Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

(A15.9)

imum value) is obtained by

(A15. 10)

(A15.11)

A in Figure 15.1. For points loping.24

n with (T1.6), (T1.8), and

(A15.12)

I Figure 15.1.

C and K for which C = 0. can write the CE line as

ratio for which the rate of 115.13) that consumption (Zo /y*) 1 /e, . By substituting in the (C,K) plane:

(A15.14)

through the coordinates

(A15.15)

I

a for KGR collapses to the usual

The Foundation of Modern Macroeconomics

By using these expressions in (A15.17) we obtain:

The impact effect is obtained by substituting (A15.20) in (A15.18):

(A15.23)

By inverting (A15.22) and noting that G{e- at , s} = 1 / (s + a) and G{T(ai, a2, t), s} al)(S a2)1 -1 we obtain the expression for the transition paths of K(t) and C(t):

C(t) C(0) + —(5.21 T(K, Ali 0, (A15.24)

[

where T(K, A1, t) is a temporary bell-shaped transition term with properties covered by the following lemma.

Lemma 1 Let T(ai, a2, t) be a single transition function of the form:

with al > 0 and a2 > 0. Then T(ai, a2, t) has the following properties: (i) (positive) T(ai, a2, t) > 0 for t E (0, oo), (ii) T(ai , 0/2, t) = 0 for t = 0 and in the limit as t oo, (iii) (single-peaked) dT(ai, cx2, t)/ dt > 0 for t E (0, t), dr(ai, a2, t)/ dt < 0 for t E (t, oo), dT(a1 , a2, t)/dt = 0 for t = t and in the limit as t oo, and dT (al, a2, 0)/dt = 1, (iv) in (a1/a2)/(al —012) ifal a2

Lemma 2 If a2 = 0 the transition function is proportional to a monotonic adjustment function: T(ai , 0, t) = (1 / ai)Mai , t), where A(ai, t) 1 —e-"lt has the following properties: 0 < A(ai, t) < A(ai, 0) = 0 and limt_,„„ A(ai, = 1, (iii) (monotonic) dA(ai, t)/dt > 0 for

dA(a brt)Iclt = 0.

It follows from Lemma A15.2 that for permanent shocks (0( = 0) (A15.24) can be rewritten as:

where the long-run effects are given by:

Derivation of (15.86i

We compute the impulse-r

where the shock vector is deterministic counterpart t (15.77) when we compute porated the rational expect from the innovation e6

In the Mathematical App the aid of the z-transform —1 < < 0 and A2 > 0, tl

where A(A2) A2/ — A and ko = 0). The impact jump it

Zly c 1 + A.21 a=-' t

1 +A2

The shock term (15.85) ca I

The z-transform for yti can •

Using (A15.31) in (A15.29) v

By substituting roc and /ix from (A15.31) that:

z — (1 + A2)

►:

p

: (i) (positive) T(ai, a2, t) > 0

C DO, (iii) (single-peaked)

N.►, dT(a , a2, t)/dt = 0 for la1/a2)/(ai —.2) ifai 0 .2 o for t* = 2t; (vi) if a t 00

ionic adjustment function: properties: 0 < A(ai, t) <

isotonic) dA(ai, t)/dt > 0 for

) (VI 5.24) can be rewritten

(A15.25)

(A15.26)

Chapter 15: Real Business Cycles

Derivation of (15.86)—(15.87)

We compute the impulse-response function associated with the innovation e6 by solving the following system:

—A [ Ct

where the shock vector is given in (15.85). The key thing to note is that (A15.27) is the deterministic counterpart to (15.77). The expectations operator, Et , can be dropped from (15.77) when we compute the impulse-response function because we have already incorporated the rational expectations assumption by substituting the path for kt that results from the innovation E6 into the shock term.

In the Mathematical Appendix we show how a system like (A15.27) can be solved with the aid of the z-transform method. Assuming that A possesses real characteristic roots, —1 < —Al < 0 and A.2 > 0, the general solution of (A15.27) is:

Zlytc, — (z/(1 + A2)) Zlytc, 1 + A2 } z — ( 1 + A2)

where A(A.2) A21 — A and we have used the fact that capital cannot jump at impact (i.e. Ro = 0). The impact jump in consumption (C0) is:

The shock term (15.85) can be written in general format as:

The z-transform for y/ can then be written as:

Using (A15.31) in (A15.29) we obtain the following expression for

By substituting qc and 11K in (A15.32) we obtain equation (15.87) in the text. We derive from (A15.31) that:

The Foundation of Modern Macroeconomics so that (A15.28) can be rewritten as:

0 (see Ogata (1995, p. 30)). This term is the discrete-time counterpart to the single transition function whose properties are similar to the ones covered in Lemma A15.1 above. A result we use in the analysis of permanent shocks is that T,(1, (1 - a2) - 1 At (a2), where At (a2 ) 1 - c4 is a discrete-time adjustment term. For purely transitory shocks we have:

(Note that in the text we combine (A15.35) and (A15.36) into (15.88).) By inverting (A15.34) we find the solution in the time domain:

By simplifying (A15.37) somewhat we find the equation (15.86) in the text.

Method of undetermined coefficients

In this subsection we show how the unit-elastic RBC model of section 15.5.1 can be solved using the method of undetermined coefficients. Following Campbell (1994, p. 470), we conjecture the following trial solution:

where rrck and gc, are coefficients to be determined. By substituting (A15.38) in the system (15.77) we obtain:

(A15.39)

[ ")/*1) zi OZt,

are the elements of A* (defined in (15.78)) and we have used the fact that EtCt± i = and Et 2t-F1 = pzZt. The system in (A15.39) gives two expressions for kt+1

a

in terms of Kt and 2t from (A15.39) we find:

0= [1 + 811 + 812 7rck - (1

+ [6 127ra + OY* - —

We use 7rck to ensure that tt manipulation we find the

6 127TA + (6 11 — 622)JTck —

where 64 are the elements 622 = -y21612 + 8;2 ). Given sa

I

7tck = — (6 1 1 — 622 )

I

(Note also that 6 127,k = 622 - of Zt in (A15.40) can be put

I

gcz

= kPz - (Y21 +

Sivrck - 622 ( 1 —

I

Once we know the coeffil row of (A15.39):

kt+1 = YrkkKt 7rkz Zt ,

where irkk 1 + 811 + 8127r, a

Computing correlation

In order to judge the empirici various correlations that ark: from an analytical viewpoin literature discussed in Chapt

26 The sign of Trck follows from can be written as On — 822) 2 + - that lAl < 0. Hence, the roots ai Hence, the discriminant is lar., root. The positive root must be st (A15.44) lies between zero and of

—8 12

• + (1 — Pz)

(A15.34)

ail = Tr(ai, a2), where Tt (.)

(A15.35)

to counterpart to the single vizi in Lemma A15.1 above.

= (1 —a2) -1 At (a2), where ,itory shocks we have:

(A15.36)

Qc(1.) By inverting (A15.34)

p

(A15.37)

n the text.

rtion 15.5.1 can be solved ^7)bell (1994, p. 470), we

(A15.38)

(A15.38) in the system

11/4,

Kt + 7rcz2t

(A15.39)

used the fact that EtCt+i = •wo expressions for Kt-f1

Chapter 15: Real Business Cycles

in terms of RI- and Zr which must hold for all (Kr , 20 combinations. By eliminating Kt-t-1 from (A15.39) we find:

We use 7rck to ensure that the term in square brackets in front of Kt is zero. After some manipulation we find the following quadratic function in 7rck:

822 = — Y21 8 12 + 822 ). Given saddle path stability, we solve (A15.41) for the positive root: 26

(Note also that 6lVrck = 822 — A2.) For this value of 7r ck, the term in square brackets in front of 2,- in (A15.40) can be put to zero by the appropriate choice of zcs:

Once we know the coefficients 7rck and 7ra , we obtain the solution for kt±i by using either row of (A15.39):

where 7rkk = 1 + 811 + 8 127rck and 7rkz 6127a + 0y*. (Note also that mkk = 1 — Al.)

Computing correlations

In order to judge the empirical performance of the unit-elastic RBC model we can compute various correlations that are implied by the theoretical model. We approach the problem from an analytical viewpoint in order to stress the link with the rational expectations literature discussed in Chapter 3. We start by computing the statistical properties of the

26 The sign of 7rck follows from saddle-point stability. First, we note that the discriminant in (A15.42)

can be written as (811 — 622)2 + 46 12 621 = (5 11 + 822)2 — 4 IAI > 0, where the sign follows from the fact that I 0 I < 0. Hence, the roots are real and distinct. Next we note that 612621 = — Y21 8i2( 1 + 81 1 ) > 0. Hence, the discriminant is larger than (Sii — 622) so that (A15.41) has one positive and one negative

root. The positive root must be selected in order to ensure that the steady state is stable, i.e. that 7rkk in (A15.44) lies between zero and one (see also Campbell, 1994, pp. 471-472).

537

The Foundation of Modern Macroeconomics capital stock. We derive from (A15.44) that:

where we have used the fact that Ekt = 0. Since 4 is covariance stationary, 27 the same holds for Kt (and all other endogenous variables). Hence, Var(Rt+i) = Var(k t ) and equation (A15.45) can be simplified to:

It is straightforward to derive from (15.83) that:

where QZ is the (constant) variance of the innovation term (i.e. al E(Er)2 ) and we have used covariance stationarity of the shock process (so that Var(2t ) = Var(Zt_i )). Similarly, we find:

where we have used the fact that (A15.44) is a stable difference equation so that n-,.(1;,Rt-T goes to zero as T becomes large. By using (A15.48) and (A15.49) we find the expression for Cov(Kt , 2t):

Cov(k t) E [Kt — EKt ] 2t = 7kz E 7rkki 1 E2t2t-i j=i

27 A stochastic process, PO, is covariance stationary if the mean is independent of time and the

sequence of autocovariance matrices, E(xt+i — Ext+i)(xt — Ext )T depends only on j but not on t. See Ljungqvist and Sargent (2000, p. 9) and Patterson (2000, ch. 3).

By substituting (A15.48) a variance of the capital s:

(1 + pi 7,

var(kt+i) = p, -

—

It follows from (A15.44,

Cov(Kt+i,kt) E[kr.

= 7rki Va

Pz

1 —

I Now that we have expi covariances of all remain we derive from (A15.38):

Var( t ) = gAVar

Cov(Ct, Kt) = 7rCk Val ,4

By using (A15.38) in (15.7 and the interest rate in t for these variables. For oi

Yt = 7Tykkt + 7ry,2t,

where 7ryk . (1)( 1 EL)

to compute the covarianc from (A15.38) and (A15.5

COV( t, Yt) = 7tck zyk

Trcz

I

Similarly, we derive from

Cov(kt, kt) = 7TykVam

Similar expressions for th4 report correlation coefia1/4.A

COV(A.

P(xt, Yr) = Nar(xt)Vai

(A15.51)

(A15.52)

(A15.47)

E(ET)2 ) and we have = Var(2t-i)). Similarly,

(A15.48)

(A15.49)

I

: ition so that n-lirkt_T 71c1 the expression for

(A15.50)

Now that we have expressions for Var(Kt), Var(2t), and Cov(Kt , 20, the variances and covariances of all remaining variables are easily obtained. For consumption, for example, we derive from (A15.38):

By using (A15.38) in (15.71)-(15.75) we can write employment, wages, output, investment, and the interest rate in terms Kt and Zt and derive expressions similar to (A15.53)-(A15.54) for these variables. For output, for example, we find the following expression:

to compute the covariances of the different variables with output. For example, it follows from (A15.38) and (A15.55) that Cov( t , kt ) is:

COV( 170 = 7cOrykVar(kt) [7cOryz Irczn'yk] Cov(Kt, Zr)

Similar expressions for the other variables are easily found. Finally, note that in the text we report correlation coefficients. These are defined as follows:

16

Intergenerational Economics,

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

1.To introduce a popular continuous-time overlapping-generations (OG) model and to show its main theoretical properties;

2.To apply this workhorse model to study fiscal policy issues and the role of debt;

3.To extend the continuous-time OG model to the cases of endogenous labour supply, age-dependent labour productivity, and the small open economy.

16.1 Introduction

In this chapter we study one of the "workhorse" models of modern macroeconomics, namely the Blanchard-Yaari model of overlapping generations. This model has proved to be quite useful because it is very flexible and contains the Ramsey model as a special case. The key element which differentiates the Blanchard-Yaari model from the Ramsey model is that the former distinguishes agents by their date of birth, whereas the latter assumes a single representative agent. By incorporating some smart modelling devices, the Blanchard-Yaari model can be solved and

analysed at the aggregate macroeconomic level, despite the fact that individual households are heterogeneous.

16.2The Blanchard—Yaari Model of Overlapping Generations

16.2.1 Yaari's lessons

One of the great certainties in life—apart from taxes—is death. After that things get fuzzy because nobody knows exactly when the Grim Reaper will make his one and

only call. In all consumptia has been ignored, however. consumption-saving mouL the Ricardian Equivalence that he/she will only live Ramsey model in which at consumption and savings the agent lives forever in ti In a seminal article, \'‘, the context of a dynamic c of the key building blocks which itself has become Yaari (1965, pp. 139-14u) odel with lifetime unc,

so is that agent's lifetime is inherently stochastic an expected utility hypotlk objective function. Seconc the time of death is sir death. In symbols, if Ak. solution procedure should Fortunately, Yaari (19: plications. First, though expected utility hypoth, tion for T. Indeed, demo b of the distribution functic time and there also seen _ So the density function fc

f (T) ?_ 0, VT > 0, j

The first property is a gel that the random variabl

T < = 1).

The consumer's lifetime

o

where U [C(r)] is instants sumption, 1 and p is till

Labour supply is taken tc part of the consumer's optima

16

COMICS)

aerations (OG) model and

ues and the role of debt;

[endogenous labour supply, i economy.

- s of modern macroeco- g generations. This model and contains the Ramsey tes the Blanchard—Yaari wishes agents by their lye agent. By incorpormodel can be solved and the fact that individual

-, fh. After that things get er will make his one and

Chapter 16: Intergenerational Economics, I

only call. In all consumption models discussed so far in this book, lifetime uncertainty has been ignored, however. Indeed, in Chapter 6 we introduced the basic two-period consumption-saving model to illustrate the various reasons for the breakdown of the Ricardian Equivalence Theorem. But in that model each agent knows exactly that he/she will only live for two periods. Similarly, in Chapter 14 we explained the Ramsey model in which an infinitely lived representative consumer makes optimal consumption and savings decisions. Again there is no lifetime uncertainty because the agent lives forever in this model.

In a seminal article, Yaari (1965) confronted the issue of lifetime uncertainty in the context of a dynamic consumption-saving model. In doing so, he provided one of the key building blocks of the Blanchard (1985) overlapping generations model which itself has become one of the workhorse models of dynamic macroeconomics. Yaari (1965, pp. 139-140) clearly identified the two complications that arise in a model with lifetime uncertainty. First, if the agent's time of death, T, is random then so is that agent's lifetime utility function. As a result the agent's decision problem is inherently stochastic and maximizing lifetime utility makes no sense. Rather, the expected utility hypothesis must be used and expected lifetime utility should be the objective function. Second, the non-negativity constraint on the agent's wealth at the time of death is similarly stochastic as it also depends on the random time of death. In symbols, if A(t) is real assets at time t, then A(T) is stochastic and the solution procedure should ensure that A(T) > 0 holds with certainty.

Fortunately, Yaari (1965) also proposed appropriate solutions to these two complications. First, though T is a random variable all we need to do to render the expected utility hypothesis operational is to postulate the probability density function for T. Indeed, demographic data can be used to obtain quite detailed estimates of the distribution function for T. Obviously, no one has a negative expected lifetime and there also seems to be a finite upper limit, T, beyond which nobody lives. So the density function for T is denoted by f (T) and it satisfies:

f (T) > 0, VT 0, f (T) dT = (16.1)

The first property is a general requirement for densities and the second one says that the random variable T lies in the interval [0, with probability 1 (i.e. Pr{0 <

T < = 1).

The consumer's lifetime utility is denoted by A (T) and is defined as follows:

where U [C(r)] is instantaneous utility (or "felicity") at time r, C(r) is private consumption, 1 and p is the pure rate of time preference. Using this notation, the

1 Labour supply is taken to be inelastically supplied. Hence, the consumption-leisure decision is not part of the consumer's optimization problem. Later on we will relax this.

541

The Foundation of Modern Macroeconomics expected lifetime utility can be written as: 2

T

where 1 – F(r) is the probability that the consumer will still be alive at time r, i.e.

The crucial thing to note about (16.3) is that the consumer's objective function is now in a rather standard format. Apart from containing some additional elements and F(r)) resulting from lifetime uncertainty, the expression in (16.3) is very

similar to the utility function of the representative consumer (namely (14.53) in Chapter 14).

The second complication identified by Yaari (1965) and discussed above can also

be easily dealt with. Assume that the household budget identity can be written as follows:

A(r) = r(r)A(r) W(r) – C(r), (16.5)

where A(r) dA(r)/ dr , r(r) is the rate of interest, and W(r) is non-interest income, all expressed in real terms (units of output). Both r(r) and W(r) are known to the consumer as lifetime uncertainty is (by assumption) the only stochastic element in

the model. The final wealth constraint, Pr{A(T) > 0} = 1, is then equivalent to: 3

A(T) = 0, C(r) W(r) whenever A(r) = 0. (16.6)

The consumer maximizes expected lifetime utility (EA(T) in (16.3)) subject to (16.5) and (16.6), the non-negativity constraint on consumption (C(r) > 0), and given

the initial wealth level (A(0)). The interior solution for this optimization problem is summarized by the following expressions:

[1 – F(r)] [C(r)] = X(r) (16.7)

where A.(r)—the co-state variable associated with (16.5)—represents the expected marginal utility of wealth. Intuitively, (16.7) says that in the interior solution the

2 In going from the first to the second line in (16.3) we have changed the order of integration.

3 Yaari (1965, pp. 142-143) shows this result as follows. We know for sure that the constraint A(r) > 0

must hold with equality for r = T, i.e. A(T) = 0. For other values of r it follows that A(r) > 0 is equivalent to A(r) = W(r) — C(r) > 0 if A(r) = 0, i.e. no dissaving is allowed if no wealth remains.

consumer equates the L.N marginal utility of wealth the optimal dynamics.

By combining (16.7) a equation in the presence

C(r) = a [C(r)J[r(7

where a [C(r)] –U' [CI elasticity (see Chapter 1 4 rate" or instantaneous pri infinitely lived consul, Euler equation.4 This is t survival leads the housel discount rate in the pr, This makes intuitive sen long enough to enjoy a discount the utility str, Up to this point we 1-L no insurance possibilr ance exist so a relevant c consumer's behaviour. 'I insurance based on so-c actuarial note can be bc consumer's death. The i rA (r) and non-zero track who buys an actuarial n the consumer during I.: sumer's death the insura estate. Reversely, a co:. loan. During the consur than the market rate of of any obligations, i.e. tl

company.

In order to determir plest possible) assumpti implied by this assume bought at time T. These consumer survives) or a

In the standard Ramse