Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

1 (17.138) we can derive the

(17.142)

ational activities (left-hand )cial benefits of these activicosts are just the value of 'efits consist of three terms.

n the expression C(Et)fP- t3 , on educational activities in

,+ line on the right-hand side )n the parent's utility. This privately optimal (internal) and third lines show the account in determining the ,'nts the effect of the parent's

.1 child with more human and a higher wage. The third n the children's incentives

arent's grandchildren). petitive allocation is subopti-

garding the lifetime utility ntuitively, this result obtains Sting their children (1994, man capital investment pro-

. - h that the parent's decision the effect on the children's ue, it is not likely that such a Id. For that reason, the instimay well achieve a welfare

• mposes a minimal level of

if)), macroeconomists have 'portant factor determining

Chapter 17: Intergenerational Economics, TI

the productive capacity of an economy. Somewhat surprisingly, however, the public capital stock has played only a relatively minor role in the literature up until recently. This unfortunate state of affairs changed dramatically a decade ago when the pathbreaking and provocative empirical research of Aschauer (1989, 1990b) triggered a veritable boom in the econometric literature on public investment (see Gramlich, 1994 for an excellent survey of this literature). Aschauer (1989) showed that public capital exerts a strong positive effect on the productivity of private capital and argued that the slowdown in productivity growth in the US since the early 1970s is due to a shortage of investment in public infrastructure. Indeed, his estimates suggest implicit rates of return on government capital of 100% or more, values which are seen as highly implausible by many commentators (see e.g. Gramlich, 1994, p. 1186). Although Aschauer's results were controversial and many subsequent studies have questioned their robustness, it is nevertheless fair to conclude that economists generally support the notion that public capital is indeed productive.

In this subsection we show how productive public capital can be introduced into the Diamond—Samuelson model. We show how the dynamic behaviour of the economy is affected if the government adopts a constant infrastructural investment policy. Finally, we study how the socially optimal capital stock can be determined. To keep things simple we assume that labour supply is exogenous, and that the government has access to lump-sum taxes. We base our discussion in part on Azariadis (1993, pp. 336-340).

Prototypical examples of government capital are things likes roads, bridges, airports, hospitals, etc., which all have the stock dimension. Just as with the private capital stock, the public capital stock is gradually built up by means of infrastructural investment and gradually wears down because depreciation takes place. Denoting the stock of government capital by Gt we have:

Gt+i — Gt = It -3GGt, (17.143)

where ./t9 is infrastructural investment and 0 < SG < 1 is the depreciation rate of public capital. Assuming that the population grows at a constant rate as in (17.21), per capita public capital evolves according to:

where gt Gt /Lt and

We assume that public capital enters the production function of the private sector, i.e. instead of (17.10) we have:

where we assume that F() is linearly homogeneous in the private production factors, Kt and L. This means that we can express per capita output (yt Yt/Lt) as follows:

The Foundation of Modern Macroeconomics

Private and public capital both feature positive (property (P1)) but diminishing marginal productivity (property (P2)). Both types of capital are essential in production, i.e. output is zero if either input is zero (property (P3)). Finally, properties (P4)-(P5) ensure that public capital is complementary with both private capital and labour. This last implication can be seen by noting that perfectly competitive firms hire capital and labour according to the usual rental expressions rt + 8 = FK (Kt, gt) and Wt = FL(Kt, Lt , gt). These can be expressed in per capita form as:

Wt = W(kt ,gt ) f(kt,gt ) - ktfk(kogt), (17.148)

where 0 < 6 < 1 is the depreciation rate of the private capital stock. We can deduce from Properties (P4)-(P5) that rk - arIakt < 0 and Wk aw lakt > 0 (as in the standard model) and rg ar/agt > 0 and Wg awiagt > 0 (public capital positively affects both the interest rate and the wage rate). To illustrate the key properties of the model we shall employ a simple Cobb-Douglas production function below of the form Yt = K LEtl s't1 , with 0 < < EL . This function satisfies properties (P1)-(P5)

and implies W(kt,gt) = ELkti-eL4 and r(kt, gt) = (1 -

To keep things simple, we assume that the representative young agent has the

where TI and TtPF1 are lump-sum taxes paid by the agent during youth and old age respectively. The consolidated budget constraint is:

where Wt is after-tax no are Cr = ciNt and C(,) 1 function can then be wr

I

St S(Wt,rt+i,Tl , 1

It follows that, ceteris whilst taxes during old is next period's stock of

St = (1 + n)kt-4-1-

The government budl infrastructural invest'; old, i.e. It = LtTtY + L1-1

We now have a com: accumulation identity constraint (17.155), and can be written in the fol (17.154): I

(1 + n)kt+ i =(1 -

Once a path for public i (17.144) and (17.156) d capital stocks. We derive and a constant public i taxes on only consequences of alter' for the reader.

The phase diagram hi representation of (17.1-1 along the line we have steady-state equilibriu The dynamics for publ.

from which we conclude the public capital stock 1 pattern has been illustra

make the following set of

(P1)

(P2)

(P3)

(P4)

(P5)

:lefty (P1)) but diminishing ▪ 9ital are essential in proxirty (P3)). Finally, properties with both private capital and

• perfectly competitive firms vressions rt +6 = FK(Kt, Lt,

form as:

(17.147)

(17.148)

apital stock. We can deduce aw lakt > 0 (as in the 0 (public capital positively

ate the key properties of the (lion function below of the satisfies properties (P1)—(P5)

■CEL g71 — 6.

ative young agent has the

(17.149)

(17.150)

(17.151)

t during youth and old age

(17.152)

Chapter 17: Intergenerational Economics, II

where Wt is after-tax non-interest lifetime income. The optimal household choices

It follows that, ceteris paribus, lump-sum taxes during youth reduce private saving whilst taxes during old age increase saving. As before, private saving by the young is next period's stock of private capital, i.e. LtSt = Kt± i. In per capita form we have:

St = (1 + n)kt-Fi. (17.154)

The government budget constraint is very simple and states that government infrastructural investment (Is) is financed by tax receipts from the young and the old, i.e. It = Lt TI + Lt_i T° which can be written in per capita form as:

We now have a complete description of the economy. The key expressions are the accumulation identity for the public capital stock (17.144), the government budget constraint (17.155), and the accumulation expression for private capital. The latter can be written in the following format by using (17.147), (17.148), and (17.153) in (17.154):

Once a path for public investment and a particular financing method are chosen, (17.144) and (17.156) describe the dynamical evolution of the public and private capital stocks. We derive the phase diagram for the case of Cobb—Douglas technology and a constant public investment policy (so that it = iG for all t) financed by taxes on only the young generations (so that 77 = iG and T° = 0 for all t). The consequences of alternative assumptions regarding financing are left as an exercise for the reader.

The phase diagram has been drawn in Figure 17.6. The GE line is the graphical representation of (17.144) for the constant public investment policy

along the line we have gt+ i = gt . The GE line is horizontal and defines a unique steady-state equilibrium value for the stock of public capital equal to g = iG I (n + The dynamics for public capital are derived from the rewritten version of (17.144):

from which we conclude that for points above (below) the GE line, gt > g g) and the public capital stock falls (rises) over time, gt+i < This (stable) dynamic pattern has been illustrated with vertical arrows in Figure 17.6.

635

The Foundation of Modern Macroeconomics

Figure 17.6. Public and private capital

The KE line in Figure 17.6 is the graphical representation of (17.156), with the constant investment policy and the financing assumption both substituted in and

imposing the steady state, kt+1 = kt . For the Cobb-Douglas technology, the KE line has the following form:

Hence, the KE line is as drawn in Figure 17.6. There are two steady-state equilibria

(at A and E0, respectively). The dynamics of the private capital stock are obtained by rewriting (17.156) as:

1 - c\

kt+ 1 - kt = (1 +0 V ) (17.160)

114-EL4 iG

and noting that a[kt+ i—kt]/agt > 0. Hence, since the wage rate increases with public capital and future consumption is a normal good, private saving increases with gt . Hence, the capital stock is increasing (decreasing) over time for points above (below)

the KE line. These dynamic forces have been illustrated with horizontal arrows in Figure 17.6.

I

It follows from the confi analysis of the linearized rn saddle point whereas the hi, the latter equilibrium it he public capital, provided L. return to Eo.

What about the steady-s typical encounters that we saddle-point equilibria, we one predetermined and predetermined variable jui example, in Chapter 4 we that K and q are, respecti' present application, howev can jump. Only if the init...

happen to lie on the saddle ally be reached given the L Appealing to the Samuelsoi remainder of this subsec

Now consider what hai r. follows from, respectively up. Clearly, the higher put of public capital, i.e. dg capital stock is ambiguous imposing the steady stated

1- 1 - c Wk

1+n

where the term in square b is outright stable around t square brackets on the rii; wage of the young hou ,

Since Wg = 71147/g, W = E steady-state private capita (> ?IQ), i.e. if public cap..,

16 Recall that for a constan - condition is satisfied around tri,

0 < akkt+, 1 1' (nc Wi <

KE: kt+1 = kt

GE: gt±i =gt

k kt

`ion of (17.156), with the on both substituted in and 'his technology, the KE line

(17.158)

I-\- and that gt reaches its ' is defined as:

(17.159)

•'•vo steady-state equilibria capital stock are obtained

(17.160)

e rate increases with public to saving increases with gt. me for points above (below)

'h horizontal arrows in

Chapter 17: Intergenerational Economics, II

It follows from the configuration of arrows (and from a formal local stability analysis of the linearized model) that the low-private-capital equilibrium at A is a saddle point whereas the high-private-capital equilibrium at E0 is a stable node. For the latter equilibrium it holds that, regardless of the initial stocks of private and public capital, provided the economy is close enough to E0 it will automatically return to E0.

What about the steady-state equilibrium at A? Is it stable or unstable? In the typical encounters that we have had throughout this book with two-dimensional saddle-point equilibria, we called such equilibria stable because there always was one predetermined and one non-predetermined variable. By letting the nonpredetermined variable jump onto the saddle path, stability was ensured. For example, in Chapter 4 we studied Tobin's q theory of private investment and showed that K and q are, respectively, the predetermined and jumping variables. In the present application, however, both K and G are predetermined variables so neither can jump. Only if the initial stocks of private and public capital by pure coincidence happen to lie on the saddle path (SP in Figure 17.6), will the equilibrium at A eventually be reached given the constant investment policy employed by the government. Appealing to the Samuelsonian correspondence principle we focus attention in the remainder of this subsection on the truly stable equilibrium at Eo

Now consider what happens if the government increases its public investment. It follows from, respectively (17.15 7) and (17.158), that both the GE and KE lines shift up. Clearly, the higher public investment level will lead to a higher long-run stock of public capital, i.e. dg 1 cliG = 1/(n + SG) > 0. The long-run effect on the private capital stock is ambiguous and depends on the relative scarcity of public capital. By imposing the steady state in (17.160) and differentiating we obtain:

where the term in square brackets on the left-hand side is positive because the model is outright stable around the initial steady-state equilibrium E0 . 16 The first term in square brackets on the right-hand side represents the positive effect on the pre-tax wage of the young households whilst the second term is the negative tax effect. Since Wg = riW 1g , W = ELY, and g = iG + 8G), it follows from (17.161) that the steady-state private capital stock rises (falls) as a result of the shock if iG ly < riEL

i.e. if public capital is initially relatively scarce (abundant).

16 Recall that for a constant level of public capital, the model is stable provided the following stability condition is satisfied around the initial steady state, Eo:

The Foundation of Modern Macroeconomics

Modified golden rules

Now that we have established the macroeconomic effects of public capital, we can confront the equally important question regarding the socially optimal amount of public infrastructure. Just as in the previous subsection on education, we study this issue by computing the public investment plan that a social planner would choose. Following Calvo and Obstfeld (1988, p. 414) and Diamond (1973, p. 219) we assume that the social welfare function takes the following Benthamite form: 17

where we assume that pG > n. Equation (17.162) is a special case of (17.131) with the generational weight set equal to X t [(1 + n)/(1 + pG )] t . This means that the social planner discounts the lifetime utility of generations at a constant rate pG which may or may not be equal to the rate employed by the agents to discount their own periodic utility (namely p). The social planner chooses sequences for consumption for young and old ({ Cr}t. 0 and {CN° 0), the per capita stocks of public and private capital (Igt+11;t 0 and Ikt+ 1 }710), in order to maximize (17.162) subject to the following resource constraint:

and taking as given ko and go. The Lagrangean associated with the social optimization problem is given by:

where bet is the Lagrange multiplier associated with the resource constraint.

17 This name for the social welfare function derives from the classical economist Jeremy Bentham (1748-1832) who argued that "it is the greatest happiness of the greatest number that is the measure of right and wrong" (quoted by Harrison, 1987, p. 226). This explains why the rate of population growth enters (17.162).

After some manipulation optimum for t = 0, ,

multipliers we find some

I

a AY Citocr

anY 00/act+1

anY(Sct)/acr _ 1 ___

anY(k_i)/ac°

where hatted variables ity in (17.169) is the sod equalization of, on the on and future consumption est factor, 1 + rt+i , where says that the socially (- that the yields on privd:t in such a way that TrG 1 = (17.170) determines the Its intuitive meaning, am planner's discount rate. porally separable preferci using Ar(xt ) U(C1) + agent's felicity function

It follows from (17.171) agent's rate of time pret U'(ar) exceeds (falls sh, (exceeds) as 3'. If pG = p, tt

of public capital, we can iocially optimal amount of 'n education, we study this al planner would choose. id (1973, p. 219) we assume

"- amite form: 17

(17.162)

wcial case of (17.131) with d t . This means that the , ns at a constant rate pG

by the agents to discount er chooses sequences for Le per capita stocks of public maximize (17.162) subject

!cl with the social optimiza-

C4t+1)

k!, go

(17.164)

resource constraint.

gal economist Jeremy Bentham ,t number that is the measure of v the rate of population growth

Chapter 17: Intergenerational Economics, II

After some manipulation we find the following first-order conditions for the social optimum for t = 0, , 00:

where xt {Cr, c?+1] . By combining (17.165)-(17.168) to eliminate the Lagrange multipliers we find some intuitive expressions characterizing the social optimum:

where hatted variables once again denote socially optimal values. The first equality in (17.169) is the socially optimal consumption Euler equation calling for an equalization of, on the one hand, the marginal rate of substitution between present and future consumption and, on the other hand, the socially optimal gross inter-

est factor, 1 + rt+ 1, where rt± i fk(kt-pi,kt+i) - S. The second equality in (17.169) says that the socially optimal stock of public capital per worker should be such that the yields on private and public capital are equalized, i.e. gt± i should be set

in such a way that rtG+i = rt+ i, where rtG+1 fg(kt+i,k+i) - 5G. Finally, equation (17.170) determines the socially optimal intratemporal division of consumption.

Its intuitive meaning, and especially the interplay between the agent's and the planner's discount rate, can best be understood by considering the case of intertemporally separable preferences (which has been used throughout this chapter). By using U(CI) + (1 p) -1 U(Ct°±1 ) we can rewrite (17.170) in terms of the agent's felicity function (U(.)) and the pure rate of time preference (p):

It follows from (17.171) that if the planner's discount rate exceeds (falls short of) the

agent's rate of time preference, PG > p (< p), then the social planner ensures that U'(4) exceeds (falls short of) U'(a°t ), and thus (since U" < 0) that Ct falls short of

(exceeds) If pG = p, the planner chooses the egalitarian solution ( = Cr) .

639

Calvo and Obstfeld (1988) have shown that with intertemporally separable preferences, the social planning problem can be solved in two stages. In the first stage, the planner solves a static problem and in the second stage a dynamic problem is solved. Their procedure works as follows. Aggregate consumption at time r, expressed per worker, is defined as:

With intertemporally separable preferences (and ignoring a constant like U(CY )) the social welfare function in period t can be rewritten as:

where the term in square brackets in (b) now contains the weighted felicity levels of old and young agents living in the same time period. The special treatment of period-t felicity of the old is to preserve dynamic consistency (see the Intermezzo above). We can now demonstrate the two-step procedure.

In the first step, the social planner solves the static problem of dividing a given level of aggregate consumption, Cr, over the generations that are alive at that time:

where U"(Cr ) is the (indirect) social felicity function. The first-order condition associated with this optimization problem is:

which is the same as (17.171). Furthermore, by differentiating (c) and using

(a) and (d) we find the familiar envelope property:

640

For the special case of is U(x) log x and th.

U(C,) = 1 4,1_,

(

(1 -

- coo + -H

In the second step th. tion and the two tyix..

subject to the initial a

Cr + ( 1+ n) [k, ,

where we have used multiplier for the res , order conditions:

(1+ n)p R, LLR+1 =

=

By using (j) for period with (17.169).

We now return to the the steady state. In the si xt = X for all t so that (17

Equation (17.172) calls the old. The first equality steady-state yield on t i the rate of time preferen

Id (1988) have shown ,(3c1 , 1 planning problem [ler solves a static problem

--heir procedure works as

-worker, is defined as:

(a)

ring a constant like - ewTitten as:

:s the weighted felicity period. The special namic consistency (see

-step procedure. problem of dividing a nerations that are alive

e first-order condition

(d)

- -tiating (c) and using

(e)

Chapter 17: Intergenerational Economics, II

For the special case of logarithmic preferences, for example, individual felicity is U(x) log x and the social felicity function would take the following form:

In the second step the social planner chooses sequences of aggregate consumption and the two types of capital in order to maximize social welfare:

subject to the initial conditions (kt and gt given) and the resource constraint:

where we have used (a) in (17.163) to get (h). Letting pR denote the Lagrange multiplier for the resource constraint in period t we obtain the following firstorder conditions:

By using (j) for period t + 1 and noting (d) and (e) we find that (i) coincides with (17.169).

We now return to the general first-order conditions (17.169)-(17.170) and study the steady state. In the steady state we have cr = cY, = C°, kt = k, gt = g, and xt = x for all t so that (17.169)-(17.170) simplify to:

Equation (17.172) calls for an optimal division of consumption over the young and the old. The first equality in (17.173) is the modified golden rule (MGR) equating the steady-state yield on the private capital stock (the steady-state rate of interest) to the rate of time preference of the social planner. There is an important difference

641

The Foundation of Modern Macroeconomics

between this version of the MGR and the one encountered in Chapter 14 in the context of the Ramsey representative-agent model. In the OLG setting, the pranner's rate of time preference features in the MGR whereas in the Ramsey model the representative agent's own rate of time preference is relevant (compare (17.173) with (14.78)).

The second equality in (17.173) is a modified golden rule for public capital that was initially derived by Pestieau (1974). It calls for an equalization of the public rate of return and the planner's rate of time preference. The two equalities in (17.173) together determine the optimal per worker stocks of public and private capital. For example, for Cobb-Douglas technology we have yt = g't1 (with /7 < EL ) so that

k/y = (1 - EL )/(PG + 1), g/y = 171(PG + SG). It follows from these results that output per worker is:

Now that we have characterized the necessary conditions for the steady-state social optimum, a relevant question concerns the decentralization of this optimum. Can the policy maker devise a set of policy tools in such a way that the private sector choices concerning consumption and private capital accumulation coincide exactly with their respective values in the social optimum? The answer is affirmative provided the policy maker has access to the right kind of policy instruments. In the present context, for example, the first-best social optimum can be mimicked in the market place if (i) the level of public investment (and thus the public capital stock) is chosen to be consistent with (17.173), and (ii) there are age-specific lump-sum taxes available (see Pestieau, 1974 and Ihori, 1996, p. 114). The latter instrument is needed to ensure that the market replicates the socially optimal mix of consumption by the young and the old (cf. (17.172)).

17.3.3 Intergenerational accounting

One of the most hotly debated concepts in macroeconomic policy circles has been the correct definition and measurements of the government budget deficit. Throughout the book we have encountered several examples pointing at a fundamental ambiguity in the concept of the deficit. For example, in Chapter 6 we saw that government investment which yields the market rate of return represents no net liability of the government. Yet, depending on the way in which these expenditures are treated by the government's accountants, they either do or do not feature in the government budget deficit (in the standard accounting approach they are typically treated as government consumption and thus feature in the deficit).

Haveman (1994, p. 95) gives an impressive list of further items for which proposals have been made to change the way in which these items are treated in the government's accounting system. They include things like federal credit and loan

1 programmes, future comp changes in the value of It is fair to say that there i

Auerbach, Gokhale, an of the problem to date or pays out is labeled in doing away with the c instead on what they labl posal is the notion that ' is the intergenerational d that the intertemporal bu attention. In words, this ( plus the present value of generations (the genera ti value of the governmer,

Auerbach et al. (1991, generational accounts I-. ational accounts are inval zero-sum feature of the generation gets will hat . can be used to study the policies.

In this subsection we f tional accounts in a sin that the population is budget identity is given b

Bt+1 = (1 + rt)Bt +

where Tt° and TtY are the and GI are pure public c, charge to, respectively, assume that these public

forwards in time yields

I

1

B =

s=o 1 +t+,r)

Rt-1,TBr+T+1 -)

where Rt _ i ,, is a discou:

= fir ( 1

s=.