = O,

(iii) wherever m(s, x) < O,

v,(s, x) = L ,

2, - k, <_ v,(s, x) <_ 2 ,

v,(s, x) = )~, - k,.

i13)

We close this section with a definition of the stationary equilibrium.

A stationary equilibrium in this economy is a set of functions {n(s, x), v(s, x), w(s, x), re(s, x), #(s, x)}, where v,(s, x) = y'v(s, x) and w,(s, x) = y'w(s, x) such that n(s, x) is given by (9), v,(s, x) solves (12), firms are value-maximizing (that is w,(s, x) satisfies (4)), migration decisions maximize worker utility (that is, re(s, x) satisfies (13)), and/~(s, x) is consistent with the transition matrix H and Eq. (10).

3. The s ta t ionary equi l ibr ium

The stationary equilibrium is constructed in the following way. We restrict attention to equilibria in which waiting-time unemployment occurs only in low

504 L King, L. Welling~Journal o f Monetary Economics 35 (1995) 499-507

product iv i ty locations. Let xn denote the equi l ibr ium popula t ion in each loca- t ion that draws an. Hence, (9) implies

n ( s , x ) = { x n if s = a n , (14) x* if s = aL.

Using (14) in (4) implies

w(s, x) = ~goanf ' (xn) if s = an, (15) (O~o if s = aL.

This al lows us to cons t ruc t the following value functions:

v,(an, x) = yt goanf(xn) + f17 [rw,(an, x) + (1 -- zr)vr(aL, x)],

v,(aL, x) = y'~oo + ~y[rw,(aL, X) + (1 - - rC)v,(an, x ) ] .

Solving these s imul taneous ly yields

v(s, x) =

[(1 -- f l?~)goanf ' (xn) + fiT(1 -- x)COo]/[1 -- fl272 -- 2xTfl(1 -- fly)]

if s = an,

[(1 -- fly~)~Oo + fly(1 -- g)goanf ' (xn)]/[1 -- f12y2 _ 2xyfl(1 -- fly)]

if s = O" L .

(16)

Since all an locat ions have the same v and all eL locat ions have the same v, and since only an locat ions experience posit ive net migra t ion and only aL locat ions experience negat ive net migra t ion , then (I 3) implies

v,(an, x) = v,(aL, X) + k,. (17)

Eqs. (16) and (17), together with the definit ion of a s ta t ionary equil ibrium, imply

goanf ' (xu) = COo + ko[1 + fly(1 - 2r0], (18)

which determines the value of Xn.

Consider now the migra t ion process. Let XL denote the average end-of-per iod popu la t ion in locat ions which d raw aL in the s ta t ionary equil ibrium. By (5), and the symmet ry of the M a r k o v t ransi t ion matr ix, this implies that 0.5XL + 0.5Xn = ~, which implies XL = 2~ -- Xn. In the t ransi t ion f rom the end of one per iod to the beginning of another , the weight of locat ions which switch f rom az to an is 0.5(1 -- r 0. These locat ions are the only ones which experience immigra t ion in the s ta t ionary equil ibrium. Their average beginning-of-per iod popu la t ion is XL = 2:~ -- Xn and their end-of-per iod popu la t ion is xn, hence the average a m o u n t of immigra t ion in locat ions which receive migrants is xn - XL = 2(Xn -- ~). Mul t ip lying this by the weight of these locat ions yields this expression for aggregate search,

S = (1 - z0 (xu - g). (19)

L King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507 505

Since the average number of workers in aL locations is XL = 2ff -- XH, and the number of workers in those locations who actually work is x*, then the average number of workers who experience waiting-time unemployment in a L locations is 2ff - xn - x*. Multiplying this amount by the weight of these locations yields this expression for aggregate waiting-time unemployment:

W = 0.5(2~ - x* - xn). (20)

Proposition. In the stationary equilibrium aggregate search is an increasing function of the growth rate, waiting-time unemployment is a decreasing function of the growth rate, and total unemployment is a decreasing function of the growth rate.

Proof From (18):

dxn kofl(l -- 2rt)

d7 go anf" (xn) > 0 . (21)

From (19), (20), and (21):

dS ( 1 - ~ ) k o f l ( l - 2tO

d7 goanf" (xn) > 0 ,

dW (0.5)kofl(1 - 2r0

d)' goanf" (xu) < 0 ,

dS dW 2(0.5 - 7t)2kofl + - - - < 0. QED

d7 d7 go an f" (xn)

Intuitively, the incentive to be in the high productivity locations is increasing in the growth rate due to the capitalization effect on workers: future expected payoffs are greater when the size of innovations is greater. The cost of moving is unaffected by the size of innovations. Hence, the equilibrium population in high productivity locations is an increasing function of 7. This larger xn applies to all high productivity locations, those that receive searching workers and those that do not. Also since xn is larger, then XL must be smaller. Locations that do receive searching workers receive more of them when 7' is large, since the number of searching workers is (on average) (Xu - XL) in these locations. Hence, aggregate search is larger when 7 is large. Also, since XL is a decreasing function of 7 and x* is not a function of 7, then the amount of waiting-time unemployment in the average low productivity location (XL -- X*) is a decreasing function of 7. Hence, so is aggregate waiting-time unemployment. Any increase in x , necessarily implies a decrease in XL, but whereas the decrease in XL has an effect on waiting-time unemployment in all low productivity locations, the increase in xu only has an effect on search in those high productivity locations that

506 1. King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507

experience a switch in productivity. Hence, economies that have larger values of will have more search and less waiting-time unemployment, but the decreased

amount of waiting-time unemployment outweighs the increased amount of search. This implies that total unemployment is a decreasing function of the growth rate. 2

4. Conclusions and discussion

In this paper we have examined the influences of the capitalization effect upon workers' decisions to search, work, or wait for conditions to improve in local labour markets when search is costly. In the absence of vacancy costs for firms, larger technological innovations imply more search, less waiting-time unemployment, and less total unemployment. With vacancy costs, Pissarides (1990), and Aghion and Howitt (1992) have found that search is decreasing in the size of innovations, in the absence of direct search costs borne by workers. In a more general setting, with both vacancy and search costs, the net effect of larger innovations upon search appears ambiguous theoretically. However, the result that waiting-time unemployment is a decreasing function of the size of innovations would not be changed in a setting with vacancy costs. We should expect, therefore, that total unemployment would be decreasing in the size of innovations in such a model.

The choice of search intensity, once agents have made the decision to search, has not been modelled in any of these studies. In the Pissarides and Aghion- Howitt models, matching is achieved by a fixed function with vacancies and searching workers as arguments. In the model presented in this paper, matching is equally mechanical. If search intensity is costly, then we should expect that, through the capitalization effect, larger technological innovations would be associated with greater search intensity on both sides and thus lower unemployment.

Finally, the results of this paper could be interpreted in a different way. If locations represent productive opportunities with specific types of human capital, where these opportunities have diminishing returns and are subject to random shocks, and k is interpreted as a training cost associated with acquiring

2Note that if zc equals one half then neither search nor waiting-time unemployment would be affected by the growth rate, because the value of the current shock in a location gives no information about the value of future shocks. In this case, a l though expected future returns are more important with larger growth rates, searching will not influence these returns. Jones and Newman (1992) consider a model in which technological change shuffles local productive opportunities and agents search using an optimal stopping rule. In that setting there is no persistence at any location and so, as in this model when n equals one half, the size of innovations does not affect equilibrium search.

1. King, L. Welling~Journal of Monetary Economics 35 (1995) 499-507 507

a new type of human capital, then choosing to collect to can be interpreted as passive search within a profession when times are bad for that profession. Hence, the proposition in this paper would imply that larger innovations would be associated with less search within a profession and more training and switching of professions)

References

Aghion, P. and P. Howitt, 1992, Growth and unemployment, Mimeo. (University of Western Ontario, London, Ont.).

Aghion, P. and P. Howitt, 1994, Growth and unemployment, Review of Economic Studies 61, 477 494.

Chari, V. and H. Hopenhayn, 1986, Vintage human capital, growth, and structural unemployment, Working paper no. 326 (Federal Reserve Bank of Minneapolis, Minneapolis, MN).

Jones, R. and G. Newman, 1992, Economic growth as a coordination problem, Mimeo. (Simon Fraser University, Burnaby, B.C.).

King, I., 1990, A natural rate model of frictional and long-term unemployment, Canadian Journal of Economics 23, 523-545.

Lucas, R. and E. Prescott, 1974, Equilibrium search and unemployment, Journal of Economic Theory 7, 188-209.

Pissarides, C., 1990, Equilibrium unemployment theory (Basil Blackwell, Oxford).

3Thanks to Peter Kennedy for offering this interpretation. See Chari and Hopenhayn (1986) for a preliminary analysis of a similar question using a different model.

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Search, unemployment, and growth