Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

(17.105)

(17.106)

i t of labour at time t, and ng) during youth. Since e have by assumption that fining during the second

by Et and equals:

(17.107)

lecision problem we must

•- st example of a training

(17.108)

ices the notion that there e production of human his initial skill level. rder to maximize lifetime

(17.105)-(17.107), and t of wages Wt , and its own ed in two steps. In the first T to maximize its lifetime

(17.109)

stment problem, taking

(17.110)

) 0

nterest of the agent not to !fling solution will hold n-increasing returns to ,Powing implication from

(17.111)

`hen the corner solution

Chapter 17: Intergenerational Economics, II

An internal solution with a strictly positive level of training is such that = 0. After some rewriting we obtain the investment equation in arbitrage

format:

This expression shows that in the interior optimum the agent accumulates physical and human capital such that their respective yields are equalized. By investing in physical capital during youth the agent receives a yield of 1 + rt±i during old age (left-hand side of (17.112)). By working a little less and training a little more during youth, the agent upgrades his human capital and gains Wt± iC(Eti ) during old age. Expressed in terms of the initial investment (foregone wages in the first period) we get the yield on human capital (right-hand side of (17.112)).

In the second step of the optimization problem the household chooses consumption for the two periods and its level of savings in order to maximize lifetime utility (17.104) subject to its lifetime budget constraint:

1 + rt±i

where 1 is now maximized lifetime income. The savings function which results from this stage of the optimization problem can be written in general form as:

In order to complete the description of the decision problem of household i we must specify its initial level of human capital at birth, i.e. H: in the training technology (17.108). Following Azariadis and Drazen (1990, p. 510) we assume that each household born in period t "inherits" (is born with) the average stock of currently available knowledge at that time, i.e. Ht = Ht on the right-hand side of (17.108). With this final assumption it follows that all individuals in the model face the same interest rate and learning technology so that they will choose the same consumption, saving, and investment plans. We can thus drop the individual index i from here on and study the symmetric equilibrium.

We assume that there is no population growth and normalize the size of the young and old populations to unity (14_1 = Lt = 1). Total labour supply in efficiency units is defined as the sum of efficiency units supplied by the young and the old, i.e.

For convenience we summarize the key expressions of the

The Foundation of Modern Macroeconomics

(simplified) Azariadis-Drazen model below.

Equation (17.115) relates saving by the representative young household to next period's stock of physical capital. Note that the capital-labour ratio is defined in terms of efficiency units of labour, i.e. kt Kt /Nt. With this definition, the expressions for the wage rate and the interest rate are, respectively (17.116) and (17.117). Equation (17.118) is labour supply in efficiency units, (17.119) is the investment equation for human capital (assuming an internal solution), and (17.120) is the accumulation for aggregate human capital in the symmetric equilibrium.

It is not difficult to show that the model allows for endogenous growth in the steady state. In the steady-state growth path the capital-labour ratio, the wage rate, the interest rate, and the proportion of time spent training during youth, are all constant over time (i.e. kr = k, Wt = W, rt = r, and Et = E). The remaining variables grow at a common growth rate y G(E) - 1. Referring the reader for a general proof to Azariadis (1993, p. 231), we demonstrate the existence of a unique steady-state growth path for the unit-elastic model for which technology is CobbDouglas (yt = 4') and the utility function (17.104) is log-linear (Al = log Cr + (1/(1 + p)) log C°±1 ). For the unit-elastic case the savings function can be written as:

By using (17.121), (17.118), and (17.120) in (17.115) and imposing the steady state we get an implicit relationship between E and k for which savings equals investment:

Similarly, by using (17.116) and (17.118) in the steady-state we get a second expression, again relating E and k, for which the rates of return on human and physical capital are equalized:

The joint determination of E and k in the steady-state growth path is illustrated in the upper panel of Figure 17.5. The portfolio-balance (PB) line is upward sloping because both the production technology and the training technology exhibit

k

h un

1

diminishing returns (f downward sloping with is downward sloping in t

k/W(k) = (1/EL)k€L wh in k. Together these res state is at E0. In the bott( of training. I The it nology (17.120) which e

a steady-state

cal skills are disembodi, they are passed on in then add to the stock of endogenous growth wou agents themselves. In thz "re-invent the wheel" t

(17.115)

(17.116)

(17.117)

(17.118)

(17.119)

)ung household to next -labour ratio is defined in is definition, the expresly (17.116) and (17.117). 17.119) is the investment )n), and (17.120) is the

,tric equilibrium. dogenous growth in the

labour ratio, the wage rate, ng during youth, are all i Et = E). The remaining Referring the reader for a e the existence of a unique rhich technology is Cobb- ' , Ig-linear (A = log Cr + function can be written as:

(17.121)

I imposing the steady state :vings equals investment:

(17.122)

ate we get a second expres- - on human and physical

(17.123)

-rowth path is illustrated e B) line is upward slopning technology exhibit

Chapter 17: Intergenerational Economics, II

Figure 17.5. Endogenous growth due to human capital formation

diminishing returns 0 and G"(E) < 0). The savings-investment (SI) line is downward sloping with Cobb-Douglas technology. The right-hand side of (17.122) is downward sloping in both k and E. With Cobb-Douglas technology we have that k/W(k) = (1/EL )kEL which ensures that the left-hand side of (17.122) is increasing in k. Together these result imply that SI slopes down. In the upper panel the steady state is at E0. In the bottom panel we relate the equilibrium growth rate to the level of training.

The engine of growth in the Azariadis-Drazen model is clearly the training technology (17.120) which ensures that a given steady-state level of training allows for a steady-state in the stock of human capital. Knowledge and technical skills are disembodied, i.e. they do not die with the individual agents but rather they are passed on in an automatic fashion to the newborns. The newborns can then add to the stock of knowledge by engaging in training. It should be clear that endogenous growth would disappear from the model if skills were embodied in the agents themselves. In that case young agents would have to start all over again and "re-invent the wheel" the moment they are born.

625

The Foundation of Modern Macroeconomics

Human capital and education

Whilst it is undoubtedly true that informal social interactions can give rise to the transmission of knowledge and skills (as in the Azariadis—Drazen (1990) model) most developed countries have had formal educational systems for a number of centuries. A striking aspect of these systems is that they are compulsory, i.e. children up to a certain age are forced by law to undergo a certain period of basic training. This prompts the question why the adoption of compulsory education has been so widespread, even in countries which otherwise strongly value their citizens' right to choose.

Eckstein and Zilcha (1994) have recently provided an ingenious answer to this question which stresses the role of parents in the transmission of human capital to their offspring. They use an extended version of the Azariadis—Drazen model and show that compulsory education may well be welfare-enhancing to the children if the parents do not value the education of their offspring to a sufficient extent. The key insight of Eckstein and Zilcha (1994) is thus that there may exist a significant intra-family external effect which causes parents to underinvest in their children's human capital. Note that such an effect is not present in the Azariadis—Drazen model because in that model the agent himself bears the cost of training during youth and reaps the benefits during old age.

We now develop a simplified version of the Eckstein—Zilcha model to demonstrate their important underinvestment result. We assume that all agents are identical. The representative parent consumes goods during youth and old age (Cr and Ct°+1 , respectively), enjoys leisure during youth (Zt ), is retired during old age, and has 1+ n children during the first period of life. Fertility is exogenous so that the number of children is exogenously given (n > 0). The lifetime utility function of the young agent at time t is given in general form as:

Al .AY(ci,Cto+1 ,zt,ot+i), (17.124)

where Ot± i ------ (1 + n)Ht± i represents the total human capital of the agent's offspring. Since the agent has 1 n kids, each child gets Ht± i in human capital (knowledge) from its parent. There is no formal schooling system so the parent cannot purchase education services for its offspring in the market. Instead, the parent must spend (part of its) leisure time during youth to educate its children and the training function is given by:

Ht+i = G(Et)Hr, , (17.125)

where Et is the educational effort per child, G(.) is the training curve (satisfying 0 < G(0) < 1, G(1) > 1, G' > 0 > G") and 0 < /3 < 1. Equation (17.125) is similar in format to (17.120) but its interpretation is different. In (17.120) lit+ i and Et are chosen by and affect the same agent. In contrast, in (17.125) the parent chooses Ht+ i and Et and the consequences of this choice are felt by both the parent and his/her offspring.

The agent has two uni inelastically to the labs of which is spent on leis

Zt + (1 + n)Et = 1. 1

The household's consul

where the left-hand sic hand side is labour incoi of labour, Nt LtHt , fr

Yr = F (Kt , Nt). The wage and rt + 8 = FK(Kr, NO-

The representative r . mize lifetime utility (17 . constraint (17.126), and ing the constraints into remaining choice variab conditions: -

aAY baci

BAY iaq+1 = 1 + r

Equation (17.128) is 0: _ tered time and again, an activities of the parent. T the net marginal bent::., benefits this term is nt activities at all (see (17.1 implies that the net mark assume that conditions A notable feature 01 .. only contains the costs

receives a higher level labour income and will d tion, however, the par ,. children and therefore I offspring. This is the

ions can give rise to the 3dis-Drazen (1990) model)

. systems for a number of ire compulsory, i.e. children in period of basic training.

-v education has been so

vvalue their citizens' right

n ingenious answer to this niccion of human capital to triadis-Drazen model and chancing to the children if a sufficient extent. The may exist a significant L-rinvest in their children's e Azariadis-Drazen model I training during youth and

model to demonstrate t all agents are identical. And old age (Cr and Ct°±1 , !'t ring old age, and has 1+n ,us so that the number of _:v function of the young

(17.124)

ital of the agent's offspring. •• man capital (knowledge) so the parent cannot pur-

. Instead, the parent must s children and the training

(17.125)

-wining curve (satisfying ivation (17.125) is similar ^ (17.120) Ht+ 1 and Et are 7.125) the parent chooses t by both the parent and

Chapter 17: Intergenerational Economics, II

The agent has two units of time available during youth, one of which is supplied inelastically to the labour market (Eckstein and Zilcha, 1994, p. 343), and the other of which is spent on leisure and educational activities:

The household's consolidated budget constraint is of a standard form:

where the left-hand side represents the present value of consumption and the righthand side is labour income. Competitive firms hire capital, Kt , and efficiency units of labour, Nt LtHt, from the households, and the aggregate production function is Yr = F(KoNt). The wage and interest rate then satisfy, respectively,

and

The representative parent chooses Cr, q+1 , Zt , Et , and Ht± i in order to maximize lifetime utility (17.124) subject to the training technology (17.125), the time constraint (17.126), and the consolidated budget constraint (17.127). By substituting the constraints into the objective function and optimizing with respect to the remaining choice variables (Cr, c?+1 , and Et) we obtain the following first-order conditions:

Equation (17.128) is the standard consumption Euler equation, which we encountered time and again, and (17.129)-(17.130) characterizes the optimal educational activities of the parent. The left-hand side appearing in (17.129)-(17.130) represents the net marginal benefit of child education. If the (marginal) costs outweigh the benefits this term is negative and the parent chooses not to engage in educational activities at all (see (17.129)). Conversely, a strictly positive (interior) choice of Et implies that the net marginal benefit of child education is zero. In the remainder we assume that conditions are such that Et > 0 is chosen by the representative parent.

A notable feature of the parent's optimal child education rule (17.130) is that it only contains the costs and benefits as they accrue to the parent. But if a child receives a higher level of human capital from its parents, then it will have a higher labour income and will thus be richer and enjoy a higher level of welfare. By assumption, however, the parent only cares about the level of education it passes on to its children and therefore disregards any welfare effects that operate directly on its offspring. This is the first hint of the under-investment problem. Loosely put, by

627

The Foundation of Modern Macroeconomics

disregarding some of the positive welfare effects its own educational activities have on its children, the parent does not provide "enough" education.

As was explained above, in our discussion regarding pension reform, there are several ways in which we can tackle the efficiency issue of under-investment in a more formal manner. One way would be to look for Pareto-improving policy interventions. For example, in the present context one could investigate whether a system of financial transfers to parents could be devised which (a) would induce parents to raise their child-educational activities and (b) would make no present or future generation worse off and at least one strictly better off. If such a transfer system can be found we can conclude that the status quo is inefficient and there is underinvestment.

An alternative approach, one which we pursue here, makes use of a social welfare function. Following Eckstein and Zilcha (1994, pp. 344-345) we postulate a specific form for the social welfare function (17.65) which is linear in the lifetime utilities of present and future agents:

After some manipulatio optimum for t = 0, ,

= At anY ac°_,

a/0' = azt

where µR, pi, and Alt-1 are the Lagrange multipliers associated with, respectively, the resource constraint, the time constraint, and the training technology.

Dynamic Consistency when using a social v importance to the sou

.41111iiiF

educational activities have ducation.

pension reform, there are of under-investment in KPareto-improving policy LouId investigate whether ecl which (a) would induce 5) would make no present letter off. If such a transfer o is inefficient and there is

• 1<es use of a social welfare

345) we postulate a specific it in the lifetime utilities

(17.131)

0► and {MN is a positive the different generations social planner chooses itocks of human and phys- ' "qrt ({EdN) in order to

- .125), the time constraint

Chapter 17: Intergenerational Economics, II

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

thus represents the socially optimal interest rate. Similarly, by using (17.134) for period t + 1 and (17.135) we obtain an expression determining the socially optimal division of consumption between old and young agents living at the same time:

• problem is given by:

1 - 8)kt]

! with, respectively, the g technology.

This expression shows that, by adopting a particular sequence of generational weights {X t }t'c o , the social planner in fact chooses the generational consumption profile between the young and the old (see Calvo and Obstfeld, 1988, p. 417).

Intermezzo

Dynamic Consistency. There are some subtle issues that must be confronted when using a social welfare function like (17.131). If we are to attach any importance to the social planning exercise we must assume that either one of

The Foundation of Modern Macroeconomics

the following two situations holds:

Commitment the policy maker only performs the social planning exercise once and can credibly commit never to re-optimize. Economic policy is a one-shot event and no further restrictions on the generational weights are needed.

Consistency the policy maker can re-optimize at any time but the generational weights are such that the socially optimal plan is dynamically consistent, i.e. the mere evolution of time itself does not make the planner change his mind.

This intermezzo shows how dynamic consistency can be guaranteed in the absence of credible commitment. We study dynamic consistency in the context of the standard Diamond-Samuelson model. The social welfare function in the planning period 0 is given in general terms by:

where A0,, is the weight that the planner in time 0 attaches to the lifetime utility of the generation born in period r (for T -1, 0, 1, 2, ...). The social planner chooses sequences for consumption during youth and old age ({C'}°° 0 and {qr.', 0 ) and the capital stock (fkr+i lc', 0) in order to maximize social welfare (a) subject to the resource constraint:

and taking the initial capital stock, ko, as given. Obviously, since past things cannot be undone, consumption during youth of the initially old generation (CY , ) is also taken as given. After some straightforward computations we find the following first-order conditions characterizing the social optimum:

where xr [Cr, C°r 4_1 1 and hats denote socially optimal values.

Now consider a planner who performs the social planning exercise at some

where 1,t,, is the weight that the planner in time t attaches to the lifetime utility of the generation born in period r (for r = t - 1, t, t + 1, t + 2, . ..). The social

planner chooses segue and {C°r }°°, t ) and the

(e) subject to the res,, consist of (c) and:

aAY(50/acr.

8AY(ir _1 )laq

The crucial thing to r interval T = t, t + 1, t -I chosen by the plant-,.. by the planner at ti m( chosen by both plan (and thus can stop social welfare functi - at time t will not nt, social plan dynamical]

Following the in _ on the admissible p. optimal social plan is that dynamic consi)...

planning period t:

AO,r -1

At,r AO,T

Condition (g) means and the planning date g is some function o; function is:

( 1

), t,r= 1 + A )

where 0 is the p the weight attached to follows necessarily, t be reverse discountit. Indeed, the dynami,

= 1 + so ;, apply this notion of re model of overlapping

i pi -inning exercise once c policy is a one-shot -ights are needed.

me but the generational nically consistent, i.e. inner change his mind.

in be guaranteed in the cistency in the context I welfare function in the

(a)

's to the lifetime utility

.). The social planner id old age ({Cr and mize social welfare (a)

(b)

kously, since past things old generation d computations we find

icial optimum:

(c)

(d)

values.

a nning exercise at some n planning period t is:

(e)

hes to the lifetime utility 1, t 2, ...). The social

Chapter 17: Intergenerational Economics, II

planner chooses sequences for consumption during youth and old age ({Crry t and {C°.}°° d and the capital stock (1kr-1-11,°'t) in order to maximize social welfare

(e) subject to the resource constraint (b) The (interesting) first-order conditions consist of (c) and:

The crucial thing to note is that conditions (d) and (f) overlap for the time interval r = t,1,+ 2, .... The tsequences {cli ctcrostrj6-=-1, and fkr+i}U are

chosen by the planner at time 0 but taken as given ("water under the bridge") by the planner at time t. But the sequences {C W°, t , {qr., t , and {kr-El t are chosen by both planners. Unless the planner at time 0 can commit to his plan (and thus can stop any future planner from re-optimizing the then relevant social welfare function), the sequences chosen by the planners at time 0 and at time t will not necessarily be the same. If they are not the same we call the social plan dynamically inconsistent (see Chapter 10).

Following the insights of Strotz (1956), Burness (1976) has derived conditions on the admissible pattern of generational weights, 4,, that ensure that the optimal social plan is dynamically consistent. Comparing (d) and (f) reveals that dynamic consistency requires the following condition to hold for any planning period t:

Condition (g) means that X t,, must be multiplicatively separable in time (r) and the planning date (t), i.e. it must be possible to write X t,, g(t)A.„ where g is some function of t. A simple example of such a multiplicatively separable function is:

where X 0 is the planner's constant discount rate. By using (h) we normalize the weight attached to the young in the planning period to unity (X t,t = 1). It follows necessarily, that in order to preserve dynamic consistency, there must be reverse discounting applied to the old generation in the planning period. Indeed, the dynamic consistency requirement (g) combined with (h) implies

1 + X so that Xt,t_i = (1 + X)Xt,t = 1 + X. Calvo and Obstfeld (1988) apply this notion of reverse discounting in the context of the Blanchard-Yaari model of overlapping generations.

631

The Foundation of Modern Macroeconomics

Finally, by using (17.135)—(17.137), and (17.141) in (17.138) we can derive the following expression:

aAY(k) G, (Eofir[(0A::) )

azt

(

/3(1+n)fit+2 (BA Y (t) ( an Y (It+i)/azt+i

In the social optimum the marginal social cost of educational activities (left-hand side of (17.142)) should be equated to the marginal social benefits of these activities (right-hand side of (17.142)). The marginal social costs are just the value of leisure time of the parent, but the marginal social benefits consist of three terms.

All three terms on the right-hand side of (17.142) contain the expression Ott)fr- t-6 which represents the marginal product of time spent on educational activities in

the production of human capital (see (17.125)). The first line on the right-hand side of (17.142) is the "own" effect of educational activities on the parent's utility. This term also features in the first-order condition for the privately optimal (internal) child-education decision, namely (17.130). The second and third lines show the additional effects that the social planner takes into account in determining the optimal level of child education. The second line represents the effect of the parent's decision on the children's earnings: by endowing each child with more human capital they will have a higher skill level and thus command a higher wage. The third line represents the impact of the parent's investment on the children's incentives to provide education for their own children (i.e. the parent's grandchildren).

Eckstein and Zilcha are able to prove that (a) the competitive allocation is suboptimal, and (b) that under certain reasonable assumptions regarding the lifetime utility function there is underinvestment of human capital. Intuitively, this result obtains because the parents ignore some of the benefits of educating their children (1994, pp. 345-346). To internalize the externality in the human capital investment process, the policy maker would need to construct a rule such that the parent's decision regarding educational activities would take account of the effect on the children's wages and education efforts. As Eckstein and Zilcha argue, it is not likely that such a complex rule can actually be instituted in the real world. For that reason, the institution of compulsory education, which is practicable, may well achieve a welfare improvement over the competitive allocation because it imposes a minimal level of educational activities on parents (1994, pp. 341, 346).

17.3.2 Public investment

At least since the seminal work by Arrow and Kurz (1970), macroeconomists have known that the stock of public infrastructure is an important factor determining

the productive capacity of lic capital stock has pla;. recently. This unfortunate the pathbreaking and r triggered a veritable boo., Gramlich, 1994 for an exc that public capital exer capital and argued that tt early 1970s is due to a s his estimates suggest more, values which are se Gramlich, 1994, p. 1186). subsequent studies have c clude that economists c. productive.

In this subsection we sh the Diamond—Samuelso: . omy is affected if the go policy. Finally, we study t To keep things simple we ernment has access to lun (1993, pp. 336-340). .1

Prototypical examples ( ports, hospitals, etc., wf capital stock, the public investment and gradua ' the stock of government

Gt±i — Gt = Ip — u,

where 1G is infrastructure public capital. Assumin, per capita public capital

(1 + n)gt+i = it + (1 -

where gt Gt/Lt and iG, We assume that pub, ic i.e. instead of (17.10) we

Yr = F(Kr, Logt),

where we assume that

Kt and Lt . This means the

Yt = f (kt, gt),