Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

.hick faces perfect mobility pecl by Matsuyama (1987), lock. To keep the model (2000) by assuming sim- ■ avoid the counterfactual ssumed that investment is

I

)bb-Douglas production ous in the private produc- (e.g. oil, 0(t)), where

<1 and EL +EK-FE0 = 1.

ssinvestment. We follow

lstallation function, 00,

..ion (T5.1), where 1(r) is maximizes the present

(16.75)

"r)ri cost function (T5.1), Tied in world markets and The resulting optimality energy demand, (T5.4), installed capital (namely - i .), is homogeneous of vv is linear-homogeneous

.f.ze q coincide, and the iavashi, 1982).

I Blanchard—Yaari model. households consume Lion is given by (16.24). there is perfect mobility

. ely, the individual and ,s taken as given by the -. Li rate of interest is also mate consumption Euler

(16.76)

!cl with relatively patient - t of (exceeds) the world ye (negative). We follow

Chapter 16: Intergenerational Economics, I

Table 16.5. The small open economy model

Y(t) = F(L(t), K(t), 0(t)) = L(t)EL K(t)EK OM"

(b)Saving subsystem

1(t) = (r + 13)H(t) — W(t)

A(t) = (r — p — p)A(t) — (p + fi)H(t) + W(t)

(c) Net foreign assets

AF (t) = A(t) — q(t)K(t)

Note: EL + cK + co = 1

Matsuyama (1987, p. 306) by restricting attention to the case of a for which r > p and thus A > 0.20

The government plays no role in the model, i.e. lump-sum taxes, public debt, and government consumption are all zero (T(t) = B(t) = G(t)). Household can hold their wealth in the form of shares in domestic firms (V(t) = q(t)K(t)) and in net foreign assets (AF(t)) so that equilibrium in the asset markets is given by equation (T5.10). By differentiating this expression with respect to time and using (T5.1)—(T5.7) we obtain:

Equation (16.77) is the current account of the balance of payments, showing the evolution of the stock of net foreign assets. The term in square brackets on the

20 See Blanchard (1985, pp. 230-231) for the analysis of both creditor (r > p) and debtor (r < p) nations in a world without physical capital. Giovannini (1988) considers both cases in a two-commodity model in which physical capital is perfectly mobile across borders.

573

The Foundation of Modern Macroeconomics

Table 16.6. The loglinearized small open economy model

(a)Investment subsystem

right-hand side of (16.77) is the trade balance consisting of domestic value added (Y(t) — Po(t)O(t)) minus domestic absorption (C(t) + /(t)).

Since aggregate consumption is given by C(t) = (p + ,B) [A(t) + H (t)] and T (t) = 0, the aggregate household budget identity (16.26) can be written as in (T5.9). Finally,

the path for human wealth is obtained by differentiating (16.20) with respect to

time, noting that dRA (t, r)/ dr = r(t) + = r + 8, and setting T(t) = 0. The resulting expression is given in equation (T5.8).

In order to study the effects of an oil price shock we loglinearize the model around an initial steady state. The resulting expressions are found in Table 16.6, where we

I

The key thing to note a total system can be subdi ics of physical capital and of human and financial v even though the full s' dynamics of (K, q) decor expressions for the two s as much as possible. By expression for aggregate

I

Y(t) W(t) = "

O(t) _ EKk( o_ i5, EL +EK

I

According to these expre energy price) boosts the cooperative in produc usage.

By using the output ( obtain a simple represe

I. ( t)

4(t) TE

The Jacobian matrix on istic polynomial:

—r

(7)

which has distinct iv shows that the investi.. and Tobin's q acting as.

21 Some authors prefer to a (see e.g. Matsuyama, 1987 an it stresses the link between. of domestic savers and in% ,);

(T6.1)

(T6.2)

(T6.3)

(T6.4)

(T6.5)

(T6.6)

(T6.7)

(T6.8)

(T6.9)

I

-are of investment, wc-ko, = 1; 'come from net foreign assests as

-f domestic value added

li t) H(t)] and T(t) = 0, n as in (T5.9). Finally, 416.20) with respect to T(t) = 0. The resulting

earize the model around -1 Table 16.6, where we qx] and x(t)

Chapter 16: Intergenerational Economics, I

The key thing to note about the model is that it can be solved recursively, i.e. the total system can be subdivided into an investment subsystem, describing the dynamics of physical capital and Tobin's q, and a savings subsystem, describing the dynamics of human and financial wealth (and thus of aggregate household consumption). So even though the full system contains four dynamic variables (K, q, H, and A), the dynamics of (K, q) decouples from that of (H, A).21 In order to find the relevant expressions for the two subsystems we first summarize the static part of the model as much as possible. By using (T6.6) and (T6.4) we obtain the quasi-reduced form expression for aggregate output, the wage rate, and energy usage:

According to these expressions, an increase in the capital stock (or a decrease in the energy price) boosts the demand for both labour and energy (because all factors are cooperative in production) and leads to an increase in output, wages, and energy

usage.

By using the output expression from (16.78) in (T6.2) and (T6.5) in (T6.1) we obtain a simple representation for the investment subsystem.

The Jacobian matrix on the right-hand side of (16.80) has the following characteristic polynomial:

shows that the investment subsystem is saddle-point stable with the capital stock and Tobin's q acting as, respectively, the predetermined and jumping variables.

21 Some authors prefer to analyse the savings subsystem by expressing it in terms of (C, AF) dynamics (see e.g. Matsuyama, 1987 and Bovenberg, 1993, 1994). We prefer the approach adopted here because it stresses the link between, on the one hand, the current account and on the other hand, the behaviour of domestic savers and investors. This makes the interpretation of the results easier.

575

The Foundation of Modern Macroeconomics

Figure 16.8. The effect of an oil shock on the investment subsystem

Similarly, the savings subsystem (T6.7)—(T6.8) can be written in a compact format as follows:

The Jacobian matrix on the right-hand side of (16.82) has the following characteristic polynomial:

from which it follows that the savings subsystem has one stable root, —Asi = r — (p

0) < 0, and one unstable root, ?4 = r + /3 > 0. 22 Financial and human wealth act as, respectively, the predetermined and jumping variables.

Investment dynamics

Let us now consider the macroeconomic effects of an unanticipated and permanent

increase in the world price of energy. We normalize the time at which the shock occurs at t = 0 and the shock to the system is represented by Po (t) = P0 for t > 0. We

[p + fJ — r]. Since tr(As) = 2r — p > 0 it follows that there is at least one positive root. Saddle-point stability requires there to be

one stable and one unstable root, i.e. As! 0 and thus (since r + > 0) that r < p + p. See Blanchard (1985, p. 230) and Matsuyama (1987, p. 305).

find the solution as follov to the shock. Next, by slit and the initial conditiu • shares) into the savings financial wealth. Finall) -, (T6.9). I

In order to explain th apparatus of Figure 16.8, investment system (16.8 which the capital stock is horizontal because Tol (below) the K(t) = Olin net investment is positil Figure 16.8. 1

The 4(t)- = 0 locus rti over time. It is downwarc marginal product of c., For points to the right (1 (high) so that part of ti Hence, 7/(t) > 0 (< 0) to arrows in Figure 16.8. 1 initial equilibrium at L

The increase in the prii energy usage is adjusted of Tobin's q, the margina and the 4(t)- = 0 line shit E1 , there is no long-run effect on the stock of cal

k" (00) = (E0) p- o

EL

It follows from (16.78) with the capital stock,

boo)= — (EL + EO

EL

At impact, the capital s from point 4, to point A

23 Note that equation (TS_ substituting this I/K value in

— K(t)= o

I

I= 0)0

lent

n a compact format

(16.82)

allowing character-

(16.83)

root, –A = r – (p + iman wealth act as,

-d and permanent t which the shock = Po for t > 0. We

a — I. Since tr(As) = requires there to be /3. See Blanchard

Chapter 16: Intergenerational Economics, I

find the solution as follows. First we solve the response of the investment subsystem to the shock. Next, by substituting the implied solution path for the wage rate ( IA7(t))- and the initial condition for financial wealth (i.e. the capital loss term on domestic shares) into the savings subsystem we obtain the solution paths for human and financial wealth. Finally, the path of net foreign assets then follows residually from (T6.9).

In order to explain the intuition behind our results, we use the diagrammatic apparatus of Figure 16.8, which is the graphical representation of the (loglinearized) investment system (16.80). The K(t) = 0 locus represents (4, K)-combinations for which the capital stock is in equilibrium, i.e. for which net investment is zero. It is horizontal because Tobin's q is constant in the steady state.23 For points above (below) the K(t) = 0 line, Tobin's q is larger (smaller) than its steady-state value, and net investment is positive (negative). This is illustrated with horizontal arrows in Figure 16.8.

The 4(t) = 0 locus represents (4, k)-combinations for which Tobin's q is constant over time. It is downward sloping because a higher capital stock leads to a fall in the marginal product of capital and thus to a lower dividend to the owners of shares. For points to the right (left) of the line the marginal product of capital is too low (high) so that part of the return on shares is explained by capital gains (losses). Hence, 4(t) to the right (left) of the line, as has been shown with vertical arrows in Figure 16.8. The arrow configuration in Figure 16.8 confirms that the initial equilibrium at E0 is saddle-point stable.

The increase in the price of energy reduces the marginal product of capital because energy usage is adjusted downward—see (16.79). To restore the equilibrium value of Tobin's q, the marginal product of capital must rise, i.e. the capital stock must fall and the 4(0 = 0 line shifts to the left. The steady-state equilibrium shifts from E0 to E1 , there is no long-run effect on Tobin's q (see above), q(oo) = 0, and the long-run effect on the stock of capital is:

It follows from (16.78) and (16.79) that output and wages fall equi-proportionally with the capital stock, k(o0) = I;V(oo) = K(oo), and that energy usage falls:

At impact, the capital stock is predetermined (K(0) = 0) and the economy jumps from point E0 to point A on the new saddle path SPi . The impact jump in Tobin's

23 Note that equation (T5.1) defines a unique steady-state value for I/K, such that 0(I/K) = S. By substituting this I/K value in (T5.6) we obtain the unique steady-state value for Tobin's q.

577

The degree of physical capital mobility, as parameterized by QA

01, is an important determinant of the transition path. Indeed, the lower is aA, the more mobile is physical capital, the more approximate is the saddle path to the k(t) = 0 line, and the higher is the adjustment speed A./.1 . In the limiting case with crA = 0 (perfect mobility), the saddle path coincides with the k(t) = 0 line, transition is immediate (),/, = oo), capital is a jumping variable, and Tobin's q is identically equal to unity precluding any capital gains or losses. 24

Wage dynamics

By using the solution path for K(t) (given in (16.87) above) in the quasi-reduced form expression (16.78) we obtain the transition path for the wage rate:

where the impact response is W(0) = -E0130 /(EL +EK ) < 0 (see (16.78)) and the longrun effect is .IA7- (oo) = -E0/30/EL < 0 (see the text below (16.84)). At impact, the wage

rate falls because the energy price increase prompts a decrease in the demand for labour. In the long run the reduction of the capital stock leads to a further decrease in the demand for labour and thus the wage.

24 See also Bovenberg (1994, p. 122) and Barro and Sala-i-Martin (1995, ch. 3) on this point. See also our discussion of the Sen and Turnovsky (1990) model in Chapter 14 above. ,,

where 1717(oo) < 0 (see alx wage income using the ai fall and the interest rate The effect on financial v of financial and human equations (T5.8)-(T5.9) ti

Since the stocks of net at impact (AF(0) = K(0) = wealth is:

A(0) = tov ij(0) < 0,

where the sign follows fi of shares in domestic fi Since human wealth i! (given in (16.88)) it exhi

171(0) = rEL r

r + 13) [(

where CV(oo) <1A7(0)- < to a weighted average a the weights depending

(4) relative to the annu and/or the adjustment C4 the weight attached to ti the capital stock and N• happens to the wage rJ opposite effect occurs, capital transition is fast

(16.87)

deed, the lower is ate is the saddle path to I Alp In the limiting case with the k(t) = 0 line,

ble, and Tobin's q is osses.24

in the quasi-reduced he wage rate:

(16.88)

(16.78)) and the long- ►). At impact, the wage - •Ise in the demand for Kis to a further decrease

7h. 3) on this point. See also

where1;17-- (oo) < 0 (see above). Steady-state human wealth is the perpetuity value of wage income using the annuity rate of interest, r + 13, for discounting. Since wages

fall and the interest rate is constant, human wealth unambiguously goes down. The effect on financial wealth is fully explained by the fact that the proportion of financial and human wealth is constant in the steady state of this model (i.e.

equations (T5.8)-(T5.9) together imply

Since the stocks of net foreign assets and physical capital are both predetermined at impact (AF (0) = K(0) = 0) it follows from (T6.9) that the impact jump in financial wealth is:

where the sign follows from (16.86). As a result of the energy price shock, owners of shares in domestic firms suffer a capital loss on their share holdings.

Since human wealth is the present value of the transition path of wage income (given in (16.88)) it exhibits a discrete jump at impact as well:

where 1717 (oo) <TA7(0)- < 0 (see above). The jump in human wealth is proportional to a weighted average of the impact and long-run effects on the wage rate with

the weights depending on the speed of adjustment in the investment subsystem

(4) relative to the annuity rate of interest (r /3). If agents are short lived (,3 high) and/or the adjustment costs of investment are severe (QA high and thus 4 low) then

the weight attached to the impact effect on wages is high. Intuitively, transition in the capital stock and wages is slow and short-lived agents mainly care about what happens to the wage rate at impact in the computation of human wealth. The opposite effect occurs, of course, if the birth rate is low (long-lived agents) and capital transition is fast (mild adjustment costs of investment).

579

The Foundation of Modern Macroeconomics

The adjustment path for human wealth displays a pattern similar to that of the wage rate (namely (16.88)):

where the key thing to note is that the adjustment speed of human capital is governed by the speed of transition in the investment system () l). Since both the impact and long-run effects on human wealth are negative it follows that H(t) < 0 for all t > 0. Furthermore, since wages decline monotonically during transition the same holds for human wealth, i.e. H(t) < 0 during transition and H(oo) <1:1(0) < 0.

The transition path for financial wealth may be non-monotonic and can be written as:

A(t) = A(o)e- * + [i - e- Alt]A(oo)

impact and in the long run and positive during transition. 25 The first two terms on the right-hand side of (16.94) show that part of the transition in the stock of financial assets is explained by a weighted average of the impact and long-run effect on financial wealth. The final term on the right-hand side of (16.94) represents the transitory effect of the energy price shock on the aggregate accumulation of financial wealth. As was explained (for a different kind of shock) by Bovenberg (1993), this transitory effect is due to a temporary, additional macroeconomic incentive to accumulate financial assets due to intergenerational distributional effects.

We complete the characterization of the macroeconomic effects of an energy price shock by determining what happens to consumption and net foreign assets. Since

It follows from (16.91)—(16.92) and (16.95) that the impact jump in consumption is negative (C (0) < 0). All existing generations at the time of the shock cut back their consumption level. Old agents, for whom financial wealth is the major wealth component, cut back consumption because they suffer a capital loss on their share possessions. Young agents, for whom human wealth is the major wealth component, cut back consumption because they suffer a capital loss on this wealth type due to the lower path of wages. By using (16.89)—(16.90) in (16.95) we find that

25 See Lemma A15.1 of the appendix to Chapter 15 for the properties of the transition term.

consumption also falls ii

rwc =

P

Since the path of final.. holds for consumption.

The long-run effect on (16.84) and (16.90) in (T

r - p

AF (00) =

RP-1-13-1

The long-run effect on ne (1997, p. 311-312) for a

I

16.5 Punchlines

In this chapter we study the continuous-time model is important not ( work with, but also be We start the chapter by consumption behaviour

lifetime (and thus his the agent's decision prob hypothesis must be empl wealth position at the tin with certainty. Yaari sh , the instantaneous probal consumption Euler equa the uncertainty of sur% . heavily.

Yaari makes the anal\ - tence of a kind of life ins, or sold by the consumer who buys an actuarial nl the consumer during 1114 sumer's death the insur estate. Reversely, a con s loan. During the consurr than the market rate of of any obligations, i.e. tt

rn similar to that of the

(16.93)

human capital is govSince both the impact s that H(t) < 0 for all

ring transition the same

.q( < C1(0)- < 0. monotonic and can be

(16.94)

subsystem (see the text term, which is zero at 1. 25 The first two terms ansition in the stock of pact and long-run effect )f (16.94) represents the accumulation of finan-

by Bovenberg (1993), °economic incentive to utional effects.

'ffects of an energy price net foreign assets. Since

(16.95)

t jump in consumption of the shock cut back

..ith is the major wealth Htal loss on their share major wealth compo- ►nss on this wealth type - 1 (16.95) we find that

of the transition term.

Chapter 16: Intergenerational Economics, I

consumption also falls in the long run:

(16.96)

(prw+cP) -cx))(= (p+ 1; —r)(r r+i3)EL 1;1-7*) <0.

Since the path of financial wealth may be non-monotonic (see above) the same

holds for consumption.

The long-run effect on the stock of net foreign assets is obtained by substituting (16.84) and (16.90) in (T6.9) and noting that q(o0) = 0:

The long-run effect on net foreign assets may be positive or negative. See Matsuyama (1997, p. 311-312) for a detailed explanation.

16.5 Punchlines

In this chapter we study one of the key models of modern macroeconomics, namely the continuous-time overlapping-generations model of Blanchard and Yaari. This model is important not only because it has proved to be quite flexible and easy to work with, but also because it nests the Ramsey (growth) model as a special case.

We start the chapter by studying the seminal insights of Yaari who studied optimal consumption behaviour in the presence of lifetime uncertainty. When an agent's lifetime (and thus his planning horizon) is uncertain two complications arise. First, the agent's decision problem becomes inherently stochastic and the expected utility hypothesis must be employed. Second, the non-negativity constraint on the agent's wealth position at the time of death is also stochastic and should be ensured to hold with certainty. Yaari showed that the key implication of uncertain lifetimes is that the instantaneous probability of death (the so-called "death hazard rate") enters the consumption Euler equation of the expected-utility maximizing agent. Intuitively, the uncertainty of survival leads the rational agent to discount the future more

heavily.

Yaari makes the analysis of terminal wealth more tractable by postulating the existence of a kind of life insurance based on actuarial notes. Such a note can be bought or sold by the consumer and is cancelled upon the consumer's death. A consumer who buys an actuarial note in fact buys an annuity which stipulates payments to the consumer during life at a rate higher than the rate of interest. Upon the consumer's death the insurance company has no further obligations to the consumer's estate. Reversely, a consumer who sells an actuarial note is getting a life-insured loan. During the consumer's life he/she must pay a higher interest rate on the loan than the market rate of interest but upon death the consumer's estate is held free of any obligations, i.e. the principal does not have to be paid back to the insurance

581

The Foundation of Modern Macroeconomics

company. Under actuarial fairness the rate of return on actuarial notes equals the rate of interest plus the death hazard. Yaari shows that households have the incentive to fully insure against the loss of life. He thus reaches the striking result that, with actuarially fair life insurance, the death hazard drops out of the consumption Euler equation altogether.

Yaari's insights lay dormant for two decades before Blanchard embedded them in his dynamic general equilibrium model with overlapping generations. In order to allow for an aggregate treatment, Blanchard made two modelling choices. First, he assumed that the death hazard is age-independent. This ensures that the optimal decision rules are "linear in the generations index" and can thus be aggregated. Second, he assumed the arrival of large cohorts of newborn agents at each instant. This ensures that frequencies and probabilities coincide. (To ensure a constant population the birth and death rates are assumed to be equal.)

The Blanchard—Yaari model has a number of important properties. First, the standard Ramsey model (based on the notion of an infinitely lived representative household) is obtained as a special case of the Blanchard—Yaari model by setting the birth rate equal to zero. Second, the steady-state capital stock is smaller in the Blanchard—Yaari model than in the Ramsey model. Due to the turnover of generations, aggregate consumption growth falls short of individual consumption growth. This means that in the steady state the interest rate exceeds the rate of time preference. It also means that the equilibrium is dynamically efficient. Third, fiscal policy, taking the form of a permanent and unanticipated lump-sum tax financed increase in government consumption, causes less (more) than one-for-one crowding out of private consumption in the short (long) run. In the short run, households do not feel the full burden of the additional taxes because they discount present and future tax liabilities at the annuity rather than the market rate of interest. As a result they do not cut back consumption by enough so that private investment is crowded out. In the long run the capital stock and output are smaller, wages are lower, and the interest rate is higher. Intuitively, the shock redistributes resources away from future generations towards present generations. Finally, Ricardian equivalence does not hold in the Blanchard—Yaari model. It is the positive birth rate (and not the agents' finite planning horizon) which causes the rejection of Ricardian equivalence.

In the second half of the chapter we show three extensions to the Blanchard—Yaari model. In the first extension we endogenize the household's labour supply decision. We use the model to study the effects of an increase in the consumption tax. In the Ramsey version of the extended model, the tax increase unambiguously leads to a decrease in the long-run capital stock because the household cuts back labour supply. With finite lives, however, the tax redistributes resources from present to future generations which tends to increase the capital stock in the long run. The net effect of the tax shock thus depends on the relative strength of the generational turnover effect vis-a-vis the factor scarcity effect.

In the second exte , 11 productivity declines notion of saving for ret number of hours du: .. valuable over time. If tl equilibrium may be youth and agents pra,.. and the economy as a I

Finally, in the third Blanchard—Yaari model.

oil price shock. model is that it can bL citizens) and debtor co Chapter 14 that in the only exists for a knife-e time preference

Further Reading

The Blanchard-Yaari mod models are presented (1987), Matsuyama (19 , closely related Weil (1989t 1994) and Nielsen and Obstfeld and Rogoff (19

Alogoskoufis and van growth into the model. neutrality. Aschauer (1990 model. On public infra , ' Scaramozzino (1995) and issues. Nielsen (1994) in the model by assuming t:. process. The International Blanchard-Yaari frames.

Appendix

Derivation of Figure

In this appendix we d, with endogenous labour approach discussed in deU