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 tend
Trang 1general 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, and so forth) This allows us to verify the
validity of the law of iterative expectations in this context For i ≥ 2, we write
Et+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 ] is itself a random variable, and we can therefore calculate its
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.
Trang 2(a) Calculate the levels of employment N b and N g that maximize the expected 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.
dis-(b) Compute the mobility cost k at which the optimal mobility decisions are consistent with a wage di fferential 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 given 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 vice versa induce the firm to adjust its labor force, while the employment level is una ffected 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 costsare potentially much wider than those illustrated in this chapter For example,Bertola (1994) discusses the implications of increased rigidity (and less efficiency) inmodels 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 acontinuous-time setting where state transitions are described as Poisson events of thetype to be introduced in Chapter 5
Economic theory can also explain why employment protection legislation isimposed despite its apparently detrimental effects Using models similar to thosediscussed here, Saint-Paul (2000) considers how politico-economic interactions canrationalize 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 moreappropriate basis
Of course, job security provisions are only one of the many institutional featuresthat 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
Trang 3ways with wage and employment rigidities in determining the dynamics of ment and unemployment across the Atlantic and within Europe For economic andempirical analyses of the European unemployment problem from an international
employ-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
Bentolila, S., and G Bertola (1990) “Firing Costs and Labor Demand: How Bad is Eurosclerosis?”
Review 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:
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
and J 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.
Trang 44 Growth in Dynamic
General Equilibrium
The previous chapters analyzed the optimal dynamic behavior of singleconsumers, firms, and workers The interactions between the decisions ofthese agents were studied using a simple partial equilibrium model (for thelabor market) In this chapter, we consider general equilibrium in a dynamicenvironment
Specifically, we discuss how savings and investment decisions by individualagents, mediated by more or less perfect markets as well as by institutionsand collective policies, determine the aggregate growth rate of an economyfrom a long-run perspective As in the previous chapters, we cannot reviewall aspects of a very extensive theoretical and empirical literature Rather, weaim at familiarizing readers with technical approaches and economic insightsabout the interplay of technology, preferences, market structure, and insti-tutional features in determining dynamic equilibrium outcomes We reviewthe relevant aspects in the context of long-run growth models, and a briefconcluding section discusses how the mechanisms we focus on are relevant
in the context of recent theoretical and empirical contributions in the field ofeconomic growth
Section 4.1 introduces the basic structure of the model, and Section 4.2applies the techniques of dynamic optimization to this base model The nexttwo sections discuss how decentralized decisions may result in an optimalgrowth path, and how one may assess the relevance of exogenous technologicalprogress in this case Finally, in Section 4.5 we consider recent models ofendogenous 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 theeconomic intuition that underlies the formal mathematical expressions, andaim 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 notdistinguish between flows that originate in the private and the public sectors
Trang 5The distinction between consumption and investment is based on theconcept of capital Broadly speaking, this concept encompasses all durablefactors of production that can be reproduced The supply of capital grows
in proportion with investments At the same time, however, existing capitalstock 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) at time t, without having to specify whether
it is measured at the beginning or the end of a period In addition, we assumethat capital depreciates at a constant rate ‰ The evolution of the supply ofcapital is therefore given by
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 Inprinciple, these stocks can be measured for any infinitesimally small timeperiod t However, a formal representation of the aggregate production
process in a single equation is normally not feasible In reality, the capitalstock consists of many different durable goods, both public and private Atthe end of this chapter we will briefly discuss some simple models that makethis disaggregate structure explicit, but for the moment we shall assume thatinvestment 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 growthrate of output and capital exceeds the growth rate of the population represents
an extremely important phenomenon: the steady increase in living standards.But in this chapter our interest in this type of growth pattern stems more fromits simplicity than from reality Even though simple models cannot capture allfeatures of world history, analyzing the economic mechanisms of a growingeconomy may help us understand the role of capital accumulation in the realworld and, more generally, characterize the economic structure of growthprocesses
Trang 64.1 Production, Savings, and Growth
The dynamic models that we consider here aim to explain, in the simplest sible way, on the one hand the relationship between investments and growth,and on the other hand the determinants of investments The productionprocess is defined by
pos-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 anindividual may depend on her physical strength, on the time and energyinvested 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 interestus
To distinguish the role of capital accumulation (which by definitiondepends endogenously on savings and investment decisions) from these otherfactors, 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 frombasic macroeconomics textbooks, but the analysis of this section is relatively
formal We assume that L (t) grows at a constant rate g ,
˙L(t) = g L(t) , L (t) = L (0)e g t ,
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 thatthe 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 toscale implies a direct relationship between the level of output and capital perunit of the non-accumulated factor,
Omitting the time index t, we can write
y = F (K , L)
L F (K /L , 1)
Trang 7which shows that the per capita production depends only on the capital/laborratio The accumulation of the stock of capital per worker is given by
Assuming that the economy as a whole devotes a constant proportion s of
output to the accumulation of capital,
simpli-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 fullycharacterize the ensuing dynamics of the aggregate and per capita income.The amount of capital per worker tends to increase when
s f (k(t)) > ( g + ‰)k(t), (4.3)and to decrease when
s f (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 s f (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 s s exists, it must satisfy
s f (k s s ) = ( g + ‰)k s s (4.6)
Trang 8Figure 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 s f (k) at k s s : for k < k s s,
˙k = s f (k) − ( g + ‰)k > 0, and the stock of capital tends to increase towards
k s s ; for k > k s s , on the contrary, ˙k < 0, and in this case k tends to decrease
towards its steady state value k s s
Figure 4.2 Steady state of the Solow model
Trang 9The 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 s s = 0, then k = K /L is approximately constant; given that
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
When k t tends to a constant k s s , as in the above figure, then d f (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
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 tion of capital, tends to accompany rather than determine the growth rate ofthe economy A once and for all increase in the savings ratio shifts the curve
accumula-s f (k) upwardaccumula-s, aaccumula-s in Figure 4.3 Aaccumula-s a reaccumula-sult, 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 ) if g = 0 and f(k) < 0 For simplicity,
consider the case in which L is constant and ‰ = 0 In that case,
Trang 10Figure 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 limk→∞ 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 How does the asymptotic behavior of ˙Y /Y depend on the value of lim k→∞ f(k)?
4.1.2 UNLIMITED ACCUMULATION
Even if f(k) is decreasing in k, 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 s s → ∞) For this to occur the followingcondition needs to be satisfied:
lim
k→∞ f(k) ≡ f(∞) ≥ g + ‰
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 state capital stock k is infinite even if lim k→∞ f(k) = 0 This does not imply
steady-that the growth rate remains high, but only steady-that the growth rate slows down
so much that it takes an infinite time period before the economy approachessomething like a steady state in which the ratio between capital and outputremains constant In fact, given that the speed of convergence is determined
Trang 11by 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 atheoretical reference point than an exact description of the final configurationfor an economy that departs from a different starting position
Nevertheless, in the long-run a positive growth rate is sustainable if theinequality in (4.8) holds strictly:
If Î is positive, the term k−Î tends to zero if k approaches infinity, and
limk→∞ f(k) = ·(·)(1/Î)−1= ·1/Î > 0: hence, this production function
sat-isfies f(∞) > 0 when 0 ≤ 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, limk→∞ f(k) = 0 For Î = 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
Îln
·kÎ+ (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 numeratorand the denominator Using the differentiation rules d ln(x)/dx = 1/x and
d y x /dx = y x ln y, the derivative of the numerator can be written as
·kÎ+ (1− ·)−1(·kÎln k) ,
Trang 12while the derivative of the denominator is equal to one Since limÎ →0kÎ
= 1,
the limit of the logarithm of f (k) is thus equal to · 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 where · = 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 introducethese 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) in each period
t Suppose that the welfare of this agent at date zero can be measured by 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 ing that ‰ = 0 too, the accumulation constraint,
Assum-f (k(t)) − c(t) − ˙k(t) = 0, (4.11)
implies that higher consumption (for a given k(t)) slows down the tion of capital and reduces future consumption opportunities At each date t,
accumula-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 tion (4.10) given the feasibility constraint (4.11) Consider the associatedHamiltonian,
func-H(t) = [u(c (t)) + Î(t)( f (k(t)) − c(t))] e −Òt ,
Trang 13where 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)eÒt where Ï(t)
measures the value at date zero The optimality conditions are given by
4.2.1 ECONOMIC INTERPRETATION AND OPTIMAL GROWTH
Equations (4.12) and (4.13) are the first-order conditions for the optimalpath of growth and accumulation In this section we provide the economicintuition 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 ) and k, leaving the time
dependence of these variables implicit
From (4.12), we have
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
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 Thisrelationship 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 ofcapital)
An economic interpretation is also available for the “transversality” tion in (4.14): it imposes that either the stock of capital, or its present value
condi-Î(t)e −Òt (or both) need to be equal to zero in the limit as the time horizonextends to infinity
Trang 14Combining the relationships in (4.15) and (4.16), we derive the followingcondition:
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 discretetime.)
Making the time dependence explicit and differentiating the function on
the left of this equation with respect to t yields
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 s s) = Ò, c s s = f (k s s),
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 s s < ∞ such that f(k s s) = Ò, then we have thesituation 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