Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

(T1.1)

(T1.2)

(T1.3)

(T1.4)

(T1.5)

(T1.6)

(T1.7)

(T1.8)

capital stock, I gross investment, - er for consumption, T lump-sum

ncy parameter of labour.

this section, however, we Eicial case of the model in

"c unity, i.e. aL = 1. This literature (see e.g. Baxter relatively easy to analyse

the data (see below). For ummarized in Table 15.1. respectively, equations

.7) is obtained by setting sing (15.12) in (15.15).

on of the phase diagram T the assumption that the t. The phase diagram is 7esents combinations in C , K) combination there points near the origin Itercept (KK) employment ''.31 stock is KGR and the points above (below) the lent is negative (positive).

.! arrows in Figure 15.1.

Chapter 15: Real Business Cycles

Figure 15.1. Phase diagram of the unit-elastic model

The C = 0 line represents (C, K) combinations for which consumption is constant over time, i.e. for which the interest rate equals the rate of time preference. Since the interest rate depends on the marginal product of capital, and production features constant returns to scale, consumption equilibrium pins down a unique capital— labour ratio and thus a unique output—capital ratio and real wage rate. It follows (from (T1.7)) that the ratio between consumption and labour supply is constant also. The C = 0 line is linear and slopes downward. Ceteris paribus the capital stock, an increase (decrease) in consumption decreases (increases) labour supply and equilibrium employment, and decreases (increases) the output—capital ratio and the rate of interest. Hence, consumption falls (rises) at points above (below) the C = 0 line. This has been indicated with vertical arrows in Figure 15.1.

It follows from Figure 15.1 that the two equilibrium loci intersect only once, at point E0. The arrow configuration shows that E0 is saddle-point stable. The saddle path associated with the steady-state equilibrium E0, denoted by SP0, is upward sloping.

15.4 Fiscal Policy

In this section we demonstrate some illustrative properties of the deterministic unitelastic model developed in the previous sections. In doing so we prepare the way for the numerical simulations of the stochastic model in the next section. In the

483

The Foundation of Modern Macroeconomics

first subsection we study the impact, transitional, and long-run effects of a permanent and unanticipated increase in government consumption. In the second subsection we study how the economy reacts to a temporary fiscal shock. Throughout this section we assume that the government finances its consumption by means of lump-sum taxes.

15.4.1 Permanent fiscal policy

This section studies the effect on the main macroeconomic variables of an increase in public consumption financed by means of lump-sum taxes. We assume that the policy shock is unanticipated and permanent, and that the economy is initially in the steady state.

Although the model in Table 15.1 may look rather complex, it was demonstrated recently by Baxter and King (1993) that the long-run effects of the policy shock can be determined in a relatively straightforward fashion. For that reason, we first study the long-run effects, before investigating the somewhat more demanding short-run and transitional effects of fiscal policy.

Long-run multipliers

Computation of the long-run "new classical multiplier" is a back-of-the-envelope exercise due to the fact that the economy is structurally characterized by a number of great ratios that are independent of public consumption (see Baxter and King, 1993, p. 319). In our model this can be demonstrated as follows. In the steady state, both the capital stock and consumption are constant, i.e. K = C = 0. Equation (T1.1) and (T1.2) in Table 15.1 then imply, respectively, that the investment-capital ratio and the rate of interest are constant, i.e. I /K = 1 and r p. The marginal productivity condition for capital, (T1.5), then pins down the equilibrium capital intensity of production, y* (Y /K)*, as a function of structural parameters only (y* (p 6)/(1 - EL)). But, since the production function, (T1.8), features constant returns to scale, the equilibrium capital intensity also determines a unique capitallabour ratio, (K/L). This, in turn, pins down the real wage and thus (by (T1.7)) the ratio between goods and leisure consumption, C/(1 - L).

The long-run constancy of the various ratios can be exploited to find the longrun effect of an increase in public consumption. By totally differentiating the goods market clearing condition, (T1.6), we obtain:

where (Di //Y = 6K/Y = 1/y*, coG C/Y, and we + (0/ + COG 1. Following the shock to public spending, eventually the various ratios will be restored. This implies

a

!le following long-run

dY(oo) dK(oo) d Y K

.:re (DLL [1- L]/L. E

we find an expression ttii

.1 similar fashion, the capital stock can be der..

A number of observati, of the labour supply dec ,leed, if labour supply 15.19) gives the immed are unchanged. Equatioi ivate by public consun lac also plays a crucial ro 7..odel developed in sc- (15.22) the long-run et:, the more elastic is labou

.iects on output, car .. effect on consumption.' I

Short-run multipliers

The impact and transit the aid of Figure 15.2. In

4 The intertemporal sut-

function suggested by Gre,- .

I

1(r) _,_ log U(r), U(r) .-

The first-order conditions t

W(r) =

Employment only depend , (r = p). It follows that fiscal out of consumption is one

i long-run effects of a per-

ITT _imption. In the second mary fiscal shock. Through- - consumption by means

1 is variables of an increase I taxes. We assume that the the economy is initially in

"nlex, it was demonstrated is of the policy shock can ir that reason, we first study --lore demanding short-run

is a back-of-the-envelope characterized by a number

• i)n (see Baxter and King, follows. In the steady state,

i.e. K = C = 0. Equation it the investment-capital and r = p. The marginal 71 the equilibrium capital structural parameters only

(T1.8), features constant srmines a unique capital- (..- and thus (by (T1.7)) the

exploited to find the long-

• differentiating the goods

(15.18)

wG 1. Following the

"I be restored. This implies

Chapter 15: Real Business Cycles

the following long-run relationships (in loglinearized form):

where cou s- [1 - /1/L. By substituting the relevant results from (15.18) into (15.19) we find an expression for dY(oo)/Y which can be rewritten in a multiplier format:

In a similar fashion, the long-run multipliers for consumption, investment, and the capital stock can be derived:

A number of observations can be made about these results. First, the endogeneity of the labour supply decision plays a crucial role for the new classical multiplier. Indeed, if labour supply is exogenous (CC = 1 so that L = 1 and wu = 0) equation (15.19) gives the immediate result that output, investment, and the capital stock are unchanged. Equation (15.21) shows that there is one-for-one crowding out of private by public consumption in that case. Second, the elasticity of labour supply (aL ) also plays a crucial role. Note that the great ratios result also holds for the general model developed in section 2 above. Indeed, by replacing coll by aLwLL in (15.20)- (15.22) the long-run effects for the more general model are obtained. Consequently, the more elastic is labour supply (i.e. the higher is aL), the larger are the long-run effects on output, capital, and investment, and the smaller is the crowding-out effect on consumption. 4

Short-run multipliers

The impact and transitional effects of the fiscal shock can be studied graphically with the aid of Figure 15.2. In this figure, CE 0 is the initial consumption equilibrium line,

4 The intertemporal substitution effect in labour supply can be eliminated by using the felicity function suggested by Greenwood et al. (1988):

(1, (r) log U(r), U(T) C(r) YL L(01+0,

( 1 4- el,

The first-order conditions for this case are:

Employment only depends on the wage rate which is pinned down by the steady-state interest rate (r = p). It follows that fiscal policy does not affect output and the capital stock either, and that crowding out of consumption is one for one. See Heijdra (1998, pp. 687-688).

485

11, •

KK K

Loglinearization. Strangely enough, loglinearization of a non-linear dynamic model often confuses people. For that reason we show in detail how we loglinearize the model in Table 15.1. Campbell (1994) and Uhlig (1999) provide further examples. We first define the variable i(t):

*(t) log [x(t)/x] x(t)/x ei(t) , (a)

where x is the steady-state value for x(t). Provided x(t) is near its steady-state value (x(t)/x ti 1 and x(t) 0) we have e (t) 1 + 540 so that it follows from

(a) that:

The Foundation of Modern Macroeconomics

Next we consider an equation like (T1.8). By taking logarithms on both sides we get:

In the steady state we have:

Deducting (e) from (d) and noting the definitions of Y(t), L(t), and i<(t) we obtain the desired expression (which no longer contains the constant log Z0):

Y(t) ELL(t) (1 — EL)K(t).

Third, we consider a linear equation like (T1.6). We derive in a few steps:

where we have used (b) in going from the first to the second line and note that in the steady state Y C + I + G.

Finally, consider an equation like (T1.7) which is loglinear in leisure (but not in labour). Indeed, we obtain in a straightforward fashion 1217- (t) + [1 — L(t)]

(t) . But in the rest of the model we work with L(t). Using (b) we can relate

The loglinearized version of the unit-elastic model is given in Table 15.2. All variables with a tilde r") are defined as proportional rates of change relative to the initial steady state, i.e. i(t) log[x(t)/x]. Variables with a tilde and a dot are time rates of change of the variable, again expressed in terms of the initial steady-state level of the variable, i.e. JO) *(t)/x. This notation is used throughout the remainder of this chapter.

7...Jle 15.2. The logline,.

=pi- (t)

=(t)

• = i-1 (t) I_(t)

= (p + 3) [Y(t) —

) = tocE. (t) + w (t) ak,u = a [CV (t) — (t)]

= EL L(t) + (1 —

Definitions: coy G/Y: outpu w, //Y: output share of ir between leisure and labour.

The state variables ar: and consumption, (t ►. model can be studied is done by expressing Labour demand (T2.4 ).

are used to compute th, and real output:

ELL(t) = (0 — 1)

W(t) = (1 — Et) [il

Y(t) = 0(1 — EL ,:

I

where 0 is a crucial para4 labour supply:

1 ± wt.

1 < = l+ coLL ( 1 i

The expressions in (15. given capital stock, a r hence employment), (.1. a given level of consum employment and raisL increase in the capital si

g logarithms on both sides

(d)

(e)

of Y(t), L(t), and K(t) we - :ins the constant log Zo):

derive in a few steps:

(t)\

G

) [1 +GM]

econd line and note that

rv.►linear in leisure (but not

ashion + [1 - L(t)] =

Using (b) we can relate

L(t) -

1-L

Chapter 15: Real Business Cycles

Table 15.2. The loglinearized model

co, I/Y: output share of investment, we + w, + (0G = 1, 6/(0/ = Y* + 6)1(1— EL). (DLL (1 — L)/L: ratio between leisure and labour. 1(t) i(t)/x,i(t) log [x(t)/x].

The state variables are the aggregate capital stock, K(t), which is predetermined, and consumption, (t), which is a jump variable. The dynamic behaviour of the model can be studied most easily by first condensing it as much as possible. This is done by expressing Y(t), e'(t), i(t), W(t), and L(t) in terms of the state variables. Labour demand (T2.4), labour supply (T2.7), and the production function (T2.8) are used to compute the conditional equilibrium levels of employment, real wages, and real output:

given in Table 15.2. All ates of change relative to the h a tilde and a dot are time ns of the initial steady-state the remainder

The expressions in (15.23)-(15.25) are easy to understand intuitively. First, for a given capital stock, a rise in the level of consumption reduces labour supply (and hence employment), drives up the real wage rate, and reduces output. Second, for a given level of consumption, a rise in the capital stock boosts labour demand and employment and raises the real wage. Output is stimulated both because of the increase in the capital stock and because of the induced effect on employment.

489

The Foundation of Modern Macroeconomics

By using the output expression (15.25) in (T2.5) and (T2.6) the conditional equilibrium results for investment and the interest rate are obtained:

p P+ 6 ) i(t) = — [i1 — 0(1 — EL)] R(t) + — 1)(t)], (15.28)

where we have incorporated the assumption (made throughout this section) that the shock in government consumption is constant over time, i.e. G(t) = O. The rate of interest depends negatively on both the capital stock and consumption. For a given level of consumption, an increase in the capital stock raises employment, as labour demand is boosted. This raises the marginal product of capital and hence the interest rate. This positive effect on the interest rate is more than offset, however, by the fact that marginal returns to capital decline as more capital is added. For a given capital stock, an increase in consumption lowers labour supply and employment, and hence lowers the marginal product of capital and the interest rate.

Of course, since there are constant returns to scale in production, there exists a unique inverse relationship between factor prices. This factor price frontier is obtained by substituting (T2.4) and (T2.5) into (T2.8):

By substituting (15.27) into (T2.1) and (15.28) into (T2.2), the dynamical system can be written in a condensed form as:

[

where A is the Jacobian matrix of the system:

Saddle-point stability of the loglinearized system can be demonstrated formally by computing the determinant and the trace of the Jacobian matrix, A. After some

I

(0)

Figure 15.3.

manipulation we find:6

IA) = - ( 9 + 8 )Ys tr(A) = p > 0.

As I A is equal to the pro there must be one negat: out this chapter the stabic

0). When written economic system (see Ch. of the characteristic roots,

The loglinearized mode earized CSE and CE schedi

wc + -1

and:

Qt)- -

(1- 0(1 -e

- 1 1

(Equations (15.35)-(15. steady state K(t) = C(t) =

6 By noting that y* (p — •

tr(A) = y*10(1- EL) - wil —

where WA pKIY =1— ct —

and (T2.6) the condition_ are obtained:

(15.27

(15.28)

roughout this section) that time, i.e. G(t) = G. The rate )ck and consumption. For a stock raises employment, as luct of capital and hence the lore than offset, however, by capital is added. For a given supply and employment,

the interest rate. production, there exists a for price frontier is obtained

(15.29)

'.2), the dynamical system

(15.30)

-

(15.32)

be demonstrated formally 'm matrix, A. After some

The Foundation of Modern Macroeconomics

We now return to the fiscal policy experiment. In the appendix to this chapter we derive a general expression for the perfect foresight solution of the model. For

the shock vector (15.32), the solution paths for consumption and the capital stock take the following form:

where C(0), C(oo), and R(oo) are given by:

AlX2

Equations (15.37)-(15.40) represent the so-called impulse-response functions for cap- ital and consumption with respect to a permanent and unanticipated shock in

government consumption which occurs at time t = 0. Equation (15.37) shows that the effect of the shock as of time t can be written as the weighted average of

the impact effect and the long-run effect with respective time-varying weights

and 1 - The impulse-response function for the remaining variables of the model (i.e. L(t), W(t), 1(t), and T(t)) are obtained by using (15.37)-(15.40) in (15.23)-(15.25) and (15.27)-(15.28).

Since the capital stock is predetermined in the impact period (K(0) = 0), the

impact effects for employment, output, the wage, and the interest rate are all proportional to C(0). The decrease in consumption causes labour supply (and employment) to increase.

As a result of the increase in employment, output also expands:

Since output expands and goods consumption falls, investment unambiguously rises:

Equation (15.43) shows the accelerator mechanism that is operative in the model (see also Baxter and King, 1993, p. 321): the impact effect on investment is proportional

to the long-run effect on t omy acting as the factor at impact, the marginal pr

Finally, the expansion oi demand for labour is dowi

dG

Of course, what happens (15.44) with the factor pri

Quantitative evidence

Now that the qualitative analytically, the next que In order to cast some information that is more The calibrated model is th long-run effects.

Essentially calibration model in such a way th ficiently robust informal, given in Tables 15.1 (in 1 parameters appearing in t of depreciation of the ca preference parameter cc.

Some of these parameti tained hypothesis that th (T1.2) that the real rate u: . erence, i.e. r = p. King an of return to capital in th On a quarterly basis this v (1.59% on a quarterly I is set at 10% per annuli, course, for buildings this years) but for machines age guess, however, it ni technology EL equals the and Rebelo set equal to t know p and 6L, we can i: •