Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

111,- tuarial notes equals the >useholds have the incen- -, the striking result that, s out of the consumption

rd embedded them in generations. In order to odelling choices. First, he ensures that the optimal n thus be aggregated. Sec- n agents at each instant. e. (To ensure a constant

- :al.)

ant properties. First, the tely lived representative d-Yaari model by setting tal stock is smaller in the o the turnover of generaual consumption growth. eds the rate of time pref-

. efficient. Third, fiscal d lump-sum tax financed Ian one-for-one crowding he short run, households , ce they discount present

:rket rate of interest. As ) that private investment :t are smaller, wages are ck redistributes resources

. Finally, Ricardian equivpositive birth rate (and he rejection of Ricardian

I

ns to the Blanchard-Yaari 'ihour supply decision.

. onsumption tax. In the unambiguously leads to old cuts back labour esources from present to ,:k in the long run. The gth of the generational

Chapter 16: Intergenerational Economics, I

In the second extension to the Blanchard-Yaari model we assume that a worker's productivity declines with age. The declining path of labour income mimics the notion of saving for retirement—even though the agent continues to work the same number of hours during life, the decline in productivity makes these hours less valuable over time. If the productivity profile declines steeply then the steady-state equilibrium may be dynamically inefficient. Because labour income is high during youth and agents practise consumption smoothing, they save a lot early on in life and the economy as a whole may oversave.

Finally, in the third extension we study a small open economy version of the Blanchard-Yaari model. We use the model to study the effects on the macroeconomy of an oil price shock. An attractive feature of the open-economy Blanchard-Yaari model is that it can be used to study both creditor nations (populated by patient citizens) and debtor countries (inhabited by impatient households). (Recall from Chapter 14 that in the corresponding Ramsey model, the steady-state equilibrium only exists for a knife-edge case in which the world interest rate equals the rate of time preference of residents.)

Further Reading

The Blanchard-Yaari model has been applied in a large number of areas. Open economy models are presented by inter alia Blanchard (1983, 1984), Frenkel and Razin (1986), Buiter (1987), Matsuyama (1987), Giovannini (1988), and Heijdra and van der Horst (2000). The closely related Weil (1989b) model is used for the analysis of tax policy by Bovenberg (1993, 1994) and Nielsen and Sorenson (1991) and for the study of current account dynamics by Obstfeld and Rogoff (1995b, pp. 1759-1764).

Alogoskoufis and van der Ploeg (1990) and Saint-Paul (1992) introduce endogenous growth into the model. Well (1991) and Marini and van der Ploeg (1988) study monetary neutrality. Aschauer (1990a) introduces endogenous labour supply in the Blanchard-Yaari model. On public infrastructure, see Heijdra and Meijdam (forthcoming). Marini and Scaramozzino (1995) and Bovenberg and Heijdra (1998, forthcoming) study environmental issues. Nielsen (1994) introduces social security into the model. Gertler (1999) generalizes the model by assuming that workers move into retirement according to a stochastic Poisson process. The International Monetary Fund's MULTIMOD model includes insights from the Blanchard-Yaari framework-see Laxton et al. (1998).

Appendix

Derivation of Figure 16.3

In this appendix we derive the phase diagram for the extended Blanchard-Yaari model with endogenous labour supply and various tax rates. In doing so, we follow the general approach discussed in detail in the appendix to Chapter 15.

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The Foundation of Modern Macroeconomics

Employment as a function of the state variables

By using labour demand (T2.5), labour supply (T2.6), and the production function (T2.7)— and dropping the time index where no confusion is possible—we obtain an expression relating (labour-market-clearing) equilibrium employment to the state variables (C and K) and the exogenous variables:

where tH (tc +OM– tL) is the tax wedge directly facing households, and r(L) is a decreasing function in the feasible interval L E [0, 1] with r/(L) = -LEL -2 [(1 - EL)(1 - L) + L] < 0 and r"(L) = (1 - EL )LEL -3 [2 – EL (1 – L)] > 0. In summary, (A16.1) shows that equilibrium

employment depends negatively on consumption and the tax wedge (via labour supply) and positively on the capital stock (via labour demand).

Capital stock equilibrium

The capital stock equilibrium (CSE) locus represents points in (C, K)-space for which k = 0 and thus 8K = Y – C. We note from (T2.5) and (T2.6) that:

We require the term in square brackets on the right-hand side to be non-negative, so that the lower bound for employment is:

By using (A16.4) and (T2.7) in (A16.3) we can write the CSE curve as follows:

Equation (A16.5) represents an implicit function, L = g(K), over the interval L E [LMIN, 1] relating K and L. In order to compute the slope of the implicit function we totally differentiate (A16.5):

Since L > LMIN > 0 the term in square brackets on the right-hand side is strictly positive so that g'(K) > 0. It follows from (A16.5) that K = 0 for L = LMIN, so as L rises from L = LMIN to L = 1, K rises from K = 0 to K = KK . 8 -1/EL > 0. We now have two zeros for the CSE line, i.e. both (K, = (0, LMIN) and (K, L) = (KK, 1) solve equation (A16.3). By using (A16.1) we find the corresponding values for C, i.e. (C, K, L) = (0, 0, LMIN) and (C, K, L) = (0, KK, 1) are both zeros for the CSE line—see Figure 16.3.

The slope of the CSE as:

C = g(K)ELK l-EL – 8K,

where L = g(K) is the (A16.7) we obtain in a ft

1

1 dC

dr‹),.<=0 [ 1- EL 01

where rig (K) is the elastic.

74(K) (Kg' (K)

g(K) g

from which it follows side of (A16.8) goes to (1 - is vertical near the origin

dC

lim = +0‘.. aK k=0

The capital stock dynan

I

For point above (below) tt is negative (positive). Th._ in Figure 16.3.

Consumption flow

The consumption equili: aggregate flow of consumi (T2.4), (T2.7) and (A16.21 ,

Y = (KL

where y Y/K is the out! (A16.13) define consumpti

- -Auction function (T2.7)— e—we obtain an expression the state variables (C and K)

(A16.1)

ieholds, and F(L) is a decreas- L*- [(1 - €L)(1 - L) + L] < 0 1) shows that equilibrium LX wedge (via labour supply)

• K)-space for which K 0

(A16.2)

(A16.3)

to be non-negative, so that

(A16.4)

carve as follows:

(A16.5)

(Per the interval L E [LMIN, 1] :)licit function we totally

( dL) (A16.6)

L

d side is strictly positive so ,-, so as L rises from L = LMIN r have two zeros for the CSE

ion (A16.3). By using (A16.1) ) and (C, K, L) = (0,

Chapter 16: Intergenerational Economics, I

The slope of the CSE line is computed as follows. We note that this line can be written as:

where L = g(K) is the implicit function defined by (A16.5). By taking the derivative of (A16.7) we obtain in a few steps:

where 7.4(K) is the elasticity of the g(.) function:

from which it follows that 74(0) = 0 so that the term in square brackets on the right-hand

side of (A16.8) goes to (1 - EL). But since g(K)/K = +oo it follows that the CSE line is vertical near the origin (see Figure 16.3):

dC

lim (— ) = +00. (A16.10)

K-±0 dK K=0

The capital stock dynamics follows by substituting (T2.7) and L = g(K) into (T2.2):

For point above (below) the CSE line, consumption is too high (low) and net investment is negative (positive). These dynamic effects have been illustrated with horizontal arrows in Figure 16.3.

Consumption flow equilibrium

The consumption equilibrium (CE) locus represents points in (C, K)-space for which the aggregate flow of consumption is in equilibrium (C = 0). By using (T2.1) in steady state, (T2.4), (T2.7) and (A16.2) we can write the CE locus as follows:

(A16.13) define consumption flow equilibrium in (K, L)-space.

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The Foundation of Modern Macroeconomics

In the representative-agent model (with /3 = 0) the CE locus represents points for which y y*. By using this in (A16.13) and (A16.1) we get after a few steps:

(1 - Ec)(1 - 61)(1 + tH)

Hence, the CE curve for the RA model (CE") is linear and downward sloping—see the dashed line from in Figure 16.3.

For the overlapping-generations model the CE line can only be described parametrically, i.e. by varying L in the feasible interval [0, 1]. We first write (A16.12) in a more convenient

where 0 > 0. Solving (A16.15) for the positive (economically sensible) root yields the equilibrium output-capital ratio for the overlapping-generations (OG) model as a function of L:

Using (A16.16) in (A16.13) yields an expression for the capital-labour ratio:

from which we derive the following limiting results:

The labour market equilibrium condition (A16.1) yields an expression for consumption:

from which we derive the following limiting results:

Hence, the CE line for the OG model has the same vertical intercept as CE as L 0 and

It is straightforward—t the origin (where L ti 1 j intercept (where L ti 0).

Uniqueness

The uniqueness of the ettl we rewrite (A16.17) as: I

It is not difficult to show t -oo. These properties er (equation (A16.22)) cross Equilibrium consumpt follows from the produLL also.

Savings dynamics in

In this appendix we sole to (16.88). We use the matical Appendix and cs state in (16.82) (H(x) = transitional effects we r: two steps. In the first st, by the first line of (16.82 wealth into the second .

Taking the Laplace tr..a s} - H(0) we obtain

£{11, s ii(0) - rE : _ s - (r

where H(0) is the impac - arising from the instate side of (A16.23) is zero is:

H(0) = reLLIW,r + r

By substituting the Lap,,, , (16.92), where we have us The transition path for r-.

;, resents points for which steps:

(A16.14)

!nwnward sloping—see the

described parametrically, i.e. 1.I2 I in a more convenient

I

(A16.15)

',- sensible) root yields the

OG) model as a function

(A16.16)

,hour ratio:

(A16.17)

(A16.18) - -.sion for consumption:

(A16.19)

(A16.20)

(A16.21)

-t as CERA as L 0 and

Chapter 16: Intergenerational Economics, I

It is straightforward—though somewhat tedious—to prove that CEO LG is horizontal near the origin (where L ti 1) and downward sloping and steeper than CERA near the vertical intercept (where L ti 0).

Uniqueness

The uniqueness of the equilibrium can be established most easily in the (K, L) we rewrite (A16.17) as:

-oo. These properties ensure that the CSE curve (equation (A16.5)) and the CE curve (equation (A16.22)) cross only once thus determining unique equilibrium values (K*, L*). Equilibrium consumption, C*, then follows from (A16.1), and equilibrium output, Y*, follows from the production function (T2.7). All other variables are determined uniquely also.

Savings dynamics in the open economy

In this appendix we solve the savings subsystem (16.82) given that wages evolve according to (16.88). We use the Laplace transform technique that is discussed in detail in the Mathematical Appendix and was also used in the appendix to Chapter 15. By imposing the steady state in (16.82) Moo) = Moo)- = 0) we find (16.89)-(16.90). To compute the impact and transitional effects we note that the savings system is itself recursive and can be solved in two steps. In the first step we solve for the dynamics of human wealth which is described by the first line of (16.82). In the second step we substitute the solution path for human wealth into the second line of (16.82) and solve the dynamics for financial wealth..

Taking the Laplace transform of the first line of (16.82) and noting that £{1:1, s} = sG{rl , s} - (0) we obtain:

where r1(0) is the impact jump in human wealth. The only way to avoid the instability arising from the instable root 04 = r + p) is to ensure that the numerator on the right-hand side of (A16.23) is zero when s = r + p. This implies that the impact jump in human capital is:

11(0) = r€LL{1217,r + (A16.24)

By substituting the Laplace transform of the transition path for wages (16.88) we obtain (16.92), where we have used the fact that 4 = r + p, ,C{e-at, s} = 1/(s + a), and G{1, s} = 1/s. The transition path for human wealth is obtained as follows. First we substitute (A16.24)

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The Foundation of Modern Macroeconomics

into (A16.23) and invert the Laplace transform:

By substituting (the Laplace transform of) (16.88) into (A16.25) we obtain the desired expression in a few steps:

where we have used (16.92) and (16.89) in going from the second to the third line.

By taking the Laplace transform of the second line of (16.82), noting that L{A, s} s.C{A, s} - ;1(0) and substituting (A16.23)-(A16.24) we obtain:

(s + AsoL{A, s};= A.(0) + rELL{V-17,s}

where we recall that -?4 r - (p + 0) < 0 is the stable root of the savings subsystem. By using the path of wages in (A16.27) we obtain (16.94) after some manipulations.

Intergen€1

I

The purpose of this char

1.To introduce ana and to show its ma

2.To apply the disct, and the macroecoi

3.To extend the m accumulation and

4.To illustrate the MI

17.1 The Diamoi

As the previous chap, framework is quite flex workhorse status. It so in a simple fashion ings. Indeed, as Blan, youth approach is that, account for life-cycle Blanchard model, a ha (first aspect) but not it! the absence of a bequ, agent to have a much agent, simply because t hazard) than the you:

A simple model whit household behaviour

(A16.25)

5.25) we obtain the desired

— (r ,8)

(A16.26)

id to the third line.. i.82), noting that L{:zi, s} =

(A16.27)

the savings subsystem. By - manipulations.

17

Intergenerational Economics, II

The purpose of this chapter is to achieve the following goals:

1.To introduce and study a popular discrete-time overlapping-generations (0G) model and to show its main theoretical properties;

2.To apply the discrete-time model to study things like (funded or unfunded) pensions and the macroeconomic effects of ageing;

3.To extend the model to account for (private versus public) human capital accumulation and public investment;

4.To illustrate the method of intergenerational accounting.

17.1 The Diamond—Samuelson Model

As the previous chapter has demonstrated, the continuous-time Blanchard—Yaari framework is quite flexible and convenient and therefore fully deserves its current workhorse status. It yields useful and intuitive macroeconomic results and does so in a simple fashion. This is not to say that the framework has no shortcomings. Indeed, as Blanchard himself points out, the main drawback of the perpetual youth approach is that, though it captures the finite-horizon aspect of life, it fails to account for life-cycle aspects of consumption (1985, p. 224). Indeed, in the standard Blanchard model, a household's age affects the level and composition of its wealth (first aspect) but not its propensity to consume out of wealth (life-cycle aspect). In the absence of a bequest motive and with truly finite lives, one would expect an old agent to have a much higher propensity to consume out of wealth than a young agent, simply because the old agent has a shorter planning horizon (a higher death hazard) than the young agent has.

A simple model which captures both the finite-horizon and life-cycle aspects of household behaviour was formulated by Diamond (1965) using the earlier insights

The Foundation of Modern Macroeconomics

of Samuelson (1958). 1 The Diamond-Samuelson model is formulated in discrete time and has been the workhorse model in various fields of economics for almost four decades. In the remainder(qof this section we describe (a simplified version of) the Diamond (1965) model in detail.

17.1.1 Households

Individual agents live for two periods. During the first period (their "youth") they work and in their second period (their "old age") they are retired from the labour force. Since they want to consume in both periods, agents save during youth and dissave during old age. We abstract from bequests and assume that the population grows at a constant rate n.

A representative young agent at time t has the following lifetime utility function:

where theSrsubscript identifies the time period and the superscript=the period of life the agent is in, with "Y" and "0" standing for, respectively, youth and old age. Hence, Cr and C°±1 denote consumption by an agent born in period t during youth and old age, respectively, and At is lifetime utility of a young agent from the perspective of his birth. As usual, p > 0 captures the notion of pure time preference and we assume that the felicity function, U(.), satisfies Inada-style conditions (U' > 0 > U", lim = +00, and U'(x) = 0).

During the first period the agent inelastically supplies one unit of labour and receives a wage Wt which is spent on consumption, cr, and savings, St . In the second period, the agent does not work but receives interest income on his savings, rt+ iSt . Principal plus interest are spent on consumption during old age, C°±1 . The household thus faces the following budget identities:

By substituting (17.3) into (17.2) we obtain the consolidated (or lifetime) budget constraint:

The young agent chooses Cr and C°±1 to maximize (17.1) subject to (17.4). The first-order conditions for consumption in the two periods can be combined after

1 An even earlier overlapping-generations model was developed by Allais (1947). Unfortunately, due to the non-trivial language barrier, it was not assimilated into the Anglo-Saxon literature.

which we obtain the t:

U (Cr) — 1 r t

Together, (17.4)-(17.5) d, St ) to the variables that key expression is the

= S( Wr, rt+ 1),

which has the following

o < sw = as— el

aWt

where 9 [x] -U"(x)x given the assumption rru inverse of 9 [x] is the in - According to (17.7), an

(17.2) and (17.3) that ix

asiawt > 0 and act°

respect to the interest rat in opposite directions the relative price of fut present consumption a! expands the budget a\ -. the agent to increase botl Equation (17.8) shows exceeds (falls short of) ui savings depend positiv:

sr o

17.1.2 Firms

The perfectly competiti% from the currently old

is formulated in discrete of economics for almost e (a simplified version of)

p

- -!od (their "youth") they re retired from the labour 's save during youth and me that the population

'ifetime utility function:

o

(17.1)

superscript the period of rlectively, youth and old u born in period t during of a young agent from the -1 of pure time preference ada-style conditions (U' >

s one unit of labour and r, and savings, St . In the t income on his savings, during old age, C°+1 . The

(17.2)

(17.3)

ited (or lifetime) budget

(17.4)

'.1) subject to (17.4). The 's can be combined after

Ulais (1947). Unfortunately, due lo-Saxon literature.

Chapter 17: Intergenerational Economics, II

which we obtain the familiar consumption Euler equation:

Together, (17.4)-(17.5) determine implicit functions relating Cr and C°t±1 (and thus St ) to the variables that are exogenously given to the agents, i.e. Wt and rt+ 1. The key expression is the savings equation:

where 0 [x] -U"(x)x/U'(x) is the elasticity of marginal utility (which is positive, given the assumption made regarding U(.) above). Recall from Chapter 14 that the inverse of 0 [x] is the intertemporal substitution elasticity, denoted by a [x] 1 1 0 [x] . According to (17.7), an increase in the wage rate increases savings. It follows from (17.2) and (17.3) that both consumption goods are normal, i.e. aCria Wt = 1 — asowt > 0 and act°±1 /awt = (1 + rt± oast /awt > 0. The response of savings with respect to the interest rate is ambiguous as the income and substitution effects work in opposite directions (see Chapter 6). On the one hand an increase in rt+ i reduces the relative price of future goods which prompts the agent to substitute future for present consumption and to increase savings. On the other hand, the increase in rt-Fi expands the budget available for present and future consumption which prompts the agent to increase both present and future consumption and to decrease savings. Equation (17.8) shows that, on balance, if the intertemporal substitution elasticity exceeds (falls short of) unity then the substitution (income) effect dominates and savings depend positively (negatively) on the interest rate:

17.1.2 Firms

The perfectly competitive firm sector produces output, Yr , by hiring capital, Kt, from the currently old agents, and labour, Lt , from the currently young agents. The

591

The Foundation of Modern Macroeconomics

production function is linearly homogeneous:

and profit maximization ensures that the production factors receive their respective marginal physical products (and that pure profits are zero):

Wt = 11(1(0 14),

rt + 3 = FK(Kt,

where 0 < 8 < 1 is the depreciation rate of capital. 2 The crucial thing to note about (17.12) is its timing: capital that was accumulated by the currently old, Kt, commands the rental rate It follows that the rate of interest upon which the currently young agents base their savings decisions (i.e. r t± i in (17.3) and (17.6)) depends on the future capital stock and labour force:

Since the labour force grows at a constant rate and we ultimately wish to study an economy which possesses a well-defined steady-state equilibrium, it is useful to rewrite (17.9)-(17.10) and (17.13) in per capita form (see Chapter 14 for details):

Yr = f (kt),

Wt = f (kr) - ktf (kr), rt+i + 3 = f'(kt+i),

where yt Yt/Lt, kt Kt/Lt, and f (kt ) F(kt ,1).

17.1.3 Market equilibrium

The resource constraint for the economy as a whole can be written as follows:

where Ct represents aggregate consumption in period t. Equation (17.17) says that output plus the undepreciated part of the capital stock (left-hand side) can be either consumed or carried over to the next period in the form of capital (right-hand side). Alternatively, (17.17) can be written as Yt = Ct +It with It AKt+1 +3Kt representing gross investment.

2 Most authors follow Diamond (1965, p. 1127) by assuming that capital does not depreciate at all (8 = 0). Since the model divides human life into two periods, each period is quite long (in historical time) and it is thus defensible to assume that capital fully depreciates with the period (6 = 1). Blanchard and Fischer (1989, p. 93) circumvent the choice of 6 by assuming that (17.10) is a net production function, with depreciation already deducted. In their formulation, 6 vanishes from the capital demand equation (17.12).

Aggregate consumF agents in period t:

Ct Lt_iCt° + /4(

I

Since the old, as a group is the sum of the undci, received from the firn consumption satisfies (1

Ltcr = WtLt - StLt . By

Ct = (rt + 8)Kt + (1 - = Yt +(1- 6)Kt -

where we have used the second line. Output is fu Finally, by combini: period's savings decisior

StLt = Kt+1.

The population is assn:..

Lt = L0(1 + n) t , n

so that (17.20), in con.

S(Wt, rt+i) = (1 -t-

The capital market is r (equation (17.16)) anc

17.1.4 Dynamics and

The dynamical behavik_ sions for Wt and supply equation (17.22 I

(1 + n)kt± i = S [ft

This expression relates tt suitable to study the obtain:

dkt+i -Sw kr •

dkt 1 + n -5,7