Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

ction is obtained:

(16.34)

pre-existing government t-value terms by present

r instantaneously. In this ic assets so that, as a result, '1. Wage flexibility ensures Is matches labour demand supply of goods equals consumption plus invest- )* equations of the model

model we now derive the that lump-sum taxes, o in the initial situation is for which the capital

14)ensure that it passes

r:ire 16.1). Golden-rule

11

A3 _

KLX K(t)

• 3ari model

Chapter 16: Intergenerational Economics, I

(GR) consumption occurs at point A2 where the K(t)

The maximum attainable capital stock, KmAx, occurs at point tion is zero and total output is used for replacement investment

6). For points above (below) the K(t) = 0 line consumption is too high (too low) to be consistent with a capital stock equilibrium and consequently net investment is negative (positive). This has been indicated by horizontal arrows in Figure 16.1.

The derivation of the C(t) = 0 line is a little more complex because its position and slope depend on the interplay between effects due to capital scarcity and those attributable to intergenerational-distribution effects. Recall from Chapter 14 that the "Keynes-Ramsey" (KR) capital stock, KKR, is such that the rate of interest equals the exogenously given rate of time preference, i.e. FK (KKR , 1) _ p. Since KGR is associated with a zero interest rate and there are diminishing returns to capital (FKK < 0), KKR lies to the left of the golden-rule point as is indicated in Figure 16.1. Furthermore, for points to the left (right) of the dashed line, capital is relatively scarce (abundant), and the interest rate exceeds (falls short of) the pure rate of time preference.

When agents have finite lives (,3 > 0) the C = 0 line is upward sloping because of the turnover of generations. Its slope can be explained by appealing directly to equations (16.28) (with A = K as we set B = 0), (16.29), and Figure 16.1. Suppose that the economy is initially on the C = 0 curve, say at point E0. Now consider a lower level of consumption, say at point B. With the same capital stock, both points feature the same rate of interest. Accordingly, individual consumption growth, C(v, t)/C(v, t) [= r — p], coincides at the two points.

Expression (16.29) indicates, however, that aggregate consumption growth depends not only on individual growth but also the proportional difference between average consumption and consumption by a newly born generation, i.e. [C(t) — C(t, t)]/C(t). Since newly born generations start without any financial capital, the absolute difference between average consumption and consumption of a newly born household depends on the average capital stock and is thus the same at the two points. Since the level of aggregate consumption is lower at B (than it is at E0), this point features a larger proportional difference between average and newly born consumption, thereby decreasing aggregate consumption growth (i.e. C(t) < 0). In order to restore zero growth of aggregate consumption, the capital stock must fall (to point C). The smaller capital stock not only raises individual consumption growth by increasing the fate of interest but also lowers the drag on aggregate consumption growth due to the turnover of generations because a smaller capital stock narrows the gap between average wealth (i.e. the wealth of the generations that pass away) and wealth of the newly born. In summary, for points above (below) the C(t) = 0 line, the capital-scarcity effect dominates (is dominated

The Foundation of Modern Macroeconomics

by) the intergenerational-redistributional effect and consumption rises (falls) over time.9 This is indicated with vertical arrows in Figure 16.1.

In terms of Figure 16.1, steady-state equilibrium is attained at the intersection of the K(t) = 0 and C(t) = 0 lines at point E 0 . Given the configuration of arrows, it is clear that this equilibrium is saddle-point stable, and that the saddle path, SP, is upward sloping and lies between the two equilibrium loci.

16.3 Applications of the Basic Model

16.3.1 The effects of fiscal policy

As a first application of the Blanchard-Yaari model we now consider the effects of a typical fiscal policy experiment, consisting of an unanticipated and permanent increase in government consumption. We abstract from debt policy by assuming that the government balances its budget by means of lump-sum taxes only, i.e. B(t) = B(t) = 0 and G(t) = T (t) in equation (T1.3). We also assume that the economy is initially in a steady state and that the time of the shock is normalized to t = 0.

In terms of Figure 16.2, the K(t) = 0 line is shifted downward by the amount of the shock dG. In the short run the capital stock is predetermined and the economy jumps from point E0 to A on the new saddle path SPi . Over time the economy gradually moves from A to the new steady-state equilibrium at E 1 . As is clear from the figure, there is less than one-for-one crowding out of private by public consumption in the impact period, i.e. -1 < dC(0)/ dG < 0. In contrast, there is more than one-for-one crowding out in the long run, i.e. dC(oo)/dG < -1.

The reason for these crowding-out results is that the change in the lump-sum tax induces an intergenerational redistribution of resources away from future towards present generations (Bovenberg and Heijdra, forthcoming). At impact, all households cut back on private consumption because the higher lump-sum tax reduces the value of their human capital. Since households discount present and future tax liabilities at the annuity rate (r(r) + /3, see (16.20)) rather than at the interest rate, existing households at the time of the shock do not feel the full burden of the additional taxes and therefore do not cut back their consumption by a sufficient amount. As a result, private investment is crowded out at impact (K(t) < 0 at point A) and the capital stock starts to fall. This in turn puts downward pressure on before-tax wages and upward pressure on the interest rate so that human capital falls over time. So, future generations are poorer than newborn generations at the

9 Since the economy features positive initial assets (as K > 0), the C = 0 line lies to the left of the dashed line representing KKR and approaches this line asymptotically as C gets large (and the intergenerational-redistribution effect gets small). If there is very little capital, the rate of interest is very high and the C = 0 line is horizontal.

Figure 14

time of the shock be( wages (since FLK < ' If the birth rate is consumer and into-: tion is one-for-one, t dynamics. In terms downward jump in

16.3.2 The non-n1

The previous subsect tional redistributic Ricardian equivalei intergeneration al ly r can be demonstratL Chapter 14). The res of government co r depends on pre-ex i (Buiter, 1988, p. 285j

.mption rises (falls) over

! at the intersection of nfiguration of arrows, it - -it the saddle path, SP, is

i.

pw consider the effects of ipated and permanent debt policy by assuming ur- p-sum taxes only, i.e.

. Nsu m e that the economy k is normalized to t = 0. -I ward by the amount of mined and the economy Over time the economy n at E1 . As is clear from the

.e by public consumption - ast, there is more than < -1.

in the lump-sum tax

..vay from future towards -(2). At impact, all house- 1 lump-sum tax reduces count present and future 'her than at the interest lot feel the full burden of r consumption by a suffiout at impact (K(t) < 0 n puts downward pressure

-.te so" that human capital

...born generations at the

be C = 0 line lies to the left of ally as C gets large (and the capital, the rate of interest is

Chapter 16: Intergenerational Economics, 1

C (t)

—dG

Figure 16.2. Fiscal policy in the Blanchard-Yaari model

time of the shock because they have less capital to work with and thus receive lower wages (since Fix < 0).

If the birth rate is zero (0 = 0) there is a single infinitely lived representative consumer and intergenerational redistribution is absent. Crowding out of consumption is one-for-one, there is no effect on the capital stock, and thus no transitional dynamics. In terms of Figure 16.2, the only effect on the economy consists of a downward jump in consumption from point B to point C.

16.3.2 The non-neutrality of government debt

The previous subsection has demonstrated that lump-sum taxes cause intergenerational redistribution of resources in the Blanchard-Yaari model. This suggests that Ricardian equivalence does not hold in this model, i.e. the timing of taxes is not intergenerationally neutral and debt has real effects. Ricardian non-equivalence can be demonstrated by means of some simple "bookkeeping" exercises (see also Chapter 14). The result that must be proved is that, ceteris paribus the time path

of government consumption (G(r) for r E [t, 00)), aggregate consumption (C(t)) depends on pre-existing debt (B(t)) and the time path of taxes (T(r) for r E [t, 00)) (Buller, 1988, p. 285).

555

The Foundation of Modern Macroeconomics

Total consumption is proportional to total wealth (see (16.24)) which can be written as follows:

A(t) H(t) K(t) B(t) H(t)

= K(t) B(t) + f [W (r) - T(r)] e-RA(t'r) dr

Note that in deriving (16.36), we have used the definition of human wealth (16.20) to go from the first to the second line and the government budget restriction (16.34)

to get from the second to the third line. In view of (16.37) and (16.34) it follows that S2(t) vanishes if and only if the birth rate is zero and RA (t, r) = R(t, r). If the birth rate is positive, S2(t) is non-zero and Ricardian equivalence does not hold.

Recall that in the Blanchard-Yaari model the birth rate of new generations is equal to the instantaneous death probability facing existing generations. As a result it is not a priori clear which aspect of the model is responsible for the failure of Ricardian equivalence. The analysis of Weil (1989b) provides the strong hint that it is the arrival rate of new generations which destroys Ricardian equivalence (see Chapter 14 above). This suggestion was formally demonstrated by Buiter (1988) who integrates and extends the Blanchard-Yaari-Weil models by allowing for differential

birth and death rates (13B and fiD) and Hicks-neutral technical change. In his model the population grows at an exponential rate n Buiter (1988, p. 285) demonstrates that a zero birth rate (3B = 0) is indeed necessary and sufficient for

Ricardian equivalence to hold.

16.4 Extensions

In this section we demonstrate the flexibility of the Blanchard—Yaari model—and thus document its workhorse status—by showing how easily it can be extended in various directions. These extensions are by no means the only ones possible—some others are mentioned in the Further Reading section of this chapter.

16.4.1 Endogenous labour supply

As we have seen throughout the book, an endogenous labour supply response often plays a vital role in the various macroeconomic theories. In Chapter 15, for

example, it was demo leisure forms one of th cycle (RBC) tradition. Blanchard-Yaari mode the households. We fol and assuming simple keep the discussion as tax in order to demo

I

Extending the model

Assume that the

E A(v, t) E.-_- f low

with 0 < Ec < 1. Lei normalized to unity) n a special case of (16.38 agent's budget identi:

A(v, r) = [r(T)+ $1

X(v, v) (1 + tc)C1

where X(v, -r) represL sumption and leisure, proportional tax on lab from the government. Following Marini an(

problem by using ttt times before in this bo Chapters 11 and 13. T models. Intuitively the how the consumer cho upon a given level of determine the optimal provided the utility t ..

In stage 1 the conk. instantaneous felicity, and conditional upon familiar first-order con

w Preferences are inte time r only depends on tirmacro literature and inde,-

(16.24)) which can be

(16.36)

(16.37)

human wealth (16.20) )udget restriction (16.34) - 1 and (16.34) it follows

I R 4 (t, r) = R(t, r). If the alence does not hold.

of new generations is ig generations. As a result risible for the failure of es the strong hint that icardian equivalence (see cd by Buiter (1988) who y allowing for differential

-al change. In his model

. Buiter (1988, p. 285)

essary and sufficient for

:lard—Yaari model—and cily it can be extended in :ily ones possible—some his chapter.

is labour supply response ,fries. In Chapter 15, for

Chapter 16: Intergenerational Economics, 1

example, it was demonstrated that the intertemporal substitutability of household leisure forms one of the key mechanisms behind most models in the real business cycle (RBC) tradition. The aim of this subsection is therefore to extend the basic Blanchard-Yaari model by allowing for an endogenous labour supply decision of the households. We follow Heijdra and Ligthart (2000) by introducing various taxes and assuming simple functional forms for preferences and technology in order to keep the discussion as simple as possible. We analyse the effects of a consumption tax in order to demonstrate some of the key properties of the model.

Extending the model

Assume that the utility function used so far (see (16.16)) is replaced by:

E A(v, t) f log [C(v, r)Ec [1 - L(v, e(P-H3)(t-r) dr, (16.38)

with 0 < Ec < 1. Leisure is defined as the consumer's time endowment (which is normalized to unity) minus labour supply, L(v, r). Note that (16.16) is obtained as a special case of (16.38) setting Cc = 1. Since labour supply is now endogenous, the agent's budget identity (16.17) is replaced by:

A(v,, -c) = [r(T) + 13] A(v, + W (T)(1 - tL) + Z(T) - X(v, (16.39)

X(v, r) == + tc)C(v,, r) + W (T)(1 - tL) [1 - L(v, r)] (16.40)

where X(v, r) represents full consumption, i.e. the sum of spending on goods consumption and leisure, tc is a proportional tax on private consumption, tL is a proportional tax on labour income, and Z(r) are age-independent transfers received from the government. The household's solvency condition is still given by (16.18).

Following Marini and van der Ploeg (1988) we solve the household's optimization problem by using two-stage budgeting. We have encountered this technique several times before in this book, albeit in the context of static models—see for example Chapters 11 and 13. The procedure is, however, essentially the same in dynamic models. Intuitively the procedure works as follows. In the first stage we determine how the consumer chooses an optimal mix of consumption and leisure conditional upon a given level of full consumption (X(v, r)). Then, in the second stage, we determine the optimal time path for full consumption itself. The procedure is valid provided the utility function is intertemporally separable. 1°

In stage 1 the consumer chooses C(v, r) and [1 - L(v, t)] in order to maximize instantaneous felicity, log [C(v, r)Ec 11 - L(v, r)] 1-EC], given the restriction (16.40) and conditional upon the level of X(v, r). This optimization problem yields the familiar first-order condition calling for the equalization of the marginal rate of

1 ° Preferences are intertemporally separable if the marginal utility of consumption and leisure at timer only depends on time r dated variables. Intertemporal separability is commonly assumed in the macro literature and indeed holds for (16.38). See also Deaton and Muellbauer (1980, p. 124).

557

The Foundation of Modern Macroeconomics

substitution between leisure and consumption and the relative price of leisure and consumption:

By substituting (16.41) into (16.40), we obtain expressions for consumption and leisure in terms of full consumption:

Since sub-felicity—the term in square brackets in (16.38)—is Cobb-Douglas and thus features a unit substitution elasticity, spending shares on consumption and leisure are constant. To prepare for the second stage we substitute

into the lifetime utility functional (16.38) to obtain the following expression:

where Pa (r) is a true cost-of-living index relating sub-felicity to full consumption:

In stage 2, the consumer chooses the path of full consumption in order to maximize (16.44) subject to the dynamic budget identity (16.39) and the solvency condition (16.18). This problem is essentially the same as the one that was solved in Section 16.2.2 above so it should therefore not surprise the reader that the solution takes the following form:

Equation (16.46) says that full consumption is proportional to total wealth (the sum of financial and human wealth) whereas (16.47) shows that optimal full consumption growth depends on the difference between the interest rate and the pure rate of time preference. Finally, (16.48) is the definition of human wealth. It differs from (16.20) because labour income is taxed at a proportional rate and because the household receives transfers.

By aggregating (16.46) and (16.47) across surviving generations and making use of expressions for aggregate consumption growth and labour supply are obtained—see equations (T2.1) and (T2.6) in Table 16.2. Compared to the basic

I

Table 16.2. The extend(

I

Y(t) = K(t) 1 WY ,

Notes: C(t) is consumption. #, • rate, Z(t) are lump-sum tran ,-

and on wage income (ta

the pure rate of time preference

Blanchard-Yaari model abstract from govern r. all tax revenues a, the government budget

simplified the product a Cobb-Douglas techno and (T1.5) yields the t

Phase diagram

The phase diagram of tl labour supply decision gram. For that reason appendix to this chapte The capital stock equi which net investment i includes various tax rat CSE line is identical to

s for consumption and

(16.42)

(16.43)

—is Cobb-Douglas and res on consumption and

. 4)stitute (16.42)-(16.43) allowing expression:

(16.44)

V

"v to full consumption:

(16.45)

ption in order to max16.39) and the solvency he one that was solved in reader that the solution

(16.46)

(16.47)

(16.48)

nal to total wealth (the that optimal full connterest rate and the pure h uman wealth. It differs al rate and because the

- ;ons and making use of owth and labour supply !. Compared to the basic

Notes: C(t) is consumption, K(t) is the capital stock, L(t) is labour supply, Y(t) is aggregate output, W(t) is the wage rate, Z(t) are lump-sum transfers, and r(t) is the interest rate. There are proportional taxes on consumption (ta) and on wage income (tL). Capital depreciates at a constant rate 8, is the birth rate (equals death rate), and p is the pure rate of time preference.

Blanchard-Yaari model we have introduced the following simplifications. First, we abstract from government spending and debt = 0) and assume that all tax revenues are rebated to households in a lump-sum fashion. As a result, the government budget identity is static—see (T2.3) in Table 16.2. Second, we have simplified the production structure of the extended model somewhat by assuming a Cobb-Douglas technology—see (T2.7). Using this specification in (T1.2), (T1.4), and (T1.5) yields the expressions (T2.2), (T2.4), and (T2.5), respectively.

Phase diagram

The phase diagram of the model is drawn in Figure 16.3. The endogeneity of the labour supply decision considerably complicates the derivation of the phase diagram. For that reason we report the details of this derivation in a mathematical appendix to this chapter and focus here on a graphical and intuitive discussion.

The capital stock equilibrium locus (CSE) represents the (C, K) combinations for which net investment is zero (K = 0). Apart from the fact that the model now includes various tax rates and government consumption is set equal to zero, the CSE line is identical to the one discussed in detail in Chapter 15. The CSE line is

559

The Foundation of Modern Macroeconomics

C (t)

K (t)

Figure 16.3. Phase diagram for the extended

Blanchard–Yaari model

concave and for points above (below) this line consumption is too high (low) and net investment is negative (positive). 11

The consumption equilibrium (CE) locus represents the (C, K) combinations for which aggregate consumption is constant (C = 0). In the representative-agent model of Chapter 15, aggregate and individual consumption coincide and CE is simply the locus of points for which the interest rate equals the rate of time preference (r = p) and the output-capital ratio is constant (see Chapter 15 for details). For convenience, the CE line for the representative-agent model is included in the figure as the dashed line connecting points A3 and A4 (see Figure 16.3).

In contrast, in the overlapping-generations model, individual and aggregate consumption do not coincide and as a result, the position and slope of the CE curve are affected by two conceptually distinct mechanisms, namely the factor scarcity effect (FS, which explains the slope of the CE curve for the representative-agent model) and the generational turnover effect (GT). The interplay between these two effects ensures that CE has the shape of a rather prominent nose. Along the lower branch,

consumption is low, equilibrium employment is close to unity (L 1), and CE is upward sloping. In contrast, along the upper branch, A2A3, consumption is high, equilibrium employment is low (L 0), and CE slopes downward. The dynamic

11 We have only drawn the upward-sloping part of the CSE line. Recall from Chapter 15 that CSE reaches a maximum for the "golden-rule" capital stock, KGR, and then becomes downward sloping.

(a) r

Figure 16.4. Fact

I forces at work can be s

C(t)

C(t) = r(t) p – fi

= r(C(t),K

where r(C, K) is short-1 on consumption and to motivate the signs ( denoted by rc and rK , r Consider Figure 16.4 tal and the labour m - the short run—say at diminishing returns to because the two factors in the labour market. T slopes downwards—a ply curve follows from upwards because (T2. , Let us now use Figu. capital stock, an increas wage rises and emplo:. for capital to the left so t

12 Normally, in static mock directions thus rendering the "problem" because

curve

= 0

K (t)

)n is too high (low)1and

I

combinations for 'sentative-agent model ie and CE is simply the f time preference (r = p) ' ails). For convenience, the figure as the dashed

and aggregate conslope of the CE curve are the factor scarcity effect esentative-agent model)

'ween these two effects

.ong the lower branch,

to unity (L 1), and CE consumption is high, ownward. The dynamic

31 , from Chapter 15 that CSE m es downward sloping.

Chapter 16: Intergenerational Economics, I

Figure 16.4. Factor markets

forces at work can be studied by writing (T2.1) as follows:

where r(C, K) is short-hand notation for the dependence of the real interest rate on consumption and the capital stock. Simple intuitive arguments can be used to motivate the signs of the partial derivatives of the r(C, K) function, which are denoted by rc and rK , respectively. Some simple graphs can clarify matters.

Consider Figure 16.4 which depicts the situation in the rental market for capital and the labour market. In panel (a), the supply of capital is predetermined in the short run—say at K0 . The demand for capital is downward sloping—due to diminishing returns to capital—and depends positively on the employment level— because the two factors are cooperative in production. Panel (b) depicts the situation in the labour market. There are diminishing returns to labour—so labour demand slopes downwards—and additional capital boosts labour demand. The labour supply curve follows from the optimal leisure-consumption choice (T2.6). It slopes upwards because (T2.6) isolates the pure substitution effect of labour supply. 12

Let us now use Figure 16.4 to deduce the signs of rc and rK . Ceteris paribus the capital stock, an increase in consumption shifts labour supply to the left so that the wage rises and employment falls. The reduction in employment shifts the demand for capital to the left so that—for a given inelastic supply of capital—the real interest

12 Normally, in static models of labour supply, the income and substitution effects work in opposite directions thus rendering the slope of the labour supply curve ambiguous. Here we do not have this "problem" because the income effect is incorporated in C. Technically speaking, (T2.6) is a so-called Frisch demand curve for leisure. See also Judd (1987b).

561

The Foundation of Modern Macroeconomics

rate must fall to equilibrate the rental market for capital, i.e. r c < 0. The thought experiment compares points E0 and A in the two panels.

An increase in capital supply—ceteris paribus consumption—has a direct effect which pushes the interest rate down (a movement along the initial capital demand schedule, KD (r, L0) from E0 to B') and an induced effect operating via the labour market. The boost in K shifts the labour demand curve to the right, leading to an increase in wages and employment and thus (in panel (a)) to an outward shift in the capital demand curve. Although this induced effect pushes the interest rate up somewhat, the direct effect dominates and rK < 0. 13 The comparison is between points E0 and B in the two panels of Figure 16.4.

We can now study the dynamical forces acting on aggregate consumption along the two branches of the CE curve in Figure 16.3. First consider a point on the lower branch of this curve (for which L ti 1). Holding capital constant, an increase in aggregate consumption leads to a small decrease in labour supply 14 and thus a small decrease in the interest rate. At the same time, however, the capital-consumption ratio falls so that aggregate consumption growth increases, i.e. C/C > 0 for points above the lower branch of CE:

Now consider a point on the upper branch of the CE curve (for which L 0). Ceteris paribus K, a given increase in C has a strong negative effect on labour supply and thus causes a large reduction in the interest rate which offsets the effect operating via the capital-consumption ratio, i.e. C/C < 0 for points above the upper branch of CE:

These dynamic effects have been illustrated with vertical arrows in Figure 16.3.

13 This follows directly from the factor price frontier, which is obtained by substituting (T2.4) and (T2.5) into (T2.7):

The boost in the wage is associated with a higher capital labour ratio and thus relatively more abundant

capital. This translates itself into a lower return to capital.

14 Holding constant the tax rates we can use (T2.6) to derive:

dL (1— L)F clW dC1 L )L1NC j .

Hence, for L ti 1 (L ti 0) the labour supply curve in Figure 16.4 is relatively steep (flat) and a given change in consumption shifts the curve by a little (a lot). This explains why the parameter toll (1 —L)/L plays

a vital role in the analysis of the loglinearized model below.

In summary, the CImodel with exogenous (the lower branch in I for the representative-a close to zero (compare Put differently, on tht. . dominates whereas on It follows from the c Figure 16.3 is saddle-pc the equilibrium occurs c factor scarcity effect d prevent the opposite oc are such that E0 lies on

I

Raising the consumptia

We now illustrate how ti on the effects of an un tax, tc . Using the me L : loglinearized along an ii collected in Table 16.3.

Table 16.3. The loglinei

= EL L(t) + (1 - E,)K(t)

Definitions: we C/Y: output si