GROWTH AND GLOBAL INDETERMINACY IN A MODEL WITH
UNIONIZED LABOR MARKETS: THE ROLE OF MONETARY POLICY IN
EQUILIBRIUM SELECTION
Luigi Bonatti * **
ABSTRACT: Combining employment and growth theory, this paper shows how the labor
market interacts with the capital market to determine the equilibrium path of the economy.
Given the presence of a mechanism causing persistence (on-the-job training) and of strategic
complementarity between investment in physical and human capital, multiple balanced growth
paths are possible and global indeterminacy can arise. If there is a cash-in-advance constraint,
this indeterminacy is not resolved under inflation targeting, while the authority’s choice of a
fixed money growth rule can be decisive in selecting the balanced growth path of the economy.
KEY WORDS: Multiple equilibria, Indeterminacy, Equilibrium selection, Insider-outsider,
Monetary policy.
JEL CLASSIFICATION NUMBERS: E52, J23, O41.
*
Dipartimento di Economia, Università di Trento. E_mail: LBONATTI@GELSO.UNITN.IT. Tel.:
39-02-6599863. Fax: 39-0461-882222.
**
I would like to thank two anonymous referees, Marina Murat, Edmund Phelps and the participants to
the workshops and conferences in which earlier versions of this paper were presented for their helpful
comments and insights. The usual disclaimer applies. This paper was written while I was fellow-inresidence at the Italian Academy for Advanced Studies in America at Columbia University. Financial
support from this institution and from the CNR (in the context of the strategic project “L’Italia in
Europa”) is gratefully acknowledged.
1 INTRODUCTION
In the light of the disappointing growth performances of the main countries of the Euro
area in the 1990s (especially vis-à-vis the robust growth of the U.S.), the so-called “dissenting”
view (see IMF, 1999) blames the “too” restrictive macroeconomic policy that dominated all
over Continental Europe in that decade. As far as monetary policy is concerned, this view set
the “obsession” with inflation typical of the “orthodox” Bundesbank and of its successor, the
ECB, against the “growth-oriented” approach of the “pragmatist” FED. The orthodox viewpoint
replies by claiming that (within limits) central banks have control on market short-term interest
rates, while the long-term real rate of interest -- the only one that can influence investment and
growth – depends on economic “fundamentals” that cannot be affected by monetary policy.
This paper aims at giving a theoretical substance to the dissenting viewpoint in a framework in
which jobs and career opportunities created by a fast-growing economy induce more
individuals to acquire the basic knowledge that is necessary to be trained on the job,1 thereby
helping to ease the upward pressure on wages exerted by the increased labor demand for
qualified workers and stimulating more investment and job creation. In this context, more than
one long-run growth path can be consistent with market fundamentals, and the model shows
that -- in the presence of a cash-in-advance constraint -- monetary policy may play a role in
selecting the long-run growth path along which the economy will actually move.
This result is obtained in a model which intends to mimic some significant features of
the functioning of the economy in the Euro area. Indeed, it is assumed that: i) a unique Central
Bank decides on the monetary policy in an economy consisting of many locations subject to
idiosyncratic shocks, ii) in each location a single union establishes the wages to be paid by the
1
Even if formal education cannot substitute for learning by doing, it is a pre-condition for it: possession of
basic formal education is necessary to be able to learn on the job. This complementarity is supported by
1
local firms, iii) labor mobility across locations is imperfect, iv) the product and the capital are
perfectly integrated across locations. Within this stylized framework, it is shown that when two
balanced growth paths are possible the Bank’s choice of a more accommodative monetary rule
can lead the economy towards the long-run equilibrium path associated with the higher growth
rate, without bringing about a permanently higher inflation rate.
The paper is organized as follows: section 2 discusses the relevant literature; section 3
presents the model; section 4 characterizes the equilibrium paths of the non-monetary economy;
section 5 studies the role of monetary policy in selecting the long-run equilibrium path of the
economy, and section 6 concludes.
2 BACKGROUND
Models analyzing the relationship between growth and unemployment in the presence
of technological progress are discussed by Aghion and Howitt (1998). In these models, an
increased pace of technological change can raise the equilibrium unemployment rate
by
accelerating the Schumpeterian process of ‘creative destruction’: more workers must move
from plants embodying old technologies to new jobs. But even if we admit that barriers to labor
mobility across occupations and industrial sectors are particularly rigid in Europe, there is no
evidence that mismatch caused by technological change has been a significant determinant of
the rise of unemployment in European regions.2
OECD (1991), which emphasizes that on average less formal schooling seems to lead to more limited
training opportunities and possibilities to augment human capital.
2
In Bean and Pissarides’ (1993) endogenous growth model, higher wages raise both unemployment and
productivity growth by redistributing income in favour of workers, who are assumed to be the savers in
the economy. Even if we accept this assumption as a plausible simplification, it is difficult to reconcile the
recent experience of the OECD countries with the existence of the positive link between unemployment
and productivity growth implied by their model.
2
In contrast, the analytical set-up presented here models growth as a self-reinforcing
process by which firms’ capital investment and job creation enable an increasing number of
individuals to be trained on the job, thus acquiring skills whose availability attracts more
investment and boosts the growth potential of the economy.3 Moreover, an important feature of
this paper is that a mechanism causing persistence (training on the job) interacts with the
existence of strategic complementary between investment in physical capital by firms and
investment in active labor-market participation by workers.4 Hence, the paper is able to avoid
the implausible implication of those models that rely on strategic complementarities according
to which it would be sufficient to "help" agents to coordinate their expectations—thereby
enabling them to converge on the desired growth path--for the undesired disparities to be
rapidly eliminated. Indeed, the existence of disparities persisting for decades prompts one to
stress the fact that endowments of certain resources strategic for growth depend on history and
cannot be easily moved across countries or regions. This is particularly the case of those assets
embodied in human beings and in communities (informal and tacit knowledge, social capital,
moral and ethical values, etc.).
As the economy can gravitate around multiple equilibrium paths and agents’ “animal
spirits” are crucial in selecting the trajectory of the economy, the proposition that efficient
capital markets are always able to coordinate intertemporal activities appropriately is
3
Empirical data seem to confirm the contribution made to total factor productivity by the learning process
which takes place on the job when machinery and technologies are used (see, for example, De Long and
Summers, 1992). There is also empirical support for the hypothesis that a shortage of qualified workers
has negative effects on productivity growth (for microeconometric evidence concerning the United
Kingdom, see Haskel and Martin, 1996).
4
The presence of strategic complementarities between investment in physical capital, in R&D, or in job
creation, on the one hand, and investment to acquire the required human capital and to conduct a job
search on the other may generate multiple equilibria and lead to coordination failures (“traps”): in the
absence of some institutional device coordinating the individual expectations and actions, decentralized
decision makers can give rise to Pareto-suboptimal outcomes (see Burdett and Smith, 1995; Acemoglu,
1996; Redding, 1996; Snower, 1996). In Saint-Paul (1994a and 1994b), the workers may prefer the low-
3
challenged (see Leijonhufvud, 1998). Consequently, policies and institutions are advocated
because of their role in averting a coordination failure and in achieving a Pareto-superior
outcome (Hahn and Solow, 1995, chapt.7; Colander and van Ees, 1996). In accordance with
this approach, the role of economic policy is to provide a consistent and reliable framework
able to enforce the convention favorable to a high growth scenario, convincing the financial
markets of its sustainability (see Ciocca and Nardozzi 1995). However, the instruments and the
mechanisms through which economic policy can select the efficient equilibrium trajectory are
never well specified in this literature. The present paper aims at filling this void by introducing
money via a cash-in-advance constraint,5 thus modeling how the choice of a monetary rule may
contribute to select the balanced growth path (BGP) to which the economy converges.6
3 THE MODEL
In the infinite-horizon economy under consideration, there are firms, investors and
workers.
Population’s dynamics
Time is discrete, and individuals are finitely lived: they have a strictly positive and
constant probability , 0<<1, of dying in each period t. Thus, the probability of dying in a
certain period is assumed to be independent of the age of the individual; and it is also assumed
that the mortality rate of each large group of individuals does not fluctuate stochastically even
though each individual's lifespan is uncertain. This implies that at the end of t a constant
skill equilibrium to the high-skill equilibrium, because the higher demand for education reduces national
savings, pushing up real interest and unemployment rates.
5
For a review of the effects of inflation on economic growth in an economy with a cash-in-advance
constraint, see Dotsey and Sarte (2000).
6
To my knowledge, no other paper focuses on the role of monetary policy in selecting the long-run
equilibrium path of the economy. For a discussion of the few papers introducing money into an
4
fraction  of individuals belonging to each group and living in location i, i[0,n], n>0, dies,
while a new cohort is born at the beginning of the following period.
The firms
In the formal economy, there is a continuum--of measure n--of locations. In each
location i there is a large number (normalized to be one) of identical firms (the “good” firms).
Locations differ with respect to the specific shock affecting them in each period. Indeed, in
each period t the representative firm located in i produces the output Y it according to the
constant-returns-to-scale technology
Yit = x it K 1it- (S it + A it ) , 0<<1, 0<<1,
(1)
where xit is a random variable taking a value in t which is specific to the i location, K it is the
physical capital that the i firm borrowed at the end of the previous period to carry out
production in t, Sit are the experienced workers (the "skilled workers") employed by the i firm
in t, Ait are the newly hired workers (the "apprentices") of the i firm in t. Note that the
apprentices are less productive than the experienced workers (<1), and that aggregate output
n

(the numeraire of the system) is given by Yt= Yit di.
0
The random variable xit is assumed to be uniformly distributed on the interval [0,n].
Moreover, it is identically distributed across locations and periods, and independently
distributed across periods. In each t, xit takes a different value in each location, with xit varying
continuously across locations. This implies that the average value of xit across locations is not a
endogenous growth model, see Chang and Lai (2000) (in their model anticipated monetary growth is
superneutral along the BGP).
5
random variable and does not fluctuate in time, even though individual firms are uncertain
about their local xit (no aggregate uncertainty).7
The period net profits  itn (net of the cost of capital) of the i firm are given by:
 itn   itg -[(1+rt)-(1-)]Kit=Yit-vitSit-eitAit-[(1+rt)-(1-)]Kit, 0<<1,
(2)
where  itg are the firm's gross profits, vit is the real wage paid by the i firm to the skilled
workers employed in t, eit is the entry wage paid by the i firm to the apprentices hired in period
t,  is a capital depreciation parameter, and rt is the (real) interest rate, i.e. the market rate at
which firms borrowed capital at the end of the previous period. Interest payment and
reimbursement of principal are due at the end of t. The interest rate is unique because capital is
perfectly mobile across locations at the end of each period, while mobility is infinitely costly
within the period: once borrowed and installed at the end of t-1, a firm's capital stock must
remain fixed until the end of t.
The investors
There is a large number (normalized to be one) of identical investors who are the firms'
owners: for simplicity and without loss of generality, we assume that all investors are entitled
to receive an equal share of the firms' net profits. Being the owners of the firms’ productive
assets, investors must decide in each t what fraction of their gross returns on wealth to spend on
consumption rather than on buying productive assets to be lent at the end of the period to firms.
Moreover, investors must finance their purchases of current consumption and productive assets
7
In
other
words,
if
Yt  n 2 K1- (S + A) , t.
Kit = K n , Sit = S n and Ait = A n , i and t,
6
then
out of money balances carried out from the previous period.8 Hence, the problem of the
representative investor amounts to deciding a contingency plan for consumption Cin
t , purchase
of productive assets Kt+1 and holding of nominal balances H dt 1 in order to maximize the
lifetime expected sequence of discounted consumption C in
t :
 

,    (1 -  ), 0    1,
E 0   t C in
t


 t 0


subject to K t 1  C in
t 
n

K t  K it di ,
0
(3)
H dt 1
H  t
H dt
 (1  rt )K t   tn  t
and to K t 1  C in
, where
t 
Pt
Pt
Pt
n
π nt

  itn di , K i0 i and H0 given. In (3),  is a time-preference parameter, Kt
0
and  tn are, respectively, aggregate capital and aggregate (net) profits, P t is the money price of
the homogeneous good and  t denotes a lump-sum monetary transfer received at the end of t
(thus, Ht+1=Ht+t is the post-transfer nominal money held at the end of t). The motion of  t is
deterministic and follows a rule known to the agents. The expectation operator Et is conditional
on the information available in t as the values taken by xit across locations are known.
Expectations are rational, in the sense that they are consistent with the model and are generated
by optimally processing the available information. Finally, for simplicity and without loss of
generality, we rule out the existence of actuarially fair annuities paid to the living investors by a
financial institution collecting their wealth as they die: the wealth of someone who dies is
inherited by some newly born individual (accidental bequests).
8
In each period, indeed, investors must buy consumption goods and productive assets before receiving
their gross return on wealth.
7
The skilled workers
Skilled workers are those who have been trained on the job while working in a good
firm for at least one period. In contrast, apprentices are workers with no work experience in the
formal economy, but who have been hired by a good firm after having invested to acquire the
required basic knowledge. In their working lives, workers never lose the general skills that they
have acquired. Being general, the skills acquired on the job are perfectly transferable. Thus, the
skilled labor force evolves according to
n
n


0
0
Mt+1=(1-)(Mt+At), M t  M it di, A t  A it di, M0 given,
(4)
where Mit are the skilled workers located in i at the beginning of period t.
As in Blanchflower and Oswald (1994), workers choose location ex ante (at the end of
t-1), while firms decide on labor input once uncertainty is resolved. As for capital, labor is
perfectly mobile across locations at the end of each period t, while mobility is infinitely costly
within one period.9
At the beginning of period t, a skilled worker located in i expects to earn in t the
following income:
~
u sk
it  E t p it v it  (1 - p it )w , v it  w ,
(5)
~
where E t is an expectation operator conditional on the information available at the beginning
of period t (as the realization of xit is not yet known), w is the monetized value of the workers’
outside option depending only on the value of informal (“home”) activities, and p it is the
fraction of the skilled workforce located in i that is employed in period t:
9
This short-term immobility implies that in period t a worker located in i does not work at all in the
formal economy if s/he is not employed in that period by a good firm of i.
8
 Sit
if Sit  M it

p it   M it
1 otherwise.

(6)
At the end of each period, a skilled worker may move to another location at no cost.
Obviously, s/he locates where the expected present value of his/her lifetime income is the
highest. Therefore, the discounted sequence of incomes that an optimizing skilled worker still
alive at the end of t and located in i can expect to earn in the rest of his/her lifetime is given by
sk
sk
Uitsk =Et [u i*t +1 + (1 -  )U i*t +1 ], 0 
  1.
(7)
In (7),  is a time-preference parameter, and i* is a location where a skilled worker can have the
sk
best lifetime prospects (a "best location"):10 U sk
i*t  U it , i.
The skilled workers working in the formal economy devote their entire income to
consume Yt:
C sk
it  S it v it ,
(8)
where C sk
it is total consumption of the good Yt by the skilled workers of location i in period t.
The unskilled workers
At the beginning of each period, an unskilled worker located in i must decide whether
to incur the cost associated with participation in the formal labor market (i.e., with the
acquisition of the basic knowledge required by the good firms 11) or to remain out of the formal
labor market: an unskilled worker can be hired by a firm only if s/he becomes «trainable». An
investment in human capital in period t yields a strictly positive probability of being employed
10
More than one location can share this status of best location. Obviously, a worker located in i* will not
move.
11
Alternatively, one may interpret this cost as due to the search of an entry job in the formal segment of
the labor market.
9
by a good firm only in that period, since the basic knowledge acquired by a person is dissipated
if it is not used on the job. Moreover, possession of the basic knowledge required by the good
firms has no value in the informal economy. Hence, the investment made in order to participate
in the formal labor market will be lost, if within one period, the worker does not find an entry
job paid at least as his/her reservation wage: after having invested in human capital, a trainable
worker will accept any job offer paying an entry wage larger than his/her reservation wage
eitmin . An unskilled worker who decides not to invest in human capital has the same lifetime
prospects as a worker who does not find an entry job after having incurred the loss entailed by
this investment. Therefore, an optimizing unskilled worker living in i can expect at the
beginning of t to get the lifetime discounted sequence of incomes associated with the best
available alternative:


~
~
~
un
un
U itun  max E t [-c  q it (e it  (1 -  )U sk
it )  (1 - q it )(w  (1 -  )U it )], E t [w  (1 -  )U it ] ,
(9)
un
where e it  e itmin  w - (1 -  )(U sk
it - U it ).
In (9), -c captures the cost of acquiring the required basic knowledge (c is the
monetized value of this disutility) and qit is the fraction of the trainable workforce located in i
which is hired in period t:
 A it
if A it  L it

q it   L it
1 otherwise,

(10)
where Lit the trainable workforce located in i.
Note in (9) that U itun is the discounted sequence of incomes that an optimizing
unskilled worker still alive at the end of t and located in i can expect to get in the rest of his/her
lifetime:
10
~
U itun = E t ( U iun
* t +1 ) ,
(11)
un
where a best location i* for an unskilled worker must be such that U iun
*t  U it , i.
The entire income earned by the apprentices is devoted to consuming Yt:
C itun  A it e it ,
(12)
where C itun is total consumption of the good Yt by the apprentices of location i in period t.
Wage determination
In each location, the wages are determined by negotiations held at the beginning of
every period between a local union unconcerned about the interests of workers with no work
experience and the local employers' association. In this context it is immaterial whether the
union is only concerned about the workers employed in the previous period, or about both the
latter and those experienced workers who were laid off in previous periods. In fact, even if the
wage setters do not care about the interests of the skilled workers on layoff, the latter put
pressure on them, insofar as they are perfect substitutes and thereby reduce the job security of
the employed.
The union operating in i negotiates the real wage that all the firms of i must pay to the
experienced workers in employment, while each individual firm takes its decisions on the
demand for labor and capital in full autonomy. This negotiation also concerns the entry wage,
which is established as the fixed fraction µ of the skilled workers' wage that firms must pay to
the apprentices (eit=µvit). It is realistic to assume that the union does not allow the wage
differential between skilled workers and apprentices fully to offset their productivity
11
differential (<µ1), so that any incentive for the employers to replace experienced workers
with apprentices is suppressed.12
The bargaining process can be represented as if each union unilaterally sets the real
wage in the awareness of its impact on the local firms' decisions. On the other hand, each union
is aware that the effects of its wage policy on the economy as a whole is negligible. Similarly,
each single firm perceives that its decisions on labor and capital input cannot influence the
wage setting process because their impact is insignificant relatively to the size of the local labor
market. Since the real wage, once negotiated, remains fixed for a certain lapse of time (a
"period"), it is reasonable to assume that the wage is set by the union before the realization of
the random variable that is relevant for that period.
In this decentralized wage setting, the local union operating in i chooses vit in order to
maximize
~
sk
E t [u sk
it + (1 -  )U it ] .
(13)
In each period the union has full control only over the current wage, if we maintain that
current union membership cannot commit the workers who will manage the union in the future
to the pursuit of policies not optimal from their own temporal perspective. In other words, a
wage policy is feasible only if it is time consistent.
A summary of the timing of events
Summarizing, in each t we have a sequence of events in the following order: i) a new
cohort is born; ii) the unions set the wage rates, the unskilled workers decide whether to invest
in order to participate actively in the labor market; iii) idiosynchratic shocks occur; iv) firms
12
Burdett and Smith (1995) emphasize that the key assumption for the existence of a low skill trap is that
an employer's profit flow is greater when employing a skilled worker than when employing an unskilled
worker. Indeed, the fact that firms lay off unskilled workers before skilled workers is difficult to reconcile
with the contention that unskilled workers are more profitable.
12
atomistically determine their demand for skilled workers and apprentices, production takes
place and apprentices are trained on the job; investors decide on how much to invest in capital
and on the amount of money to hold for next-period purchases, workers are paid and output is
sold; v) firms reimburse the principal and pay the interest on the capital borrowed at the end of
the previous period, firms also pay the dividends to the shareholders, new capital is borrowed
by the firms for carrying out production in the next period; money transfers are distributed to
the investors; vi) a fraction  of each group of population dies at the end of the period, while
the surviving individuals decide where to locate in the next period and possibly move into
another location.
4 CHARACTERIZATION OF AN EQUILIBRIUM PATH
Equilibrium conditions in the markets for product and physical capital
Considering (2), (3), (8) and (12), one can easily derive the conditions for equilibrium
both in the product market and in the market for productive assets:
n
Yt + (1 -  )K t = K t +1 + C + C  C , C   C di, C
in
t
sk
t
un
t
sk
t
sk
it
0
n
un
t
  Citun di,
(14a)
0
K st +1 = K dt +1 ,
(14b)
H dt 1  H t 1 ,
(14c)
where K st +1 are the assets held by the investors and K dt +1 are those demanded by the firms at the
end of t.
Firms' optimality condition for capital accumulation
Firms of i determine their demand for capital at the end of t by satisfying the optimality
condition
13
 g ( x it+1 , M it+1 , k it+1 , sit+1 , v it+1 )
Et (
)  [(1  rt+1 )  (1 -  )] ,
K it+1
(15)
where the firms’ (gross) profit function g(.) is given in (A3).
This optimality condition defines kit+1, i.e., the physical capital/skilled labor ratio in
the firms of i, as an implicit function of the trainable labor/skilled labor ratio of i, the wage and
the interest rate:
f(kit+1, sit+1, vit+1)= rt+1+, f1 0, f20 and f30,
f (.) 
where

(16)
 (1  sit 1 ) (1 -  )n
(1 -  )v it2 1 (  / ) 2 [1 - (1  s it 1 ) 2  ] - 1
2n  (2 -  )k it2 -1
and
2kit 1
vit+1
is
determined by the union operating in i according to the time-invariant wage rule (see the
Appendix):
U
sk
t


u
sk
(
k
t

1
,
v
(k
t

1
,
w
t

1
),
w
vit+1=v(kit+1)=
t

1
)

(2 -  )nk 1it-1
2


U
sk
t

1
w
.
2
,
(17)
General equilibrium path
Using (17) and (A5), one obtains the equation governing the lifetime utility of a skilled
worker:
sk
sk
U sk
t  u ( k t 1 , v(k t 1 ))  U t 1 ,    (1 -  ) ,
(18)
where the subscripts denoting the location are dropped. Indeed, an equilibrium pair
s  , k  , which satisfies (16)-(18), and (A7) for an exogenously given trajectory of rt,

t 1

t 1
 
and which converges to a steady state s , k
(the bar denotes the steady-state value of a
variable), depends on structural parameters assumed to be equal across locations. Therefore,
different locations display equal physical capital/skilled labor and trainable labor/skilled labor
ratios even if they are endowed with different stocks of physical capital and skilled labor.
14
Hence, local unions are induced to set the same wage in all locations and (ex ante) workers can
be indifferent among locations, expecting the same income everywhere.13 Thus, it is reasonable
to assume that at the beginning of each t the skilled workers are evenly distributed across
locations:
Mit=Mt/n, t>0.
(19)
Using (A7), one can rewrite (18) as
(s t 1, k t 1, s t , k t ) 
c
q(v(k t ), k t , s t )
-
v(k t ) w sk
c
 - u (v(k t 1), k t 1)  v(k t 1)  0,


q(v(k t 1), k t 1, s t 1)
(20)
where q(.) is given by (A6).
To determine the time profile of the interest rate, one must consider the optimizing
problem of the representative investors. Hence, a general equilibrium path must be such that:
(1  rt 1 ) 
1   t 1

2
, t 
Pt 1  Pt 14
,
Pt
lim  t K t  0 ,
(21a)
(21b)
t 
where along an equilibrium trajectory:
1   t 1 
(1   t 1 )(C in
t 1  K t  2 )
(C in
t 2
C in
t  K t 1 
 K t 3 )
, t 
H t 1 - H t 15
,
Ht
Mt
G(v(k t ), k t , s t ) ,16
n
(22)
(23)
13
In other words, the equilibrium solution is symmetric across locations.
14
The cash-in-advance constraint is always binding if the following condition is satisfied:   1   t 1 .
15
 t is governed by a deterministic rule known to all agents.
16
From (3), (4), (17), (19), (A3) and (A6) one has:
2
2  -1 
(.) 
G(.
v(K t ) 
kt
2t (2 -  )

1    (1  s t )2 
   
-nv(kt)(1+st)+(1-)nkt.
15

- 

2


k1t- n 2
(1  s t ) 2
M t 1  (1   M t )M t ,  M t 
M t 1 - M t
  (k t , s t )  (1   )[1  s t q(v(k t ), k t , s t )]  1 . (24)
Mt
One can see in (21a) that the intertemporal trade-off faced by the investors involves
three periods (instead of two as in a non-monetary economy), since the decision of hoarding
money at time t affects the possibility of accumulating capital at t+1, on which depends
consumption at t+2. This implies that the real rate of return on capital increases with the cost of
holding money, that is with the inflation rate: the return on capital must compensate the
investors for the cost that they incur by holding the liquidity required to accumulate capital.
Hence, money is not superneutral since cash is required for investment (see Stockman, 1981).
Given (16), (21a), (22), (23) and (24), the condition for equilibrium in the capital
market can be written as
(kt+2,st+2,kt+1,st+1,t+1)=f(kt+1,st+1,v(kt+1))-
(1   t 1 )G (v(k t 1 ), k t 1 , s t 1 )
 2 [1   (k t 1 , s t 1 )]G (v(k t  2 ), k t  2 , s t  2 )
 1 -  =0.(25)
Therefore, a general equilibrium path of kt and st must satisfy (20), (21b) and (25),
with money growing according to a deterministic rule. Along this path, the dynamics of k t and
st.does not depend on the size of the skilled workforce Mt or on the stock of physical capital
Kt. By setting t+1=, kt+2=kt+1=k and st+2=st+1=s in (25), one can obtain the condition
that must hold along a balanced growth path (BGP) for equilibrium in the capital market:
f(k,s,v(k))+1-=
(1  )
 [1   (k, s)]
2
.
It is apparent in (26) that the rate of money growth affects long-run real growth.
16
(26)
5 THE NON-MONETARY ECONOMY (FRIEDMAN RULE)
The path of the economy is the same as it is in the absence of the cash-in-advance
constraint if the inflation rate is such that
1   t 1   , t .
(27)
A monetary policy consistent with (27) follows the Friedman rule:17
1   t 1 
 [1   (k t 1 , s t 1 )]G(v(k t 2 ), k t 2 , s t 2 )
G(v(k t 1 ), k t 1 , s t 1 )
.
(28)
Applying rule (28), (25) becomes
(kt+1,st+1)=f(kt+1,st+1,v(kt+1))--1+1-=0.
(29)
Balanced growth path
The system satisfying (20), (21b) and (29) characterizes an equilibrium path of the
economy, i.e. an equilibrium path of the economy in the absence of the cash-in-advance
constraint. By setting kt=kt+1=k and st=st+1=s in (20) and (29), one can obtain the steady-state
values of the physical capital/skilled labor ratio and of the trainable labor/skilled labor ratio.
Indeed, a steady-state pair ( k, s) must satisfy the system:
 ( k, s)  0 ,
(30a)
 (k , s)  0 ,
(30b)
where (30a) must hold in order to ensure long-term equilibrium in the (trainable) labor market,
and (30b) must hold in order to ensure long-term equilibrium in the (physical) capital market.
If kt and st reach their steady-state values, output (Yt), employment in the formal
economy (St+At) and skilled labor (Mt) follow their BGP. Along this path, Yt, St+At and Mt
grow at their steady-state rate, which is determined only by the parameters of the model:
17
As monetary policy follows the Friedman rule, the cash-in-advance constraint is just binding.
17
 Y   M  SA  (1 -  )[1  sq(v(k), k, s)] - 1 ,  Yt 
where:
Yt =
Yt 1 - Yt
S  A t 1 - S t - A t
,  St  A t  t 1
,(31)
Yt
St  A t
 Y
 Y
 0,
 0, St  At = M t [ p(v( k t ), k t )  st q(v(k t ), k t , st )] ,18
s
k
Mt
n
2
2 -

] - [v( k t , w)]2 n 2 k1t- (1  s t )
 [ v( k t , w)] [1 - (1  s t )


2

2 (2 -  ) 2 k1t-


 19
.


Multiple BGP
Equation (30a) implicitly defines s as an increasing function of k (see fig. 1):
s=a(k), a’ >0.
(32a)
Given the number of skilled workers existing in the economy, more physical capital is
necessary to induce an increasing number of unskilled workers to invest in (formal) labor
market participation. Other things being equal, a larger number of unskilled workers investing
in basic knowledge depresses any single trainable worker’s probability of being hired. Thus,
this larger s needs to be accommodated by a higher capital stock, which entails both a higher
probability of being hired and better lifetime prospects for any single worker if hired.
Similarly, equation (32b) implicitly defines s as an increasing function of k:
s=b(k), b’ >0.
18
Along
(32b)
a BGP, the Friedman rule dictates 1     (1   Y ) and it is never the case that
1     (1   Y ) . An intuitive explanation for the non-existence of a BGP with 1     (1   Y ) is
obtained by using the argument in Abel (1985), p.58. Suppose this BGP exist, and consider consuming
one unit less at time t and holding Pt more units of money. This money can be used to buy (1+t+1)-1
units of consumption at time t+1. Hence, this will change the net present value of consumption by
 1   (1   t 1 ) 1 . Since along a BGP 1+t+1= (1+)(1+Y)-1, the change in the net present value of
1
consumption is [ (1  ) (1   Y )  1] . If 1     (1   Y ) then this change is strictly positive
and the growth path could not have been optimal.
19
Consistently with the model, one can rule out the possibility that the steady-state rate of growth of the
skilled labor force is higher than the steady-state rate of growth of the workers’ population by
endogeneizing the birth rate of the workers’ population, i.e. by assuming that in the long run it responds to
economic conditionst, and/or (if the “economy” does not coincide with the world economy) by allowing
immigration of unskilled workers.
18
Indeed, a rise in s has a positive effect on the firms’ expected profits: at any level of the capital
stock, the increment in expected profits due to a marginally higher k increases with s. This
creates the potential for multiple BGP, since the existence of multiple BGP requires that the
expected marginal profitability of investment in physical capital does not fall with a larger k
because of the increased number of workers who are induced to invest in human capital by the
better job opportunities that the larger k brings about. Therefore, one can have two steady-state
pairs ( k h , s h ) and ( k l , s l ) , such that k h  k l and s h  s l (see figure 1).20 This can be seen in
the (k,r) plane by considering the condition that must be satisfied in the capital market for longterm equilibrium. Indeed, one can use (29), (30) and (32a) to see that a steady-state value of k
must satisfy:
f(k,a(k),v(k))-=-1-1,
(33)
where for optimality on the demand side of the capital market one must have
f(k,a(k),v(k))-=r, and for optimality on the supply side of the capital market one must have
-1-1=r (see figure 2). Equation (33) states the condition that must be satisfied for long-run
equilibrium in the capital market only in terms of k by considering only those values of s that
are consistent for given values of k with long-run equilibrium of the trainable-labor market. In
figure 3 we have a steady state associated with lower k and a steady state associated with higher
k that are both consistent with the same steady-state level of the interest rate: the dynamic
externalities created by an increase in capital investment offset the tendency of the marginal
productivity of capital to fall at higher k.
20
As numerical example, let c=.0123832, w=.1,   .25, n=1, =.2681247, =.03, =.75, =.4906097,
=.8247422 and =.2290823. Given these parameters’ values, one obtains: k  .1, s  .2 ,
h
 Yh =.0083379, k l  .0986489, s l  .1 and  Yl =-.0106358.
19
h
Given (31), the rate of growth of employment and output is permanently higher at
( k h , s h ) than it is along the BGP associated with ( k l , s l ) :  Yh  Sh A   Yl  Sl  A .
Moreover, given the cumulative nature of the growth process, the levels of output and (formal)
employment of an economy moving along the high growth BGP diverge over time from the
employment and output levels of a structurally similar economy following the BGP
characterized by ( k l , s l ) . Finally, it should be emphasized that a high-growth equilibrium path
is always Pareto superior than a low growth path. Indeed, in a high growth regime i) firms’
owners (investors) benefit from the higher propensity of the unskilled workers to invest in
human capital, ii) the skilled workers are able to exploit a more favorable trade-off between
real wage and the probability of being employed by a good firm, thus increasing their expected
lifetime sequence of incomes, iii) the apprentices’ wages are higher, iv) a larger number of
workers have the opportunity to be trained on the job and increase their human wealth, and v)
the lifetime prospects of an unskilled worker remain unchanged.
Global indeterminacy
The system governing kt and st reduces to a single first-order difference equation, since
(20) can be rewritten as a first-order difference equation in kt only by using the fact that (29)
implicitly defines st as a function of kt. Given that the motion of kt and st is completely
governed by forward-looking expectations, the initial condition on k0 does not play any role in
determining the dynamics of kt and st for t1, and in t=1 the system can jump to ( k h , s h ) or to
20
one among the continuum of paths converging to ( k l , s l ) in a neighborhood of it 21 (see figure
3). This implies that -- depending on the “animal spirits” of capital-market participants -structurally similar economies starting with equal initial endowments of K0 and M0 may grow
at different steady-state rates of growth (global indeterminacy).
FIGURE 1
FIGURE 2
FIGURE 3
6 INFLATION TARGETING VERSUS FIXED RATE OF MONEY GROWTH
In this economy, the Friedman rule is an example of inflation targeting since it entails a
monetary rule keeping t+1=-1 t. More in general, inflation targeting amounts to the
adoption of the following monetary rule:
1   t 1 
(1   )[1   (k t 1 , s t 1 )]G(v(k t  2 ), k t  2 , s t  2 )
,
G(v(k t 1 ), k t 1 , s t 1 )
(34)
where    - 1 . As the rule (34) is adopted, an equilibrium path of kt and st is governed by
(20) and by
21
Linearizing the first-order difference equation (20) around k yields the characteristic equation



 K t  s t k 

s   0,
 
where all derivatives are evaluated at ( k, s ) . Since at ( k h , s h ) one has

k 
 K t 1  s t 1

s 

 k  k t
  t 1
 s  s
t
 t 1


 k t
    k , while at ( k l , s l ) one has   k t 1
 s  s

s
t 1
t



    k , it is straightforward

s

that >1 if (20) is linearized around ( k h , s h ) and <1 if it is linearized around ( k l , s l ) . Indeed, in the
numerical example of the previous note, the characteristic root obtained by linearizing (20) around
k h  .1 is =1.0210701, while the characteristic root obtained by linearizing (20) around
k l =.0986489 is =.9812596.
21
(k t 1 , s t 1 ,  ) =f(kt+1,st+1,v(kt+1))+1--
(1   )
2
 0.
(35)
It is apparent that    - 1 implies a higher cost of capital than under the Friedman
rule because of the increase in the cost of holding money. Moreover, the existence of multiple
BGP tends to be preserved when the growth rate of money supply is adjusted so as to keep
 t 1   t . This is because under inflation targeting monetary policy sterilizes any effect of
changes in the economy’s growth rate on the equilibrium cost of capital. In this case, there are
no movements in rt which amplify possible differentials in real growth rates by lowering the
cost of capital when growth accelerates or by raising it when the opposite happens. Hence, both
a high-growth path and a low-growth path are consistent with the long-run equilibrium of the
economy.
In contrast, under a fixed money growth rule,22 rt is a negative function of the
economy’s growth rate, and a rise in  Yt is amplified by the fall in the cost of capital brought
about by this rise. Hence, only one BGP tends to be consistent with the structural parameters of
the economy and with the money growth rule selected by the authorities. This implies that
when the non-monetary economy may converge to two different BGP no matters what its initial
state is, this global indeterminacy is not resolved by monetary policy under inflation targeting,
while the monetary authority can select one of these long-term equilibrium paths and lead the
economy to converge toward it by choosing the appropriate fixed rate of money growth. In this
case, the choice of a larger  can lead the economy to grow at a higher steady-state rate,
In this case, an equilibrium path is governed by (20) and (25), where  t 1   t . Again, the motion
of kt and st is completely governed by forward-looking expectations and the initial condition on k0 does
not play any role in determining the dynamics of kt and st for t1.
22
22
without bringing about a higher steady-state rate of inflation.23 In other words, the choice of
more restrictive monetary policy (smaller  ) would have led the economy toward the lowgrowth BGP, without any long-term benefit in terms of lower inflation.
7 CONCLUSION
The general equilibrium dynamic model presented in this paper combines employment
theory and growth theory in order to improve understanding of the mechanisms through which
monetary policy may be relevant in shaping the long-run behavior of the economy when its
future evolution depends on the “animal spirits” of capital-market participants. With regard to
this point, the paper is motivated by the debate on the recent experience of Continental Europe.
As more than one equilibrium path is possible, it is not necessarily the case that a longterm trade-off between employment and real wage emerges: along a long-run equilibrium path
characterized by a higher rate of growth, both the employment level and the average real wage
tend to be higher than when the economy follows a low-growth balanced growth path. Also
23
For instance, suppose that c=.0123832, w=.1,   .25, n=1, =.2681247, =.03, =.75, =.4906097,
=.8247422 and =.2290823. Given these parameters’ values, in the non-monetary economy there exist
h
l
h
h
l
two BGP characterized by k  .1, s  .2 ,  Y =.0083379 and by k  .0986489, s  .1 and
 Yl =-.0106358. The existence of both these BGP is preserved under inflation targeting
h
h
( 1   t  1    .8 t ). In contrast, under the fixed rule  t 1     (1   Y )  .8066703 t ,
there exists only one BGP (the high-growth BGP of the non-monetary economy). Linearizing the system
consisting of (20) and (25) about its unique steady state ( ,1. 
h
k
s h  .2 ), one can obtain the
following characteristic roots: 1=3.6264 and 2=.1838: the linearized system characterizes a continuum
l
l
of paths converging to its BGP. Similarly, under  t 1     (1   Y )  .7914913 t , there exists
only one BGP (the low-growth BGP of the non-monetary economy). Again, linearizing the system about
its unique steady state ( k  .0986489, s  .1 ), one can obtain the following characteristic roots:
l
l
1=3.6904 and 2=.1594: the linearized system characterizes a continuum of paths converging to its
BGP. Note that along both BGP the inflation rate is the same ( 1    .8 ).
23
firms’ profits and investors’ consumption are higher along a high-growth balanced growth
path. Hence, it is highly desirable to lead the economy toward a long-run equilibrium path
along which the economy expands at a higher rate. It is shown in this paper that in the presence
of a cash-in-advance constraint and of global indeterminacy, i.e. in a situation in which more
than one balanced growth path is consistent with economic fundamentals and the current state
of the economy, monetary policy has the power to select the Pareto-superior balanced growth
path. In this case, a monetarist rule dictating a higher fixed rate of money growth leads the
economy to converge to the high-growth balanced growth path, while a rule dictating a lower
fixed rate of money growth leads the economy to converge toward the low-growth balanced
growth path. Moreover, the steady-state rate of inflation is the same under the two alternative
rules: the higher rate of money creation allows to meet the faster increase in the demand for
nominal balances brought about by the more rapid expansion of economic activity.
APPENDIX
1. Derivation of the firms’ (gross) profit function g(.)
Given the perfectly transferable nature of the general skills acquired by an apprentice, each employer is
aware that there is no guarantee that a newly hired worker will remain with his/her firm in the future. This
is why an employer does not consider the future returns accruing from the on-the-job training of an
apprentice: since the forthcoming benefit of adding a skilled worker to the stock of human capital
available to the economy as a whole cannot be appropriated privately, the employer can ignore it as an
insignificant externality. Therefore, the selection of the optimal labor policies by a firm amounts in each t
to solving the static decision problem of maximizing (2) with respect to S it and Ait. Given its optimal
labor policies, a firm is able to determine at the end of t-1 the amount of Kit to borrow and install. It may
be the case that the aggregate demand for either trained labor or apprentices by firms in location i is
rationed. In the aggregate, it is always the case that:
Sit  Mit,
Ait  Lit.
(A1a)
(A1b)
When labor demand happens to be rationed, it is reasonable to assume that the scarce supply of labor is
evenly distributed among firms of the same location. Note that the union wages are not determined at the
24
firm level and that employers cannot compete for labor in short supply by raising the relevant wages in
order to keep and poach workers, even if skills are perfectly transferable among firms (see Soskice, 1990).
Therefore, with one as the normalized number of firms of location i, we can take (A1) to be the constraints
faced by each individual firm as the union wages induce all the available skilled and trainable workers to
accept a job offer. Hence, the firm's choice of the labor inputs amounts to solving the static decision
problem of maximizing (2) subject to (1) and (A1), from which we derive optimal labor policies:
1

(1 )



x
v it kit 1
K
M it k it  it 
, k it  it
if x it 
Sit  S ( x it , M it , k it , v it )  

M it
 v it 

(A2a)
M it otherwise,

v k 

x
 0 if



  x  

) = M k 

 v 

 Mitsit otherwise, s

1
it
it
it
1
Ait  A(x it , M it , k it , sit , v it
it
it
it
it
(1 )
v it
v it (1  sit )1-
M it
 x it 
 if k1it
k 1it
L
it .
(A2b)
it 
M
it
The firms' net profits are an increasing function of xit. In fact, using (1), (2) and (A2), one has:
 itn   g ( x it , M it , k it , s it , v it )  (1 + rt )  (1 -  )K it ,
(A3)
where


 1  (1 )

 x it 
v it k it 1


1


M
k
if
x



it it 
it

 v it 




 1
 1
 x k1 M  M v if v it k it  x  v it k it

it
it it
it
 g (.)   it it




1 1 



 x it 
1  s it 1 v it
M it v it    
v it
 1   M it k it 

if

x



k1it it
k1it

 v it 




1
 x it k it M it 1  s it   v it M it  v its it M it otherwise.
2. Derivation of the equilibrium condition for the trainable labor market and of the wage rule
Having the optimal demand for skilled labor in (A2a), one can compute the probability of a skilled worker
located in i (before the realization of xit ) being employed in period t:
p(v it , k it ) = 1 
v it kit -1
.
n (2 -  )
(A4)
25
By using (A4), one can rewrite (5) as:

v k  1 
v k  1
u sk ( v it , k it )  1  it it  v it  it it
w.
n (2 -  )
 n (2 -  ) 
(A5)
Similarly, one can use (A2b) to compute the probability that a trainable worker located in i (before the
realization of xit) will be hired in period t:
q(v it , k it , sit ) = 1 


v it kit -1 1 + sit 2  1
, q1 < 0, q2 > 0 and q3 < 0 .
n (2 -  ) 2sit
(A6)
Note that q(.) diminishes as a larger number of inexperienced workers become trainable, remaining
constant both the size of the skilled workforce and the stock of capital located in i. In equilibrium, the
number of inexperienced workers who become trainable in location i must be such that an inexperienced
worker is indifferent between investing in basic knowledge or staying in the informal economy:
un
c  q( v it , k it , s it )[v it - w  (1 -  )(U sk
it - U it )] ,
(A7a)
where along an equilibrium path
U itun = w  U itun1 ,    (1 -  ).
(A7b)
The expected income of a skilled worker depends on the real wage and on the physical capital/skilled
labor ratio, which is a predetermined variable when an union sets the wage. Given the forward looking
behavior of firms and unskilled workers captured by (16) and (A7), the current wage policy of an union
could affect the union’s future policy and the future well-being of its members only if it had a significant
impact on the investors’ behavior. However, this is not the case because of the continuum of unions
operating in the economy. Thus, each single union solves the sequence of static problems
max u sk (v it , k it ) , obtaining the time-invariant wage rule (17).
v it
REFERENCES
Abel, A.B., 1985. Dynamic Behavior of Capital Accumulation in a Cash-in-advance Model,
Journal of Monetary Economics 16, 55-71.
26
Acemoglu, D., 1996. A Microfoundation for Social Increasing Returns in Human Capital,
Quaterly Journal of Economics 111, 779-804.
Aghion, P., Howitt, P., 1998. Endogenous Growth Theory, MIT Press, Cambridge Mass.
Bean, C.R., Pissarides, C.A., 1993. Unemployment, Consumption and Growth, European
Economic Review, 37, 837-854.
Bentolila, S., Jimeno, J., 1998. Regional unemployment persistence (Spain 1976-94), Labour
Economics, 5, No.1, 25-52.
Blanchard, O.J., 1997. The Medium Run, Brookings Papers on Economic Activity, 2, 89-158.
Blanchflower, D., Oswald, A., 1994. The Wage Curve, MIT Press, Cambridge Mass.
Burdett, K., Smith, E., 1995. The Low-Skill Trap. Discussion Paper No. 95-40, Department of
Economics, University of British Columbia.
Chang, W., Lai, C., 2000. Anticipated inflation with Endogenous Growth, Economica 67, 399417.
Ciocca, P., Nardozzi, G., 1995. The High Price of Money: an Interpretation of World Interest
Rates, Oxford University Press, London.
Colander, D., van Ees, H., 1996. Post Walrasian macroeconomic policy. In: Colander, D. (Ed.),
Beyond Microfoundations: Post Walrasian Macroeconomics, Cambridge University Press,
Cambridge.
De Long, J.B., Summers, L.H., 1992. Equipment Investment and Economic Growth: How
Strong is the Nexus?, Brookings Papers on Economic Activity, 2, 157-211.
Dotsey, M., Sarte, P.D., 2000. Inflation uncertainty and growth in a cash-in-advance economy,
Journal of Monetary Economics 45, 631-655.
Hahn, F., Solow, R., 1995. A Critical Essay on Modern Macroeconomic Theory, MIT Press,
Cambridge Mass.
Haskel, J., Martin, C., 1996. Skill Shortages, Productivity Growth and Wage Inflation. In:
Booth, A.L., Snower, D.J. (Eds.), Acquiring Skills, Cambridge University Press, Cambridge.
IMF, 1999. World Economic Outlook, May, International Monetary Fund, Washington.
Leijonhufvud, A., 1998. Three Items for the Macroeconomic Agenda, Kyklos, 51,197-218.
OECD, 1991. Employment Outlook 1991, OECD Publications, Paris.
27
Redding, S., 1996. The Low-Skill, Low-Quality Trap: Strategic Complementarities between
Human Capital and R&D, Economic Journal, 106, 458-470.
Saint-Paul, G., 1994a. Unemployment and Increasing Private Returns to Human Capital. CEPR
Discussion Paper.
Saint-Paul, G., 1994b. Unemployment, Wage Rigidity and the Returns to Education, European
Economic Review, 38, 891-898.
Snower, D.J., 1996. The Low-skill, Bad-job Trap. In: Booth, A.L., Snower, D.J. (Eds.),
Acquiring Skills, Cambridge University Press, Cambridge.
Soskice, D., 1990. Wage Determination: The Changing Role of Institutions in Advanced
Industrialized Countries, Oxford Review of Economic Policy, 6, No.4, 36-61.
Stockman, A.C., 1981. Anticipated Inflation and the Capital Stock in a Cash-in-advance
Economy, Journal of Monetary Economics 8, 387-393.
FIGURE 1
28
s
b(k)
a(k)
sh
sl
k
kl
kh
FIGURE 5
FIGURE 2
r
 -1 - 1
r
f (.) - 
k
k
kl
h
FIGURE 3
kt+1
kh
kl
45°
kt
k
l
k
h
29