Working Paper 99-11December 1999

Equilibrium Search withHuman Capital Accumulation

Henning Bunzel, Bent J. Christensen,Nicholas M. Kiefer, and Lars Korsholm

Published by

Centre for Labour Market and Social ResearchUniversitetsparken, Building 350

8000Aarhus C, Denmark

Editor: Peder J. Pedersen

Copyrights: Henning Bunzel, Bent J. Christensen, Nicholas M. Kiefer,and Lars Korsholm

ISSN 0908-8962

Equilibrium Search withHuman Capital Accumulation

Henning Bunzel, Bent J. Christensen, Nicholas M. Kiefer,and Lars Korsholm

Abstract: The homogeneous search equilibrium model of Mortensen ( 1990) and Burdettand Mortensen (1998) is extended to allow on-the-job wage growth. The extension allowsan improved empirical fit to the cross-section wage distribution. The estimation problemis nonregular but tractable. Estimated parameters are sensible and interpretable. The fitto the wage distribution is improved relative to the homogeneous model but still notentirely satisfactory.

Keywords: Dynamic programming, human capital, equilibrium search

JEL classifications: C23, C41, C51, J24, J41, J64

Acknowledgements: The authors are grateful to the Centre for Labour Market and SocialResearch (CLS), Aarhus, and to the Danish Social Science Research Council, for researchsupport.

Addresses: Henning Bunzel: CLS and Department of Economics, University of Aalborg.Bent Jesper Christensen: CLS and School of Economics and Management, University ofAarhus. Nicholas M. Kiefer: CLS and Department of Economics, Cornell University. LarsKorsholm: CLS and Department of Biostatistics, University of Odense.

Equilibrium Search with Human Capital Accumulation

Henning Bunzel, Bent J. Christensen, Nicholas M. Kiefer and Lars Korsholm

The homogeneous search equilibrium model of Mortensen (1990) and Burdett andMortensen (1998) is extended to allow on—the-job wage growth. The extension allowsan improved empirical fit to the cross-section wage distribution. The estimation problemis nonregular but tractable. Estimated parameters are sensible and interpretable. Thefit to the wage distribution is improved relative to the homogeneous model but still notentirely satisfactory.

1. INTRODUCTION

In this paper, we introduce a new framework for analysis of the rate of return to humancapital. Information imperfections are modelled using search theory. The central assetpricing implications (see (6), (10) and (21) below) are key to this development.The homogeneous search model of Mortensen (1990) and Burdett and Mortensen (1998)

builds on the insight that explicit modeling of worker flows among firms allows generationof endogeneous wage dispersion for homogeneous workers. This basic model and a numberof extensions allowing for productivity differences among firms or for measurement errorhave been fit to data for several countries, see (Kiefer and Neumann 1993, Koning, Ridderand van den Berg 1995, van den Berg and Ridder 1998). The specifications are comparedusing the same data set by Bunzel, Christensen, Jensen, Kiefer, Korsholm, Muus, Neu-mann and Rosholm (1999). The homogeneous model explains flows, but does not give anadequate fit to the cross-section distribution of wages. That fit is improved dramaticallyby the addition of measurement error in wages or heterogeneity in productivity of firms.This paper considers a different extension of the homogeneous search model, allowing

for on-the-job wage growth. This has the economic interpretation of a return to the humancapital appreciation associated with accumulated experience. Although our specificationis quite simple, it does represent an alternative to specification of exogeneously givenmeasurement error or exogeneously set productivity differences across firms. This paperdevelops and extends the preliminary analysis given in Bunzel et al. (1999).In Section 2 we describe the equilibrium search model with wage growth on the job.

The corresponding empirical model and estimation procedure is described in Section 3.The data and estimation results that we use are described in Section 4. The estimationproblem is nonregular but tractable using the theory of local cuts. Estimates are presentedfor 18 groups of workers - grouped to control nonparametrically for variations in observablecharacteristics (age, education, etc.). Section 5 concludes.

2. EQUILIBRIUM SEARCH WITH WAGE GROWTH

The economy consists of a homogeneous population of workers and a homogeneouspopulation of firms. Workers conduct job search both when unemployed and when em-ployed. Let λ0 denote the offer arrival rate when unemployed, and λ1 the correspondingrate relevant for on-the-job search. Unemployment income net of search costs is givenby b. When employed, workers with work experience c yield productivity p + αc in firmproduction and are laid off at rate δ. Here, α ≥ 0 is the rate at which human capital isaccumulated while employed. It is assumed that the worker reaps the total benefits of theproductivity increase, i.e. the wage is w = w0 + αc, where w0 is the initial reemploymentwage. All experience is lost when a worker is laid off, reflecting the inexperience of newentrants and allowing an interpretation of lay-offs in terms of retirement of older workersthat are replaced by new entrants. The Mortensen (1990) model is the case α = 0 (nohuman capital accumulation). Thus, the economy is described by the structural parameterθ = (α,λ0,λ1, δ, p, b) ∈ <5+ ×<.Consider first an unemployed worker receiving benefits b per period and discounting

future income at rate ρ. At times ρ may be included in θ, too, although we shall not doso in the empirical analysis below. With T0 > 0 denoting the waiting time for the firstjob offer x, the value of being unemployed is

V0 = E

ÃZ T0

0be−ρtdt+ e−ρT0max {V0, V1 (x, 0)}

!, (1)

where V1 (x, 0) is the value of becoming employed at wage x per period. In the sequel,we shall write V1 (x, c) for the value of employment for a worker who has been employedfor c periods, with reemployment wage x and thus current wage x + αc. To assess thevalue of a constant flow we evaluateZ T0

0e−ρtdt =

"−1ρe−ρt

#T00

=1

ρ

³1− e−ρT0

´. (2)

Offers arrive according to a Poisson process with intensity λ0 > 0. Thus, T0 ∼ exp (λ0),i.e. the density of the waiting time is

f0 (t) = λ0e−λ0t, t > 0, (3)

the exponential distribution with arrival rate λ0. Then the expected value of the randomdiscount factor in (1) is

Ee−ρT0 =Z ∞0e−ρtf0 (t) dt =

λ0λ0 + ρ

Z ∞0(λ0 + ρ) e−(λ0+ρ)tdt =

λ0λ0 + ρ

. (4)

Combining with (2), this allows calculating the expected value of the integral in (1) as

EZ T0

0be−ρtdt = bE

1

ρ

³1− e−ρT0

´=b

ρ

Ã1− λ0

λ0 + ρ

!=

b

λ0 + ρ. (5)

Offers to inexperienced workers are distributed according to F , independently of thepreceding waiting times, so (1) is recast as

V0 =b+ λ0

Rmax {V0, V1 (x, 0)} dF (x)

λ0 + ρ. (6)

A firm paying w to inexperienced workers receives a profit equal to (p+ αc)−(w + αc) =p−w per worker with experience c. Note that this does not depend on c, by assumption.Consider now a worker with experience c employed in a firm paying w to inexperiencedworkers. Offers now arrive at rate λ1 with first arrival at T1 ∼ exp (λ1) . Lay-offs occurat rate δ, so the job dissolves at Tδ ∼ exp (δ). Thus, the current job is kept at least untilTm = min {T1, Tδ}. If T1 < Tδ i.e. an offer from a firm paying x to inexperienced workersis received before lay-off, the worker chooses between continuing in the current job, ormoving to the new job. If Tδ < T1, the worker returns to unemployment (i.e. value V0 )at Tm = Tδ. We therefore have

V1 (w, c) = E

ÃZ Tm

0(w + α (c+ t)) e−ρtdt

!+ (7)

P (T1 < Tδ)E³e−ρT1max {V1 (w, c0, T1) , V1 (x, c+ T1)} |T1 < Tδ

´+P (Tδ < T1)E

³e−ρTδ |Tδ < T1

´V0.

We have

P (Tm ≤ t) = 1− P (Tm > t) (8)

= 1− P (T1 > t)P (Tδ > t)= 1− e−(λ1+δ)t

and so Tm ∼ exp (λ1 + δ). From (5) the part of the first expectation in (7) which involvesw + αc is (w + αc) / (λ1 + δ + ρ). The remaining part of the first expectation in (7) isevaluated by integration by parts,

E

ÃZ Tm

0αte−ρtdt

!= αE

"−1ρte−ρt

#Tm0

+Z Tm

0

1

ρe−ρtdt

= −α

ρ

Z ∞0te−ρt (λ1 + δ) e−(λ1+δ)tdt

+α

ρ2

³1− E

³e−ρTm

´´=

α

(λ1 + δ + ρ)2

To evaluate (7), we need in addition

P (Tδ < T1)E³e−ρTδ |Tδ < T1

´=

Z ∞0

Z ∞se−ρsλ1e−λ1tδe−δsdtds (9)

=δ

λ1 + δ + ρ

Z ∞0(λ1 + δ + ρ) e−(λ1+δ+ρ)sds

=δ

λ1 + δ + ρ.

Finally, (7) is given by

V1 (w, c) =w + αc+ δV0λ1 + δ + ρ

+α

(λ1 + δ + ρ)2(10)

+P (T1 < Tδ)E³e−ρT1 max {V1 (w, c+ T1) , V1 (x, c+ T1)} |T1 < Tδ

´.

An unemployed worker who receives an offer x chooses between values V0 and V1 (x, 0).Since V1 (x, 0) is increasing in x, the worker becomes employed if and only if x ≥ r, wherethe reservation wage r is given by V0 = V1 (r, 0). Substituting V1 (r, 0) for V0 in the integralin (6) allows writing it asZ

max {V0, V1 (x, 0)} dF (x) =Zmax {V1 (r, 0) , V1 (x, 0)} dF (x) (11)

= P (x ≥ r)E (V1 (x, 0)− V1 (r, 0) |x ≥ r) + V1 (r, 0)=

Z h

r(V1 (x, 0)− V1 (r, 0)) dF (x) + V1 (r, 0)

where h is the upper bound of the support of F. Using (11) in (6) yields

V0 =b+ λ0

R hr (V1 (x, 0)− V1 (r, 0)) dF (x) + λ0V1 (r, 0)

λ0 + ρ. (12)

Writing V0 = V1 (r, 0) on the left side of (12) and isolating V1 (r, 0), we get

V1 (r, 0) =b

ρ+

λ0ρ

Z h

r(V1 (x, 0)− V1 (r, 0)) dF (x) . (13)

Evaluating (10) at (w, c) = (r, 0) and rewriting the integral as in (11) produces

V1 (r, 0) =r + δV0

λ1 + δ + ρ+

α

(λ1 + δ + ρ)2(14)

+P (T1 < Tδ)E³e−ρT1max {V1 (r, T1) , V1 (x, T1)} |T1 < Tδ

´=

r + δV1 (r, 0)

λ1 + δ + ρ+

α

(λ1 + δ + ρ)2

+Z ∞0

Z ∞te−ρtλ1e−λ1tδe−δs

ÃZ h

r(V1 (x, t)− V1 (r, t)) dF (x) + V1 (r, t)

!dsdt

=r + δV1 (r, 0)

λ1 + δ + ρ+

α

(λ1 + δ + ρ)2

+λ1

Z ∞0e−(λ1+δ+ρ)t

ÃZ h

r(V1 (x, t)− V1 (r, t)) dF (x) + V1 (r, t)

!dt

Equating (13) and (15) yields

r =

Ã1 +

λ1ρ

!Ãb+ λ0

Z h

r(V1 (x, 0)− V1 (r, 0)) dF (x)

!(15)

− α

λ1 + δ + ρ− (λ1 + δ + ρ)

Z ∞0e−(λ1+δ+ρ)tÃZ h

r(V1 (x, t)− V1 (r, t)) dF (x) + V1 (r, t)

!dt

By integration by parts,Z h

r(V1 (x, t)− V1 (r, t)) dF (x) = [(V1 (x, t)− V1 (r, t))F (x)]hr −

Z h

rV 0

1x (x, t)F (x) dx

= V1 (h, t)− V1 (r, t)−Z h

rV 01x (x, t)F (x) dx

=Z h

rV 01x (x, t) (1− F (x)) dx, (16)

where V 01x denotes the derivative of V1 in the first argument. From (10) ,

V 01x (w, c) =1

λ1 + δ + ρ+ P (x ≤ w)

Z ∞0

Z ∞te−ρtλ1e−λ1tδe−δsV 01x (w, c+ t) dsdt(17)

=1

λ1 + δ + ρ+ λ1F (w)

Z ∞0e−(λ1+δ+ρ)tV 01x (w, c + t) dt

From (17) it is clear that V 01x (w, c) does not depend on c (the equation characterizingthe cross derivative is solved by the zero function) and hence V 01x can come out of theintegral leading to the explicit formula

V 01x (w, c) = 1/ (λ1 (1− F (w)) + δ + ρ) .

Since this is positive, the value of employment is increasing in the initial wage and areservation wage policy is optimalTo close the model, note that in (7), a worker employed in a firm paying the highest

possible wage h to inexperienced workers will not leave the firm before Tδ, i.e. there willnot be a better offer from any other firm. Therefore, in analogy with the derivation of(10),

V1 (h, c) = E

ÃZ Tδ

0(h+ α (c + t)) e−ρtdt

!(18)

+E³e−ρTδ

´V0

=h+ αc

δ + ρ+

α

(δ + ρ)2+

δ

δ + ρV1 (r, 0)

We can now compute

V1 (x, 0) =Z x

rV 01x (w, 0) dw + V1 (r, 0) (19)

=Z x

r

1

λ1 (1− F (w)) + δ + ρdw + V1 (r, 0)

From (13), we get

V1 (r, 0) =b

ρ+

λ0ρ

Z h

r

Z x

rV 01x (w, 0) dwdF (x) (20)

=b

ρ+

λ0ρ

Z h

r

Z x

r

1

λ1 (1− F (w)) + δ + ρdwdF (x) .

We now have V1 (r, 0) in (18), and thereby V1 (h, c). Hence,

V1 (r, c) = V1 (h, c)−Z h

rV 01x (x, c) dx

= V1 (h, c)−Z h

r

1

λ1 (1− F (x)) + δ + ρdx.

In particular, V1 (r, t) in (15) is now available, as is the integral in (15), using (16) and(17). This means that (16) gives the relationship between the reservation wage r forunemployed workers and the other parameters, in particular b. Combining all the results,

r =

Ã1 +

λ1ρ

!Ãb+ λ0

Z h

r

Z x

r

1

λ1 (1− F (w)) + δ + ρdwdF (x)

!

− α

λ1 + δ + ρ− (λ1 + δ + ρ)

Z ∞0e−(λ1+δ+ρ)t.ÃZ h

r

1− F (x)λ1 (1− F (x)) + δ + ρ

dx+h+ αt

δ + ρ+

α

(δ + ρ)2

+δ

δ + ρ

Ãb

ρ+

λ0ρ

Z h

r

Z x

r

1

λ1 (1− F (w)) + δ + ρdwdF (x)

!

−Z h

r

1

λ1 (1− F (x)) + δ + ρdx

!dt

Simplifying, the reservation wage is

r =(ρ+ λ1) (δ + ρ)− δ

ρ (δ + ρ)

Ãb+ λ0

Z h

r

Z x

r

1

λ1 (1− F (w)) + δ + ρdwdF (x)

!

− α

λ1 + δ + ρ− h

δ + ρ− α

(λ1 + δ + ρ)− α

(δ + ρ)2

−Z h

r

F (x)

λ1 (1− F (x)) + δ + ρdx. (21)

The case (ρ,α) = (0, 0) was considered byMortensen and Neumann (1988) andMortensen(1990).When employed, any wage higher than that currently received is accepted, i.e. the on-

the-job reservation wage is the current wage. It is this extension to on-the-job search thatallows an equilibrium derivation of firm behavior and, in particular, the offer distributionF . The balancing condition which equates supply and demand is that firms will offerhigher wages if and only if they can expect to get an additional number of workers tocover the lower per worker profits. Higher wages attract more workers to a firm and allowfirms to retain workers longer.We are now in a position to describe the equilibrium distributions. Profit per worker

in a firm paying inexperienced workers w is (p+ αc)− (w + αc) = p− w. Thus, the firmearns equal profit for all employed workers regardless of their experience. Therefore thetotal amount of profit within this firm is

π (w) = (p− w) l (w)

with l(w) being the total employment in a given firm that offers the starting wage w toany worker without experience. This amount is characterized by the number of firms inthis group and the number of workers employed in these firms. Therefore let G0 be thedistribution function of earned wages transformed to wages for workers with no experience.In other words all earned wages at a given point of time is on some wage line wt = w0+αt.

Since α is a global parameter this line is determined only by w0, and G0 is the distributionof the number of workers on these different lines. The density g0 of G0 is just saying howmany workers are employed in firms paying w0 as a starting wage to the workers withoutexperience. Recall that F is the distribution function of starting wage offers made byfirms to inexperienced persons, i.e. F (w0) is the probability that an offer is made by afirm paying wages along the line w0 + αt or any lower line (lower w0). Further, m is thenumber of workers and u the number of unemployed. Then l(w) is given by:

l (w) = lim²↓0G0 (w + ²)−G0 (w)F (w + ²)− F (w) (m− u)

=g0 (w)

f (w)(m− u) .

The last building block of the model is the steady state conditions. Since the distrib-ution of workers across firms is determined by F and G0, it is natural to come up withsteady state conditions on these two distributions. Once we have determined G0 it isfairly easy to compute the distribution of current earned wages; see below. We focus ona steady state where the distribution of workers across firms is constant.We have the following two steady state conditions:

1. The flow into and out of unemployment are equal:

λ0u = δ(m− u)

2. The flow into and out of the group of workers with an associated starting wageabove w are equal:

λ1 (m− u)G0 (w) F (w) + λ0uF (w) = δ (m− u) G0 (w)for all w in the support of F , where F = 1− F .

The model is closed with the equilibrium assumption of equal profit across firms insteady state; i.e.

π(w) = π(r)

for all initial wages w.Then we see that this is just the standard homogeneous model in terms of F0 and G0,

and we have the explicit solution:

(i) Distribution function of offered initial wages:

F (w) =δ + λ1λ1

Ã1−

sp− wp− r

!

(ii) Distribution function of earned initial wages:

G0 (w) =F (w)

1 + λ1δF (w)

=δ

λ1

Ãsp− rp− w − 1

!

(iii) Density function of offered initial wages:

f (w) =1

2

δ + λ1λ1

1√p− r√p− w

(iv) Density function of earned initial wages:

g0 (w) =1

2

δ

λ1

√p− r (p− w)−3/2 ,

where all distribution and density functions are both increasing and convex on theinterval [r;h]. The only difference with the addition of human capital accumulationis that the support of the functions is moved to the left. Since r is smaller than

that in the homogeneous model, hence h = p − (p − r)³1 + λ1

δ

´−2is smaller too.

Otherwise, the functions have exactly the same characteristics.

The cross-section distribution of earned wages, on the other hand, is quite different fromthat in the homogeneous search model. In the basic model, which corresponds to α = 0,this was just g0. The distribution is relevant since it is from this we have observations.Since we know g0, the density of the branch/line on which his income lies, we only needthe distribution of the time he has been employed since last unemployment spell. This isgiven in the waiting time paradox.To understand the waiting time paradox, let us consider a Poisson process with intensity

λ started at time zero with value zero. Let t be given and let U(t) be the time between tand the last arrival and let V (t) be the waiting time until next arrival. Even though thetime between two arrivals in a Poisson process is exponentially distributed with parameterλ, we also have that V (t) is exponentially distributed with parameter λ and independentof U(t). This is the lack of memory property in the exponential distribution. U(t) hasthe distribution function

FU(t) (s) =

(1− e−λs s < t1 s ≥ t .

However, if we assume that our process has been running for ever or that we have reacheda steady state, we see that U(t) is exponentially distributed with parameter λ.With the insight from this paradox in mind, we see that in steady state the distribution

of seniority in the model at a given point of time (U(t)) is simply exponential withparameter δ. Hence the density of the distribution of earned wages is given by:

g (w) =Z ∞0g0 (w − αt) δe−δtdt.

The intuition is that a worker earning w today and who ended his last unemploymentspell t periods of time ago (with density δe−δt) is now in a firm paying w0 = w − αt asstarting wage to workers without experience (with density g0(w−αt)). The integral is inprinciple from zero to infinity. Since g0 is positive on [r;h] only and t is positive we have

r ≤ w − αt ≤ min (h, w) .

Inserting the limits in the integral and changing the integration variable from t toy = w − αt, we have

g (w) =δ

αe−

δαwZ min(h,w)

reδαyg0

Ãy;

δ

λ1

!dy

= K

Ãw;

δ

α,δ

λ1

!δ

αe−

δαw . (22)

The function K(w; δα, δλ1) is given by

R wr g0

³y; δ

λ1

´eδαydy for w ≤ h, and for w ≥ h the

functionK(w; δα, δλ1) = K( δ

α, δλ1) is simply the Laplace transform at δ

αof a random variable

with density g0(·; δλ1).

We now report some properties of the density of the earned wage distribution (proofsare in Appendix A). The first obvious thing to note is that the support of g is the halfline (r,∞) and that g is zero at r, the left endpoint of the support. Since the density g isthe product of two continuous functions it becomes continuous itself.

Proposition 1 The density of earned wages g is differentiable in the wage for all w 6= h.Moreover, h is the unique mode point and the density is convex (concave) at r if p− r ≤(≥) δ/α.

These results imply that r may be estimated as a lower support point of earned wagesbut h is now identified as the mode point. Furthermore, we see that the tail of g tendsto zero at an exponential rate. To help generate intuition, note that wage growth occursfrom two sources: job changes, which occur with probability declining with the initialwage, and wage growth due to human capital accumulation. Figure 1 illustrates theseproperties of g.

Proposition 2 For w ≥ h the density of earned wages is log linear, i.e. for some constantC

log g (w) = C − δ

αw .

Proof. Obvious from (22)

Let us summarize what we have deduced. The density g of the distribution of earnedwages in steady state is positive on (r,∞). It has a single (non—differentiable) maximumpoint h, and it has a decreasing tail of the form K( δ

α, δλ1)e−

δαw. Thus, the tail is heavier

than a normal tail and lighter than a lognormal tail.We now consider practical problems in calculating the integral K(w; δ

α, δλ1). Again,

the details are given in Appendix A. First, K(w) is similar to the incomplete digammafunction, which relates to a χ2 distribution. We give the following formula.

Proposition 3 Let Φ(·) denote the standard normal distribution function and let χ2(a, f)denote the tail probability at a in a χ2-distribution with f degrees of freedom. Then for

r ≤ w ≤ h we have

K (w) =δ

λ1

(sp− rp− we

wδ/α − erδ/α)− δ

λ1

qπδ/α

√p− r : epδ/α

×hχ2 (2 (δ/α) (p− w) , 1)− χ2 (2 (δ/α) (p− r) , 1)

i=

δ

λ1

(sp− rp− we

wδ/α − erδ/α)− 2 δ

λ1

qπδ/α

√p− r : epδ/α

×·Φµq2 (δ/α) (p− r)

¶− Φ

µq2 (δ/α) (p− w)

¶¸and for w > h we simply have K(w) = K(h).

Since every computer has accurate and fast calculation procedures for the standardnormal distribution function and approximations of tails probabilities in the χ2(1) distri-bution with one degree of freedom, we can indeed handle K(w). To increase the accuracy,other numerical methods may be used. The point is that one should not use any kind ofTaylor expansion to approximate K(w).Figure 1 gives an example of a possible cross-section wage distribution for parameters.

It is clear that this model allows a much better fit to observed wages than the upwardsloping density implied by the simple model.

3. ESTIMATION

Consider now estimation in the extended model with human capital accumulation froma cross section sample. If we have current earned wage data alone we expect, in general,to identify δ/α, δ/λ1, r, and h, or in terms of the structural parameters δ/α, δ/λ1, b, andp.If in addition to wage data we have data on employment duration (since end of last

unemployment spell) we are additionally able to identify δ, and hence the five structuralparameters α, δ, λ1, b, and p can be estimated.The last structural parameter λ0 will always be estimated from duration data on the

length of the unemployment spell, independently of the other parameters.Assume that we have independent observations (ti, wi) i = 1, . . . , n, where wi is the

current wage and ti is the total length of current employment spell for an individual. Forthis setting of observations the likelihood function becomes:

L (α, δ,λ1; (ti, wi)ni=1) =

nYi=1

fW,Tδ (ti, wi;α, δ,λ1)

=nYi=1

fW |Tδ (wi|ti;α, δ,λ1) fTδ (ti;α, δ,λ1)

=nYi=1

g0 (wi − αti; δ,λ1) fTδ (ti; δ) ,

where we again use the waiting time paradox. The parameter range is λ1 > 0, δ > 0,α ≥ 0 and r < wi − αti < h for all i.

The question of inference is nonstandard as seen in the original model (α = 0) where rand h are estimated by the minimum respectively the maximum of the wage observations{wi|1 ≤ i ≤ n}, and where only h is known to be the maximum likelihood estimate.Whether the estimate for r is the maximum likelihood estimate is an open question, butChristensen and Kiefer (1997) show that it is asymptotically equivalent to the maximumlikelihood estimator. For α fixed, it is natural to consider (implied) initial wages definedas w0i := w

0i (α) = wi − αti. On the basis of these pseudo observations we might estimate

r, h, δ,λ1 as in the original model to obtain r(α), h(α), δ(α), λ1(α). The parameter p inthe expression for g0 is given in terms of the other parameters as the linear combinationp = ah+ (1− a)r, where a = (1+c1)2

(1+c1)2−1with c1 = λ1/δ. When we insert our estimates andconsider the ’asymptotic profile likelihood’ for α, the non differentiability of p(α) as afunction of α is carried through and actual computations strongly suggest that this is thecase in particular at the maximum. Hence the model is non-regular and the method isunattractive, though feasible. Further, we would expect a regular estimate of α obtainedin this way to be of order

√n. However, as we show below, an estimate converging at a

faster rate is in fact attainable.From the inequality r ≤ wi − αti ≤ h and the functional form of the likelihood we see

that the parameters r, h, and α alone determine the support of the simultaneous distrib-ution of (Tδ,W ), which is sketched in Figure 2. Support estimation has been extensivelystudied, see e.g. (Korostelev and Tsybakov 1993) chapter 7. The idea is now to apply thistheory to the search model and study the properties of the resulting estimators. Since thesimultaneous density is bounded from below by a strict positive constant on [0;K]×< for afixed constantK, we estimate the support as if the observations were uniformly distributedon the parametric set B(r, h,α) of the form { (t, w) | r + αt ≤ w ≤ h+ αt, 0 ≤ t ≤ K }.Of course, it may seem unmotivated to switch from the true likelihood function to theuniform, but we shall show that a super—consistent estimator nonetheless obtains fromthis approach. Hence, we propose the following pseudo maximum likelihood estimator:

³r, h, α

´= argmax

r,h,α

Yi

1{(wi,ti)∈B(r,h,α)}|B (r, h,α)|

= argminr,h,α

|B (r, h,α)| ,

where the minimum is taken over the sets B(r, h,α) which contains all the data points(ti, wi) with ti ≤ K, and |A| denotes the Lebesgue measure of a set in <2. Direct calcu-lations show that |B(r, h,α)| = K(h− r). Hence³

r, h, α´= argmin

r,h,α{h− r | r + αti ≤ wi ≤ h+ αti, ti ≤ K, ∀i}

= argminr,h,α

{h− r | r = rn (α) , h = hn (α) , ti ≤ K, ∀i} , (23)

where rn(α) = min{wi − αti | ti ≤ K}, and hn(α) = max{wi − αti | ti ≤ K}. The firstis a simple linear programing problem, the second is to minimize the difference betweena convex and a concave function, which is simple too. The difference hn − rn convergesto a constant at rate 1/ logn for α 6= α0 and to the difference of the true parametersh0−r0 at rate n for α = α0, see Appendix B. The second estimation method is illustrated

graphically in Figure 3 by plotting (α, rn(α)) and (α, hn(α)). Then α is chosen as thevalue at which the vertical distance between the two graphs is minimized. The resultbelow only holds for a fixed known value of K independent of the sample size. However,in practice, one typically uses all the observations to determine r, h, and α, which willlead to an even better estimate. When α is known (without loss of generality we maythen assume that α = 0, which is equivalent to assuming the original BM—model) weknow that r and h converge at rate n. Thus, we expect the rate n convergence (provenbelow) to be optimal within this setting.

Proposition 4 The proposed estimate in (23) is of order n, i.e.

|r − r| = OP³n−1

´,¯h− h

¯= OP

³n−1

´, and |α− α| = OP

³n−1

´.

Proof. Order the observations into Y1, . . . , YN(n) such that Yi ∈ [0;K] × < for all1 ≤ i ≤ N(n), {Y1, . . . , YN(n)} ⊆ {(Tδ1,W1), . . . , (Tδn,Wn)}, and N(n) is maximal. ThenY1, . . . , YN(n) constitute an independent and identically distributed sample with densityCg0(y

2i − αy1i ; δ,λ1)fTδ(y

1i ; δ), where C is the finite constant such that

RCg0fTδdy = 1.

The random number N(n) is the number of observations that contribute to the estimator;it satisfies N(n)→∞ a.s. and

N (n)

n→ P (Tδ ≤ K) > 0 .

Here, for convenient notation, let B denote the true support B(r0, h0,α0) and let Bdenote the estimated support B(r, h, α). The proof is built on the following inequality.By considerations case by case one finds that there exists a positive constant D such that¯

B\B¯+¯B\B

¯≥ 2D

h|r − r|+

¯h− h

¯+ |α− α|

i.

By the maximum likelihood property of B all observations must be in B∩B, so we deducethat

ni | Yi ∈

³B\B

´∪³B\B

´ois empty and that¯B\B

¯≤¯B\B

¯,

since B is the smallest set containing all data. Let R be a fixed positive number to bechosen later. Then we have

P³|r − r|+

¯h− h

¯+|α− α| > Rn−1)

≤ P³¯B\B

¯+¯B\B

¯> 2DRn−1

´≤ P

³¯B\B

¯> DRn−1

´= P

³Yi /∈ B\B : 1 ≤ i ≤ N (n) ,

¯B\B

¯> DRn−1

´=n1− P

³Y1 ∈ B\B,

¯B\B

¯> DRn−1

´oN(n)≤1− Cg0 (r) δe−δKDR

N(n)n

N (n)

N(n)

≤ e−c0R .

In the fourth line we use that no observations fall in B\B. The sixth line follows bythe fact that for any subset A ⊆ B we have that P (Y1 ∈ A) ≥ Cg0(r)δe

−δK|A|. Theconstant Cg0(r)δe−δK equals the minimum of the simultaneous density of Y1 over the setB. The number c0 is the limit of the denominator in the fraction in the sixth line, thatis Cg0 (r) δe−δKD

N(n)n→ Cg0 (r) δe

−δKDP (Tδ ≤ K) = c0 > 0. Choosing the constant Rsufficiently large the probability vanishes.From proposition 4 the estimates r, h, and α for the corresponding parameters operate

as local cuts, i.e. r, h and α may be estimated marginally since the distribution of thestatistics br, bh and bα does not depend on (δ,λ1) asymptotically, and we may draw inferenceon δ and λ1 conditionally on br, bh and cα, see (Christensen and Kiefer 1994). Thus, we setα = argmin{hn(α) − rn(α)}, r = rn(α), h = hn(α), and compute implied initial dataw0i = wi − αti. Then the likelihood for δ and λ1 is

L³α, r, h, δ,λ1; (ti, w

0i )ni=1

´=

nYi=1

g0

Ãw0i ;

δ

λ1

!fTδ (ti; δ)

and the estimation is carried out as in the original model, see (Christensen and Kiefer1997).

4. ESTIMATES

The data we consider is a sample from the Danish population with a total of 9351observations. We observe duration time t in number of weeks and average wage per houron the job:

x =1

t

tXj=1

w0 + α (j − 1)

or equivalently x = w0 + α(t− 1)/2. Thus we transform the duration time t→ (t− 1)/2and enter the setting from above with the same interpretation for all the parameters.Hence α denotes the numeral increase per week in the hourly wage, i.e. an α of 0.1 saysthat over a year the person is paid approximately 5,000 DKK more than if he stayed atthe same hourly wage as at the end of the previous year in full time employment. Wewill not discuss the data set in detail here but refer to Bunzel et al. (1999) for a detaileddescription. We present summary statistics in Table 1. The job offer arrival rate whenunemployed, λ0, is estimated from the unemployment spells.The results from the estimation appear in Table 2. Standard errors are included in

parentheses for the√n-consistent parameters, but are left out for the super-consistent

human capital accumulation parameter α. Note that bp is a simple function of br, bh, bλ1andbδ (see Section 3), whereas bb is given in (21), using the explicit equilibrium solution forF (x) to complete the integrals, and substituting in bp (in F ) and all the other estimatesabove, along with bα and an assumed value for ρ (the Table 2 results correspond to ρ = 0).The results for the structural parameters λ0,λ1, δ, p, b make sense and are in fact quite

similar to those reported in Bunzel et al. (1999) for the measurement error and heteroge-neous productivity models. This is expected for the three flow parameters λ0,λ1, δ sinceour model extension does not alter the dynamic flow between states in the model. For thehuman capital accumulation parameter α we observe that it has a tendency to increase

with age and for the male workers to be less for the high school groups than the others.This can be given meaningful economic interpretations. The older the age group, thehigher is the possibility to enter a job with career opportunities. In particular, for the51-76 male groups, the high estimate of α appears to be necessary for the model to fit thelong right tail in the earnings distribution. For women below the high school level, weinstead observe an initial acceleration in the relationship between wage growth and age,followed by an eventual drop in level, which is consistent with productivity growth beingthe result of ’learning by doing’ for these groups.The strongest evidence of positive human capital accumulation is for the groups with

education below the high school level and for groups with at least Bachelor degrees. Wemay cautiously give this an occupational interpretation and conjecture that blue collarworkers build up skills on the job that are rewarded, whereas a transition to a white collarjob through basic high school education is insufficient to make for a career path offeringsimilar human capital investment opportunities — these are only generated by completionof higher degrees. The estimates make sense since in Denmark, high school educationis not considered a direct qualification for labour market, but a prerequisite for highereducation.The estimated cross-section wage densities for those groups with nonzero wage growth

are reported in Figures 4. Although these are substantially better than the upward-slopingdensities implied by the homogeneous equilibrium search model, they are not as realisticas those usually obtained by other heterogeneous specifications.

5. CONCLUSION

We have introduced a new dynamic dimension to the original Burdett—Mortensen equi-librium search model by allowing for human capital accumulation. We have studied themodel and shown that this allows for cross sectional wage distributions with a more sat-isfactory shape. We provide means of estimating the wage growth parameter, which weprove to be super—consistent with a rate of at least n and such that the established rate nconvergence for the support parameters r and h in the Burdett—Mortensen model remainstrue. Hence estimation of the remaining parameters is carried out with the well knownmethods for this type of models. The model is fit to Danish data. The estimated para-meters are sensible and common parameters are comparable to those found in previousspecifications. While the human capital model allows a much better fit to wage data thanthe homogeneous search model, and is intellectually appealing because the heterogeneityin the model is endogenous, it does not describe wage data as well as other specifica-tions with exogenous heterogeneity via measurement error or productivity heterogeneity,see (Bunzel et al. 1999). Further refinement of models with endogenous heterogeneity isclearly in order.

Appendix AProof of proposition 1 and 3Proof. Direct calculation of the derivative of g with respect to w is given by

g0 (w) =

( − δαg (w) if w > h

− δαg (w) + δ

αg0 (w) if r ≤ w < h

Since both δ/α and g0(h) are positive we see that g is not differentiable at h. Furthermore,from the expression of g0 we see that h is the unique mode if g(w) < g0(w) for all w in[r;h].Since g0 is strictly increasing we have

g (w) = e−βwZ w

rg0 (y) βe

βydy

< g0 (w) e−βw

Z w

rβeβydy

= g0 (w) e−βw heβyiw

r

= g0 (w)h1− e−β(w−r)

i< g0 (w) .

Hence we see that g is increasing from r to h. The last assessment follows since g00(r) ≤0⇔ p− r ≤ δ/α.

Appendix B.The asymptotic distribution of rn (α) and hn (α)Let w01, . . . , w

0n and t1, . . . , tn be independent random variables with density g0 and an

exponential distribution with parameter δ, respectively. Let us assume that we observean unknown linear combination of each pair (wi, ti) = (w0i + α0ti, ti) as in our extendedjob search model. Define

rn (α) = min {wi − αti}hn (α) = max {wi − αti}

where the optimum is over 1 ≤ i ≤ n. For α ≤ α0 we have that rn(α)→ r and for α > α0that rn(α) → ∞, and hn(α) → ∞ for α < α0 and hn(α) → h for α ≥ α0 almost surely.Furthermore, the rate of convergence is as follows. For α < α0:

(log n)−1 hn (α) → α0 − α

δin pr.s

δ

2 (α− α0)g0 (r)

√n (rn (α)− r) d→ Y2 .

For α = α0:

n

g0 (h)(hn (α0)− h) d→ −Y1

n

g0 (r)(rn (α0)− r) d→ Y1 .

And for α > α0:sδ

2 (α− α0)g0 (h)

√n (hn (α)− h) d→ −Y2

(log n)−1 rn (α) → α− α0δ

in pr.

where Y1 and Y2 are positive random variables with Y1 having an exponential distributionwith parameter one, and Y2 having a continuous distribution function given by

F3 (x) =

(0 x < 0

1− e−x2 x ≥ 0 .

Proof. The idea here is very simple and takes the following form. For α fixed,consider the identically distributed random variables Zi =wi−αti =w0i + (α0−α)ti. Thedistribution of Zi is the convolution of g0 and an exponential density with parameterλ = δ

α0−α .For α < α0, Z has the density function

fZ (z) =

0 z ≤ rλe−λz

R zr e

λxg0 (x) dx r ≤ z ≤ hλe−λz

R hr e

λxg0 (x) dx h ≤ z,

where g0 is given in (iv) of Section 2 and depends on the parameter δ/λ1. In terms of theZ’s we have hn(α) = max{Zi|1 ≤ i ≤ n}, and

P (hn (α) ≤ xn) =Ã1− n [1− FZ (xn)]

n

!nwhere FZ is the continuous distribution function for Z. If we take xn = x log n and denotethe integral in fZ for z > h by K(λ) then

n [1− FZ (x log n)] = nZ ∞x logn

λe−λzK (λ) dz

= K (λ)n1−λx →0 λ−1 < xK (λ) λ−1 = x∞ λ−1 > x

for n→∞. Hence³1− n[1−FZ(x logn)]

n

´n → e−∞ = 0 for x < λ−1 and³1− n[1−FZ(x logn)]

n

´n →e−0 = 1 for λ−1 < x. Thus hn(α)

log nconverges in distribution and hence in probability to a

degenerate random variable concentrated at λ−1.With respect to rn(α) we first see that

P (rn (α) ≤ xn) = 1− P (Zi > xn)n = 1−Ã1− n [FZ (xn)− FZ (r)]

n

!n.

Set xn = r + x√nfor x > 0 fixed. Since F 0Z(r) = fz(r) = 0 and f

0Z(r) = λg0(r) we obtain

by L’Hospital’s rule that

n

"FZ

Ãr +

x√n

!− FZ (r)

#= x2

FZ³r + x√

n

´− FZ (r)³

x√n

´2 → λ

2g0 (r) x

2 .

Hence we have that

P

sλ

2g0 (r)

√n {rn (α)− r} ≤ x

→ 1− e−x2 .

For α = α0 matters simplify, since Zi =w0i and then fZ = g0. Considering the maximumwe have

P (hn (α) ≤ xn) = (G0 (xn))n =Ã1− n [G0 (h)−G0 (xn)]

n

!nLet x > 0 be positive and set xn = h− x

n. Then by L’Hospital’s rule we find that

n·G0 (h)−G0

µh− x

n

¶¸= x

G0 (h)−G0³h− x

n

´xn

→ xg0 (h) .

Hence

P

Ãn

g0 (h){hn (α)− h} ≤ −x

!→ e−x .

Considering the minimum rn(α) we obtain

P (rn (α) ≤ xn) = 1− P (Zi > xn)n = 1−Ã1− n [G0 (xn)−G0 (r)]

n

!n.

Here we let xn = r + xn, where x > 0 is positive. In a similar fashion we obtain

P

Ãn

g0 (r){rn (α)− r} ≤ x

!→ 1− e−x .

Finally, for α > α0 let Zi = w0i − (α − α0)ti, i.e. Zi is the convolution of g0 and minusan exponential distribution with parameter λ = δ

α−α0 . The density function for Z is

fZ (z) =

λeλz

R hr e

−λxg0 (x) dx z ≤ rλeλz

R hz e

−λxg0 (x) dx r ≤ z ≤ h0 h ≤ z

Note here that fZ(h) = 0 and that the derivative at h is f 0Z(h) = −λg0(h) < 0. As abovewe use

P (hn (α) ≤ xn) =Ã1− n [FZ (h)− FZ (xn)]

n

!nwith xn = h− x√

nfor x > 0 positive. By applying L’Hospital’s rule we conclude that

n

"FZ (h)− FZ

Ãh− x√

n

!#= x2

FZ (h)− FZ³h− x√

n

´³x√n

´2→ λ

2g0 (h) x

2 .

Hence

P

sλ

2g0 (h)

√n {hn (α)− h} ≤ −x

→ e−x2

.

With respect to rn(α) we conclude as follows:

P (rn (α) ≤ xn) = 1− P (Zi > xn)n = 1−Ã1− nFZ (xn)

n

!nwith xn = −x logn for x > 0. Then we have

nFz (−x log n) = nZ −x logn−∞

λeλzK (−λ) dz

= K (−λ)n1−λx →0 λ−1 < xK (−λ) λ−1 = x∞ λ−1 > x

for n → ∞. As above we conclude that rn(α)logn

converges in distribution and hence inprobability to a degenerate random variable concentrated at λ−1.

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98-14 Dale T. Mortensen: Equilibrium Unemployment with Wage Posting: Burdett-Mortensen Meet Pissarides

98-15 Helena Skyt Nielsen: Child Labor and School Attendance: Two Joint Decisions.

98-16 Paul Bingley and Niels Westergård-Nielsen: Three Elements of PersonnelPolicy: Worker Flows, Retention and Pay

98-17 Trine Filges and Birthe Larsen: Active Labour Market Policy and EndogenousSearch

98-18 Nabanita Datta Gupta, Ronald L. Oaxaca and Nina Smith: Wage Dispersion,Public Sector Wages and the Stagnating Danish Gender Wage Gap

98-19 Peder J. Pedersen and Nina Smith: Low Incomes in Denmark, 1980 - 1995.

99-01 Paul Bingley and Gauthier Lanot: Labour Supply and the Incidence of IncomeTax on Wages.

99-02 Tim Callan, Shirley Dex, Nina Smith and Jan Dirk Vlasblom: Taxation ofSpouses: A Cross-Country Study of the Effects on Married Women’s LabourSupply.

99-03 Michael Svarer Nielsen and Michael Rosholm: Wages, Training, and JobTurnover in a Search-Matching Model.

99-04 N. Westergaard-Nielsen, Esben Agerbo, Tor Eriksson and Preben BoMortensen: Mental Illness and Labour Market Outcomes: Employment andEarnings.

99-05 Peter Jensen, Michael Svarer Nielsen and Michael Rosholm: The Effects of Bene-fits, Incentives, and Sanctions on Youth Unemployment.

Universitetsparken, Bygn. 350 Phone: +45 8942 2350 Email: cls@cls.dk

DK-8000 Århus C Fax: +45 8942 2365 WWW: http://www.cls.dk

99-06 Peter Jensen and Michael Svarer Nielsen: Short- and Long-Term Unemployment:How do Temporary Layoffs Affect this Distinction?

99-07 Trine Filges: Organization of the Labour Market.

99-08 Trine Filges: Wage Setting in Democratic Labour Unions.

99-09 Paul Bingley, Tor Eriksson, Axel Werwatz, and Niels Westergård-Nielsen:Beyond “Manucentrism” - Some Fresh Facts About Job and Worker Flows.

99-10 Mark Y. An, Bent Jesper Christensen, and Nabanita Datta Gupta: A BivariateDuration Model of the Joint Retirement Decisions of Married Couples.

99-11 Henning Bunzel, Bent J. Christensen, Nicholas M. Kiefer, and Lars Korsholm:Equilibrium Search with Human Capital Accumulation.

ISSN 0908-8962

CENTRE FOR LABOUR MARKET ANDSOCIAL RESEARCH

University Park, Building 350, University ofAarhus, 8000 Aarhus C, Denmark

Phone: +45 8942 2350 Fax: +45 8942 2365 Email: cls@cls.dk

WWW: http://www.cls.dk

Financial support from the Danish National ResearchFoundation is gratefully acknowledged