Models for dynamic macroeconomics phần 6 pdf

LABOR MARKET 127
general not relevant for the probability of x
t+i
= x
g
from the viewpoint of t +1.From
the viewpoint of period t, the probabilities of the same event can be written as
P
t,t+i
=(P
t+1,t+i
|x
t+1
= x
b
) · P(x
t+1
= x
b
|I
t
)
+(P
t+1,t+i
|x
t+1
= x
g
) · P(x
t+1
= x

g
|I
t
) (3.A7)
(where P(x
t+1
= x
g
|I
t
)=1− p if x
t
= x
g
,andsoforth).Thisallowsustoverifythe
validity of the law of iterative ex pectations in this context. For i ≥ 2, we write
E
t+1
[x
t+i
]=x
b
+(x
g
− x
b
)P
t+1,t+i
. (3.A8)
At date t + 1, the probability on the right-hand side of (3.A8) is given, while at time t

it is not possible to evaluate this probability with certainty: it could be (P
t+1,t+i
|x
t+1
=
x
b
), or (P
t+1,t+i
|x
t+1
= x
g
), depending on the realization of x
t+1
. Given the uncertainty
associated with this realization, from the point of view of time t the conditional
expectation E
t+1
[x
t+1+i
]isitselfarandom variable, and we can therefore calculate its
expected value:
E
t
[E
t+1
[x
t+i
]] = P(x

t+1
= x
b
|I
t
)E
t+1
[x
t+i
|x
t+1
= x
b
]
+P(x
t+1
= x
g
|I
t
)E
t+1
[x
t+i
|x
t+1
= x
g
].
Inserting (3.A8), using (3.A7), and recalling (3.A6), it follows that

E
t
[x
t+i
]=x
b
+(x
g
− x
b
)P
t,t+1
= E
t
[E
t+1
[x
t+i
]].
EXERCISES
Exercise 30 Consider the production function
F (k, l; ·)=(k + l)· −
‚
2
l
2
−
„
2
k

2
.
(a) Suppose a firm with that production function has given capital k =1, can hire l
costlessly, pays g iven wage w =1,andmustpayF =1for each unit of l fired. If ·
t
takes the values 4 or 2 with equal probability p =0.5, and future cash flows are
discounted at rate r =1, what is the optimal dynamic employment policy?
(b) Suppose capital depreciates at rate ‰ =1and can be costlessly adjusted to ensure
that its marg inal product is equal to the cost of funds r + ‰. Does capital adjust-
ment change the optimal employment pattern? What are the optimal levels of
capital whe n ·
t
=4and when ·
t
=2?
Exercise 31 Consider a labor market in which firms have a linear demand curve for labor
subject to parallel oscillations, Ï(N, Z)=Z − ‚N. As in the main text, Z can take two
values, Z
b
and Z
g
> Z
b
, and oscillates between these values with transition probability
p. Also, the wage oscillates between two values, w
b
and w
g
>w
b

, and the oscillations of
the wage are synchronized with those of Z.
128 LABOR MARKET
(a) Calculate the levels of employment N
b
and N
g
that maximize the expected dis-
counted value of the revenues of the firm if the discount rate is equal to r and if the
unit hiring and firing costs are given by H and F respectively.
(b) Compute the mobility cost k at which the optimal mobility decisions are consistent
with a wage differential w = w
g
− w
b
when workers discount their future
expected income at rate r.
(c) Assume that the labor market is populated by 1,000 workers and 100 firms of
which exactly half are in a good state in each period. What levels of the wage w
b
are compatible with full employment (with w
g
= w
b
+ w as above), under the
hypothesis that labor mobility is instantaneous?
Exercise 32 Suppose that the marginal productivity of labor is given by Ï(Z, N)=Z −
‚N, and that the indicator Z
t
can assume three rather than two values {Z

b
, Z
M
, Z
g
},
with Z
b
< Z
M
< Z
g
, where the realizations of Z
t
are independent, while the wage rate
is constant and equal to
¯
w in each period. Finally, hiring and firing costs are give n by H
and F respectively. What form does the recursive relationship
Î(Z
t
, N
t
)=Ï(Z
t
, N
t
) −
¯
w + E

t
[Î(Z
t+1
, N
t+1
)]
take if the parameters are such that only fluctuations from Z
b
to Z
g
or v ice versa induce
the firm to adjust its labor force, while the employment level is unaffected for fluctuations
from and to the average level of labor demand (from Z
b
to Z
M
or vice versa, or from Z
M
to Z
g
or vice versa)? Which are the two employment levels chosen by the firm?

FURTHER READING
Theoretical implications of employment protection legislation and firing costs
are potentially much wider than those illustrated in this chapter. For example,
Bertola (1994) discusses the implications of increased rigidity (and less efficiency) in
models of growth like the ones that will be discussed in the next chapter, using a two-
state Markov process similar to the one introduced in this chapter but specified in a
continuous-time setting where state transitions are described as Poisson events of the
type to be introduced in Chapter 5.

Economic theory can also explain why employment protection legislation is
imposed despite its apparently detrimental effects. Using models similar to those
discussed here, Saint-Paul (2000) considers how politico-economic interactions can
rationalize labor market regulation and resistance to reforms, and Bertola (2004)
shows that, if workers are risk-averse, then firing costs may have beneficial effects:
redundancy payments not only can remedy a lack of insurance but also can foster
efficiency if they allow forward-looking mobility decisions to be taken on a more
appropriate basis.
Of course, job security provisions are only one of the many institutional features
that help explain why European labor markets generate lower employment than Amer-
ican ones. Union behavior and taxation play important roles in determining high-
wage, low-employment outcomes. And macroeconomic shocks interact in interesting
LABOR MARKET 129
ways with wage and employment rigidities in determining the dynamics of employ-
ment and unemployment across the Atlantic and within Europe. For economic and
empirical analyses of the European unemployment problem from an international
comparative perspective, see Bean (1994), Alogoskoufis et al. (1995), Nickell (1997),
Nickell and Layard (1999), Blanchard and Wolfers (2000), and Bertola, Blau, and Kahn
(2002), which all include extensive references.

REFERENCES
Alogoskoufis, G., C. Bean, G. Bertola, D. Cohen, J. Dolado, G. Saint-Paul (1995) Unemployment:
Choices for Europe, London: CEPR.
Bean, C. (1994) “European Unemployment: A Survey,” Journal of Economic Literature, 32,
573–619.
Bentolila, S., and G. Bertola (1990) “Firing Costs and Labor Demand: How Bad is Eurosclerosis?”
Revie w of Economic Studies, 57, 381–402.
Bertola, G. (1990) “Job Security, Employment and Wages,” European Economic Review, 34,
851–886.
(1992) “Labor Turnover Costs and Average Labor Demand,” Journal of Labor Economics,

10, 389–411.
(1994) “Flexibility, Investment, and Growth,” Journal of Monetary Economics, 34, 215–238.
(1999) “Microeconomic Perspectives on Aggregate Labor Markets,” in O. Ashenfelter
and D. Card (eds.), Handbook of Labor Economics , vol. 3B, 2985–3028, Amsterdam: North-
Holland.
Bertola, G. (2003) “A Pure Theory of Job Security and Labor Income Risk,” Revie w of Economic
Studies, 71(1): 43–61.
F. D. Blau, and L. M. Kahn (2002) “Comparative Analysis of Labor Market Outcomes:
Lessons for the US from International Long-Run Evidence,” in A. Krueger and R. Solow (eds.),
The Roaring Nineties: Can Full Employment Be Sustained? New York: Russell Sage, pp. 159–218.
and A. Ichino (1995) “Wage Inequality and Unemployment: US vs Europe,” in B. Bernanke
andJ.Rotemberg(eds.),NBER Macroeconomics Annual 1995, 13–54, Cambridge, Mass.: MIT
Press.
and R. Rogerson (1997) “Institutions and Labor Reallocation,” European Economic Review,
41, 1147–1171.
Blanchard, O. J., and J. Wolfers (2000) “The Role of Shocks and Institutions in the Rise of
European Unemployment: The Aggregate Evidence,” Economic Journal, 110: C1–C33.
Nickell, S. (1997) “Unemployment and Labor Market Rigidities: Europe versus North America,”
Journal of Economic Perspectives, 11(3): 55–74.
and R. Layard (1999) “Labor Market Institutions and Economic Performance,” in
O. Ashenfelter and D. Card (eds.), Handbook of Labor Economics, vol. 3C, 3029–3084, Amster-
dam: North-Holland.
Saint-Paul, G. (2000) The Political Economy of Labour Market Institutions,Oxford:Oxford
University Press.
4
Growth in Dynamic
General Equilibrium
The previous chapters analyzed the optimal dynamic behavior of single
consumers, firms, and workers. The interactions between the decisions of
these agents were studied using a simple partial equilibrium model (for the

labor market). In this chapter, we consider general equilibrium in a dynamic
environment.
Specifically, we discuss how savings and investment decisions by individual
agents, mediated by more or less perfect markets as well as by institutions
and collective policies, determine the aggregate growth rate of an economy
from a long-run perspective. As in the previous chapters, we cannot review
all aspects of a very extensive theoretical and empirical literature. Rather, we
aim at familiarizing readers with technical approaches and economic insights
about the interplay of technology, preferences, market structure, and insti-
tutional features in determining dynamic equilibrium outcomes. We review
the relevant aspects in the context of long-run growth models, and a brief
concluding section discusses how the mechanisms we focus on are relevant
in the context of recent theoretical and empirical contributions in the field of
economic growth.
Section 4.1 introduces the basic structure of the model, and Section 4.2
applies the techniques of dynamic optimization to this base model. The next
two sections discuss how decentralized decisions may result in an optimal
growth path, and how one may assess the relevance of exogenous technological
progress in this case. Finally, in Section 4.5 we consider recent models of
endogenous growth. In these models the growth rate is determined endoge-
nously and need not coincide with the optimal growth rate.
The problem at hand is more interesting, but also more complex, than those
we have considered so far. To facilitate analysis we will therefore emphasize the
economic intuition that underlies the formal mathematical expressions, and
aim to keep the structure of the model as simple as possible. In what follows
we consider a closed economy. The national accounting relationship
Y (t)=C (t)+I (t) (4.1)
between the flows of production (Y ), consumption (C), and investment
therefore holds at the aggregate level. Furthermore, for simplicity, we do not
distinguish between flows that originate in the private and the public sectors.

EQUILIBRIUM GROWTH 131
The distinction between consumption and investment is based on the
concept of capital. Broadly speaking, this concept encompasses all durable
factors of production that can be reproduced. The supply of capital grows
in proportion with investments. At the same time, however, existing capital
stock is subject to depreciation, which tends to lower the supply of capital. As
in Chapter 2, we formalize the problem in continuous time. We can therefore
define the stock of capital, K (t)attimet, without having to specify whether
it is measured at the beginning or the end of a period. In addition, we assume
that capital depreciates at a constant rate ‰. The evolution of the supply of
capital is therefore given by
lim
t→0
K (t + t) − K (t)
t
≡
dK(t)
dt
≡
˙
K (t)=I (t) − ‰K (t).
The demand for capital stems from its role as an input in the productive
process, which we represent by an aggregate production function,
Y (t)=F (K (t), ).
This expression relates the flow of aggregate output between t and t + t
to the stocks of production factors that are available during this period. In
principle, these stocks can be measured for any infinitesimally small time
period t. However, a formal representation of the aggregate production
process in a single equation is normally not feasible. In reality, the capital
stock consists of many different durable goods, both public and private. At

the end of this chapter we will briefly discuss some simple models that make
this disaggregate structure explicit, but for the moment we shall assume that
investment and consumption can be expressed in terms of a single good as in
(4.1). Furthermore, for simplicity we assume that “capital” is combined with
only one non-accumulated factor of production, denoted L (t).
In what follows, we will characterize the long-run behavior of the economy.
More precisely, we will consider the time period in which per capita income
grows at a non-decreasing rate and in which the ratio between aggregate
capital K and the flow of output Y tends to stabilize. The amount of capital
per worker therefore tends to increase steadily. The case in which the growth
rate of output and capital exceeds the growth rate of the population represents
an extremely important phenomenon: the steady increase in living standards.
Butinthischapterourinterestinthistypeofgrowthpatternstemsmorefrom
its simplicity than from reality. Even though simple models cannot capture all
features of world history, analyzing the economic mechanisms of a growing
economy may help us understand the role of capital accumulation in the real
world and, more generally, characterize the economic structure of growth
processes.
132 EQUILIBRIUM GROWTH
4.1. Production, Savings, and Growth
The dynamic models that we consider here aim to explain, in the simplest pos-
sible way, on the one hand the relationship between investments and growth,
and on the other hand the determinants of investments. The production
process is defined by
Y (t)=F (K (t), L (t)) = F (K (t), A(t)N(t)), (4.2)
where N(t) is the number of workers that participate in production in period
t and A(t) denotes labor productivity; at time t each of the N(t)workers
supplies A(t) units of labor. Clearly, there are various ways to specify the
concept of productive efficiency in more detail. The amount of work of an
individual may depend on her physical strength, on the time and energy

invested in production, on the climate, and on a range of other factors. How-
ever, modeling these aspects not only complicates the analysis, but also forces
us to consider economic phenomena other than the ones that most interest
us.
To distinguish the role of capital accumulation (which by definition
depends endogenously on savings and investment decisions) from these other
factors, it is useful to assume that the latter are exogenous. The starting point
of our analysis is the Solow (1956) growth model. This model is familiar from
basic macroeconomics textbooks, but the analysis of this section is relatively
formal. We assume that L(t) grows at a constant rate g ,
˙
L(t)=gL(t), L(t)=L (0)e
gt
,
and for the moment we abstract from any economic determinant for the level
or the growth rate of this factor of production. Furthermore, we assume that
the production function exhibits constant returns to scale, so that
F (ÎK , ÎL )=ÎF (K , L )
for any Î. The validity of this assumption will be discussed below in the light
of its economic implications. Formally, the assumption of constant returns to
scale implies a direct relationship between the level of output and capital per
unit of the non-accumulated factor,
y(t) ≡ Y (t)/L (t)andk(t) ≡ K (t)/L(t).
Omitting the time index t,wecanwrite
y =
F (K , L )
L
=
LF(K /L , 1)
L

= f (k),
EQUILIBRIUM GROWTH 133
which shows that the per capita production depends only on the capital/labor
ratio. The accumulation of the stock of capital per worker is given by
˙
k(t)=
d
dt

K (t)
L(t)

=
˙
K (t)L (t) −
˙
L(t)K (t)
L(t)
2
=
˙
K (t)
L(t)
−
˙
L(t)
L(t)
K (t)
L(t)
.

Since
˙
K (t)=I (t) − ‰K (t)and
˙
L(t)=gL(t), we thus get
˙
k(t)=
I (t)
L(t)
− ( g + ‰)k(t).
Assuming that the economy as a whole devotes a constant proportion s of
output to the accumulation of capital,
C(t)=(1− s )Y (t), I (t)=sY(t),
then I(t)/L(t)=sY(t)/L(t)=sy(t)=sf(k(t)), and thus
˙
k(t)=sf(k(t)) − ( g + ‰)k(t).
The main advantage of this expression, which is valid only under the simpli-
fying assumptions above, is that it refers to a single variable. For any value of
k(t), the model predicts whether the capital stock per worker tends to increase
or decrease, and using the intermediate steps described above one can fully
characterize the ensuing dynamics of the aggregate and per capita income.
The amount of capital per worker tends to increase when
sf
(k(t)) > ( g + ‰)k(t), (4.3)
and to decrease when
sf(k(t)) < ( g + ‰)k(t). (4.4)
Having reduced the dynamics of the entire economy to the dynamics of
a single variable, we can illustrate the evolution of the economy in a simple
graph as shown in Figure 4.1. Clearly, the function sf(k) plays a crucial role
in these relationships. Since f (k)=F (k, 1) and F (·) has constant returns to

scale, we have
f (Îk)=F (Îk, 1) ≤ F (Îk, Î)=ÎF (k, 1) = Îf (k) for Î > 1, (4.5)
where the inequality is valid under the hypothesis that increasing L,the
second argument of F (·,
·), cannot decrease production. Note, however, that
the inequality is weak, allowing for the possibility that using more L may leave
production unchanged for some values of Î and k.
If the inequality in (4.5) is strict, then income per capita tends to increase
with k, but at a decreasing rate, and f (k) takes the form illustrated in the
figure. If a steady state k
ss
exists, it must satisfy
sf(k
ss
)=(g + ‰)k
ss
. (4.6)
134 EQUILIBRIUM GROWTH
Figure 4.1. Decreasing marginal returns to capital
4.1.1. BALANCED GROWTH
The expression on the right in (4.3) defines a straight line with slope ( g + ‰).
In Figure 4.2, this straight line meets the function sf(k)atk
ss
: for k < k
ss
,
˙
k = sf(k) − ( g + ‰)k > 0, and the stock of capital tends to increase towards
k
ss

; for k > k
ss
,onthecontrary,
˙
k < 0, and in this case k tends to decrease
towards its steady state value k
ss
.
Figure 4.2. Steady state of the Solow model
EQUILIBRIUM GROWTH 135
The speed of convergence is proportional to the vertical distance between
the two functions, and thus decreases in absolute value while k approaches its
steady-state value. In the long-run the economy will be very close to the steady
state. If k ≈ k
ss
=0,thenk = K /L is approximately constant; given that
d
dt
K (t)
L(t)
=

˙
K (t)
K (t)
−
˙
L(t)
L(t)


K (t)
L(t)
≈ 0 ⇒
˙
K (t)
K (t)
≈
˙
L(t)
L(t)
,
the long-run growth rate of K is close to the growth rate of L. Moreover, since
F (K , L ) has constant returns to scale, Y (t) will grow in the same proportion.
Hence, in steady state the model follows a “balanced growth” path, in which
the ratio between production and capital is constant. For the per capita capital
stock and output, we can use the definition that L(t)= A(t)N(t). This yields
Y (t)
N(t)
=
Y (t)
L(t)
L(t)
N(t)
= f (k
t
)A(t),
K (t)
N(t)
= k
t

A(t).
In terms of growth rates, therefore, we get the expression
(d/dt)[Y(t)/N(t)]
Y (t)/N(t)
=
(d/dt) f (k
t
)
f (k
t
)
+
˙
A(t)
A(t)
.
When k
t
tends to a constant k
ss
, as in the above figure, then df(k
t
)/dt =
f

(k
t
)
˙
k tends to zero; only a positive growth rate

˙
A(t)/A(t) can allow a long-
run growth in the levels of per capita income and capital. In other words, the
model predicts a long-run growth of per capita income only when L grows
over time and whenever this growth is at least partly due to an increase in A
rather than an increase in the number of workers N.
If we assume that the effective productivity of labor A(t) grows at a positive
rate g
A
, and that
g ≡
˙
L
L
=
˙
A
A
+
˙
N
N
= g
A
+ g
N
,
then the economy tends to settle in a balanced growth path with exogenous
growth rate g
A

: the only endogenous mechanism of the model, the accumula-
tion of capital, tends to accompany rather than determine the growth rate of
the economy. A once and for all increase in the savings ratio shifts the curve
sf(k) upwards, as in Figure 4.3. As a result, the economy will converge to a
steady state with a higher capital intensity, but the higher saving rate will have
no effect on the long-run growth rate.
In particular, the accumulation of capital cannot sustain a constant growth
of income (whatever the value of s )ifg =0and f

(k) < 0. For simplicity,
consider the case in which L is constant and ‰ =0.Inthatcase,
˙
Y
Y
=
f

(k)
˙
k
f (k)
= sf

(k), (4.7)
136 EQUILIBRIUM GROWTH
Figure 4.3. Effects of an increase in the savings rate
and an increase in k clearly reduces the growth rate of per capita income.
Asymptotically, the growth rate of the economy is zero if lim
k→∞
f


(k)=0,
or it reaches a positive limit if for k →∞the limit of f

(k)=∂ F (·)/∂ K is
strictly positive.
Exercise 33 Retaining the assumption that s is constant, let ‰ > 0.Howdoesthe
asymptotic behavior of
˙
Y /Y depend on the value of lim
k→∞
f

(k)?
4.1.2. UNLIMITED ACCUMULATION
Even if f

(k)isdecreasingink, nothing prevents the expression on the left
of (4.3) from remaining above the line ( g + ‰)k for all values of k, implying
that no finite steady state exists (k
ss
→∞). For this to occur the following
condition needs to be satisfied:
lim
k→∞
f

(k) ≡ f

(∞) ≥

g + ‰
s
, (4.8)
so that the distance between the functions does not diminish any further when
k increases from a value that is already close to infinity.
Consider, for example, the case in which g = ‰ = 0: in this case the steady-
state capital stock k is infinite even if lim
k→∞
f

(k) = 0. This does not imply
that the growth rate remains high, but only that the growth rate slows down
so much that it takes an infinite time period before the economy approaches
something like a steady state in which the ratio between capital and output
remains constant. In fact, given that the speed of convergence is determined
EQUILIBRIUM GROWTH 137
by the distance between the two curves in (4.2), which tends to zero in the
neighborhood of a steady state, the economy always takes an infinite time
period to attain the steady state. The steady state is therefore more like a
theoretical reference point than an exact description of the final configuration
for an economy that departs from a different starting position.
Nevertheless, in the long-run a positive growth rate is sustainable if the
inequality in (4.8) holds strictly:
lim
k→∞
f

(k) ≡ f

(∞) >

g + ‰
s
.
If L is constant, and if there is no depreciation (‰ = 0), the long-run growth
rate is
˙
Y
Y
= sf

(∞) > 0,
and it is dependent on the savings ratio s and the form of the production
function.
Consider, for example, the case of a constant elasticity of substitution (CES)
production function:
F (K , L )=[·K
Î
+(1−·)L
Î
]
1/Î
, Î ≤ 1. (4.9)
In this case we have
f (k)=[·k
Î
+(1−·)]
1/Î
and thus
f


(k)=[·k
Î
+(1−·)]
(1/Î)−1
·k
Î−1
= ·[· +(1−·)k
−Î
]
(1−Î)/Î
.
If Î is positive, the term k
−Î
tends to zero if k approaches infinity, and
lim
k→∞
f

(k)=·(·)
(1/Î)−1
= ·
1/Î
> 0: hence, this production function sat-
isfies f

(∞) > 0when0≤ Î < 1.
The production function (4.9) is also well defined for Î < 0. In this case,
the term in parentheses tends to infinity and, since its exponent (1 − Î)/Î is
negative, lim
k→∞

f

(k)=0.ForÎ = 0 the functional form (4.9) raises unity
to an infinitely large exponent, but is well defined. Taking logarithms, we get
ln( f (k)) =
1
Î
ln

·k
Î
+(1−·)

.
The limit of this expression can be evaluated using l’Hôpital’s rule, and is equal
to the ratio of the limit of the derivatives with respect to Î of the numerator
and the denominator. Using the differentiation rules d ln(x)/dx =1/x and
dy
x
/dx = y
x
ln y, the derivative of the numerator can be written as

·k
Î
+(1−·)

−1
(·k
Î

ln k),
138 EQUILIBRIUM GROWTH
while the derivative of the denominator is equal to one. Since lim
Î→0
k
Î
=1,
the limit of the logarithm of f (k)isthusequalto· ln k, which corresponds to
the logarithm of the Cobb–Douglas function k
·
.
Exercise 34 Interpret the limit condition in terms of the substitutability between
K and L . Assuming ‰ = g =0, analyze the growth rate of capital and production
in the case where Î =1, and in the case w here · =1.
4.2. Dynamic Optimization
The model that we discussed in the previous section treated the savings ratio
s as an exogenous variable. We therefore could not discuss the economic
motivation of agents to save (and invest) rather than to consume, nor could
we determine the optimality of the growth path of the economy. To introduce
these aspects into the analysis, we will now consider the welfare of a repre-
sentative agent who consumes an amount C (t)/N(t) ≡ c (t)ineachperiod
t. Suppose that the welfare of this agent at date zero can be measured by the
following integral
U =

∞
0
u(c(t))e
−Òt
dt. (4.10)

The parameter Ò is the discount rate of future consumption; given Ò > 0, the
agent prefers immediate consumption over future consumption. The function
u(·) is identical to the one introduced in Chapter 1: the positive first derivative
u

(·) > 0 implies that consumption is desirable in each period; however, the
marginal utility of consumption is decreasing in consumption, u

(·) < 0,
which gives agents an incentive to smooth consumption over time.
The decision to invest rather than to consume now has a precise economic
interpretation. For simplicity, we assume that g = 0, so that normalizing by
population as in (4.10) is equivalent to normalizing by the labor force. Assum-
ing that ‰ = 0 too, the accumulation constraint,
f (k(t)) − c(t) −
˙
k(t)=0, (4.11)
implies that higher consumption (for a given k(t)) slows down the accumula-
tion of capital and reduces future consumption opportunities. At each date t,
agents thus have to decide whether to consume immediately, obtaining utility
u(c(t)), or to save, obtaining higher (discounted) utility in the future.
This problem is equivalent to the maximization of objective func-
tion (4.10) given the feasibility constraint (4.11). Consider the associated
Hamiltonian,
H(t)=
[
u(c(t)) + Î(t)( f (k(t)) − c(t))
]
e
−Òt

,
EQUILIBRIUM GROWTH 139
where the shadow price is defined in current values. This shadow price
measures the value of capital at date t and satisfies Î(t)=Ï(t)e
Òt
where Ï(t)
measures the value at date zero. The optimality conditions are given by
∂ H
∂c
=0, (4.12)
−
∂ H
∂k
=
d

Î(t)e
−Òt

dt
, (4.13)
lim
t→∞
Î(t)e
−Òt
k(t)=0. (4.14)
4.2.1. ECONOMIC INTERPRETATION AND OPTIMAL GROWTH
Equations (4.12) and (4.13) are the first-order conditions for the optimal
path of growth and accumulation. In this section we provide the economic
intuition for these conditions, which we shall use to characterize the dynamics

of the economy. The advantage of using the present-value shadow price Î(t)is
that we can draw a phase diagram in terms of Î (or c)andk, leaving the time
dependence of these variables implicit.
From (4.12), we have
u

(c)=Î. (4.15)
Î(t) measures the value in terms of utility (valued at time t) of an infinitesimal
increase in k(t). Such an increase in capital can be obtained only by a reduc-
tion of current consumption. The loss of utility resulting from lower current
consumption is measured by u

(c). For optimality, the two must be the same.
In addition, we also have the condition that
˙
Î =(Ò − f

(k)) Î, (4.16)
which has an interpretation in terms of the evaluation of a financial asset:
the marginal unit of capital provides a “dividend” f

(k)Î, in terms of utility,
and a capital gain
˙
Î. Expression (4.16) implies that the sum of the “dividend”
and the capital gain are equal to the rate of return Ò multiplied by Î.This
relationship guarantees the equivalence of the flow utilities at different dates,
and we can interpret Î as the value of a financial activity (the marginal unit of
capital).
An economic interpretation is also available for the “transversality” condi-

tion in (4.14): it imposes that either the stock of capital, or its present value
Î(t)e
−Òt
(or both) need to be equal to zero in the limit as the time horizon
extends to infinity.
140 EQUILIBRIUM GROWTH
Combining the relationships in (4.15) and (4.16), we derive the following
condition:
d
dt
u

(c)=(Ò − f

(k)) u

(c).
Along the optimal path of growth and accumulation, the proportional
growth rate of marginal utility is equal to Ò − f

(k), the difference between
the exponential discount rate of utility and the growth rate of the available
resources arising from the accumulation of capital. This condition is a Euler
equation, like that encountered in Chapter 1. (Exercise 36 asks you to show that
it is indeed the same condition, expressed in continuous rather than discrete
time.)
Making the time dependence explicit and differentiating the function on
the left of this equation with respect to t yields
du


(c(t))/dt = u

(c(t))dc(t)/dt.
Thus, we can write (omitting the time argument)
˙
c =

u

(c)
−u

(c)

( f

(k) − Ò). (4.17)
Since the law of motion for capital is given by
˙
k = f (k) − c, (4.18)
we can therefore study the dynamics of the system in c, k-space.
4.2.2. STEADY STATE AND CONVERGENCE
The steady state of the system of equations (4.17) and (4.18) satisfies
f

(k
ss
)=Ò, c
ss
= f (k

ss
),
if it exists. For the dynamics we make use of a phase diagram as in Chapter 2.
On the horizontal axis we measure the stock of capital k (which now refers
to the economy-wide capital stock rather than the capital stock of a single
firm). On the vertical axis we measure consumption, c, rather than the shadow
price of capital. (The two quantities are univocally related, as was the case
for q and investment in Chapter 2.) If f (·) has decreasing marginal returns
and in addition there exists a k
ss
< ∞ such that f

(k
ss
)=Ò,thenwehavethe
situation illustrated in Figure 4.4.
Clearly, more than one initial consumption level c(0) can be associated with
a given initial capital stock k(0). However, only one of these consumption
levels leads the economy to the steady state: the dynamics are therefore of the
saddlepath type which we already encountered in Chapter 2. Any other path
EQUILIBRIUM GROWTH 141
Figure 4.4. Convergence and steady state with optimal savings
leads the economy towards points where c =0,orwherek = 0 (which in turn
implies that c =0if f (0) = 0 and if capital cannot become negative). Under
reasonable functional form restrictions the solution is unique, and one can
show that only the saddlepath satisfies (4.14).
Exercise 35 Repeat the der ivation, supposing that g
A
=0but ‰ > 0,g
N

> 0.
Show that the system does not converge to the capital stock associated with
maximum per capita consumption in steady state.
4.2.3. UNLIMITED OPTIMAL ACCUMULATION
In the above diagram the accumulation of capital cannot sustain an indefinite
increase of labor productivity and of per capita consumption. However, as in
the Solow model, the hypothesis that F (·) has constant returns to scale in
capital and labor does not necessarily imply that k
ss
< ∞. In these cases one
cannot speak about a steady state in terms of the level of capital, consumption,
and production. However, it is still possible that there exists a steady state in
terms of the growth rates of these variables—that is, a situation in which the
economy has a positive and non-decreasing long-run growth rate even in the
absence of exogenous technological change. Suppose for instance that f

(k)=
b, which is constant and independent of k for all the relevant values of the
capital stock. If the elasticity of marginal utility is constant, so that
u

(c)
c
u

(c)
= −Û
142 EQUILIBRIUM GROWTH
for all values of c, then we can rewrite (4.17) as
˙

c(t)
c(t)
=
b − Ò
Û
, (4.19)
and consumption increases (or decreases, if b < Ò and agents can disinvest)
at a constant exponential rate. The utility function considered here is of the
constant relative risk aversion (CRRA) type, given by
u(c)=
c
1−Û
1 − Û
, u

(c)=c
−Û
, u

(c)=−Ûc
−Û−1
. (4.20)
The conditions u

(·) > 0, u

(·) < 0 are satisfied if Û > 0. If Û = 1, the func-
tional form (4.20) is not well defined, but the marginal utility function u

(x)=

x
−1
(which completely characterizes preferences) coincides with the derivative
of log(x): hence, for Û =1wecanwriteu(c)=log(c). Given f

(k)=b,wecan
write f (k)=bk + Ó with Ó a constant of integration. From the law of motion
for capital,
˙
k(t)= f (k(t)) − c(t)=Ó + bk(t) − c (t),
we can derive
˙
k(t)
k(t)
=
Ó
k(t)
+ b −
c(t)
k(t)
.
If we focus on the case in which k(t) tends to infinity and Ó/k(t)tozero,
we have
lim
t→∞
˙
k(t)
k(t)
= b − lim
t→∞

c(t)
k(t)
. (4.21)
The proportional growth rate of k then tends to a constant if k(t) tends to
grow at the same (exponential) rate as c(t).
One can show that this condition is necessarily true if the economy satisfies
the transversality condition (4.14). With equation (4.20) for u(·), we get
Î(t)=u

(c(t))=(c(t))
−Û
.
Given that c(t) grows at a constant exponential rate, Ï(t)=Î(t)e
−Òt
has expo-
nential dynamics.
Now, consider (4.21). If c(t)/k(t) diminishes over time, then k(t)grows
at a more than exponential rate and the limit in (4.14) does not exist. If, on
the contrary, c(t)/k(t)isgrowing,
˙
k(t)/k(t) becomes increasingly negative.
As a result, k(t) will eventually equal zero, and production, consumption, and
accumulation will come to a halt—which is certainly not optimal, since for
the case of (4.20) we have u

(0) = ∞.
The first case corresponds to paths that hit or approach the vertical axis in
Figure 4.4; the second corresponds to paths that hit or approach the horizontal
axis. Hence, as in the case of the phase diagram, there is only one initial level
EQUILIBRIUM GROWTH 143

of consumption that satisfies the transversality condition. (In fact, the phase
diagram remains valid in a certain sense; however, the economy is always arbit-
rarily far from the steady state.) The consumption/capital ratio is therefore
constant over time under our assumptions. Imposing
˙
k
k
=
˙
c
c
in (4.19) and in (4.21), we get
c(t)=
(Û − 1)b + Ò
Û
k(t). (4.22)
Equation (4.22) implies that the initial consumption is an increasing function
of b, the intertemporal rate of transformation, if Û > 1. In this case the income
effect of a higher b dominates the substitution effect, which induces capital
accumulation and hence tends to reduce the level of consumption. For Û =1,
equation (4.20) is replaced by u(c )=ln(c), and the ratio c/k is equal to Ò and
does not depend on b.
Since y(t)=bk(t), savings are a constant fraction of income as in the Solow
model:
s =1−
(Û − 1)b + Ò
bÛ
.
Nonetheless, in the model with optimization, the savings ratio s is constant
only if u(·) is given by (4.20) and if f (·)=bk, and not in more general cases.

Moreover, s is not a given constant as in the Solow model. The savings ratio
depends on the parameters that characterize utility (Û and Ò) and technol-
ogy (b).
Having shown that capital grows at an exponential rate, we now return
to (4.14). In order to satisfy this transversality condition, the growth rate of
capital needs to be smaller than the rate at which the discounted marginal
utility diminishes along the growth path. We thus have
d
dt

ln(c(t)
−Û
e
−Òt
)

= −Û
b − Ò
Û
− Ò = −b.
Since in the case considered here Ï(t)=e
−Òt
c(t)
−Û
and f

(k)=b,thisisa
reformulation of condition (4.13).
In addition, we have
˙

k = sy = sbkwhere s is the savings ratio. The transver-
sality condition is therefore satisfied if
˙
k
k
= sb <
d
dt
|

ln(c(t)
−Û
e
−Òt
)

| = b,
144 EQUILIBRIUM GROWTH
or equivalently if s < 1. Hence, the propensity to save s =1−C/Y ,whichis
implied by (4.22), must be smaller than one: this leads to the condition that
0 < 1 − s =
c(t)
y(t)
=
c(t)
bk(t)
=
(Û − 1)b + Ò
bÛ
,

which is equivalent to
(1 − Û)b < Ò. (4.23)
If the parameters of the model violated (4.23), the steady state growth
path that we identified would not satisfy (4.14). But in that case the optimal
solution would not be well defined since the objective function (4.10) could
take an infinite value: although technically speaking consumption could grow
at rate b, the integral in (4.10) does not converge when (1 − Û)b − Ò > 0.
The steady-state growth path describes the optimal dynamics of the econ-
omy without any transitional dynamics if f (k)=bk for each 0 ≤ k ≤∞.We
should note, however, that the constant b is not allowed to be a function of
L if F (ÎK , ÎL )=ÎF (K , L ). Hence, F (K , L )=bK =
˜
F (K ), and the non-
accumulated factor L cannot be productive for the economy considered, that
grows at a constant rate in the absence of any (exogenous) growth in L ,if
the production function has constant returns to scale in K and L together.
Alternatively, the economy may converge asymptotically to the steady-state
growth path if lim
k→∞
f

(k)=b > 0eventhough f

(k) < 0 for any 0 ≤
k < ∞. In this case the marginal productivity of L can be positive for each
value of K and L, but the productive role of the non-accumulated factor
becomes asymptotically negligible (in a sense that we will make more precise
in Section 4.4). In both cases we have or are approaching a steady-state growth
path: the economy grows at a positive rate if b > Ò, and (less realistically) at a
negative rate if b < Ò. With ‰ > 0, it is not difficult to prove that the economy

can grow indefinitely if lim
k→∞
f

(k) > ‰.
4.3. Decentralized Production and
Investment Decisions
The analysis of the preceding section proceeded directly from the maximiza-
tion of the objective function of a representative agent (4.10), subject to tech-
nological constraint represented by the production function. Under certain
conditions, the optimal solution coincides with the growth of an economy
in which the decisions to save and invest are decentralized to households
and firms. In order to study this decentralized economy, we need to define
the economic nature and the productive role of capital in greater detail. Let us
assume for now that K is a private factor of production. The property rights of
EQUILIBRIUM GROWTH 145
this factor are owned by individual agents who in the past saved part of their
disposable income.
The economy is populated by infinitely lived agents, or “households,” which
for the moment we assume to be identical. The typical household, indexed
by i, owns one unit of labor. For simplicity, we assume that the growth rate
of the population is zero. In addition, each household owns a
i
(t) units of
financial wealth (measured in terms of output, consumption, or capital) at
date t. Moreover, individual agents or households take the wage rate w(t)and
the interest rate r (t) at which labor and capital are compensated as given. (In
other words, agents behave competitively on all markets.)
Family i maximizes
U =


∞
0
u(c
i
(t))e
−Òt
dt, (4.24)
subject to the budget constraint
w(t)+r (t)a
i
(t)=c(t)+
˙
a
i
(t).
The flow income earned by capital and labor is either consumed, or added to
(subtracted from, when negative) the family’s financial wealth.
Production is organized in firms. Firms hire the production factors from
households and offer their goods on a competitive market. At each date t,the
firm indexed by j produces F (K
j
(t), L
j
(t)) using quantities K
j
(t)andL
j
(t)
of the two factors, in order to maximize the difference between its revenues

andcosts.Sinceallpricesareexpressedintermsofthefinalgood,firmssolve
the following static problem:
max
K
j
,L
j
(F (K
j
, L
j
) −rK
j
− wL
j
).
Given that F (·, ·) has constant returns to scale, we can write
max
K
j
,L
j

L
j
f

K
j
L

j

−rK
j
− wL
j

,
where f (·) corresponds to the output per worker defined in the previous
section. The first-order conditions of the firm are therefore given by
f


K
j
(t)
L
j
(t)

= r(t),
f

K
j
(t)
L
j
(t)


−
K
j
(t)
L
j
(t)
f

(K
j
(t)/L
j
(t)) = w(t),
which are valid for each t and each j.
Since all firms face the same unit costs of capital and labor, every firm
will choose the same capital/labor ratio, K
j
/L
j
≡ k. In equilibrium firms
therefore can differ only as regards the scale of their operation: if L is the
146 EQUILIBRIUM GROWTH
aggregate stock of labor (or the number of households), we can index the scale
of individual firms by Ó
j
so that

j
Ó

j
= 1, and denote L
j
= Ó
j
L.Thanksto
the assumption of constant returns to scale, we can assume that F (·, ·)has
the same functional form as at the aggregate level. We can then immediately
derive a simple expression for the aggregate output of the economy:
Y ≡

j
F (K
j
, L
j
)=

j
L
j
f (K
j
/L
j
)=


j
Ó

j

Lf(k)=F (K , L).
Hence if the production function has constant returns to scale and if all
markets are competitive, the number of active firms and the scale of their
operation is irrelevant.
36
Atthispointwenotethat

j
L
j
= L = AN = A


N
j =1
1

.Hence,the
same factor of labor efficiency A is applied to each individual unit of labor
that is offered on the labor market. Moreover, we notice that in equilibrium
the profits of each firm are equal to zero. It is therefore irrelevant to know
which family owns a particular firm and at which scale this firm operates.
Let us now return to the household. The dynamic optimization problem of
the household is expressed by the following Hamiltonian:
H(t)=e
−Òt
[u(c
i

(t)) + Î
i
(t)(w(t)+r (t)a
i
(t) − c
i
(t))].
The first-order conditions are analogous to (4.12)–(4.14), and can be rewrit-
ten as
d
dt
c
i
(t)=
−u

(c
i
(t))
u

(c
i
(t))

r (t) − Ò

,
lim
t→∞

e
−Òt
u

(c
i
(t))a
i
(t)=0.
Exercise 36 Compare this optimality condition with
u

(c
t
)=
1+r
1+Ò
u

(c
t+1
),
also known as a Euler equation, which holds in a deterministic environment with
discrete time. Complete the parallel between the consumption problems studied
here and in Chapter 1 by deriving a version of the cumulated budget restriction
in continuous time.
³⁶ For simplicity, we suppose that the stock of capital may vary without adjustment costs. The
following derivations would remain valid if, as in some of the models studied in Chapter 2, returns
to scale were constant in adjustment as well as in production, implying that—at least in the long
run—the size of firms is irrelevant.

EQUILIBRIUM GROWTH 147
4.3.1. OPTIMAL GROWTH
We close the model by imposing the restriction that the total wealth of house-
holds must equal the aggregate stock of capital. Inter-family loans and debts
cancel out on aggregate, and in any case there is no reason why such loans and
debts should exist if households are identical and start with the same initial
wealth: a
i
(t)=a(t). From
L

i=1
a
i
(t)=La(t)=K (t),
we get
a
i
(t)=a(t)=k(t).
Furthermore given that
r = f

(k),
it is easy to verify that optimality conditions for the accumulation of financial
wealth coincide with those for the accumulation of capital along the path of
aggregate growth that maximizes (4.10). (This also remains true if g > 0, if
‰ > 0, and even if n > 0 — where we should note that, in the presence of
population growth, the per capita rate of return on capital a is given by r − n,
and that if capital depreciates we have r = f


(k) − ‰.)
Hence, the growth path of a market economy will coincide with the optimal
growth path if the following conditions are satisfied.
(A) Production has constant returns to scale.
(B) Markets are competitive.
(C) Savings and consumption decisions are taken by agents who independ-
ently solve identical problems.
Conditions (A) and (B) guarantee that r (t)= f

(k). The savings of an
individual household are compensated according to the ag gregate marginal
productivity of capital. Moreover, given conditions (A) and (B), the market
structure is very simple and the entire economy behaves as a “representa-
tive” firm.
Hypothesis (C) allows us to represent the savings decisions in terms of the
optimization of a single “representative agent.” Most differences between indi-
vidual agents on the market are made irrelevant by the presence of a perfectly
competitive capital market (as implicitly assumed above). For example, the
supply of labor may follow different dynamics across households, but access
to a perfectly competitive capital market may prevent this from having any
effect at the aggregate level: individuals or households whose labor income is
temporarily low can borrow from households that are in the opposite position,
with no aggregate effects as long as total labor supply in the economy is
148 EQUILIBRIUM GROWTH
fixed. This is an application of the permanent income hypothesis discussed
in Chapter 1.
It is also useful to note that differences in individual consumption have no
impact at the aggregate level if agents have a common utility function with a
constant elasticity of substitution as in (4.20). In this case the growth rate of
consumption is the same for all households, so that

˙
C
C
=

i
˙
c
i

i
c
i
=

i
r (t)−Ò
Û
c
i

i
c
i
=
r (t) − Ò
Û
.
Functional form (4.20) thus has two advantages. On the one hand, this func-
tional form is compatible with a steady-state growth path (as we saw above).

On the other hand, it allows us to aggregate the individual investment deci-
sions, even in the case in which agents consume different amounts, because
the interest rate r (t) is the same for all agents.
4.4. Measurement of “Progress”: The Solow Residual
The hypotheses of constant returns to scale and perfectly competitive markets
(realistic or not) not only are crucial for the equivalence between the opti-
mization at the aggregate and decentralized levels, but also make it possible to
measure the technological progress that may allow unlimited growth of labor
productivity when k
ss
< ∞.
Differentiating the production function Y (t)=F (K (t), L(t)), we get
˙
Y (t)=F
K
(·)
˙
K (t)+F
L
(·)
˙
L(t)=F
K
(·)
˙
K (t)+F
L
(·)

˙

N(t)A(t)+N(t)
˙
A(t)

,
where F
L
(·)andF
K
(·) denote the partial derivatives with respect to the pro-
duction factors, which are measured in current values. The second equality
exploits our definition of labor supply L (t) ≡ N(t)A(t). Rewriting the above
expression in terms of proportional growth yields
˙
Y
Y
=
F
K
(·)K
Y
˙
K
K
+
F
L
(·)AN
Y
˙

N
N
+
F
L
(·)N
Y
˙
A, (4.25)
where we have omitted the time dependence. Now, if labor markets are per-
fectly competitive, we have w = ∂ F (·)/∂ N = AF
L
(·).Wecanthuswrite
F
L
(·)AN
Y
=
wN
Y
≡ „,
which expresses labor’s share of national income, which is in general observ-
able, in terms of a derivative of the aggregate production function. Moreover,
given that the production technology has constant returns to scale in K and
L,theentirevalueofoutputwillbepaidtotheproductionfactorsifthese
EQUILIBRIUM GROWTH 149
are paid according to their marginal productivity. In fact, for each F (·, ·) with
constant returns to scale,
F


K
Y
,
L
Y

= 1 with Y = F (K , L ).
Using Euler’s Theorem, we therefore have
1=F

K
Y
,
L
Y

=
∂ F (K , L )
∂ K
K
Y
+
∂ F (K , L )
∂ L
L
Y
. (4.26)
Hence,
F
K

(·)K
Y
=1−
AF
L
(·)N
Y
=1− „,
and (4.25) implies
„
˙
A
A
=
˙
Y
Y
− (1 − „)
˙
K
K
− „
˙
N
N
. (4.27)
If accurate measures of „ (the income share of the non-accumulated factor N)
and the proportional growth rate of Y, K ,andN are available, then (4.27)
provides a measure known as “Solow’s residual,” which indicates how much
of the growth in income is accounted for by an increase in the measure of

efficiency A(t) (which as such is not measurable) rather than by an increase in
the supply of productive inputs.
If the production function has the Cobb–Douglas form,
Y = F (K , L )=F (K , AN)=K
·
(AN)
1−·
, (4.28)
or, equivalently, if
Y =
˜
A
˜
F (K , N)=
˜
AK
·
N
1−·
, where
˜
A = A
1−·
, (4.29)
then „ is constant and equal to 1 − ·. The Cobb–Douglas function is therefore
convenient from an analytic point of view, and also because it does not attach
any practical relevance to the difference between a labor-augmenting technical
change as in (4.28) and a neutral technological change as in (4.29). In fact, the
Solow residual defined in (4.27) corresponds to the rate of growth of
˜

A.
Exercise 37 Verify that, if
˙
K /K =
˙
A/A +
˙
N/N, the income shares of capital
and labor are constant as long as the production function has constant returns to
scale, even if it does not have the Cobb–Douglas form.
Unfortunately, the functional form (4.28) implies that
lim
k→∞
f

(k) = lim
k→∞
˜
A·k
·−1
=0
if · < 1, that is if „ > 0 and labor realistically receives a positive share of
national income. Given that the labor share is approximately constant (around
150 EQUILIBRIUM GROWTH
60% in the long-run), the empirical evidence does not seem supportive of
unlimited growth with constant returns to scale.
More generally, for each case in which the aggregate production function
F (·, ·) has constant returns to scale and
lim
k→∞

f

(k) = lim
k→∞
∂ F (K , L )
∂ K
= b > 0,
then F
L
(·)L /F (·) tends to zero when K and k approach infinity for a con-
stant L.Itsuffices to take the limit of expression (4.26) with K →∞(and
thus L/Y = L /F (K , L ) → 0), which yields
1=F

lim
K →∞
K
Y
, 0

= b lim
K →∞
K
F (K , L )
+lim
K →∞

∂ F (K , L )
∂ L
L

F (K , L )

, (4.30)
l’Hôpital’s rule then implies (as in exercise 33 above) that
lim
K →∞
K
F (K , L )
=1

lim
K →∞
∂ F (K , L )
∂ K

=
1
b
.
Hence, the first term on the right-hand side of (4.30) tends to one, and the
second term (the income share of the non-accumulated factor) therefore has
to tend to zero.
In sum, the income share of the non-accumulated factor „ needs to decline
to zero with the accumulation of an infinite amount of capital if
(i) the accumulation of capital allows the economy to grow indefinitely,
and
(ii) the production function has constant returns to scale.
This conclusion is intuitive in light of the reasoning that led us to draw a
convex production function in Figure 4.1, and to identify a steady state in
Figure 4.2; if we have equality rather than a strict inequality in (4.5), that is if

f (Îk)=F (Îk, 1) = F (Îk, Î)=ÎF (k, 1) = Î f (k)
for Î =1,thenoutputisproportionaltoK and increasing L will not have
any effect on output. If the increase in the stock of capital tends to have pro-
portional effects on output, then both marginal productivity and the income
share of the non-accumulated factor must steadily decrease.
Exercise 38 Verify this result for the case of a function in the form (4.9).
Naturally, equation (4.27) and its implications are valid only under the twin
assumptions that the production technology exhibits constant returns to scale
and that production factors are paid according to their marginal productivity.
EQUILIBRIUM GROWTH 151
From a formal point of view, nothing would prevent us from considering
models in which either assumption is violated. As illustrated in the exercise
below, in that case it does not make much sense to measure
˙
A/A by inserting
labor’s income share „ in (4.27).
Exercise 39 Consider a Cobb–Douglas production function with increasing
returns to scale,
Y = AN
·
K
‚
, · + ‚ > 1.
Suppose, in addition, that wages are below the marginal productivity of labor,
AF
N
(·)=
w
1 − Ï
,

where Ï > 0 can be interpreted as a monopolistic mark-up. What does the Solow
residual measured by (4.27) correspond to in this case?
The above hypotheses correspond to conditions (A) and (B) in the previous
section, which allowed us to connect the macroeconomic dynamics to the
savings and consumption decisions of individual agents. Constant returns to
scale allowed us simply to aggregate the production functions of the individual
firms. And the remuneration of production factors equal to their marginal
product (which in turn followed from the assumption that all markets are
characterized by free entry and perfect competition) ensured that the dynamic
path of the economy maximized the welfare of a hypothetical representative
agent. In the rest of this chapter we consider models for which the macro-
economic dynamics are well-defined (but not necessarily optimal from the
aggregate point of view) in the absence of perfectly competitive markets and
in the presence of increasing returns to scale.
4.5. Endogenous Growth and Market Imperfections
To obtain an income share for the non-accumulated factor that is not reduced
to zero in the long-run and at the same time allow for an endogenous growth
rate that is determined by the investment decisions of individual agents, we
need to reconsider the assumption of constant returns to scale. Henceforth
we will consider steady-state growth paths only in the absence of exogenous
technological change. We know that, in order to sustain long-run (propor-
tional) growth, the economy needs to exhibit constant returns to capital: from
now on we therefore assume that f

(k)=b,withb independent of k.Ifthat
condition is satisfied, and if the productivity of the non-accumulated factor L
is positive, aggregate production is characterized by increasing returns to scale.