Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

. We recall that E N,„, > 0

-es (is dominated by) the

ithe Aaron condition is statutory tax rate decreases

ld derive the fundamental he approach of Ihori (1996, er of agents (Li ) no longer n (Lt Nt ). By redefining the ons for the wage and the mice frontier is still as given - •31 in the next period, i.e.

) we get:

(17.77)

we note that the labour nctional form as:

(17.78)

(17.79)

)wing expression:

t E

it-1-2),xt+1- (17.80)

contains the terms kr, kt+1,

1-order difference equation

!t point out, however, the Into the model, i.e. since

pends on Nt+2 which itself is on the entire sequence

I though we assume perfect Since we assume that the over the indeterminacy

in overlapping-generations ,cussion and Reichlin (1986)

Chapter 17: Intergenerational Economics, II

The steady state

We study two pertinent aspects of the steady state. First, we show how the endogeneity of labour supply affects the welfare effect of the PAYG pension. Second, we show that in the unit-elastic model the pension crowds out capital in the long run. As before, the long-run welfare analysis makes use of the indirect utility function which is defined as follows:

Retracing our earlier derivation we can derive the following properties of the indirect utility function:

The effect of a marginal change in the statutory tax rate on steady-state welfare is now easily computed:

dA Y aAY dW aAY- dr aAY

where we have used (17.82)—(17.84) in going from the first to the second line and (17.53) and (17.77) in going from the second to the third line. There are two noteworthy conclusions that can be drawn on the basis of (17.85). First, if the economy is initially in the golden-rule equilibrium (r = n), then a marginal change in tL does not produce a first-order welfare effect on steady-state generations. Intuitively, the

labour supply decision is not distorted because the effective tax on labour is zero in that case (tt = tL (r — n) (1 + r) = 0). Second, if the economy is not in the golden-

rule equilibrium (r n), then the sign of the welfare effect is determined by the sign of the term in square brackets on the right-hand side of (17.85). Just as for the case with lump-sum contributions (see (17.54)), the PAYG pension affects welfare through lifetime resources (first term in brackets) and via the capital-labour ratio (second term). It turns out, however, that with endogenous labour supply the sign of dr I dt-L (and thus the sign of dAY/c/h ) is ambiguous (Ihori, 1996, p. 237).

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The Foundation of Modern Macroeconomics

Matters are simplified quite a lot if Cobb-Douglas preferences are assumed, i.e. if (17.66) is specialized to:

At log Cr + A, log[1 - Aid + 11 logC°,, (17.86) p

where p is the rate of time preference and Ac (>0) regulates the strength of the labour supply effect. The following solutions for the decision variables are then obtained by maximizing (17.86) subject to (17.71):

implicit function, kr-Fi = we obtain:

4

Since A is positive (as A < 1). As a result, the la negative in the unit-e1

dk gt

gk <0.

crences are assumed, i.e. if

(17.86)

"- e strength of the labour

.:- iables are then obtained

(17.87)

(17.88)

(17.89)

. In the unit-elastic model, 400ds and labour supply rice! out. Since the current imount of labour (Nt+i =

to:

(17.90)

m payments during old age ively on the interest rate-

'mic preferences are used. sitively on the interest rate edition is violated (holds)

odel is recovered by setting qual to unity.

s crowded out by the PAYG I difference equation for the

p

(17.91)

itutes a first-order difference -elastic model the indeterove) disappears because the

I \YG system on the capital- s' he partial derivatives of the

The Foundation of Modern Macroeconomics

TE

EE

—N

Figure 17.3. Deadweight loss of taxation

and let the household choose current and future consumption in order to maximize:

subject to the constraint CY + C°/(1 + r) = X. This yields the following expressions:

In the right-hand panel of Figure 17.3 the line EE relates old-age consumption to lifetime income. In that panel the value of consumption during youth can be deduced from the fact that it is proportional to lifetime income.

By substituting the expressions (17.97) into, respectively, the utility function (17.86) and the budget constraint (given in (17.81)) we obtain:

In the second stage, the household chooses its labour supply and lifetime income in order to maximize (17.98) subject to (17.99). The solution to this second-stage problem is, of course, that N takes the value indicated in (17.89) and X follows from the constraint. The second-stage optimization problem is shown in the left-hand panel of Figure 17.3. In that panel, TE represents the budget line (17.99) in the absence of taxation (t r, = 0). It is upward sloping because we measure minus N on the horizontal axis. The indifference curve which is tangent to the pre-tax budget

line is given by IC and till the EE line gives the co:

Now consider what hal happens in the right-h, - a counter-clockwise fast.. the origin. We know that in labour supply cancel I Hence, the new equilibria line in a parallel fashi curve we find that the p shift from E0 to E2 (the i vertical distance OB rei . to restore it to its pre-tax What is the tax revenue we draw a line, that is pat point E2. This line has an two expressions for lines 1

X + W(1 — tf)(-N) =

AB in Figure 17.3. By d represents the tax reveni OB the DWL of the tax is

I

As a number of authors pension system has impo improving reform (see e references to more reck discussion at the end PAYG to a fully funded s resources cannot be fu,.. reform without making the PAYG system repre'. (1993) point out, provi„, during the transition can be achieved in a Pare tortionary to a non-disto can be used to compen •

15 The distortive nature of Li. Demmel and Keuschnigg ( 10QC which is exacerbated by tht sion reform reduces unemplog

tion in order to maximize:

(17.96)

s the following expressions:

(17.97)

ald-age consumption to lifeng youth can be deduced

lively, the utility function

( 1 + r ) 11(1±P)

R)2+,9 (17.98)

(17.99)

and lifetime income Aution to this second-stage 17.89) and X follows from is shown in the left-hand budget line (17.99) in the -e we measure minus N on _:ent to the pre-tax budget

Chapter 17: Intergenerational Economics, II

line is given by IC and the initial equilibrium is at E 0 . In the right-hand panel E0 on the EE line gives the corresponding optimal value for old-age consumption.

Now consider what happens if a positive effective tax is levied (if > 0). Nothing happens in the right-hand panel but in the left-hand panel the budget line rotates in a counter-clockwise fashion. The new budget line is given by the dashed line from the origin. We know that in the unit-elastic model income and substitution effects in labour supply cancel out so that labour supply does not change (see (17.89)). Hence, the new equilibrium is at E1 in the two panels. By shifting the new budget line in a parallel fashion and finding a tangency along the pre-tax indifference curve we find that the pure substitution effect of the tax change is given by the shift from E0 to E2 (the income effect is thus the shift from E2 to E1). Hence, the vertical distance OB represents the income one would have to give the household to restore it to its pre-tax indifference curve. We call this hypothetical transfer Zo. What is the tax revenue which is collected from the agent? To answer that question we draw a line, that is parallel to the pre-tax budget line, through the compensated point E2. This line has an intercept with the vertical axis at point A. We now have two expressions for lines that both pass through the compensated point E2, namely X + W (1— tt)( — N) = Z0 and X + W(—N) = Zo — T, where T is the vertical distance AB in Figure 17.3. By deducting the two lines we find that T = tEWN so that AB represents the tax revenue collected from the agent. Since the required transfer is OB the DWL of the tax is given by the distance OA.

Reform

As a number of authors have recently pointed out, the distorting nature of the pension system has important implications for the possibility of designing Paretoimproving reform (see e.g. Homburg, 1990, Breyer and Straub, 1993, and the references to more recent literature in Belan and Pestieau, 1999). Recall from the discussion at the end of section 17.2.1 that a Pareto-improving transition from PAYG to a fully funded system is not possible in the standard model because the resources cannot be found to compensate the old generations at the time of the reform without making some future generation worse off. Matters are different if the PAYG system represents a distorting system. In that case, as Breyer and Straub (1993) point out, provided lump-sum (non-distorting) contributions can be used during the transition phase, a gradual move from a PAYG to a fully funded system can be achieved in a Pareto-improving manner. Intuitively, by moving from a distortionary to a non-distortionary scheme, additional resources are freed up which can be used to compensate the various generations (Belan and Pestieau, 1999). 15

15 The distortive nature of the PAYG scheme does not have to result from endogenous labour supply. Demmel and Keuschnigg (1999), for example, assume that union wage-setting causes unemployment which is exacerbated by the pension contribution. Efficiency gains then materialize because pension reform reduces unemployment. In a similar vein, Belan et al. (1998) use a Romer-style (1986,

617

The Foundation of Modern Macroeconomics

Table 17.1. Age composition of the population

17.2.3 The macroeconomic effects of ageing

Up to this point we have assumed that the rate of population growth is constant and equal to n (see equation (17.21) above). This simplifying assumption of course means that the age composition of the population is constant also. A useful measure to characterize the economic impact of demography is the so-called (old-age) which is defined as the number of retired people divided by the working-age population. In our highly stylized two-period overlapping-generations

model the number of old and young people at time t are, respectively, L t_ 1 and Lt = (1 + n)Lt_i so that the dependency ratio is 1/(1 + n).

Of course, as all members of the baby-boom generation will surely know, the assumption of a constant population composition, though convenient, is not a particularly realistic one. Table 17.1, which is taken from Weil (1997, p. 970), shows that significant demographic changes have taken place between 1950 and 1990 and are expected to take place between 1990 and 2025.

The figures in Table 17.1 graphically illustrate that throughout the world, and particularly in the group of OECD countries and in the US, the proportion of young people (0-20 years of age) is on the decline whilst the fraction of old people (65 and over) steadily increases. Both of these phenomena are tell-tale signs of an ageing population.

1989) endogenous growth model and show that reform may be Pareto-improving because it helps to internalize a positive externality in production. See also Corneo and Marquardt (2000).

In this subsection we st position changes can be model. We only stress soi action between dem,

is referred to Weil (1997) of ageing.

In the absence of distinct sources, namely period overlapping-ger. _ fixed but we can neverth population growth, ii. In a demographic shock \\ of a variable growth rate

Lt = (1 +

Assuming a constant cor equals Zt = (1 + nt )T. Re( the following fundamc: .

S(Wt , rt+i ,nt+i,T) =

where the savings fun...

Ceteris paribus, saving b population growth, nt _

(as 4+1 = (1 + nt-FOT). pension and lifetime it future consumption and hand side of (17.101) sh it possible to support a h saving.

Following the solution (17.101) defines an imp 0 < gk < 1 (see equation

gn = i 1 + ant±

It follows that a permah,

n1, gives rise to an incr,

The transition path of 1. In that figure, the dasher tion path with social sc _ at impact so that, if the tion path is the dotted 1

•tion growth is constant ring assumption of course Cant also. A useful mea-

r is the so-called (old-age) people divided by the , verlapping-generations

kre, respectively, L t_ 1 and

:I will surely know, the

• zh convenient, is not a ( 1997, p. 970), shows tNveen 1950 and 1990 and

ughout the world, and t h e proportion of young , n of old people (65 and !1-tale signs of an ageing

., roving because it helps to Auardt (2000).

Chapter 17: Intergenerational Economics, II

In this subsection we show how the macroeconomic effects of demographic composition changes can be analysed with the aid of a simple overlapping-generations model. We only stress some of the key results, especially those relating to the interaction between demography and the public pension system. The interested reader is referred to Weil (1997) for an excellent survey of the literature on the economics of ageing.

In the absence of immigration from abroad, population ageing can result from two distinct sources, namely a decrease in fertility and a decrease in mortality. In the twoperiod overlapping-generations model used so far the length of life is exogenously fixed but we can nevertheless capture the notion of ageing by reducing the rate of population growth, n. In order to study the effects on allocation and welfare of such a demographic shock we first reformulate the model of subsection 17.2.1 in terms of a variable growth rate of the population, nt . Hence, instead of (17.21) we use:

Lt = (1 + (17.100)

Assuming a constant contribution rate per person (Tt = T), the pension at time t equals Zt = (1 + nt )T . Redoing the derivations presented in subsection 17.2.1 yields the following fundamental difference equation of the model:

where the savings function is the same as in (17.40) but with nt+i replacing n. Ceteris paribus, saving by the young depends negatively on the (expected) rate of

population growth, n t+i , because the pension they receive when old depends on it (as Zt+1 = (1 + nt+i )T). An anticipated reduction in fertility reduces the expected pension and lifetime income, and causes the agent to cut back on both present and future consumption and to increase saving. Hence, 0. The righthand side of (17.101) shows that a decrease in the population growth rate makes it possible to support a higher capital-labour ratio for a given amount of per capita saving.

Following the solution method discussed in subsection 17.2.1, we can derive that

It follows that a permanent reduction in the population growth rate, say from n o to n1, gives rise to an increase in the long-run capital stock, i.e. dk/dn = gn /(1 gk) < 0. The transition path of the economy to the steady state is illustrated in Figure 17.4. In that figure, the dashed line labelled "kt + 1 = g(kt , no)" reproduces the initial transition path with social security in Figure 17.2. The reduction in fertility boosts saving at impact so that, if the economy starts out with a capital stock ko, the new transition path is the dotted line from B to the new equilibrium at E 1 . During transition

619

The Foundation of Modern Macroeconomics

k kt+1-kt

t+1

Figure 17.4. The effects of ageing

the wage rate gradually rises and the interest rate falls. The intuition behind the long-run increase in the capital-labour ratio is straightforward. As a result of the demographic shock there are fewer young households, who own no assets, and more old households, who own a lot of assets which they need to provide income for their retirement years (Auerbach and Kotlikoff, 1987, p. 163).

The effect of a permanent reduction in fertility on steady-state welfare can be

computed by differentiating the indirect utility function (17.47) with respect to n, using (17.50)-(17.51) and (17.54), and noting that 0A /0n = T (anY iacY) /(1 + r):

In a dynamically efficient economy (for which r > n holds) there are two effects which operate in opposite directions. The first term in square brackets on the righthand side of (17.103) represents the effect of fertility on the long-run interest rate. Since dr /dn = r'dk/dn > 0, a fall in fertility raises long-run welfare on that account. The second term in square brackets on the right-hand side of (17.103) is the PAYGyield effect. If fertility falls so does the rate of return on the PAYG contribution. Since the yield effect works in the opposite direction to the interest rate effect, the overall effect of a fertility change is ambiguous. If the PAYG contribution is very

(T 0) and the ea a drop in fertility raises I( Although our results 2. fled) model, they nevert,

Auerbach and Kotlikoft eral equilibrium model results: wages rise, the ii (see their Table 11.3). In is endogenous, producti% endogenous, taxes are

17.3Extensions

17.3.1Human capita

Human capital and grow

Following the early colauthors have drawn atte for the theory of ecor interest in human capita subsection we show h( can be extended by in...

households. We show hi Lucas (1988) model can Chapter 14 above).

As in the standard mc periods, but we deviate works full-time during th and training during ye worker's level of skill at I at time t by Hti and ass will thus pay a skill-dep‘A in Chapter 14 above). A

The lifetime utility t general terms by:

This expression incorpol and attaches no utility va in improving its skills be

k,

f

k,+1= g (kt, ni )

kt+i = g (kt, n0)

k,

e intuition behind the )(ward. As a result of the who own no assets, and v need to provide income

163).

1 \- -state welfare can be ( 17.47) with respect to n,

= T (anY/acY) + r):

(17.103)

1s) there are two effects

-e brackets on the righthe long-run interest rate.

Thl fare on that account.

4 (17.103) is the PAYG- I the PAYG contribution. ke terest rate effect, the G contribution is very

Chapter 17: Intergenerational Economics, II

small (T ti 0) and the economy is not close to the golden-rule point (r >> n), then a drop in fertility raises long-run welfare.

Although our results are based on a highly stylized (and perhaps oversimplified) model, they nevertheless seem to bear some relationship to reality. Indeed, Auerbach and Kotlikoff (1987, ch. 11) simulate a highly detailed computable general equilibrium model for the US economy and find qualitatively very similar results: wages rise, the interest rate falls, and long-run welfare increases strongly (see their Table 11.3). In their model, households live for 75 years, labour supply is endogenous, productivity is age-dependent, households' retirement behaviour is endogenous, taxes are distorting, and demography is extremely detailed.

17.3Extensions

17.3.1Human capital accumulation

Human capital and growth

Following the early contributions by Arrow (1962) and Uzawa (1965), a number of authors have drawn attention to the importance of human capital accumulation for the theory of economic growth. The key papers that prompted the renewed interest in human capital in the 1980s are Romer (1996) and Lucas (1988). In this subsection we show how the Diamond-Samuelson overlapping-generations model can be extended by including the purposeful accumulation of human capital by households. We show how this overlapping-generations version of the celebrated Lucas (1988) model can give rise to endogenous growth in the economy (see also Chapter 14 above).

As in the standard model, we continue to assume that households live for two periods, but we deviate from the standard model by assuming that the household works full-time during the second period of life and divides its time between working and training during youth. Following Lucas (1988) human capital is equated to the worker's level of skill at producing goods. We denote the human capital of worker i at time t by H: and assume that producers can observe each worker's skill level and will thus pay a skill-dependent wage (just as in the continuous-time model discussed in Chapter 14 above).

The lifetime utility function of a young agent who is born at time t is given in general terms by:

A t AY(CY'i t+i (17.104)

This expression incorporates the notion that the household does not value leisure and attaches no utility value to training per se. The household is thus only interested in improving its skills because it will improve its income later on in life. The budget

621

The Foundation of Modern Macroeconomics

where Wt denotes the going wage rate for an efficiency unit of labour at time t, and N: is the amount of time spent working (rather than training) during youth. Since the agent has one unit of time available in each period we have by assumption that Nti+ , = 1 (there is no third period of life so no point in training during the second period). The amount of training during youth is denoted by Et and equals:

To complete the description of the young household's decision problem we must specify how training augments the agent's skills. As a first example of a training technology we consider the following specification:

where G' > 0 > G" and G(0) = 1. This specification captures the notion that there are positive but non-increasing returns to training in the production of human capital and that zero training means that the agent keeps his initial skill level.

The household chooses CY'i , Ct°+'il , St, Nti , and Et in order to maximize lifetime utility A Y'` (given in (17.104)) subject to the constraints (17.105)-(17.107), and given the training technology (17.108), the expected path of wages Wt, and its own initial skill level The optimization problem can be solved in two steps. In the first step the household chooses its training level, Et, in order to maximize its lifetime income, 4, i.e. the present value of wage income:

The first-order condition for this optimal human capital investment problem, taking explicit account of the inequality constraint (17.107), is:

This expression shows that it may very well be in the best interest of the agent not to pursue any training at all during youth. Indeed, this no-training solution will hold if the first inequality in (17.110) is strict. Since there are non-increasing returns to training (so that G'(0) > G'(Eti ) for Et > 0) we derive the following implication from (17.110):

Wt+ 1

If the training technology is not very productive (G'(0) low) then the corner solution will be selected.

I

An internal solut: d4/dEti = 0. After some format:

Et > 0= 1 + r; _

This expression shows and human capital suc physical capital duri (left-hand side of (17.1; youth, the agent III _ Expressed in terms of .4 get the yield on human In the second step . tion for the two periods (17.104) subject to it ,

4

where 1t is now max. from this stage of the u

St = S(rt+i, (1 - Et

In order to comp,, must specify its initial ogy (17.108). Follov. household born in per, available knowledge at With this final assun., interest rate and learn' tion, saving, and invt here on and study the We assume that th. - e and old populations to is defined as the sum

Nt = (1 Et)Ht + Ht . ,