Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

(16.3)

II be alive at time r, i.e.

111

(16.4)

-'s objective function is ome additional elements

--2ssion in (16.3) is very ner (namely (14.53) in

"scussed above can also ientity can be written as

(16.5)

) is non-interest income, I t"(T) are known to the " . stochastic element in then equivalent to: 3

k 416.3)) subject to (16.5) 1 (C(r) > 0), and given s optimization problem

(16.7)

(16.8)

presents the expected interior solution the

-order of integration.

-the constraint A(T) > 0

:mom,s that A(r) > 0 is ed if no wealth remains.

Chapter 16: Intergenerational Economics, I

consumer equates the expected marginal utility of consumption to the expected marginal utility of wealth. Equation (16.8) is the standard expression summarizing the optimal dynamics.

By combining (16.7) and (16.8) we obtain the household's consumption Euler equation in the presence of lifetime uncertainty:

rate" or instantaneous probability of death at time T. Compared to the case of an infinitely lived consumer, the hazard rate is the additional term appearing in the Euler equation.4 This is the first lesson from Yaari (1965, p. 143): the uncertainty of survival leads the household to discount the future more heavily, i.e. the subjective discount rate in the presence of lifetime uncertainty is p + p(r) rather than just p. This makes intuitive sense. If there is a positive probability that you will not live long enough to enjoy a given planned future consumption path, then you tend to discount the utility stream resulting from it more heavily.

Up to this point we have studied the optimal behaviour of the consumer when no insurance possibilities are available. But in reality various forms of life insurance exist so a relevant question is how this institutional feature would change the consumer's behaviour. Yaari (1965, pp. 140-141) suggests a particular kind of life insurance based on so-called actuarial notes issued by the insurance company. An actuarial note can be bought or sold by the consumer and is cancelled upon the consumer's death. The instantaneous rate of interest on such notes is denoted by r^ (r) and non-zero trade in such notes only occurs if rA (r) exceeds r (O. 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 company.

In order to determine the rate of return on actuarial notes, Yaari makes the (sim plest possible) assumption of actuarial fairness. To derive the expression for rA (r) implied by this assumption, assume that one dollar's worth of actuarial notes is bought at time T. These notes are either redeemed with interest at time r +dr (if the consumer survives) or are cancelled (if the consumer dies between r and r

4 In the standard Ramsey model no lifetime uncertainty exists. See e.g. Chapters 14 and 15.

The Foundation of Modern Macroeconomics

1 — F(t)

where the equality holds as dr 0. The right-hand side of (16.10) shows the yield if the dollar is invested in regular market instruments whereas the left-hand side shows the yield on the actuarial note purchase. The term in round brackets is less than unity and corrects for the fact that the consumer may pass away between r and t +dr . By solving for rA (r) and taking the limit as dr —> 0 we obtain the following—rather intuitive—no-arbitrage equation between the two kinds of financial instruments: 5

rA (r) = r(r) + 13(4

Recall that t is not only the time index but also stands for the age of the consumer

so (16.11) has the sensible implication that rA (r) oo as t —> T. The closer the consumer gets to the maximum possible age T, the higher will be the instantaneous

probability of death and thus the higher will be the required excess yield on actuarial notes.

Let us now return to the consumer's choice problem. As Yaari (1965, p. 145) points out, the consumer will always hold his/her financial assets in the form of actuarial

notes, i.e. he/she will fully insure against the loss of life and the budget identity will be:

A(r) = rA (r)A(r) + W (r) — C(r). (16.12)

Hence the restriction on the terminal asset position is trivially met as all actuarial notes are automatically cancelled when the consumer dies. The intuition behind this full-insurance result is best understood by looking at the two cases. If the consumer has positive net assets at any time then they will be held in the form of actuarial notes because these yield the highest return (which is all the consumer is interested in in the absence of a bequest motive). Conversely, if the consumer had any negative outstanding net assets in other than actuarial notes, he/she would be violating the constraint on terminal assets mentioned above (i.e. the requirement that Pr{A(T) > 01 = 1).

We are not out of the forest of complications yet as we also need to ensure that the consumer is unable to beat the system by engaging in unlimited borrowing (sales of actuarial notes) and covering the ever increasing interest payments with

5 Equation (16.11) is derived as follows. We note that (16.10) can be rewritten as:

By letting dr 0, the first term on the right-hand side goes to r(r) and the second approaches /3(r) f (r)/ [1 — F(r)].

yet further borrowings. T 1965, p. 146 for a detaile

Intuitively, the conditio must be equal to the su and future non-interest notes for discounting. The consumer maxim the solvency condition (C(r) > 0). The interior

Euler equation:

(.r)

C(r) = [C(r)] [r

= a [C (T )1 -

where we have used (16 thing to note about (16. equation with fully in , when no lifetime uncer sumption levels will LI possibility frontier will

16.2.2 Turning Iessc

Yaari's crucial insights 1 them the core elem, which subsequently b economics. Blanchard probability density fu in (16.1) is specified as:

I e- PT f (T) = 0

so that 1 — F(r) J : instead of assuming ai did—Blanchard assu consumer's age. This ai optimal consumption We are thus able to mai

(1(.10)

116.10) shows the yield if s the left-hand side shows und brackets is less than way between r and T+dr.

-1 the following—rather f financial instruments:5

the age of the consumer s r D. The closer the will be the instantaneous vcess yield on actuarial

(1965, p. 145) points in the form of actuarial i the budget identity will

(16.12)

met as all actuarial

s.The intuition behind two cases. If the con-

be held in the form of

. is all the consumer is ely, if the consumer had otes, he/she would be (i.e. the requirement

.o need to ensure that in unlimited borrowing -,.tereq payments with

, ritten as:

second approaches /3(T)

Chapter 16: Intergenerational Economics, I

yet further borrowings. This prompts the consumer's solvency condition (see Yaari, 1965, p. 146 for a detailed derivation):

Intuitively, the condition says that the present value of the consumption stream must be equal to the sum of initial financial assets plus the present value of current and future non-interest income (i.e. "human wealth"), using the rate on actuarial

notes for discounting.

The consumer maximizes expected lifetime utility (EA(T) in (16.3)) subject to the solvency condition (16.13) and the non-negativity constraint on consumption (C(r) > 0). The interior solution to this problem is characterized by the following Euler equation:

= [C(r)][t.A (r) – p – s(01

C(r)

= a iC(r)ii ) – ,

where we have used (16.11) in going from the first to the second line. The striking thing to note about (16.14)—and thus Yaari's second lesson—is the fact that the Euler equation with fully insured lifetime uncertainty is identical to the Euler equation when no lifetime uncertainty exists! It should be observed, however, that the consumption levels will differ between the two scenarios as the lifetime consumption possibility frontier will differ between the two cases.

16.2.2 Turning lessons into a workhorse

Yaari's crucial insights lay dormant for twenty years until Blanchard (1985) made them the core elements of his continuous-time overlapping-generations model which subsequently became one of the workhorse models of modern macroeconomics. Blanchard simplified the Yaari setup substantially by assuming that the probability density function for the consumer's time of death is exponential, i.e. f(T) in (16.1) is specified as:

so that 1 – F(r) f(T)dT = f (r)/ 13 and p(r) f(r)/ [1 – Fer)] = 13. Hence, instead of assuming an age-dependent instantaneous death probability—as Yaari

did—Blanchard assumes that the hazard rate is constant and independent of the consumer's age. This approach has several advantages. First and foremost, it leads to optimal consumption rules that are easy to aggregate across households (see below). We are thus able to maintain a high level of aggregation in the model despite the fact

The Foundation of Modern Macroeconomics

that the underlying population of consumers is heterogeneous by age. Second, it follows from (16.15) that the expected remaining lifetime of any agent is equal to 1//3. By setting /3 = 0, the Blanchard model thus coincides with the representative-agent model studied extensively in Chapters 14 and 15 above. 6

Individual households

The first task at hand is to derive the expressions for consumption and savings for an individual household at an arbitrary time during its life. Assume that the utility function at time t of a consumer born at time v t is given by EA (v, t):

where we have used the property of the exponential distribution in (16.15) to deduce that 1 — f (r — t) = efi(t- T) . Furthermore, in going from (16.3) to (16.16) we have assumed a logarithmic felicity function (featuring a unit intertemporal substitution elasticity), and we have added indexes for the agent's date of birth (v) and the time to which the decision problem refers (t). Consequently, C(v, r) stands for planned consumption at time r by an agent born at time v. The agent's budget identity is:

)1(v, r) = [r(r) + 13] A(v, r) W (r) — T (r) — C(v, r), (16.17)

where r(r), is the interest rate, W(r) is the wage rate, T (r) is the lump-sum tax levied by the government, and A(v, r) are real financial assets. Equation (16.17) incorporates the Yaari notion of actuarially fair life-insurance contracts and is a straightforward generalization of (16.12) with (16.11) substituted in. Specifically, during life agents receive /3A (v, r) from the life-insurance company but at the time of the agent's death the entire estate A(v , r) reverts to that company. To avoid the agent from running a Ponzi game against the life-insurance company, the following solvency condition must be obeyed.

By combining (16.17) and (16.18) the household's lifetime budget restriction is obtained:

6 Of course, the modelling simplification does not come without a price tag. The main disadvantage of assuming a constant instantaneous death probability is that it leads to a consumption model that— like the representative-agent model—is at odds with the typical life-cycle consumption pattern observed in empirical studies. We will return to this issue below. _

11 where H(t) is the huma lifetime after-tax wage Di

H(t) f [W (r) —

t

Equation (16.19) is du, present value of the ho financial and human

=

Intermezzo

Intuition behind tht the household's sol note that (16.17) call

--- [^ h^ r)«-R-

where we have use— going from the first ti interval [t, oo) we obi

dA(v , r)e -h

lim e-RA(t •T ) A( ` T DO

where we have ust companies will ensu. to as "terminal assets est of the consume: motive and does no . of consumption rt minal assets will b,.

(1(.18). By using it restriction (16.19) - mai discussion of ti control problem.

c_."Ous by age. Second, it my agent is equal to 1/p. the representative-agent

umption and savings for Assume that the utility

n by EA (v, t):

(16.16)

In in (16.15) to deduce 6.3) to (16.16) we have -temporal substitution birth (v) and the time „ r) stands for planned

t's budget identity is:

(16.17)

is the lump-sum tax ssets. Equation (16.17) ce contracts and is a tuted in. Specifically, 'mpany but at the time )mpany. To avoid the mpany, the following

(16.18)

I

re budget restriction is

(16.19)

tag. The main disadvantage , nsumption model thatption pattern observed

Chapter 16: Intergenerational Economics, I

where H(t) is the human wealth of the agents consisting of the present value of lifetime after-tax wage income using the annuity factor, RA (t, r), for discounting:

Equation (16.19) is the counterpart to (16.13) above. Intuitively, it says that the present value of the household's consumption plan must be equal to the sum of financial and human wealth.

Intermezzo

Intuition behind the household's solvency condition. The intuition behind the household's solvency condition (16.18) can be explained as follows. We note that (16.17) can be premultiplied by e-RA(t'T) and rearranged to:

going from the first to the second line. By integrating both sides of (a) over the interval [t, cc) we obtain:

) —RA(t,r) ____ f W(r) - T(r) C(v, r)] e -RA t'r) dr

dA( t

where we have used (16.20) and have noted that e- RA(t,t) = 1. The insurance companies will ensure that the limit on the left-hand side of (b) (loosely referred to as "terminal assets") will be non-negative. Similarly, it is not in the best interest of the consumer to plan for positive terminal assets as he/she has no bequest motive and does not get satiated from consuming goods (as the marginal felicity of consumption remains strictly positive—see (16.16)). Hence, planned terminal assets will be strictly equal to zero. This yields the solvency condition (16.18). By using it in (b) the expression for the household's lifetime budget restriction (16.19) is obtained. See Chiang (1992, pp. 101-103) for a more formal discussion of the transversality condition in an infinite-horizon optimal control problem.

547

The Foundation of Modern Macroeconomics

The consumer maximizes expected lifetime utility (16.17) subject to its lifetime budget restriction (16.19). The first-order conditions are (16.19) and:

where A (t), the Lagrange multiplier associated with the lifetime budget restriction (16.19), represents the marginal expected lifetime utility of wealth. 7 Intuitively, the optimality condition (16.21) instructs the consumer to plan consumption at each time to be such that the appropriately discounted marginal utility of consumption (left-hand side) and wealth (right-hand side) are equated (see also the discussion following). By using (16.21) for the planning period (r = t) we see that C(v, t) = 1/A(t). Using this result and (16.19) in (16.21) we can express C(v, t) in terms of total wealth:

Optimal consumption in the planning period (r = t) is proportional to total wealth, and the marginal propensity to consume out of total wealth is constant and equal to the "effective" rate of time preference, p + p.

Aggregate households

Now that we know what the consumption rules for individual households look like, the next task at hand is to describe the demographic structure of the Blanchard model. To keep things simple, Blanchard assumes that at each instant in time a large cohort of new agents is born. The size of this cohort of newborns is P(r, r) = 13P(r), where P(r) stands for the aggregate population size at time T. These newborn agents start their lives without any financial assets as they are unlinked to any existing agents and thus receive no bequests, i.e. A (r , r) = 0. Of course, at each instant in time a fraction of the existing population dies. Since each individual agent faces an instantaneous probability of death equal to ,8 and the number of agents P(r) is large, "frequencies and probabilities coincide" and the number of deaths at each instant will be equal to fil)(r). Since births and deaths exactly match, the size of the population is constant and can be normalized to unity (P(r) =

7 Note that by differentiating (16.21) with respect to r we obtain the household's Euler equation:

C(v, r) = r(r) — p. C(v, r)

In the OG model we also need to solve for the consumption level in the planning period.

8 Net population change can easily be incorporated in the Blanchard model by allowing the birth and death rates to differ—see Buiter (1988).

Another very useful co exactly trace the size of at time v will be of size t members will have died i cohort it is possible to V■ ing the consumption 1ev following expression for

C(t) fi f es( `'-t) C

Of course, (16.23) is simp because the optimal cor. of total wealth which is i gives rise to a very simple

C(t) ,B- el3(")(p

oo

=Go + fi)[0

=(p + P) [A(t) --t-

where aggregate financial (given in (16.23)). It ca: lows from the assumptio probability (see (16.16)). (1965) model—then the ( generation-independent exact aggregation is imp() What does the aggrt

we have that A(t) –= of Leibnitz's rule):

A(t) = ,8A(t, t) — 13A

where the first term on ti 0), the second term is the in assets of existing agent obtain the aggregate as ,

A(t) – _$A(t)± 13 f,

=—$A(t) + [r(t )

=r(t)A(t) +

I7) subject to its lifetime

16.19) and:

(16.21)

-time budget restriction wealth. ? Intuitively, the 'in consumption at each utility of consumption (see also the discussion t) we see that C(v, t) = (press C(v, t) in terms of

(16.22)

rtiorial to total wealth, :1 is constant and equal

i dual households look ructure of the Blanchard '1 instant in time a large vborns is P (r , r) = pP(r), r . These newborn agents

linked to any existing purse, at each instant in h individual agent faces number of agents P(r) is tmber of deaths at each

match, the size of the

• = 1). 8

-, ehold's Euler equation:

-"Inning period.

!el by allowing the birth

Chapter 16: Intergenerational Economics, I

Another very useful consequence of the large-cohort assumption is that we can exactly trace the size of any particular cohort over time. For example, a cohort born at time v will be of size peo (v-t) at time t > v, because ,6 [1 – e00/ -0 ] of the cohort members will have died in the time interval [v, t]. Since we know the size of each cohort it is possible to work with aggregate variables. For example, by aggregating the consumption levels of all existing agents in the economy we obtain the following expression for aggregate consumption at time t:

Of course, (16.23) is simply a definition and is not of much use in and of itself. But because the optimal consumption rule (16.22) features a propensity to consume out of total wealth which is independent of the generations index v, equation (16.23) gives rise to a very simple aggregate consumption rule:

where aggregate financial wealth is defined analogously to aggregate consumption (given in (16.23)). It cannot be overemphasized that the aggregation property follows from the assumption that each agent faces a constant instantaneous death probability (see (16.16)). If instead the hazard rate varies with age—as in the Yaari (1965) model—then the optimal household consumption rule no longer features a generation-independent marginal propensity to consume out of total wealth and exact aggregation is impossible.

What does the aggregate asset accumulation identity look like? By definition we have that A(t) ,8 ft A(v, t)eo (v- t) dv from which we derive (by application of Leibnitz's rule):

where the first term on the right-hand side represents assets of newborns (A(t, t) =

0), the second term is the wealth of agents who die, and the third term is the change in assets of existing agents. By substituting (16.17) into (16.25) and simplifying we obtain the aggregate asset accumulation identity:

A(t) = – pA(t) + p f [[r(t) + p] A(v, t) + W (t) – T (t) – C(v, Ne13(") dv

The Foundation of Modern Macroeconomics

Whereas individual wealth attracts the actuarial interest rate, r(t)+ 13, for agents that stay alive (see 16.17), equation (16.26) shows that aggregate wealth accumulates at the rate of interest, r(t). The amount ,BA(t) does not represent aggregate wealth accumulation but is a transfer—via the life-insurance companies—from those who die to those who remain alive.

In the formal analysis of the` model it is useful to have an expression for the "aggregate Euler equation". It follows from (16.23) that:

According to (16.22) newborn agents consume a fraction of their human wealth at birth, i.e. C(t, t) = (p+$)H(t). Equation (16.24) shows that aggregate consumption is proportional to total (human and financial) wealth, i.e. C(t) = (p + s)

Finally, it follows from (16.21) that individual households' consumption growth satisfies C(v, t)/C(v, t) = r(f') – p for r E [t, oo) (see footnote 7). By using all these results in (16.27) we obtain the aggregate Euler equation modified for the existence of overlapping generations of finitely lived agents:

Equation (16.28) has the same form as the Euler equation for individual households except for the correction term due to the distributional effects caused by the turnover of generations. Optimal consumption growth is the same for all generations (since they face the same interest rate) but older generations have a higher consumption level than younger generations (since the former generations are wealthier). Since existing generations are continually being replaced by newborns who hold no financial wealth, aggregate consumption growth falls short of individual consumption growth. The correction term appearing on the right-hand side of (16.28) thus represents the difference in average consumption and consumption by newborns, i.e. (16.28) can be re-expressed as in (16.29).

Firms

The production sector is characterized by a large number of firms that produce an identical good under perfect competition. Output, Y(t), is produced according to a linearly homogeneous technology with labour, L(t), and physical capital, K(t), as homogeneous factor inputs which are rented from households:

Table 16.1. The Blar,,_

e-(t) = [r(t) – t

= F(K(t), L

h(t) = r(t)a(t)

r(t) + 3 = FK (K(t).

W (t) = FL (K(t), L•

L(t) = 1

Notes: C(t) is consumption . A sum taxes, and r(t) is the ir • rate), and p is the pure rate

I where F(.) satisfies th( value of the represen...

V(t) = fco [Y(r) —

t

The firm chooses labs

production function (1

a

k(t) = I(t) – 3K(:

where I (t) denotes gro capital. There are no a conditions imply that producer costs of th Table 16.1. Finally, ss

equal to the replaceme

A

The government and a

The government buc„, consumes G(t) units o Government debt is Like the private sector game condition like:

ate, r(t)+ 0, for agents that re wealth accumulates at epresent aggregate wealth mpanies—from those who

ave an expression for the

(16.27)

of their human wealth at 4regate consumption is Cu) = (p fi) [A(t) H(t)].

Ads' consumption growth note 7). By using all these modified for the existence

-)n for individual housetional effects caused by the ", e same for all generations :s have a higher consump- ‘7,..nerations are wealthier). I coy newborns who hold no ort of individual consump- 4-hand side of (16.28) thus sumption by newborns,

- of firms that produce an 's produced according to a physical capital, K(t), as

seholds:

Chapter 16: Intergenerational Economics, I

Table 16.1. The Blanchard–Yaari model

Notes: C(t) is consumption, K(t) is the capital stock, B(t) is government debt, W(t) is the wage rate, T(t) is lumpsum taxes, and r(t) is the interest rate. Capital depreciates at a constant rate 3, p is the birth rate (equals death rate), and p is the pure rate of time preference.

where F(.) satisfies the usual Inada conditions (see Chapter 14). The stockmarket value of the representative firm is:

The firm chooses labour and capital in order to maximize (16.31) subject to the production function (16.30) and the capital accumulation constraint:

where I(t) denotes gross investment, and 8 is the constant rate of depreciation of capital. There are no adjustment costs associated with investment. The first-order conditions imply that the marginal productivity of labour and capital equal the producer costs of these factors—see, respectively, equations (T1.4) and (T1.5) in Table 16.1. Finally, we recall from Chapter 14 that the market value of the firm is equal to the replacement value of its capital stock, i.e. V(t) = K(t).

The government and market equilibrium

The government budget identity is given in (T1.3) in Table 16.1. The government consumes G(t) units of the good and levies lump-sum taxes on households T(t). Government debt is B(t) so that r(t)B(t) is interest payments on outstanding debt. Like the private sector, the government must remain solvent and obey a no-Ponzi- game condition like:

The Foundation of Modern Macroeconomics

By using (T1.3) and (16.33) the government budget restriction is obtained:

Intuitively, government solvency means that if there is a pre-existing government debt (positive left-hand side) it must be covered in present-value terms by present and future primary surpluses (right-hand side).

At each instant of time, factor and goods markets clear instantaneously. In this closed economy households can only accumulate domestic assets so that, as a result, financial market equilibtium requires that A(t) = K(t)+B(t). Wage flexibility ensures that the aggregate supply of labour 1) by households matches labour demand by firms. Goods market equilibrium is obtained when the supply of goods equals aggregate demand, which consists of private and public consumption plus investment: Y(t) = C(t) + I(t) + G(t). For convenience, the key equations of the model have been gathered in Table 16.1.

The phase diagram

In order to illustrate some of the key properties of the model we now derive the phase diagram in Figure 16.1. We assume for simplicity that lump-sum taxes, government consumption, and public debt are all zero in the initial situation (T(t) = G(t) = B(t) = 0). The K(t) = 0 line represents points for which the capital stock is in equilibrium. The Inada conditions (see Chapter 14) ensure that it passes through the origin and is vertical there (see point A l in Figure 16.1). Golden-rule

C(t)

Figure 16.1. Phase diagram of the Blanchard—Yaari model

(GR) consumption occw

I

The maximum attainabll tion is zero and total ou 8). For points above (bell be consistent with a cap negative (positive). This The derivation of the and slope depend on the attributable to intergent

the "Keynes-Ramsey" the exogenously given To is associated with a zc - (Fla < 0), KKR lies to tL. Furthermore, for points scarce (abundant), anc preference.

When agents have fini the turnover of genera t. tions (16.28) (with A = l< economy is initially on t of consumption, say at p same rate of interest.

coincides at t„ Expression (16.29) in depends not only on ind average consumption ar

C(t, t)]/C(t). Since nev, absolute difference bet born household depenc the two points. Since tl is at E0), this point feat newly born consumpt. C(t) < 0). In order to n stock must fall (to poi!" consumption growth 1- : aggregate consumption F capital stock narrows t erations that pass awa above (below) the C(t) =