Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

`‘le Ramsey model as can entity (in per capita form) e government must also )ndition of the following

(14.116)

rrent budget restriction:

(14.117)

' - ht (b(0) > 0), solvency f future primary surpluses. d g(t) (and hence for the

in (14.55). It is modified agent:

(14.118)

,!ncy condition (14.56), but with a tax-modified "

(14.119)

.117) into (14.119), the

(14.120)

le expression for human cular path for lump-sum ble to the representative

t affected either.

cnression we obtain:

the government solvency 14.117) is obtained.

Chapter 14: Theories of Economic Growth

By using (14.120) in (14.57), the household budget restriction can be written as:

This expression shows clearly why Barro (1974) chose the title he did for his pathbreaking article. Under Ricardian equivalence, government debt should not be seen as household wealth, i.e. b(0) must be deducted from total financial wealth in order to reveal the household's true financial asset position, as is in fact done in (14.121).

14.5.8 Overlapping generations of infinitely lived dynasties

In the previous section we saw that the Ramsey model yields classical conclusions vis-a-vis fiscal policy and implies the validity of Ricardian equivalence. The question which arises is which aspect of the model can be considered the prime cause for these classical results. In this subsection we show that once we allow for "disconnectedness of generations", debt neutrality no longer holds despite the fact that individual generations live forever.

Up to now we have introduced population growth by assuming growth of the dynastic family. Suppose now, however, that individual agents are infinitely lived (as the dynastic family is in the Ramsey model) but that population growth takes the form of new agents gradually entering the economy. These new agents also have infinite lives but are not linked to any of the agents already alive at the time of their birth. This is the setting of infinitely lived overlapping generations (OLG) suggested by Weil (1989b). Loosely put, growth of the population now occurs at the "extensive" rather than the "intensive" margin.

Since different agents enter life at different moments in historical time, both the birth rate (generational index) and the calendar date must be distinguished. We assume that a representative agent of generational cohort v < t chooses a path for consumption in order to maximize lifetime utility,

A (v, f log c(v, r)eP(t- r ) dr, (14.122)

subject to the budget identity:

and the intertemporal solvency condition:

where c(v, r) and a(v, r) denote, respectively, consumption and financial assets at timer of the representative household of vintage v. The felicity function appearing in (14.122) features an intertemporal substitution elasticity of unity. Equation

443

The Foundation of Modern Macroeconomics

(14.123) shows that the household supplies a single unit of labour inelastically to the competitive labour market and receives a wage, W(r), which is age-independent. A new generation is born without any financial assets:

In view of the simple structure of preferences, the level and time profile of consumption by a representative household of vintage v are easily computed:

and r(r) represents lump-sum taxes per agent. This tax is, by assumption, the same for all agents and thus does not feature a generations index either.

At the beginning of time, the economy starts out with N(0) agents, so that with a constant exponential population growth rate of n, the population size at time t is:

N(t) = N (0)ent (14.129)

The instantaneous arrival rate of new generations is dN (t) I dt so we know that the number of agents of generation v < t is given by N(v, v) dN(v)/dv. This suggests that aggregate per capita variables can be measured as follows:

where we have used (14.126) plus the fact that human wealth is age-independent. The aggregate counterpart to (14.123) is:

and equation (14.128) implies that human wealth (per agent) accumulates according to:

Human and non-human wealth accumulate at different rates because newborns have no financial wealth, and thus drag down aggregate per capita financial wealth

Table 14

E(t) = [r

t■ =

,t) = f

Notes: See

accumuL )b, pp.

Equation for th,

ca)

c(t)

=

I

The first t- consumpt e7 .7ionsat: -Lot (b(t) = non-hum,: The mo.L. marginal p b, ri

The phas in Figure 1 = sw:

Lion 5.6 ab O tt

equation (1 Ain Fi_ .re

individual , n (14.: uiily on in consume.

eve betwe

There al the capital st

inelastically to the s age-independent. A

(14.125)

and time profile of !a si ly computed:

(14.126)

(14.127)

(14.128)

assumption, the same her.

agents, so that with a

• on size at time t is:

(14.129)

co we know that the v)/dv. This suggests

(14.130)

as follows:

(14.131)

is age-independent.

(14.132)

-,cumulates accord-

(14.133)

because newborns ;ta financial wealth

Notes: See Table 14.1 for definitions of the variables.

accumulation, whilst all generations have the same level of human wealth (see Weil, 1989b, pp. 187-188).

Equations (14.131)-(14.133) can be combined to yield the Euler equation modified for the existence of overlapping generations:

The first term on the right-hand side of these expressions represents individual consumption growth, whilst the second term indicates how the arrival of new generations affects per capita consumption growth. In the absence of initial government debt (b(t) = 0), equilibrium in the financial capital market implies that household non-human wealth is held in the form of productive capital, i.e. a(t) = k(t). 27 The model is completed by the capital accumulation identity (14.113) and the marginal productivity condition (T1.3). For convenience the key equations have been collected in Table 14.4.

The phase diagram for the overlapping-generations (OLG) model has been drawn in Figure 14.12 for the case with zero initial debt and government consumption (b(t) = g(t) = 0). The k(t) = 0 line has already been discussed extensively in section 5.6 above. The W) = 0 line is obtained by combining (T4.3) and (T4.1) and invoking the steady state. The slope of the . (t) line can be explained by appealing to equation (14.135) and Figure 14.12. Suppose that the economy is initially at point A in Figure 14.12 and consider point B which lies directly above it. With the same amount of capital per worker, both points feature the same interest rate so that individual consumption growth, 4v, t)/c(v, t), coincides at the two points. Equation (14.135) indicates, however, that aggregate consumption growth depends not only on individual growth but also on the proportional difference between average consumption and consumption by a newly born generation, i.e. [c(t) - c(t, t)]/c(t). Since newly born generations start without any financial capital, the absolute difference between average and new-born consumption depends on the average capital stock (i.e. c(t) - c(t, t) = pk(t)) and is thus the same at points A and B. Since the level

27 There are no adjustment costs of investment so that the stockmarket value of the firm is equal to the capital stock. See (14.72) above.

445

The Foundation of Modern Macroeconomics

c(t) = 0

c ( t)

kKR

Figure 14.12. Fiscal policy in the overlapping-generations model

of aggregate consumption is larger at B, this point features a smaller proportional difference between average and new-born consumption, thereby raising aggregate consumption growth. In order to restore zero growth of aggregate per capita consumption, the capital stock must rise (to point C, which lies to the right of B). The larger capital stock not only reduces individual consumption growth by decreasing the rate of interest but also raises the drag on aggregate consumption growth due to the arrival of new dynasties because a larger capital stock widens the gap between average wealth and wealth of a newly born.

This argument also explains that for points above (below) the 4t) = 0 line, consumption rises (falls). This has been indicated with vertical arrows in Figure 14.12. The intersection of the W) = 0 and k(t) = 0 lines yields a unique, saddle-point stable, equilibrium at point E0 . The capital stock per worker associated with point E0 is kMKR, where "MKR" stands for modified-Keynes—Ramsey rule. It is clear from the diagram that kMKR kKR and that the steady-state interest rate exceeds the rate of time preference:

p < rMKR = fr (kMKR) 6 < p n. (14.136)

The upward-sloping time profile of individual consumption that is implied by (14.136) and (14.127) ensures that new generations, like old generations, accumulate capital. Note that there is nothing preventing the capital-labour ratio from being larger than the golden-rule ratio. Indeed, )n.

The "normal ca exceeds the rate 197). This is ind It is straightfo model. Indeed,

b(t) = f r

where we now al (14.128) we can

a(t) h(t)

By comparing (1 right-hand side new unconnectl rate of generatic (14.131)) aggr,, deficits and debt

The intuition ment of taxati( generations beta rather because t ations to which from the anal \ mines the valiu, in-and-of itself 1

The OLG moo does, however, sumption on . follows. Assume that the govern] ment consumi

28 See Weil (196J approach in Chapte

k (t)

rations model

!s a smaller proportional reby raising aggregate 12gregate per capita con- , to the right of B). The on growth by decreasing Timption growth due to dens the gap between

..-) the c(t) = 0 line, con- 1 arrows in Figure 14.12. a unique, saddle-point associated with point ,sey rule. It is clear from st rate exceeds the rate

(14.136)

-)n that is implied by old generations, accu-

nital-labour ratio from

,koR rMKR < > ),.

Chapter 14: Theories of Economic Growth

The "normal case" appears to be, however, that the rate of pure time preference exceeds the rate of population growth (p > n), so that rMKR > n (see Weil, 1989b, p. 197). This is indeed the case that we will restrict attention to.

It is straightforward to demonstrate that Ricardian equivalence fails in the OLG model. Indeed, the government budget restriction can be written as:

where we now allow there to be a non-zero initial debt. By substituting (14.137) into (14.128) we can derive the following expression for aggregate per capita wealth:

a(t) + h(t) k(t) + b(t) + h(t)

=k(t) + b(t) +f [147 (r) — g (T)] e-R(trir) dr

—[b(t) +[g(r)f]— r (0] er(t ) dr

=k(t) +f [W (r) — g(0] e-R(tic) dr

By comparing (14.138) to (14.137) it is clear that the term in square brackets on the right-hand side of (14.138) only vanishes if there is no population growth, i.e. if no new unconnected generations enter the economy (n = 0). With a positive arrival rate of generations (n > 0) Ricardian equivalence fails because total wealth and (by (14.131)) aggregate per capita consumption are both affected by the path of primary deficits and debt.

The intuition behind this result is provided by Weil (1989b, p. 193). A postponement of taxation which is financed by means of government debt makes all existing generations better off not because they don't have to pay taxes in the future, but rather because the future tax base will be larger as it includes newly arrived generations to which the present generations are not linked. An important conclusion from the analysis is that it is the economic identity of future taxpayers which determines the validity of Ricardian equivalence. Whether or not agents have finite lives in-and-of itself has no implication for Ricardian equivalence. 28

The OLG model thus generally refutes the notion of Ricardian equivalence. It does, however, yield classical predictions regarding the effects of government consumption on output, consumption, and capital etc. This can be demonstrated as follows. Assume that lump-sum taxation is used and that there is no initial debt, so that the government budget identity reduces to g(t) = (t). An increase in government consumption shifts the k(t) = 0 line down by the amount of the shock (dg),

28 See Weil (1989b) and Buiter (1988) on this point. We return to the different versions of the OLG approach in Chapter 16 below.

447

The Foundation of Modern Macroeconomics

so that the long-run equilibrium shifts from E0 to E1 in Figure 14.12. In contrast to what happens in the Ramsey model, not only consumption but also the capital stock per worker is reduced in the long run:

where c*, k*, and r* are the initial steady-state levels of consumption, capital per worker, and the interest rate, respectively. The denominator appearing in (14.139)- (14.140) is negative by saddle-point stability. 29

Since the model is saddle-point stable, the economy jumps at impact from Ea0 to E', and consumption falls but by less than one-for-one in the impact period:

where A2 > 0 is the unstable root of the Jacobian matrix of the linearized system (footnote 29 shows that A2 > r* - p). Existing generations do not bear the full burden of taxation because they know that future generations will eventually expand the tax base. As a result, present generations cut back consumption by too little and thus save too little to maintain the capital stock per worker at its old level. Over time the capital stock falls, as does the wage. Gradually, new generations are born with a lower level of human wealth due to the decreasing wage. This explains why aggregate per capita consumption falls during transition.

14.6 Endogenous Growth

Up to now we have exclusively worked with a production structure which satisfies the Inada conditions (See (P2) and (P3) for the properties). Although these conditions facilitate the construction of the phase diagrams they are not innocuous (in an economic-theoretic sense) because they imply the existence of diminishing returns to both factors of production. This, in turn, ensures that economic

29 The dynamical system (T4.1)—(T4.3) can be linearized around the initial steady state:

[k(t) [ r* — n —1 ][k(t) — k*

where we have used the fact that r* r(k*) — 8. The determinant of the Jacobian matrix on the righthand side is I A I . (r* — n)(r* — p) + c*f"(k*)— np = r* [r* — (p + n)] + c*f"(k*) < 0, where the sign follows from the fact that p < r* < p + n (see (14.136)). The characteristic roots of A are —Ai < 0 and A2 > 0, respectively. Since tr(A) --= A2 — Al = 2t* — (p + n) we find that A2 — (r* — p) = Al + (r* — n) > 0, where the final inequality follows from the assumption of dynamic efficiency (r* > n).

growth eventually se state capital-labour given population g, As was pointed out and are certainly d the production funct alone, an investigati conditions seems a realm of so-called "e

14.6.1 "Capitall

The aspect of trad, to its exogenously gi capital. As k(t) rises,

d [f (k(t))/k(t)] dk(t) 1

where the term in s, is positive (see (14.; steady-state capital- sy(t) 1 k(t) and (5 + ► by l'HOpital's rule

lim f (k(t)) - k(t)—>o k(t)

lim f (k(t)) = k(t)-÷oo k(t)

Equation (14.144) S ratio becomes very 1, capital-labour ratio

I

Easy substitution be

As was already weL tion functions w' production funcu

3° See the symposi

Perspectives. See also 1: 31 See e.g. Burmeist,:

re 14.12. In contrast on but also the capital

(14.139)

(14.140)

umption, capital per appearing in (14.139)-

1 ,

Is at impact from E0 to e impact period:

(14.141)

the linearized system bear the full burden eventually expand the +i on by too little and r at its old level. Over bigenerations are born e. This explains why

Pticture which satisfies Although these coney are not innocuous stence of diminishnsures that economic

-0 steady state:

P

hian matrix on the right- 0, where the sign follows are -h < 0 and

7. Al + (r* - n) > 0, where

r7).

Chapter 14: Theories of Economic Growth

growth eventually settles down to a constant. In terms of Figure 14.4, the steadystate capital—labour ratio is constant and growth equals the sum of exogenously given population growth and technological progress (see equation (14.16)).

As was pointed out above, the Inada conditions have no obvious intrinsic appeal and are certainly difficult to test empirically since they deal with the curvature of the production function for very low and very high levels of capital. For this reason alone, an investigation of the consequences of abandoning (some of) the Inada conditions seems a worthwhile endeavour. As it turns out, this brings us into the realm of so-called "endogenous growth" models. 3° •

14.6.1 "Capital-fundamentalist" models

The aspect of traditional growth models which ensures that growth settles down to its exogenously given steady-state rate is the existence of diminishing returns to capital. As k(t) rises, the average product of capital falls:

where the term in square brackets denotes the marginal product of labour, which is positive (see (14.74)). This is not enough to ensure the existence of a constant steady-state capital—labour ratio, however, because this requires equality between sy(t)/k(t) and (3+ n) in the Solow model. Provided (P2) and (P3) hold, we can derive by l'HOpital's rule that:

Equation (14.144) shows that sy(t)/k(t) goes to zero (infinity) as the capital—labour ratio becomes very large (small). This ensures the existence of a constant steady-state capital—labour ratio and thus a balanced growth path.

Easy substitution between capital and labour

As was already well known in the 1960s, 31 there are perfectly legitimate production functions which violate the results in (14.143)—(14.144). Consider the CES production function given in (14.24), for which the average product of capital

3° See the symposium on new growth theory in the Winter 1994 issue of the Journal of Economic Perspectives. See also Barro and Sala-i-Martin (1995).

31 See e.g. Burmeister and Dobell (1970, pp. 30-36), and indeed Solow (1956).

449

The Foundation of Modern Macroeconomics

equals:

It is clear from this expression that two separate cases must be distinguished, depending on the ease with which capital and labour can be substituted in production. If substitution is difficult (so that 0 < am, < 1) then the average product of capital satisfies:

The average product of capital goes to zero as more and more capital is added but near the origin it attains a finite value, i.e. while (14.144) is still satisfied (14.143) no longer holds. It is therefore not even guaranteed that the average product of capital around the origin is high enough to exceed so that a situation as illustrated in Figure 14.13 is a distinct possibility. (In that figure, we assume that 0 < aKL < 1 and saaKI, OWL -1) < An economy characterized by Figure 14.13 would never be able to accumulate any capital nor would it be able to produce any output (as both product factors are essential in production). Alternatively, if this economy were to start out with the initial capital-labour ratio k o (say because it

Figure 14.13. Difficult substitution between labour and capital

experienced a

;..e origin.

Matters are - exceeds

lim - ° km-0

f

lim -

The average p t it appre..._

no longer hot

(6 + n)/s), so

situation i

sestaliata. - 1)

- houtbou: long run the r

Lion (am_ > 1►.

01 effective) simply substit

of the capita.

Y * =So'

This growth r

n, paramL stark contrast

sections 2-3 It is not din,

nous growth. Aare of cap.. endogenous g

odels

and in the lin, is, of course,

The AK model

An even moil

'AK" model most ruuair

Lion function

(14.145)

-list be distinguished, be substituted in prothe average product of

(14.146)

(14.147)

capital is added but still satisfied (14.143) e average product of s), so that a situation figure, we assume that - ized by Figure 14.13 be able to produce any ). Alternatively, if this atio k0 (say because it

k (t)

n)k(t)

Our

Chapter 14: Theories of Economic Growth

experienced a higher savings rate in the past), then it would slowly decline towards the origin.

Matters are radically different if capital can be easily substituted for labour, i.e. if an exceeds unity. In that case, the average product of capital satisfies:

The average product of capital starts out very high (as the Inada conditions require) but it approaches a positive limit as more and more capital is added, i.e. (14.144) no longer holds. It follows that the average product of capital may not fall below ((8 + n)/ s), so that a steady-state capital-labour ratio may not exist. This is indeed the situation illustrated in Figure 14.14. (In that figure, we assume that aKL > 1 and seKL (GKL -1) > 8 + n.) Starting from an initial value k0, the capital-labour ratio grows without bounds. Despite the fact that there are diminishing returns to capital, in the long run the production factors are very much alike and substitute well in production (aKL > 1). This means that if capital grows indefinitely the constant growth rate of (effective) labour never becomes a binding constraint. Relatively scarce labour is simply substituted for capital indefinitely. The long-run "endogenous" growth rate of the capital-labour ratio and the output-labour ratio is:

y* = saaKL 1(aKL -1) - (8 + n) > 0. (14.150)

This growth rate is called "endogenous" because it is affected not only by exogenous parameters (a, 8, and n) but also by the savings rate (s) , a result which is in stark contrast to the predictions of the standard Solow-Swan model discussed in sections 2-3 above.

It is not difficult to understand that with this kind of labour-substituting endogenous growth, labour becomes less and less important and eventually the income share of capital goes to unity and that of labour goes to zero. This is why this endogenous growth model is an example of the "capital-fundamentalist" class of models (King and Levine, 1994). With cr ia, > 1, labour is not essential in production and in the limit it is possible to produce with (almost) only capital. This prediction is, of course, at odds with the stylized facts (SF3) and (SF5).

The AK model

An even more radical example of a capital-fundamentalist model is the so-called "AK" model proposed by Romer (1986), Barro (1990), Rebelo (1991), and others. In its most rudimentary form, the AK model eliminates (raw) labour from the production function altogether and assumes constant returns to scale on a broad measure

451

The Foundation of Modern Macroeconomics

(t)

sf (k (t))— (6 + n) k (t)

k0 k (t)

Figure 14.14. Easy substitution between labour and capital

of capital. Hence, equation (14.6) is replaced by:

Y(t) = AK(t), (14.151)

which of course clearly violates the Inada conditions. The first task at hand is to derive the behavioural equations of firms and households under the technology (14.151).

Following the analysis in section 5.2, we assume that the representative (perfectly competitive) producer chooses its output and investment plans in order to maximize the discounted value of its cash flows, taking the production function (14.151) and the capital accumulation identity (14.4) as given. After some manipulation we find that the rental rate of capital depends on the constant marginal product of capital

(A) and both the level and the time change in the investment subsidy.

The representative household is assumed to be infinitely lived. As in section 5.1, it maximizes lifetime utility subject to its accumulation identity:

where a is the constant intertemporal substitution elasticity (see (14.65)), tc is a consumption tax, Z(t) is a lump-sum transfer from the government (or tax if it is

negative), A(t) rep' analytical meths...

equation can be de

4

C(t)C(t) = a [r(t

Since the model the only financial

A(t) = K(t). The

in Table 14.5. Equations (TS.1►

accumulation ide production functic tence of perpetual case for which bu „ by agents to be) co rate is constant a: At the same time. to show that the is optimal for the consumption and Hence, the comr._ investment is give!

C(t)

Y = C(t) =

The striking conch affected by the ir. growth model disc

Table 14.5. The Li,

t(t) = Q [r(t) — p —

r = (A — 6)K (t) —

tqt)— —

Notes: C(t) is consw investment subsidy, p

32 The key point to r

growth rate as this 0 ,,

33 Since there is n, wealth.