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Education and the distribution of unemployment
ELSEVIER
European Journal of Political Economy Vol. 12 (1996) 113-132
EuropeanJournalof POLITICAL ECONOMY
Education and the distribution of unemployment C.J. M c K e n n a * Department of Economics, Universityof Guelph, Guelph, Ont., NIG 2W1, Canada Accepted 15 January 1995
Abstract This paper presents a two-sector model of job matching in which jobs are of two types distinguished by their accessibility to uneducated workers and by their productivity. Education confers two related benefits to workers: broader access to jobs, and higher lifetime earnings. The model determines the aggregate level of education in the economy, the aggregate human capital requirements of industry and the distribution of unemployment between educated and uneducated workers. The paper then addresses the issues of equilibrium unemployment and the incidence of unemployment in the steady-state.
JEL classification: 121; J24 Keywords: Job matching; Unemployment;Education
1. Introduction
We study the long-run equilibrium of an economy with endogenous human capital. The formal framework is a simple two-sector equilibrium matching model in which both workers and firms make decisions that determine the amount of investment in education, the rate of unemployment and the distribution of unemployment (or underemployment) between educated and uneducated workers. The formal model is in the spirit of Pissarides (1985), Pissarides (1990), and Mortensen (1989) in that random matchings are a source of employment friction and determine, along with a wage-determination rule, the relative number of active
* Tel.: (519) 824-4120; fax: (519) 763-8497; e-mail: [email protected].
0176-2680/96/$15.00 Published by Elsevier Science B.V. SSDI 0176-2680(95)00040-2
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searchers. There is then a feedback from the profile of searchers to the matching process. The paper is one of a small number of two-sector equilibrium models. Others include Davidson et al. (1987), Davidson et al. (1988), Hosios (1990a), and Hosios (1990b). An early paper by Welch (1976) studies a formal two-sector model which distinguishes between skilled and unskilled workers, though there is no matching friction there. In this paper, we present a specific model designed to explain some facts about the relationships between human capital and unemployment. Unlike Hosios (1990a) and Hosios (1990b) we permit on-the-job search, which is an important mechanism for allocating workers to jobs. In this model, temporary 'mismatches' arise as part of the equilibrium search strategy of workers and the hiring strategy of firms. On-the-job search arises because characteristics of firms and workers acquired through investments cause mismatches which may not be rectified by wage adjustment. Several empirical studies have indicated a two-way relationship between unemployment and education. The education investment decision is determined in part by unemployment, since spells of unemployment affect the relative rate of return. At the same time education, by affecting the supply of imperfectly-substitutable types of labour, may influence the incidence of unemployment. Some common findings appear to be: 1. education reduces the incidence and possibly the duration of unemployment for an individual i (Ashenfelter and Ham, 1979; Devine and Kiefer, 1991; Kiefer, 1985; Nickell, 1979), 2 2. unemployment, by reducing the probability of employment and hence the rate of return to participation, increases the demand for education (Kodde, 1988; Pissarides, 1981). The theoretical basis for the second of these findings is of particular interest, since in an equilibrium model where both the demand for education and the unemployment rate are jointly determined the causal links between the two are going to be conditioned by the set of exogenous variables. Indeed we find that the sign of any correlation between unemployment and education will depend on the causes of the changes in unemployment. In this model education expands employment opportunities since educated workers are productive in all jobs whereas uneducated workers are productive in only some jobs. However, not all jobs are equally attractive to educated workers, though the wage structure is such that less attractive jobs may be taken temporarily until a preferred alternative is found. The empirical counterpart to the tempo-
Incidence may be thoughtof as the probabilityof an individual with a given set of characteristics entering unemployment,while duration refers to the length of an unemploymentspell conditionalon it occurring. 2 Table 4, Chapter 6 of Layard et al. (1991) also contains a summary of unemploymentrate by education level for a number of OECD countries.
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rary mismatch of an educated worker to a low output job is often referred to as 'trading-down' or 'overeducation'. Empirically, these workers tend to: 1. earn less than similarly qualified workers in jobs requiring the same level of schooling (i.e. 'correctly-matched' workers), but 2. earn more than workers with less schooling in similar jobs (Sicherman, 1991), and 3. have higher quit rates than do correctly-matched workers (Hersch, 1991). The approach is very simple. We model the markets for educated and uneducated workers in the presence of matching frictions. Given a wage structure which, among other things, makes temporary job-holding by educated workers an optimal strategy, we derive the equilibrium rate of human capital acquisition and the economy unemployment rate jointly. The aggregate unemployment rate is decomposed into unemployment rates for the two types of workers and so the overall unemployment rate is sensitive to the distribution of unemployment between educated and uneducated workers. The empirical work on education and labour market experience points to many interesting regularities. However, there appears to be no unified theoretical framework which studies the intricate relationship between unemployment and education. This paper provides such a framework by studying a simple equilibrium model.
2. Preliminaries 2.1. Main assumptions
The components of the model are the education investment opportunities, the main players; workers and finns, the matching process, and the wage-determination rule. The basis of the model is the education investment decision by workers. Education is available to all at a cost and confers the following benefits: Education.
1. greater accessibility to jobs, 2. higher earnings. Education is highly stylized in the model. It is an 'all-or-nothing' investment and (because the model looks only at stationary steady-states) is acquired entirely at the beginning of a worker's 'life'. We do not consider on-the-job productivity enhancements or job-specific training. This helps keep the market dynamics and
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the wage structure relatively simple. Finally, education is potentially productivityenhancing and observable.
Workers. Workers differ in the cost of acquiring education but are otherwise identical. A large number of workers, N, are 'born' in each period and immediately make the decision to educate or not by comparing the net returns to education and to no education. The distribution of idiosyncratic education costs, c, is assumed to be continuous with distribution function F(c), defined for c ~ E+. Workers who choose not to educate remain uneducated for their lifetime, and similarly for those who opt for education. Educated and uneducated workers are indexed by e and u respectively. The N new births replace the workers who leave the economy permanently. The survival rate for all workers regardless of education level, employment status, or occupation is s ~ (0, 1), and so the (expected) lifetime of an individual is (1 - s ) -~. After making the education investment decision, workers search for employment subject to the following restrictions: 1. uneducated workers may qualify for only one type of job while educated workers qualify for all jobs; 2. all workers search with a fixed intensity; 3. educated workers search the two sectors sequentially, while uneducated workers search only one sector; 4. it is optimal for educated workers who take jobs for which the uneducated are qualified to continue to search on the job for an improved match.
Firms. Firms are of two generic types indexed h and l. Uneducated workers are completely unproductive in h-jobs and educated workers are just as productive as uneducated workers in /-jobs. Firms do not exercise choice over their type but latent h-firms and /-firms make the decision to open up vacancies in response to the market opportunities in their respective sectors. Each firm represents one job which at any point in time is either filled by a qualified worker or is vacant. Vacancies are costly, the vacancy cost for an i-finn being k i for i ~ {h, l}. The per period output produced by an h-firm (resp. /-firm) is Yh (resp. Yt) with (Yh > Y~)- Note that Yt is independent of the education level of the employee. Firms are infinitely-lived and discount future profits at the rate r > 0. A firm's current match breaks up with probability 1-s representing the permanent exit of a worker. The matching process. Educated workers search in both sectors sequentially. We restrict the search process in such a way that in each period an educated worker searches in the market for h-jobs, and, if unsuccessful after a single attempt, switches to search in the market for /-jobs at no additional cost. If both search attempts are unsuccessful, search begins again in the following period in the h-sector. The optimality of this strategy implies certain restrictions on the payoffs
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which we come to presently. Uneducated workers search only for /-jobs. 3 The payoff structure guarantees the optimality of educated workers taking an /-job, if found, and continuing search on-the-job. It is clear that this search process forces an interdependence between the two sectors in terms of the level of search activity. This is one source of externalities between sectors which makes the model a non-trivial extension of single-sector models. Let Ph be the probability that an educated worker locates an h-job in any period. Since sectoral search is sequential for educated workers, this probability depends only on the total number of educated workers searching the h-sector, S h, and the number of vacancies in the h-sector, V h. In particular, we assume that Ph depends on the ratio of vacancies to searchers: Ph=P(Sh'Vh)=P
--~h ~ ( 0 , 1 ) ,
(1)
where (see below) the probability of success in finding a match is increasing in the number of vacancies but decreasing in the total number of searchers, as we might expect. Searchers for h-jobs are all educated but are composed of two groups. There are those who are unemployed, U e and those who are employed in the /-sector, E~e. This last group are the on-the-job searchers in this economy. Hence we have, S h = U e -~- El. The aggregate number of unemployed searchers in the /-sector is the sum of uneducated searchers and 'unsuccessful' educatedsearchers: S t -= U" + (1 - Ph)U e. Firms in this sector hire the first worker they encounter and so do not discriminate between educated and uneducated workers. 4 The probability of success for a searcher in the/-sector is therefore independent of education level but does depend on the aggregate number of searchers, S t, and the number of /-vacancies V~, or, Pl = p ( U " + (1 -- P h ) U e, Vt)
(0, l).
(2)
(3)
(Note that in the context of /-sector search by educated workers, pz has the interpretation as the conditional probability of finding an /-sector job given that an h-sector job has not been found.) The source of one potential externality may be seen immediately in (2), where the probability of success in the h-sector
3 The search intensity is not a control variable for either type of worker. H o w e v e r , conditional on being unsuccessful in finding an h-job, an educated w o r k e r has the same probability o f locating an /-job in a n y period as does an u n e d u c a t e d worker. In this sense the m o d e l gives educated workers a search a d v a n t a g e over u n e d u c a t e d workers. 4 F o r this to be a sensible policy for the firm requires some restrictions on h o w small v a c a n c y costs c a n be. W e turn to this later.
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determines the probability of success in the/-sector. We expect that an increase in the aggregate number of searchers in the /-sector reduces this success probability whereas an increase in the number of /-firms increases this probability. We assume that the matching functions in each sector are identical, and that the matching probability for searchers in each sector depends only on the ratio of vacancies to searchers, so p l ( X ) =ph(X) =p(X) for all X ~ (0, ~ ~). Firms with open vacancies also face a stochastic matching environment. Let qh be the probability that an h-finn finds an e-worker in any period. Since h-firms only hire e-workers, the following identity holds:
PhSh -- qhVh,
(4)
which simply states that the number of matches found in this sector must be the number of successful workers or equivalently the number of successful firms. Similarly, /-firms are assumed to have a per period probability qt of finding a worker. In the /-job sector the matching probabilities are related by
ptSt =- qtV,.
(5)
Throughout, we will use the following notation for the sectoral v a c a n c y searcher ratios: 0 - -Vh -, Sh
0 ~ [X, X,]
(6)
/z=~,
/z~
(7)
It is clear from (2) that 0 affects Iz, given U" and Vr 5 However, the equilibrium value of Ix is independent of 0, with the effect being offset fully by adjustments in U u and V/. In view of (4) and (5) and the common matching technology we have
p(X)=Xq(X),
X ~ { O , tz}.
(8)
We expect q ' ( X ) < 0 and p ' ( X ) > 0 since a firm will have more difficulty filling a vacancy if there are large numbers of competing vacancies relative to available workers. Similarly, the presence of a relatively large number of vacancies facilitates the process of workers locating a job. These sign restrictions are satisfied if the elasticity of q with respect to X is no greater than unity in absolute value. 6 We will invoke this assumption in what follows.
5 The relevant derivative is: p'(O)Vev, ~'(0) = [vU + ( l _ p( O))Ve]2 >_o. 6 Note that p(X) = Xq(X) for all X, so p'(X) = q'(X)X + q(X). Note that the implied elasticity of p with respect to X is also less than unity.
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Wages. There are three wages in the model: the wage received by (educated) workers in the h-sector, Wh; the wage received by (mismatched) educated workers in the /-sector, wT; and the wage received by uneducated workers in the /-sector, w~'. In addition, workers receive a money compensation of b, per period of unemployment independent of their type. 2.2. An outline of the model Each period, a number N of new workers are born and immediately take the decision to educate or not by comparing market returns of being uneducated with the returns to education net of education costs. We are concerned with a stationary environment which means that investments in human and physical capital are made immediately or never. After this decision, all workers search for job openings until successful. There is no re-entry of employed workers into the unemployment pool. Firms enter the market in response to profit opportunities. Firms do not exercise choice over their type but are able to determine whether to enter the labour market with a vacancy or not. In the steady-state equilibrium we determine the unemployment rate in each sector, and the amount of human capital formation.
3. The
model
The building blocks of the model are the labour market returns, flows of workers, firm entry decisions, and the determination of wages.
3.1. Labour market returns Let u e be defined as the expected lifetime earnings of an unemployed, educated worker. Given that such an individual searches once in the h-market and, if unsuccessful, once in the l-market each period, we may define u e recursively by
ut=b+sp(O') +s(1-
Wh
-( 1 -- s )
e Ue + S(1 -- p( Ot))p (tZt) [max{ u;,t+ l ,+ l}]
p( I.tt) ) ( 1 - p( Ot) )u et+l.
(9)
That is, the worker receives one payment of b (net of any cost of unemployment search activity) and searches for an h-job. If successful, w h is received in each period of the worker's lifetime. 7 If an h-job is not found then the /-market is
7 All jobs are assumed to start in the period following their discovery and so the survival probability, s precedes all payments. Workers have a finite expected lifetime and do not discount future income.
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searched. If successful, the worker chooses between taking the /-job, receiving expected lifetime earnings of uT, and remaining unemployed, receiving u e. If search attempts in both markets fail, then the worker remains unemployed and, conditional on survival, receives u e next period. It is easy to see that the interaction between the two sectors arises only if e-workers ever accept/-jobs. We will therefore assume that u, < uT, t V t , which is the condition for on-the-job search to be optimal. In the sequel, we consider the case where in steady-state u e = u7 and assume that the worker tips the balance in favour of working. This condition is obtained by a simple parameter restriction below. The formula for u~ is given recursively by Wh
e = w7 _ ,~ + s p ( O,) l _ s + s ( 1 - p ( o , ) ) u ~ , , + , , ut,,
where s: is the per period cost of on-the-job search. In steady-state, w;u; =
K+ sp( O ) w h / ( 1 - s) l - ~(1 - p ( O ) )
( lO)
This may be substituted in (9) to give in steady-state, ~ ) ( w ; - K ) / [ 1 - s(1 - p ( 0 ) ) ]
b + s(1 - p ( O ) ) p (
Ue
+
1 -s(1 -p( 0))(1 -p(/z)) ~p(0) wh [l-s(1-p(0))]
=
b + w(O)wh/(l 1-s(1-p(O))
(l-s) -
~) '
(11)
where the last equality follows from the fact that u e = u~ is implied by w[ - s: = b, so that 'mismatched' workers are paid their 'reservation wage'. That is, workers are indifferent between working, engaging in on-the-job search and remaining unemployed, receiving b each period. This is discussed below. The return to being unemployed for an uneducated worker is simply given by
w~" (1 -~)
U t = b -}- s p ( I d ~ t ) -
ql_ s ( l
- - p ( /.Lt))ut+ i ,
(12)
where only one search attempt in the /-sector is permitted for the uneducated worker. Hence, we have, in steady-state, b + s p ( I x ) w ~ / ( 1 -- s)
u" =
I - s(1 - p ( ~ ) )
(13)
The steady-state decision to educate is based on the calculation, max{u u, u ~ - c}.
(14)
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12t
We will denote by ~ the unique 8 solution to = II e -- U u,
(15)
and the proportion of new workers who choose to educate is F(~). Thus 6 has the property that workers with a c-value less than ~ will choose to educate while those with education costs greater than ~ will choose not to invest in education. In view of the fact that u e depends on 0, and u" depends on p,, the proportion of workers who choose to educate depends on labour market conditions in each sector. Specifically, ~( 0, ~[z) = u e ( 0 )
- b/u(~).
(16 I)
The wage rates are independent of 0 and ~z in this model, and so we can verify, using (11) and (13), that
sp'( O)(w h - b ) ~0 = u~ = [1 - s(1 - p ( 0 ) ) ]
2 > 0,
(17)
sp'( t z ) ( w ~ - b ) e" = -"
u=_
2 -< 0 ,
(18)
so that investment in education is increased by an increase in 0 and reduced by an increase in I~. This is intuitive, since an increase in 0 (resp. I~) leads to an increase in expected return to education (resp. no education).
3.2. Labour market flows The flow of unemployed educated workers is determined by the difference equation Ut e ~
F(ct) N + s(1 - p ( / z t ) ) ( 1 - p(
Ot))Ute
l.
(19)
The unemployment pool in the educated sector is added to each period by newly-arrived educated workers, and these are joined by those previously unemployed educated workers who remain unsuccessful in finding a job, after attempts in both markets. The steady-state stock is
F(~)N v e =
1 - s(1 - p ( / . Q )
(1
-p(O))
(20)
The flow of unemployed uneducated workers is determined by thedifference equation Utu = [1 - F ( ~ t ) ] N + s(1 - p ( Ixt))U," ,.
(21)
8 Existence and uniqueness of such a ~ follows from the facts that c ~ [0, c¢) and in non-trivial equilibria, u ~ - u u is positive, and independent of c. See the appendix.
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The stock of unemployed uneducated workers is added to each period by new arrivals who choose not to educate and by the number of unsuccessful searchers from the previous period. The steady-state stock is [1 - F ( ( ) ] N Uu=
1 -- s(1 - - p ( ~ ) )
.
(22)
The flows of employment for educated and uneducated workers are given by three difference equations: E~, = s(1 - p ( O t ) ) E ~ , , _ , + s p ( / ~ t ) ( l
- p ( O,) )U,~ ,,
(23)
E~.,=sE~,.,_, + sp(O,)Sh.,_ , ,
(24)
E; = sE;_ 1 + sp( /~t)U,~ ,.
(25)
Employment of educated workers in the low-output sector in each period consists of those so employed in the previous period who failed to find a high-output sector job plus those who were unemployed in the previous period who were unsuccessful in locating a high-output job, but who succeeded in finding a low-output job. Similarly, the stock of employed in the high output sector consists of survivors from the previous period and successful searchers from the previous period. The employment dynamic of uneducated workers has a similar interpretation. These give the steady-state stocks as sp(/~)(1-p(O))U e E[ =
E~
,
1 -s(l-p(0))
(26)
sp( O) Sh
- - , 1-s
(27)
u
Eu=
1-s
,
(28)
where Sh = E 7 + U e. Note that the steady-state equilibrium number of mismatched workers, Et~, is strictly positive. It is easily checked that the total population of this economy in the steady-state, the sum of (20), (22), (26), (27), and (28), is N / ( 1 - s) as required. We define the respective unemployment rates among educated and uneducated workers as Ue VRe( O, ~ )
= S h -[- E~
(29)
1 --S
l - 41
-e(0))
'
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123
VU VR"(
-
V" + E"' 1 - s
(30)
[1-s(1-p(/x))]" The unemployment rate among educated people is decreasing in both 0 and Ix, and the unemployment rate among the uneducated is decreasing in ix. For all admissible values of IX and 0, the unemployment rate among educated workers is less than the unemployment rate among uneducated workers. 9 This is a consequence of educated workers having at most two search attempts in each period compared to at most one attempt permitted for uneducated workers. It is also apparent that an increase in ~ lowers the unemployment rate among the uneducated more than among the educated. The economy aggregate unemployment rate is simply a weighted average of the sectoral rates Ue + Uu
UR( ~, O, ~)
-
N/(1 - s )
(31)
= F(~) URe( O, ~) + [1 -- F( ~)] URU(I~), and we should note, from (16) that ~ is itself a function of 0 and ~. Using this fact, we see that the economy unemployment rate is a decreasing function of 0. Not only does an increase in 0 reduce the unemployment rate among educated workers, but it also increases the supply of educated workers, thus placing greater weight on the lower unemployment rate in the aggregate index. On the other hand, an increase in I~ has an indeterminate effect on the economy unemployment rate. This is because both the sectoral rates are reduced by an increase in IX, but fewer workers undertake education, thus shifting greater weight onto the higher unemployment rate in the aggregate. Without further restrictions the net effect of these factors is unknown. 10 The expected durations of unemployment in each sector i, D i are given by
D e= [ 1 - s ( 1 - p ( / ~ ) ) ( l - p ( 0 ) ) ]
',
D u = [1 - s ( 1 - p (
and a simple calculation show that for all realizations of 0 and IX, we have D e < D u. Thus our model exhibits at least two of the stylized facts noted earlier: educated workers have lower unemployment rates and lower expected durations of 9 This is easily established by comparing (29) and (30). 1o The expression for UR. is
URu = F(~)UR~ + [1 - F(~)]UR~ + f ( O ) ~ [ URe - UR"]. The first two terms are negative and the last is positive. The unemployment rate is more likely to increase when i~ increases the higher is the responsiveness of the demand for education to I~.
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unemployment. This, of course, is a consequence of the increased range of job opportunities for educated workers under our specification and the fact that educated workers sample from two, not one, job pool. We may also use (26)-(28) to define a measure of job mis-match as the ratio of mis-matched workers to all employed educated workers:
El E;+E
M- - -
sp(
[1 - s(1
-p( O)) - p ( 0 ) ) ] [p(/x)(1 -p(O)) + p ( 0 ) ]
where it is straightforward to show that an increase in Ix increases mis-match and an increase in 0 reduces mis-match (given s).
3.3. Wages We keep wage-determination as straightforward as possible, by assuming a simple sharing rule. Specifically, each match rewards the worker with a wage which is an average of the output in that match and the worker's reservation wage. Thus, we have
Wh=aYh+(1--a)b,
ce ~ (0, 1),
(32)
w~=fly,+(l-[3)b,
/3 ~ (0, 1),
(33)
w;=~y,+(l-~)(b+K),
~
(0, 1),
(34)
for constants ¢~, [3, and 6- This makes the analysis much more tractable than assuming a Nash bargain between firms and workers, whilst maintaining a symmetry in the treatment of each wage. ~1 Moreover, the weights in each case have a 'bargaining power' interpretation. Since, by assumption, Yh > Y~ > b, the wages w h and w~' are increasing in o~ and [3 respectively, so these may be interpreted as measures of worker bargaining power. Note that under this specification, wages are independent of 0 and Ix. Although special, this does enable us to focus on the relationship between education and unemployment for a given wage structure. Having wages additionally depend on 0 and Ix complicates the returns Eqs. (10), (11) and (13) but adds little to the story of the relationship between education and unemployment. The wage structure chosen appears to reflect some of the observed patterns discussed in the introduction, and is also consistent with educated workers taking temporary jobs.
I1 The weakness of this approach is that all wages in the decentralized equilibrium are insensitive to labour market tightness. The loss in generality is quite small, since even the Nash rule fails to internalize externalities in general. See Hosios (1990a).
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Imposing the parameter restriction Yl = b + K gives the result that w 7 = b + K. This implies that, in this economy, mismatched workers are as productive as in their worst alternative, and l-firms earn zero rents from mismatches. Since Yt > w~ we see that 1-firms would prefer to have a vacancy filled by an uneducated worker with certainty, than by an educated worker. Further, we have that Yt > b implies w~ > b; w h > W7 is implied by ot(y h b) > K; and w~ > w~' is implied by 13( Yt - b) < K. Then the desired wage structure is obtained by assuming et( Yh -b) > 13(yl - b), and that search costs fall into the interval -
-
K E (Or( Yh -- b ) , / 3 ( Y t - b ) ) . Restricting our attention to this parameter space, the complete ranking of wages is therefore w h > w~ = b + K > w ' / > b,
(35)
and the wage difference between well-matched educated and uneducated workers is measured by w h-w~=o~(yh-b)+flK>O,
(36)
while the wage difference between educated and uneducated workers in the /-sector 12 is w;-w~
= K(1 - / 3 ) > 0 .
(37)
The wage differential between well-matched workers is increasing in et, Yh and f3 and decreasing in b. The wage differential between educated and uneducated workers in the /-sector is increasing in K, and decreasing in 13. Thus, the differentials behave in an intuitive way. Increased bargaining power of one group of workers changes the differential in that group's favour. 3.4. F i r m e n t r y
Firm entry in each sector is determined by a zero expected profit condition. Potential firms are able to enter only one of the sectors and are unable to change their type. However there is no restriction on the number of potential firms of each type. Entry therefore, is based only on a comparison of the returns in the relevant sector relative to returns elsewhere (assumed to be zero). In equilibrium, firms are indifferent e x a n t e between operating in either sector. A firm in sector j ~ {h, l} incurs a cost kj for each period with an unfilled vacancy. There are no set-up costs. All firms have the same discount factor,
J2 We may also define the empirically-relevant measure of wage differences as the difference between the wage in the h-sector and the average wage in the /-sector as Wh
w~E; + w~E" E7 + E u
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126
p = (1 + r) ~, where r is the interest rate. Denote by wj, j ~ {h, l} the expected profit of being a new entrant firm in sector j. For an h-firm "rrh is defined recursively by
"rch,, = --k h + psq( Ot)
( Yh -- Wh) 1
--
+ p(1 - q( O,) )Trh.t+ ~
ps
+ p ( l - s) q ( 0 , ) 7rh,,+ , .
(38)
Setting ~h = 0 as the equilibrium condition, letting p --* 1 and substituting for w h gives S
kh -- 1 -- s q(O)(1 -- a ) ( Y h -- b).
(39)
Since, k h, Yh, b, c~ and s are exogenous, the zero profit condition determines 0. Now consider the expected profits of an /-firm. Define h as the conditional probability that a located worker is uneducated. So 1 - h is the conditional probability that a located worker is educated. Then, expected profits of an l-finn are given recursively by
7rt't= - k t + psq( tz'){ ( 1 - A)~r['+I + A ( y ' -~wp ~s ) + O(1 - q( P,t))Trz.,+ ,.
(401
where "rrl,t+ e I is the expected discounted profit for an l-firmemploying an e-worker given by
"TTel., ( y t - w[)[1 + p s ( 1 - p ( O,)) + p2s2(1 - p ( 0 , ) ) ( 1 - p ( O,+ ,)) =
+ ...1 + o(1
+
+ o 2S277-
,,,+3+...]
+ Psi p(Ot+ ,)vr,,,+, + ps(1 -p(O,+ l))p(O,+217r,.,+ 2 + . . . ] . (411 Eq. (40) uses the assumption that it is better for the firm to fill a vacancy with an e-worker than to leave the vacancy unfilled, or, 'rr~t.t+J > "rrt,t+ j Vt. Using the steady-state zero profit condition, rr t = 0, the fact that w 7 = y~, substituting for w;' and letting p ~ 1 in (40) and (41) gives S
k, = ~ - _ s q ( / x ) A(1 - f l ) ( y , - b ) .
(42)
In general we expect h to depend on the proportion of searchers in the /-sector who are uneducated. In particular, we may write A=A
,
A'(.)>0.
(43)
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127
Alternatively, we note that Uu St
1 1 d-(1 - p ( O ) ) ( u e / u
u)
where U e and U" in turn depend on ~, 0 and Ix, from (20) and (22). Thus k = k(~, 0, Ix) with k e < 0, k 0 > 0 and k~ _< 0.
4. The decentralized equilibrium Equilibrium in this economy is a triple {~, 0, IX} satisfying Eqs. (16), (39) and (42). 4.1. Multiple equilibria
The model has the potential for generating multiple equilibria. This is easily seen by looking at conditions (16) and (42). First note that 0 is determined uniquely by (39). Given 0, (16) defines a function ~(IX)I06) with c~[06) < 0. Using (42) and the properties of k we have a second function O(IX)I(4/) with ~](42) < 0. Thus, depending on the precise shapes of these two functions, multiple equilibria cannot be ruled out. For example, suppose ~(IX) given by (18) is convex in tx (i.e. uU(ix) is concave in Ix, while OIx defined implicitly by (42) is concave in IX, then we have at most two equilibria as illustrated in Fig. 1. Clearly, the equilibria show a positive correlation between levels of education and rates of unemployment in the /-sector. Point A, for example is associated with a 'high' rate of education but also a 'high' rate of unemployment among the uneducated. 4.2. A special case
To provide a focus for comparative-static results, we look at'a leading special case. Here, the general model is simplified substantially by assuming that, over the
Fig. 1.
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C.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
relevant range, h is insensitive to changes in 2, 0 and Ix. Note that the zero-profit conditions and the exogeneity of h imply that 0 and Ix are independent and constant. The appendix discusses the formalities of existence and uniqueness of an equilibrium. Even in this simple model, we see the interdependence between education decisions and unemployment. The influence of labour market tightness on education investment is apparent in (16), while (31) clearly indicates the role of education in the composition and magnitude of the economy's unemployment rate. A variety of preliminary results are now available relating to the effects of parameter changes on 0 and Ix. These are summarized in the following. The proofs are obtained by differentiation of (39) and (42) respectively. Proposition 1. (i) Each o f the following increases 0: an increase in s or Yh, a decrease in kh, OL and b. (ii) Each o f the foUowing increases Ix: an increase in s, X or Yl, a decrease in k t, f3 and b.
These are all intuitive. An increase in sectoral output, a reduction in recruitment costs, and a reduction in workers' bargaining power all increase the profit associated with vacancy creation and so more vacancies are created. An increase in s also increases the expected value of a job slot in both sectors. In the/-sector, an increase in the probability of locating an uneducated worker also increases expected profit, since l-firms only earn rent from uneducated workers. These results lead directly to the following effects on the sectoral unemployment rates, (29) and (30). Proposition 2. (i) Each o f the following reduces the unemployment rate among educated workers: an increase in s, Yh, h or Yl, a decrease in kh, kl, or, ~, or b. (ii) Each o f the following reduces the unemployment rate among uneducated workers: an increase in s, k or Yt, a decrease in k t, f3, or b.Again, these are all
intuitive, but notice how improvements in the /-sector benefit the educated workers in terms of lowering their sectoral unemployment rate. On the other hand, productivity improvements (an increase in yh), for example, in the h-sector have no direct benefit for uneducated workers in terms of their unemployment rate. Thus sector-specific productivity changes have quite different implications depending on which sector experiences the improvement. As a first pass, given a choice, a policy which exploited the spillover and encouraged productivity improvements in the /-sector might be preferred to one which encouraged improvements in the h-sector, at least in terms of the effect on sectoral unemployment rates. This conclusion is erroneous. The overall unemployment rate in the economy will not necessarily be reduced if improvements in the /-sector are accompanied by a sufficiently large reduction in human capital investment. If this happens, then a higher proportion of uneducated workers will increase the overall expected
c.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
129
duration of unemployment and so increase the average unemployment rate. To see this formally, we take the following steps. We may use (11), (13) and (16) to derive results for changes in human capital investment. The following are the most immediate results.
Proposition 3. Each o f the following increases the proportion o f educated workers (i.e. increases ~): an increase in k t or Yh, a decrease in kh, h or Yl.
Each of these increases the lifetime income of educated workers relative to that of uneducated workers, and so encourages greater investment in human capital. Changes in costs and in l-firms' perceptions about the availability of uneducated workers increase expected income through their effect on vacancy creation and hence matching probabilities. Increases in productivity also work through this route, but also directly through the effect on wages in the sharing rule. Other effects are ambiguous in general because their indirect effects on expected income, through the matching probabilities, work against the direct effects through wages. An increase in h-workers' bargaining power, o~, for example, has the effect of reducing 0 (Proposition 1) and lowering the probability of an educated worker finding an h-job. On the other hand, an increase in a has the direct effect of increasing the wage in the bargain. Notice here the effect referred to earlier of productivity improvements in the /-sector reducing human capital investment. We may combine the results for the sectoral unemployment rates and the rate of human capital accumulation to deduce the effect of various changes on the aggregate unemployment rate, using (31). The following are immediate results:
Proposition 4. An increase in Yh or a decrease in k h will reduce the overall unemployment rate.
Both of these changes have the effect of both stimulating education investment, increasing the overall search efficiency of the labour market, and of reducing the sectoral unemployment rate for educated workers. Other effects on the aggregate unemployment rate are less clear. For example, the productivity improvement in the /-sector has several effects. First, from Proposition 3, an increase in yt reduces the proportion of educated workers, both because it increases the wage and because it promotes vacancy creation in the /-sector, thus increasing lifetime earnings from being uneducated relative to being educated. This change in the composition of the labour force towards less educated workers tends to increase the average unemployment rate. Secondly, by stimulating vacancy-creation in both sectors, both sectoral unemployment rates fall. The net effect on the aggregate unemployment rate is ambiguous.
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5. Conclusion We have proposed a two-sector equilibrium model in which the rate of human capital investment and the unemployment rate are co-determined. The model emphasises the incidence of unemployment among educated and uneducated workers. To the, extent that educated workers may choose to work in jobs requiring less skill they experience lower unemployment rates and have shorter unemployment durations. Asymmetric spillover effects between the two sectors mean that the source, as well as the size of productivity or other changes affect the incidence of unemployment between educated and uneducated workers. Productivity changes also affect the education decision. Because the average unemployment rate depends on both the sectoral unemployment rates and on the amount of human capital investment, the overall effect of productivity changes needs to take account of the education decision. An increase in productivity in the high-output sector, whist having no positive spillover into the low-output sector, does have the effect of reducing the overall unemployment rate. On the other hand, an increase in productivity in the low output sector, despite having a positive spillover effect in reducing both sectoral unemployment rates does not necessarily lead to an overall reduction in the aggregate unemployment rate. Some aspects of the model make it rather special in the sense that we have made assumptions so as to create some interdependencies between the markets for educated and less educated workers, without considering all the possible sources of interdependence. In particular, the wage rules and the probability (for a firm) that a located worker is uneducated, are insensitive to labour market conditions for the purpose of our exercises. This enables us to abstract from issues of multiple equilibria and feedback effects into relative (labour) prices from the search process. These maybe of second-order importance, but the model as it stands seems consistent with available evidence and contains some testable hypotheses on the effects of various sources of productivity improvements. The model is a useful starting point to explore these issues and generalizations of it may serve to explain a variety of other issues.
Acknowledgements An earlier version of this paper was presented at the 17th Canadian Economic Theory Meetings, Universit6 de Montr6al and to the Theory Workshop at the Australian National University. I am grateful to the participants for helpful comments. The paper has also benefited from the comments of two anonymous referees. Financial support from a Social Science and Humanities Research Council of Canada grant (410-90-0151), and a B.C. Matthews Fellowship are acknowledged. The work was completed while I was a Visiting Fellow in the
C.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
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Department of Economics, Faculties, at A.N.U. I am solely responsible for remaining errors and omissions.
Appendix A Let p(X) be a continuous, non-decreasing function on [X, ~'] and let p(X) ==Xq(X) for X ~ {0, p~}. Assume; p(X)>0,
p(~')
q(X)
q(.~)>_0
while
and q(X) is non-increasing on [X, ~']. (See footnote 5.) First consider (16). Since u e is continuous and non-decreasing in 0, and u u is continuous and non-decreasing in p~ and since c ~ [0, oo) then necessary and sufficient condition for existence and uniqueness of ~ is that e _ Umax u >0 Umin where Uein -- ue(X) and u~i . - uU(X). We restrict our parameter space so that this is so. From (39) the right-hand side is continuous and non-increasing in 0 while the left-hand side is a positive constant. Existence and uniqueness therefore follow from the assumptions that (1 - s)k h < q(X)(1 - a ) ( Yh -- b) and (1 - s)k h > q( ~')(1 - a ) ( y h - - b). Similarly from (42) we require; (1 - s)k t < sq(X)A(1 - f l ) ( y l -
b)
and
(1 - s ) k I > sq( X)A(1 - / 3 ) ( Y l -
b).
References Ashenfelter, O.C. and J. Ham, 1979, Education, unemployment, and earnings, Journal of Political Economy 87, $99-Sl15. Davidson, C., L. Martin and S. Matusz, 1987, Search, unemployment, and the production of jobs, The Economic Journal 97, 857-876. Davidson, C., L. Martin and S. Matusz, 1988, The structure of simple general equilibrium models with frictional unemployment, Journal of Political Economy 96, 1267-1293.
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Devine, T.J. and H.M. Kiefer, 1991, Empirical labor economics: The search approach (Oxford University Press, Oxford). Hersch, J., 1991, Education match and job match, Review of Economics and Statistics, 140-144. Hosios, A.J., 1990a, Factor market search and the structure of simple general equilibrium models, Journal of Political Economy 98, 325-355. Hosios, A.J., 1990b, On the efficiency of matching and related models of search and unemployment, Review of Economic Studies 57, 279-298. Kiefer, N.M., 1985, Evidence on the role of education in labor turnover, Journal of Human Resources 20, 445-452. Kodde, D.A., 1988, Unemployment expectations and human capital formation, European Economic Review 32, 1645-1660. Layard, R., S. Nickell and R. Jackman, 1991, Unemployment: Macroeconomic performance and the labour market (Oxford University Press, Oxford). Mortensen, D.T., 1989, The persistence and indeterminacy of unemployment in search equilibrium, Scandinavian Journal of Economics 91,347-370. Nickell, S., 1979, Education and lifetime patterns of unemployment, Journal of Political Economy 87, Sl17-S131. Pissarides, C.A., 1981, Staying on at school in England and Wales, Economica 48, 345-363. Pissarides, C.A., 1985, Short-run equilibrium dynamics of unemployment vacancies and real wages, American Economic Review 75, 676-690. Pissarides, C.A., 1990, Equilibrium unemployment theory (Blackwell, Oxford). Sicherman, N., 1991, Overeducation in the labor market, Journal of Labor Economics 9, 101-122. Welch, F., 1976, Employment quotas for minorities, Journal of Political Economy 84, S105-S139.
European Journal of Political Economy Vol. 12 (1996) 113-132
EuropeanJournalof POLITICAL ECONOMY
Education and the distribution of unemployment C.J. M c K e n n a * Department of Economics, Universityof Guelph, Guelph, Ont., NIG 2W1, Canada Accepted 15 January 1995
Abstract This paper presents a two-sector model of job matching in which jobs are of two types distinguished by their accessibility to uneducated workers and by their productivity. Education confers two related benefits to workers: broader access to jobs, and higher lifetime earnings. The model determines the aggregate level of education in the economy, the aggregate human capital requirements of industry and the distribution of unemployment between educated and uneducated workers. The paper then addresses the issues of equilibrium unemployment and the incidence of unemployment in the steady-state.
JEL classification: 121; J24 Keywords: Job matching; Unemployment;Education
1. Introduction
We study the long-run equilibrium of an economy with endogenous human capital. The formal framework is a simple two-sector equilibrium matching model in which both workers and firms make decisions that determine the amount of investment in education, the rate of unemployment and the distribution of unemployment (or underemployment) between educated and uneducated workers. The formal model is in the spirit of Pissarides (1985), Pissarides (1990), and Mortensen (1989) in that random matchings are a source of employment friction and determine, along with a wage-determination rule, the relative number of active
* Tel.: (519) 824-4120; fax: (519) 763-8497; e-mail: [email protected].
0176-2680/96/$15.00 Published by Elsevier Science B.V. SSDI 0176-2680(95)00040-2
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searchers. There is then a feedback from the profile of searchers to the matching process. The paper is one of a small number of two-sector equilibrium models. Others include Davidson et al. (1987), Davidson et al. (1988), Hosios (1990a), and Hosios (1990b). An early paper by Welch (1976) studies a formal two-sector model which distinguishes between skilled and unskilled workers, though there is no matching friction there. In this paper, we present a specific model designed to explain some facts about the relationships between human capital and unemployment. Unlike Hosios (1990a) and Hosios (1990b) we permit on-the-job search, which is an important mechanism for allocating workers to jobs. In this model, temporary 'mismatches' arise as part of the equilibrium search strategy of workers and the hiring strategy of firms. On-the-job search arises because characteristics of firms and workers acquired through investments cause mismatches which may not be rectified by wage adjustment. Several empirical studies have indicated a two-way relationship between unemployment and education. The education investment decision is determined in part by unemployment, since spells of unemployment affect the relative rate of return. At the same time education, by affecting the supply of imperfectly-substitutable types of labour, may influence the incidence of unemployment. Some common findings appear to be: 1. education reduces the incidence and possibly the duration of unemployment for an individual i (Ashenfelter and Ham, 1979; Devine and Kiefer, 1991; Kiefer, 1985; Nickell, 1979), 2 2. unemployment, by reducing the probability of employment and hence the rate of return to participation, increases the demand for education (Kodde, 1988; Pissarides, 1981). The theoretical basis for the second of these findings is of particular interest, since in an equilibrium model where both the demand for education and the unemployment rate are jointly determined the causal links between the two are going to be conditioned by the set of exogenous variables. Indeed we find that the sign of any correlation between unemployment and education will depend on the causes of the changes in unemployment. In this model education expands employment opportunities since educated workers are productive in all jobs whereas uneducated workers are productive in only some jobs. However, not all jobs are equally attractive to educated workers, though the wage structure is such that less attractive jobs may be taken temporarily until a preferred alternative is found. The empirical counterpart to the tempo-
Incidence may be thoughtof as the probabilityof an individual with a given set of characteristics entering unemployment,while duration refers to the length of an unemploymentspell conditionalon it occurring. 2 Table 4, Chapter 6 of Layard et al. (1991) also contains a summary of unemploymentrate by education level for a number of OECD countries.
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rary mismatch of an educated worker to a low output job is often referred to as 'trading-down' or 'overeducation'. Empirically, these workers tend to: 1. earn less than similarly qualified workers in jobs requiring the same level of schooling (i.e. 'correctly-matched' workers), but 2. earn more than workers with less schooling in similar jobs (Sicherman, 1991), and 3. have higher quit rates than do correctly-matched workers (Hersch, 1991). The approach is very simple. We model the markets for educated and uneducated workers in the presence of matching frictions. Given a wage structure which, among other things, makes temporary job-holding by educated workers an optimal strategy, we derive the equilibrium rate of human capital acquisition and the economy unemployment rate jointly. The aggregate unemployment rate is decomposed into unemployment rates for the two types of workers and so the overall unemployment rate is sensitive to the distribution of unemployment between educated and uneducated workers. The empirical work on education and labour market experience points to many interesting regularities. However, there appears to be no unified theoretical framework which studies the intricate relationship between unemployment and education. This paper provides such a framework by studying a simple equilibrium model.
2. Preliminaries 2.1. Main assumptions
The components of the model are the education investment opportunities, the main players; workers and finns, the matching process, and the wage-determination rule. The basis of the model is the education investment decision by workers. Education is available to all at a cost and confers the following benefits: Education.
1. greater accessibility to jobs, 2. higher earnings. Education is highly stylized in the model. It is an 'all-or-nothing' investment and (because the model looks only at stationary steady-states) is acquired entirely at the beginning of a worker's 'life'. We do not consider on-the-job productivity enhancements or job-specific training. This helps keep the market dynamics and
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the wage structure relatively simple. Finally, education is potentially productivityenhancing and observable.
Workers. Workers differ in the cost of acquiring education but are otherwise identical. A large number of workers, N, are 'born' in each period and immediately make the decision to educate or not by comparing the net returns to education and to no education. The distribution of idiosyncratic education costs, c, is assumed to be continuous with distribution function F(c), defined for c ~ E+. Workers who choose not to educate remain uneducated for their lifetime, and similarly for those who opt for education. Educated and uneducated workers are indexed by e and u respectively. The N new births replace the workers who leave the economy permanently. The survival rate for all workers regardless of education level, employment status, or occupation is s ~ (0, 1), and so the (expected) lifetime of an individual is (1 - s ) -~. After making the education investment decision, workers search for employment subject to the following restrictions: 1. uneducated workers may qualify for only one type of job while educated workers qualify for all jobs; 2. all workers search with a fixed intensity; 3. educated workers search the two sectors sequentially, while uneducated workers search only one sector; 4. it is optimal for educated workers who take jobs for which the uneducated are qualified to continue to search on the job for an improved match.
Firms. Firms are of two generic types indexed h and l. Uneducated workers are completely unproductive in h-jobs and educated workers are just as productive as uneducated workers in /-jobs. Firms do not exercise choice over their type but latent h-firms and /-firms make the decision to open up vacancies in response to the market opportunities in their respective sectors. Each firm represents one job which at any point in time is either filled by a qualified worker or is vacant. Vacancies are costly, the vacancy cost for an i-finn being k i for i ~ {h, l}. The per period output produced by an h-firm (resp. /-firm) is Yh (resp. Yt) with (Yh > Y~)- Note that Yt is independent of the education level of the employee. Firms are infinitely-lived and discount future profits at the rate r > 0. A firm's current match breaks up with probability 1-s representing the permanent exit of a worker. The matching process. Educated workers search in both sectors sequentially. We restrict the search process in such a way that in each period an educated worker searches in the market for h-jobs, and, if unsuccessful after a single attempt, switches to search in the market for /-jobs at no additional cost. If both search attempts are unsuccessful, search begins again in the following period in the h-sector. The optimality of this strategy implies certain restrictions on the payoffs
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which we come to presently. Uneducated workers search only for /-jobs. 3 The payoff structure guarantees the optimality of educated workers taking an /-job, if found, and continuing search on-the-job. It is clear that this search process forces an interdependence between the two sectors in terms of the level of search activity. This is one source of externalities between sectors which makes the model a non-trivial extension of single-sector models. Let Ph be the probability that an educated worker locates an h-job in any period. Since sectoral search is sequential for educated workers, this probability depends only on the total number of educated workers searching the h-sector, S h, and the number of vacancies in the h-sector, V h. In particular, we assume that Ph depends on the ratio of vacancies to searchers: Ph=P(Sh'Vh)=P
--~h ~ ( 0 , 1 ) ,
(1)
where (see below) the probability of success in finding a match is increasing in the number of vacancies but decreasing in the total number of searchers, as we might expect. Searchers for h-jobs are all educated but are composed of two groups. There are those who are unemployed, U e and those who are employed in the /-sector, E~e. This last group are the on-the-job searchers in this economy. Hence we have, S h = U e -~- El. The aggregate number of unemployed searchers in the /-sector is the sum of uneducated searchers and 'unsuccessful' educatedsearchers: S t -= U" + (1 - Ph)U e. Firms in this sector hire the first worker they encounter and so do not discriminate between educated and uneducated workers. 4 The probability of success for a searcher in the/-sector is therefore independent of education level but does depend on the aggregate number of searchers, S t, and the number of /-vacancies V~, or, Pl = p ( U " + (1 -- P h ) U e, Vt)
(0, l).
(2)
(3)
(Note that in the context of /-sector search by educated workers, pz has the interpretation as the conditional probability of finding an /-sector job given that an h-sector job has not been found.) The source of one potential externality may be seen immediately in (2), where the probability of success in the h-sector
3 The search intensity is not a control variable for either type of worker. H o w e v e r , conditional on being unsuccessful in finding an h-job, an educated w o r k e r has the same probability o f locating an /-job in a n y period as does an u n e d u c a t e d worker. In this sense the m o d e l gives educated workers a search a d v a n t a g e over u n e d u c a t e d workers. 4 F o r this to be a sensible policy for the firm requires some restrictions on h o w small v a c a n c y costs c a n be. W e turn to this later.
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determines the probability of success in the/-sector. We expect that an increase in the aggregate number of searchers in the /-sector reduces this success probability whereas an increase in the number of /-firms increases this probability. We assume that the matching functions in each sector are identical, and that the matching probability for searchers in each sector depends only on the ratio of vacancies to searchers, so p l ( X ) =ph(X) =p(X) for all X ~ (0, ~ ~). Firms with open vacancies also face a stochastic matching environment. Let qh be the probability that an h-finn finds an e-worker in any period. Since h-firms only hire e-workers, the following identity holds:
PhSh -- qhVh,
(4)
which simply states that the number of matches found in this sector must be the number of successful workers or equivalently the number of successful firms. Similarly, /-firms are assumed to have a per period probability qt of finding a worker. In the /-job sector the matching probabilities are related by
ptSt =- qtV,.
(5)
Throughout, we will use the following notation for the sectoral v a c a n c y searcher ratios: 0 - -Vh -, Sh
0 ~ [X, X,]
(6)
/z=~,
/z~
(7)
It is clear from (2) that 0 affects Iz, given U" and Vr 5 However, the equilibrium value of Ix is independent of 0, with the effect being offset fully by adjustments in U u and V/. In view of (4) and (5) and the common matching technology we have
p(X)=Xq(X),
X ~ { O , tz}.
(8)
We expect q ' ( X ) < 0 and p ' ( X ) > 0 since a firm will have more difficulty filling a vacancy if there are large numbers of competing vacancies relative to available workers. Similarly, the presence of a relatively large number of vacancies facilitates the process of workers locating a job. These sign restrictions are satisfied if the elasticity of q with respect to X is no greater than unity in absolute value. 6 We will invoke this assumption in what follows.
5 The relevant derivative is: p'(O)Vev, ~'(0) = [vU + ( l _ p( O))Ve]2 >_o. 6 Note that p(X) = Xq(X) for all X, so p'(X) = q'(X)X + q(X). Note that the implied elasticity of p with respect to X is also less than unity.
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Wages. There are three wages in the model: the wage received by (educated) workers in the h-sector, Wh; the wage received by (mismatched) educated workers in the /-sector, wT; and the wage received by uneducated workers in the /-sector, w~'. In addition, workers receive a money compensation of b, per period of unemployment independent of their type. 2.2. An outline of the model Each period, a number N of new workers are born and immediately take the decision to educate or not by comparing market returns of being uneducated with the returns to education net of education costs. We are concerned with a stationary environment which means that investments in human and physical capital are made immediately or never. After this decision, all workers search for job openings until successful. There is no re-entry of employed workers into the unemployment pool. Firms enter the market in response to profit opportunities. Firms do not exercise choice over their type but are able to determine whether to enter the labour market with a vacancy or not. In the steady-state equilibrium we determine the unemployment rate in each sector, and the amount of human capital formation.
3. The
model
The building blocks of the model are the labour market returns, flows of workers, firm entry decisions, and the determination of wages.
3.1. Labour market returns Let u e be defined as the expected lifetime earnings of an unemployed, educated worker. Given that such an individual searches once in the h-market and, if unsuccessful, once in the l-market each period, we may define u e recursively by
ut=b+sp(O') +s(1-
Wh
-( 1 -- s )
e Ue + S(1 -- p( Ot))p (tZt) [max{ u;,t+ l ,+ l}]
p( I.tt) ) ( 1 - p( Ot) )u et+l.
(9)
That is, the worker receives one payment of b (net of any cost of unemployment search activity) and searches for an h-job. If successful, w h is received in each period of the worker's lifetime. 7 If an h-job is not found then the /-market is
7 All jobs are assumed to start in the period following their discovery and so the survival probability, s precedes all payments. Workers have a finite expected lifetime and do not discount future income.
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searched. If successful, the worker chooses between taking the /-job, receiving expected lifetime earnings of uT, and remaining unemployed, receiving u e. If search attempts in both markets fail, then the worker remains unemployed and, conditional on survival, receives u e next period. It is easy to see that the interaction between the two sectors arises only if e-workers ever accept/-jobs. We will therefore assume that u, < uT, t V t , which is the condition for on-the-job search to be optimal. In the sequel, we consider the case where in steady-state u e = u7 and assume that the worker tips the balance in favour of working. This condition is obtained by a simple parameter restriction below. The formula for u~ is given recursively by Wh
e = w7 _ ,~ + s p ( O,) l _ s + s ( 1 - p ( o , ) ) u ~ , , + , , ut,,
where s: is the per period cost of on-the-job search. In steady-state, w;u; =
K+ sp( O ) w h / ( 1 - s) l - ~(1 - p ( O ) )
( lO)
This may be substituted in (9) to give in steady-state, ~ ) ( w ; - K ) / [ 1 - s(1 - p ( 0 ) ) ]
b + s(1 - p ( O ) ) p (
Ue
+
1 -s(1 -p( 0))(1 -p(/z)) ~p(0) wh [l-s(1-p(0))]
=
b + w(O)wh/(l 1-s(1-p(O))
(l-s) -
~) '
(11)
where the last equality follows from the fact that u e = u~ is implied by w[ - s: = b, so that 'mismatched' workers are paid their 'reservation wage'. That is, workers are indifferent between working, engaging in on-the-job search and remaining unemployed, receiving b each period. This is discussed below. The return to being unemployed for an uneducated worker is simply given by
w~" (1 -~)
U t = b -}- s p ( I d ~ t ) -
ql_ s ( l
- - p ( /.Lt))ut+ i ,
(12)
where only one search attempt in the /-sector is permitted for the uneducated worker. Hence, we have, in steady-state, b + s p ( I x ) w ~ / ( 1 -- s)
u" =
I - s(1 - p ( ~ ) )
(13)
The steady-state decision to educate is based on the calculation, max{u u, u ~ - c}.
(14)
C.J. McKenna / European Journal o f Political Economy 12 (1996) 1 1 3 - 1 3 2
12t
We will denote by ~ the unique 8 solution to = II e -- U u,
(15)
and the proportion of new workers who choose to educate is F(~). Thus 6 has the property that workers with a c-value less than ~ will choose to educate while those with education costs greater than ~ will choose not to invest in education. In view of the fact that u e depends on 0, and u" depends on p,, the proportion of workers who choose to educate depends on labour market conditions in each sector. Specifically, ~( 0, ~[z) = u e ( 0 )
- b/u(~).
(16 I)
The wage rates are independent of 0 and ~z in this model, and so we can verify, using (11) and (13), that
sp'( O)(w h - b ) ~0 = u~ = [1 - s(1 - p ( 0 ) ) ]
2 > 0,
(17)
sp'( t z ) ( w ~ - b ) e" = -"
u=_
2 -< 0 ,
(18)
so that investment in education is increased by an increase in 0 and reduced by an increase in I~. This is intuitive, since an increase in 0 (resp. I~) leads to an increase in expected return to education (resp. no education).
3.2. Labour market flows The flow of unemployed educated workers is determined by the difference equation Ut e ~
F(ct) N + s(1 - p ( / z t ) ) ( 1 - p(
Ot))Ute
l.
(19)
The unemployment pool in the educated sector is added to each period by newly-arrived educated workers, and these are joined by those previously unemployed educated workers who remain unsuccessful in finding a job, after attempts in both markets. The steady-state stock is
F(~)N v e =
1 - s(1 - p ( / . Q )
(1
-p(O))
(20)
The flow of unemployed uneducated workers is determined by thedifference equation Utu = [1 - F ( ~ t ) ] N + s(1 - p ( Ixt))U," ,.
(21)
8 Existence and uniqueness of such a ~ follows from the facts that c ~ [0, c¢) and in non-trivial equilibria, u ~ - u u is positive, and independent of c. See the appendix.
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The stock of unemployed uneducated workers is added to each period by new arrivals who choose not to educate and by the number of unsuccessful searchers from the previous period. The steady-state stock is [1 - F ( ( ) ] N Uu=
1 -- s(1 - - p ( ~ ) )
.
(22)
The flows of employment for educated and uneducated workers are given by three difference equations: E~, = s(1 - p ( O t ) ) E ~ , , _ , + s p ( / ~ t ) ( l
- p ( O,) )U,~ ,,
(23)
E~.,=sE~,.,_, + sp(O,)Sh.,_ , ,
(24)
E; = sE;_ 1 + sp( /~t)U,~ ,.
(25)
Employment of educated workers in the low-output sector in each period consists of those so employed in the previous period who failed to find a high-output sector job plus those who were unemployed in the previous period who were unsuccessful in locating a high-output job, but who succeeded in finding a low-output job. Similarly, the stock of employed in the high output sector consists of survivors from the previous period and successful searchers from the previous period. The employment dynamic of uneducated workers has a similar interpretation. These give the steady-state stocks as sp(/~)(1-p(O))U e E[ =
E~
,
1 -s(l-p(0))
(26)
sp( O) Sh
- - , 1-s
(27)
u
Eu=
1-s
,
(28)
where Sh = E 7 + U e. Note that the steady-state equilibrium number of mismatched workers, Et~, is strictly positive. It is easily checked that the total population of this economy in the steady-state, the sum of (20), (22), (26), (27), and (28), is N / ( 1 - s) as required. We define the respective unemployment rates among educated and uneducated workers as Ue VRe( O, ~ )
= S h -[- E~
(29)
1 --S
l - 41
-e(0))
'
C.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
123
VU VR"(
-
V" + E"' 1 - s
(30)
[1-s(1-p(/x))]" The unemployment rate among educated people is decreasing in both 0 and Ix, and the unemployment rate among the uneducated is decreasing in ix. For all admissible values of IX and 0, the unemployment rate among educated workers is less than the unemployment rate among uneducated workers. 9 This is a consequence of educated workers having at most two search attempts in each period compared to at most one attempt permitted for uneducated workers. It is also apparent that an increase in ~ lowers the unemployment rate among the uneducated more than among the educated. The economy aggregate unemployment rate is simply a weighted average of the sectoral rates Ue + Uu
UR( ~, O, ~)
-
N/(1 - s )
(31)
= F(~) URe( O, ~) + [1 -- F( ~)] URU(I~), and we should note, from (16) that ~ is itself a function of 0 and ~. Using this fact, we see that the economy unemployment rate is a decreasing function of 0. Not only does an increase in 0 reduce the unemployment rate among educated workers, but it also increases the supply of educated workers, thus placing greater weight on the lower unemployment rate in the aggregate index. On the other hand, an increase in I~ has an indeterminate effect on the economy unemployment rate. This is because both the sectoral rates are reduced by an increase in IX, but fewer workers undertake education, thus shifting greater weight onto the higher unemployment rate in the aggregate. Without further restrictions the net effect of these factors is unknown. 10 The expected durations of unemployment in each sector i, D i are given by
D e= [ 1 - s ( 1 - p ( / ~ ) ) ( l - p ( 0 ) ) ]
',
D u = [1 - s ( 1 - p (
and a simple calculation show that for all realizations of 0 and IX, we have D e < D u. Thus our model exhibits at least two of the stylized facts noted earlier: educated workers have lower unemployment rates and lower expected durations of 9 This is easily established by comparing (29) and (30). 1o The expression for UR. is
URu = F(~)UR~ + [1 - F(~)]UR~ + f ( O ) ~ [ URe - UR"]. The first two terms are negative and the last is positive. The unemployment rate is more likely to increase when i~ increases the higher is the responsiveness of the demand for education to I~.
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unemployment. This, of course, is a consequence of the increased range of job opportunities for educated workers under our specification and the fact that educated workers sample from two, not one, job pool. We may also use (26)-(28) to define a measure of job mis-match as the ratio of mis-matched workers to all employed educated workers:
El E;+E
M- - -
sp(
[1 - s(1
-p( O)) - p ( 0 ) ) ] [p(/x)(1 -p(O)) + p ( 0 ) ]
where it is straightforward to show that an increase in Ix increases mis-match and an increase in 0 reduces mis-match (given s).
3.3. Wages We keep wage-determination as straightforward as possible, by assuming a simple sharing rule. Specifically, each match rewards the worker with a wage which is an average of the output in that match and the worker's reservation wage. Thus, we have
Wh=aYh+(1--a)b,
ce ~ (0, 1),
(32)
w~=fly,+(l-[3)b,
/3 ~ (0, 1),
(33)
w;=~y,+(l-~)(b+K),
~
(0, 1),
(34)
for constants ¢~, [3, and 6- This makes the analysis much more tractable than assuming a Nash bargain between firms and workers, whilst maintaining a symmetry in the treatment of each wage. ~1 Moreover, the weights in each case have a 'bargaining power' interpretation. Since, by assumption, Yh > Y~ > b, the wages w h and w~' are increasing in o~ and [3 respectively, so these may be interpreted as measures of worker bargaining power. Note that under this specification, wages are independent of 0 and Ix. Although special, this does enable us to focus on the relationship between education and unemployment for a given wage structure. Having wages additionally depend on 0 and Ix complicates the returns Eqs. (10), (11) and (13) but adds little to the story of the relationship between education and unemployment. The wage structure chosen appears to reflect some of the observed patterns discussed in the introduction, and is also consistent with educated workers taking temporary jobs.
I1 The weakness of this approach is that all wages in the decentralized equilibrium are insensitive to labour market tightness. The loss in generality is quite small, since even the Nash rule fails to internalize externalities in general. See Hosios (1990a).
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Imposing the parameter restriction Yl = b + K gives the result that w 7 = b + K. This implies that, in this economy, mismatched workers are as productive as in their worst alternative, and l-firms earn zero rents from mismatches. Since Yt > w~ we see that 1-firms would prefer to have a vacancy filled by an uneducated worker with certainty, than by an educated worker. Further, we have that Yt > b implies w~ > b; w h > W7 is implied by ot(y h b) > K; and w~ > w~' is implied by 13( Yt - b) < K. Then the desired wage structure is obtained by assuming et( Yh -b) > 13(yl - b), and that search costs fall into the interval -
-
K E (Or( Yh -- b ) , / 3 ( Y t - b ) ) . Restricting our attention to this parameter space, the complete ranking of wages is therefore w h > w~ = b + K > w ' / > b,
(35)
and the wage difference between well-matched educated and uneducated workers is measured by w h-w~=o~(yh-b)+flK>O,
(36)
while the wage difference between educated and uneducated workers in the /-sector 12 is w;-w~
= K(1 - / 3 ) > 0 .
(37)
The wage differential between well-matched workers is increasing in et, Yh and f3 and decreasing in b. The wage differential between educated and uneducated workers in the /-sector is increasing in K, and decreasing in 13. Thus, the differentials behave in an intuitive way. Increased bargaining power of one group of workers changes the differential in that group's favour. 3.4. F i r m e n t r y
Firm entry in each sector is determined by a zero expected profit condition. Potential firms are able to enter only one of the sectors and are unable to change their type. However there is no restriction on the number of potential firms of each type. Entry therefore, is based only on a comparison of the returns in the relevant sector relative to returns elsewhere (assumed to be zero). In equilibrium, firms are indifferent e x a n t e between operating in either sector. A firm in sector j ~ {h, l} incurs a cost kj for each period with an unfilled vacancy. There are no set-up costs. All firms have the same discount factor,
J2 We may also define the empirically-relevant measure of wage differences as the difference between the wage in the h-sector and the average wage in the /-sector as Wh
w~E; + w~E" E7 + E u
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p = (1 + r) ~, where r is the interest rate. Denote by wj, j ~ {h, l} the expected profit of being a new entrant firm in sector j. For an h-firm "rrh is defined recursively by
"rch,, = --k h + psq( Ot)
( Yh -- Wh) 1
--
+ p(1 - q( O,) )Trh.t+ ~
ps
+ p ( l - s) q ( 0 , ) 7rh,,+ , .
(38)
Setting ~h = 0 as the equilibrium condition, letting p --* 1 and substituting for w h gives S
kh -- 1 -- s q(O)(1 -- a ) ( Y h -- b).
(39)
Since, k h, Yh, b, c~ and s are exogenous, the zero profit condition determines 0. Now consider the expected profits of an /-firm. Define h as the conditional probability that a located worker is uneducated. So 1 - h is the conditional probability that a located worker is educated. Then, expected profits of an l-finn are given recursively by
7rt't= - k t + psq( tz'){ ( 1 - A)~r['+I + A ( y ' -~wp ~s ) + O(1 - q( P,t))Trz.,+ ,.
(401
where "rrl,t+ e I is the expected discounted profit for an l-firmemploying an e-worker given by
"TTel., ( y t - w[)[1 + p s ( 1 - p ( O,)) + p2s2(1 - p ( 0 , ) ) ( 1 - p ( O,+ ,)) =
+ ...1 + o(1
+
+ o 2S277-
,,,+3+...]
+ Psi p(Ot+ ,)vr,,,+, + ps(1 -p(O,+ l))p(O,+217r,.,+ 2 + . . . ] . (411 Eq. (40) uses the assumption that it is better for the firm to fill a vacancy with an e-worker than to leave the vacancy unfilled, or, 'rr~t.t+J > "rrt,t+ j Vt. Using the steady-state zero profit condition, rr t = 0, the fact that w 7 = y~, substituting for w;' and letting p ~ 1 in (40) and (41) gives S
k, = ~ - _ s q ( / x ) A(1 - f l ) ( y , - b ) .
(42)
In general we expect h to depend on the proportion of searchers in the /-sector who are uneducated. In particular, we may write A=A
,
A'(.)>0.
(43)
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127
Alternatively, we note that Uu St
1 1 d-(1 - p ( O ) ) ( u e / u
u)
where U e and U" in turn depend on ~, 0 and Ix, from (20) and (22). Thus k = k(~, 0, Ix) with k e < 0, k 0 > 0 and k~ _< 0.
4. The decentralized equilibrium Equilibrium in this economy is a triple {~, 0, IX} satisfying Eqs. (16), (39) and (42). 4.1. Multiple equilibria
The model has the potential for generating multiple equilibria. This is easily seen by looking at conditions (16) and (42). First note that 0 is determined uniquely by (39). Given 0, (16) defines a function ~(IX)I06) with c~[06) < 0. Using (42) and the properties of k we have a second function O(IX)I(4/) with ~](42) < 0. Thus, depending on the precise shapes of these two functions, multiple equilibria cannot be ruled out. For example, suppose ~(IX) given by (18) is convex in tx (i.e. uU(ix) is concave in Ix, while OIx defined implicitly by (42) is concave in IX, then we have at most two equilibria as illustrated in Fig. 1. Clearly, the equilibria show a positive correlation between levels of education and rates of unemployment in the /-sector. Point A, for example is associated with a 'high' rate of education but also a 'high' rate of unemployment among the uneducated. 4.2. A special case
To provide a focus for comparative-static results, we look at'a leading special case. Here, the general model is simplified substantially by assuming that, over the
Fig. 1.
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C.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
relevant range, h is insensitive to changes in 2, 0 and Ix. Note that the zero-profit conditions and the exogeneity of h imply that 0 and Ix are independent and constant. The appendix discusses the formalities of existence and uniqueness of an equilibrium. Even in this simple model, we see the interdependence between education decisions and unemployment. The influence of labour market tightness on education investment is apparent in (16), while (31) clearly indicates the role of education in the composition and magnitude of the economy's unemployment rate. A variety of preliminary results are now available relating to the effects of parameter changes on 0 and Ix. These are summarized in the following. The proofs are obtained by differentiation of (39) and (42) respectively. Proposition 1. (i) Each o f the following increases 0: an increase in s or Yh, a decrease in kh, OL and b. (ii) Each o f the foUowing increases Ix: an increase in s, X or Yl, a decrease in k t, f3 and b.
These are all intuitive. An increase in sectoral output, a reduction in recruitment costs, and a reduction in workers' bargaining power all increase the profit associated with vacancy creation and so more vacancies are created. An increase in s also increases the expected value of a job slot in both sectors. In the/-sector, an increase in the probability of locating an uneducated worker also increases expected profit, since l-firms only earn rent from uneducated workers. These results lead directly to the following effects on the sectoral unemployment rates, (29) and (30). Proposition 2. (i) Each o f the following reduces the unemployment rate among educated workers: an increase in s, Yh, h or Yl, a decrease in kh, kl, or, ~, or b. (ii) Each o f the following reduces the unemployment rate among uneducated workers: an increase in s, k or Yt, a decrease in k t, f3, or b.Again, these are all
intuitive, but notice how improvements in the /-sector benefit the educated workers in terms of lowering their sectoral unemployment rate. On the other hand, productivity improvements (an increase in yh), for example, in the h-sector have no direct benefit for uneducated workers in terms of their unemployment rate. Thus sector-specific productivity changes have quite different implications depending on which sector experiences the improvement. As a first pass, given a choice, a policy which exploited the spillover and encouraged productivity improvements in the /-sector might be preferred to one which encouraged improvements in the h-sector, at least in terms of the effect on sectoral unemployment rates. This conclusion is erroneous. The overall unemployment rate in the economy will not necessarily be reduced if improvements in the /-sector are accompanied by a sufficiently large reduction in human capital investment. If this happens, then a higher proportion of uneducated workers will increase the overall expected
c.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
129
duration of unemployment and so increase the average unemployment rate. To see this formally, we take the following steps. We may use (11), (13) and (16) to derive results for changes in human capital investment. The following are the most immediate results.
Proposition 3. Each o f the following increases the proportion o f educated workers (i.e. increases ~): an increase in k t or Yh, a decrease in kh, h or Yl.
Each of these increases the lifetime income of educated workers relative to that of uneducated workers, and so encourages greater investment in human capital. Changes in costs and in l-firms' perceptions about the availability of uneducated workers increase expected income through their effect on vacancy creation and hence matching probabilities. Increases in productivity also work through this route, but also directly through the effect on wages in the sharing rule. Other effects are ambiguous in general because their indirect effects on expected income, through the matching probabilities, work against the direct effects through wages. An increase in h-workers' bargaining power, o~, for example, has the effect of reducing 0 (Proposition 1) and lowering the probability of an educated worker finding an h-job. On the other hand, an increase in a has the direct effect of increasing the wage in the bargain. Notice here the effect referred to earlier of productivity improvements in the /-sector reducing human capital investment. We may combine the results for the sectoral unemployment rates and the rate of human capital accumulation to deduce the effect of various changes on the aggregate unemployment rate, using (31). The following are immediate results:
Proposition 4. An increase in Yh or a decrease in k h will reduce the overall unemployment rate.
Both of these changes have the effect of both stimulating education investment, increasing the overall search efficiency of the labour market, and of reducing the sectoral unemployment rate for educated workers. Other effects on the aggregate unemployment rate are less clear. For example, the productivity improvement in the /-sector has several effects. First, from Proposition 3, an increase in yt reduces the proportion of educated workers, both because it increases the wage and because it promotes vacancy creation in the /-sector, thus increasing lifetime earnings from being uneducated relative to being educated. This change in the composition of the labour force towards less educated workers tends to increase the average unemployment rate. Secondly, by stimulating vacancy-creation in both sectors, both sectoral unemployment rates fall. The net effect on the aggregate unemployment rate is ambiguous.
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C.J. Mc Kenna / European Journal of Pol#ical Economy 12 (1996) 113-132
5. Conclusion We have proposed a two-sector equilibrium model in which the rate of human capital investment and the unemployment rate are co-determined. The model emphasises the incidence of unemployment among educated and uneducated workers. To the, extent that educated workers may choose to work in jobs requiring less skill they experience lower unemployment rates and have shorter unemployment durations. Asymmetric spillover effects between the two sectors mean that the source, as well as the size of productivity or other changes affect the incidence of unemployment between educated and uneducated workers. Productivity changes also affect the education decision. Because the average unemployment rate depends on both the sectoral unemployment rates and on the amount of human capital investment, the overall effect of productivity changes needs to take account of the education decision. An increase in productivity in the high-output sector, whist having no positive spillover into the low-output sector, does have the effect of reducing the overall unemployment rate. On the other hand, an increase in productivity in the low output sector, despite having a positive spillover effect in reducing both sectoral unemployment rates does not necessarily lead to an overall reduction in the aggregate unemployment rate. Some aspects of the model make it rather special in the sense that we have made assumptions so as to create some interdependencies between the markets for educated and less educated workers, without considering all the possible sources of interdependence. In particular, the wage rules and the probability (for a firm) that a located worker is uneducated, are insensitive to labour market conditions for the purpose of our exercises. This enables us to abstract from issues of multiple equilibria and feedback effects into relative (labour) prices from the search process. These maybe of second-order importance, but the model as it stands seems consistent with available evidence and contains some testable hypotheses on the effects of various sources of productivity improvements. The model is a useful starting point to explore these issues and generalizations of it may serve to explain a variety of other issues.
Acknowledgements An earlier version of this paper was presented at the 17th Canadian Economic Theory Meetings, Universit6 de Montr6al and to the Theory Workshop at the Australian National University. I am grateful to the participants for helpful comments. The paper has also benefited from the comments of two anonymous referees. Financial support from a Social Science and Humanities Research Council of Canada grant (410-90-0151), and a B.C. Matthews Fellowship are acknowledged. The work was completed while I was a Visiting Fellow in the
C.J. McKenna / European Journal of Political Economy 12 (1996) 113-132
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Department of Economics, Faculties, at A.N.U. I am solely responsible for remaining errors and omissions.
Appendix A Let p(X) be a continuous, non-decreasing function on [X, ~'] and let p(X) ==Xq(X) for X ~ {0, p~}. Assume; p(X)>0,
p(~')
q(X)
q(.~)>_0
while
and q(X) is non-increasing on [X, ~']. (See footnote 5.) First consider (16). Since u e is continuous and non-decreasing in 0, and u u is continuous and non-decreasing in p~ and since c ~ [0, oo) then necessary and sufficient condition for existence and uniqueness of ~ is that e _ Umax u >0 Umin where Uein -- ue(X) and u~i . - uU(X). We restrict our parameter space so that this is so. From (39) the right-hand side is continuous and non-increasing in 0 while the left-hand side is a positive constant. Existence and uniqueness therefore follow from the assumptions that (1 - s)k h < q(X)(1 - a ) ( Yh -- b) and (1 - s)k h > q( ~')(1 - a ) ( y h - - b). Similarly from (42) we require; (1 - s)k t < sq(X)A(1 - f l ) ( y l -
b)
and
(1 - s ) k I > sq( X)A(1 - / 3 ) ( Y l -
b).
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