Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

appendix to this chapter t solution of the model. For 'tion and the capital stock

(15.37)

(15.38)

(15.39)

(15.40)

►e-response functions for capnd unanticipated shock in 0. Equation (15.37) shows the weighted average of time-varying weights e --Alt - 1-naining variables of the using (15.37)—(15.40) in

- t period (R(0) = 0), the interest rate are all propor- r supply (and employment)

(15.41)

(15.42)

vestment unambiguously

(15.43)

-,erative in the model (see vestment is proportional

Chapter 15: Real Business Cycles

to the long-run effect on the capital stock, with the adjustment speed of the economy acting as the factor of proportionality. Because the output—capital ratio expands at impact, the marginal product of capital and thus the real interest rate rises:

Y ( dr (0)	(p + 8)(0 — 1) [A.2+ (p + 3)(0 — 1)]	> 0.	(15.44)
dG	A2(4) + we — 1)

Finally, the expansion of labour supply implies that the real wage rate falls as the demand for labour is downward sloping:

L dW (0)	(1 — 60(0	— 1) 1A2 + (P + 8)(0 — 1 )1	< 0.	(15.45)
dG		A2(0 + we — 1)

Of course, what happens to the wage rate can also be determined by combining (15.44) with the factor price frontier (15.29).

Quantitative evidence

Now that the qualitative effects of the fiscal shock have been fully characterized analytically, the next question concerns the quantitative size of the various effects. In order to cast some light on this issue we must now calibrate the model by using information that is more or less plausible for a typical advanced market economy. The calibrated model is then used to compute the various impact, transitional, and long-run effects.

Essentially calibration amounts to choosing the parameters of the theoretical model in such a way that the model replicates certain outcomes about which sufficiently robust information is available. Take, for example, the unit-elastic model given in Tables 15.1 (in levels) and 15.2 (in loglinearized format). The structural parameters appearing in that model are the pure rate of time preference p, the rate of depreciation of the capital stock 3, the efficiency parameter of labour EL, the preference parameter Ec, and the general productivity parameter Zo.

Some of these parameters are not hard to guess. For example, under the maintained hypothesis that the economy is in (or near) a steady state, it follows from (T1.2) that the real rate of interest must be (nearly) equal to the rate of pure time preference, i.e. r = p. King and Rebelo (1999, p. 953) suggest that the average real rate of return to capital in the US has been 6.5% per annum over the period 1948-1986. On a quarterly basis this would give us the estimate r = p = (1.065) 1 /4 —1 = 0.0159 (1.59% on a quarterly basis). The annual rate of depreciation of the capital stock is set at 10% per annum by King and Rebelo, i.e. 8 = (1.1) 1 /4 — 1 = 0.0241. Of course, for buildings this figure is far too high (most buildings last longer than ten years) but for machines (e.g. personal computers) it may be far too low. As an average guess, however, it may not be too widely off the mark. With Cobb-Douglas technology EL equals the share of labour income in output (see (T1.4)) which King and Rebelo set equal to two-thirds, i.e. EL = 2/3 (1999, p. 954). But now that we know p and EL , we can infer the implied estimate for the equilibrium output—capital

493

The Foundation of Modern Macroeconomics

ratio from (T1.5), i.e. y* (17 /K)* = (p + 8)/(1 - EL) = 3(0.0159 + 0.0241) = 0.12. By imposing the steady state in (T1.1) we obtain the implied investment share of

output, i.e. 04 //Y = 8/y* = 0.0241/0.12 = 0.201. Baxter and King (1993, p. 320) suggest that the average postwar share of government consumption in output was 20% in the US, i.e. (0G = 0.2. We now have estimates for almost all parameters of interest. By using (T1.6) we observe that the consumption share in output is we C/Y = 1 - w1 - LOG = 0.599. By combining (T1.4) and (T1.6) we derive:

	LOLL	1 - L = coc) (1 - Ec)			(15.46)
		L	EL )	cc

so choosing EC implies choosing L (and thus (0u) and vice versa. King and Rebelo suggest that 20% of total available time has been dedicated to working in the postwar period in the US, i.e. L = 0.2 and LOLL = 4, so that it follows from (15.46) and the other estimates that EC = wc/[wc + ELwu] = 0.183. Finally, we observe that Zo is a "free parameter" in the sense that it merely fixed the scale of the economy. In the next section we shall set Zo = 1 but here we normalize Zo such that output is unity in the initial steady state, i.e. we set Zo = L-EL (y*) 1-EL = 1.442. 7

In summary, we have now calibrated the model using the following values for the

	structural parameters:
	p = 0.0159 8 = 0.0241 €L = 2/3		(15.47)
	Ec = 0.183 Zo = 1.442	(0G = 0.2

The resulting initial steady state is given by:

	Y = 1	C = 0.599 I=0.201 G = T = 0.2		(15.48)
	r = 0.0159	L = 0.2	K = 8.337 W = 3.333

Using these calibration values in (15.31)-(15.32) and noting (15.26) we obtain the implied estimates for the Jacobian matrix, A, and the shock terms, yK(t) and yc(t): 8

	A [ 0.06156 -0.20892			(15.49)
		-0.01142 -0.04569
[		[ -0.02399	G.	(15.50)
Yc(t)	-	0

The characteristic roots of A are, respectively, -Ai = -0.0646 and A2 = 0.0805. What do these figures mean? Recall that Al represents the adjustment speed in the economy—see (15.37). Using the reasoning explained in Chapter 14, the half-life of the adjustment process in the economy is t112 (1/A1) log 2 = 10.7. Since we

7 Most numerical solution algorithms work best when the endogenous variables are all of the same

order of magnitude. For that reason it is wise to normalize output such that this is indeed the case.

8 We present the actual numerical estimates here not to test the reader's patience but rather to enable replication and to give a 'feel' for the magnitudes and dimensions involved.

Table

mul -

Vanah

dY

cr6 dC

d'

(dL)7

(dr,

(dis

W

have calibrated on quarto rate on capital, this fi t ....

non-jumping variable Oh the shock occurred.

Using the information (15.37)-(15.45) we obta effects on the different There is severe crowding c $1 of extra government Because the representati% more hours to the labour (saving equals) investm,_ consumption so that the I some of the other ma, spending gives rise to a I rate. The interest rate rise rate rises by 0.0082 percL.

In the long run the intern and the capital-labour ues. For a 1% increase in 0.211%. In the long-run no is restored. Consumptkii. the output multiplier is a

In summary, the res',.. to permanent govemmen agent model. The mec :n

494

Chapter 15: Real Business Cycles

	0159 + 0.0241) = 0.12.	Table 15.3. Government consumption
	nplied investment share of
		multipliers
	\ ter and King (1993, p.
	ent consumption in output	Variable		Impact effect Long-run effect
	a tes for almost all parame-	dY
	.mption share in output is	dG
	1.0291.054
	ind (T1.6) we derive:	dC		—0.539	—0.158
		dG
		dl		0.568	0.212
		dGdq 1
			( dG)	0	0.211
	ce versa. King and Rebelo	1K ) i	G )
	d to working in the postwar	cIL)/	(wdG )	0.309	0.211
	)%vs from (15.46) and the	dr\(dG/		0.518	0
	ily, we observe that Z0 is a	r )	G
	le of the economy. In the	dW)	1 ( cIG	—0.103	0
	such that output is unity	W ) 1 k, G
	1.442. 7	have calibrated on quarterly observations for the interest rate and the depreciation
	he following values for the
	I	rate on capital, this figure means, for example, that half of the adjustment in the
		non-jumping variable (the capital stock) is completed almost eleven quarters after
	(15.47)	the shock occurred.
		Using the information from (15.49)–(15.50) in the various analytical expressions
		(15.37)–(15.45) we obtain the numerical estimates for the impact and long-run
		effects on the different variables. These results have been summarized in Table 15.3.
	(15.48)	There is severe crowding out of private by public consumption at impact. For every
		$1 of extra government consumption private consumption falls by $0.54 at impact.
	ig (15.26) we obtain the	Because the representative agent cuts back on leisure consumption—by supplying
		more hours to the labour market—household labour income rises. The additional
	xk terms, MK (t) and yc(t): 8	(saving equals) investment at impact is $0.57 out of every $1 of extra government
		consumption so that the output multiplier exceeds unity at impact. Let us look at
	(15.49)	some of the other magnitudes involved. At impact a 1% increase in government
		spending gives rise to a 0.3% increase in employment and a 0.1% fall in the wage
	(15.50)	rate. The interest rate rises proportionally by 0.5%, i.e. in absolute terms the interest
		rate rises by 0.0082 percentage points from 1.587% to 1.595% on a quarterly basis.
		In the long run the interest rate, the wage rate (see the factor price frontier (15.29)),
	0.0646 and )1/4.2 = 0.0805.	and the capital–labour ratio all return to their respective initial equilibrium val-
	e adjustment speed in the	ues. For a 1% increase in government consumption the capital stock increases by
	Chapter 14, the half-life	0.211%. In the long-run net investment ceases as the initial investment–capital ratio
	► log 2 = 10.7. Since we	is restored. Consumption crowding out remains but is less severe than at impact and
	ous variables are all of the same	the output multiplier is a little higher than at impact.
		In summary, the results in this subsection show that large output multipliers due
	- hat this is indeed the case.	to permanent government consumption are quite possible in the representative-
	patience but rather to enable
	. ed.	agent model. The mechanism behind the multiplier is, however, quite classical and
					495

- Y*(0GO)

The Foundation of Modern Macroeconomics

originates from the dynamic interaction of the supply of labour and capital (Baxter and King, 1993, pp. 323-324). The additional lump-sum taxes make people poorer which leads them to increase labour supply both at impact and in the long run. In the long run the capital-labour ratio is restored so that the capital stock rises also. In the short run the investment accelerator (see (15.43)) explains that the public consumption shock is accompanied by an investment boom.

15.4.2 Temporary fiscal policy

One of the recurrent themes in the study of fiscal policy is the difference between the effects of temporary and permanent policy. Baxter and King, for example, employ numerical methods to study to what extent the impact multiplier for output depends on the duration for which the fiscal policy impulse is in operation (1993, p. 315). In this subsection we show how a temporary (but unanticipated) fiscal spending shock affects the economy. To keep things simple we assume that the government raises its consumption level unexpectedly at some time to = 0 and then gradually lets it fall back to the initial level. In terms of (15.32), the shock term for the CE line is unaffected (i.e. yc (t) 0 because lump-sum taxes continue to be used in this experiment) but the shock term affecting the CSE line is changed to:

Mt) —e-4.Kty*(0G6,

(15.51)

where 4'1‹ > 0 is the exponential rate at which government consumption returns to its initial level. At impact the shock is the same as before (since MO) =

but eventually the shock vanishes (lirnt--,00yK (t) = 0). Since agents in the economy are assumed to know the path of government consumption they will condition their behaviour accordingly and will change their plans optimally. Note that 4K parameterizes the persistence of the shock. For example, if K ti 0 then the shock is highly persistent and yK(t) falls only very slowly towards zero. In contrast, if K is large, then yK(t) drops off rapidly as time goes by and the shock is very transitory.

The time path for yK(t) is illustrated in Figure 15.4 for different values of 4•K, ranging from K = 0 (permanent shock) to 4K = 0.5 (very transitory shock).

Using the methods explained in the appendix to this chapter, the perfect foresight solution of the model is obtained:

k(t) [ 0 e _Ait K — (p + 8)(0 - 1)

C(t) C(0) (p+8)[i -q(1 - EL)]

[

		X MK, A1, t)	Y* COGb	(15.52)
			), 2
where the impact effect on consumption, C(0), is:
e(0)	=	Xz+ (p+	- 1)) ( WG G	(15.53)
		(oc + 0 - 1	< 0
			)k,2 + 4k)

0.8

0.6

G

0.4

0.2

0

Time

Figure 15.4. The

and where T(K, Xi, t) is a I

T(K, x1, t)

to- '

Before developing the c, and the capital stock, as shape of the temporary ti shape of this term for a ri of the economy is set at I the text below equation

We observe from Figun term is a non-negative be both at the time of the s the value of K, the later and the slower is the der with 4.1( = 0, the shock is adjustment term of the f function is not bell-shape

We are now in a posit I of a temporary public sp the impulse-response d results given in (15.52)- diagram presented in Fig

496

1 - e-)`lt .

ibour and capital (Baxter ixes make people poorer and in the long run. In he capital stock rises also. explains that the public

)om.

is the difference between r and King, for example, impact multiplier for out- y impulse is in operation orary (but unanticipated) gs simple we assume that y at some time to = 0 and

(15.32), the shock term c, im taxes continue to be LSE line is changed to:

(15.51)

t consumption returns to e (since yK(0) = -y*coG G)

:e agents in the economy on they will condition optimally. Note that 4K 0 then the shock is zero. In contrast, if 4K is shock is very transitory. - nt values of K , ranging

y shock).

"ter, the perfect foresight

(15.52)

(15.53)

Chapter 15: Real Business Cycles

Figure 15.4. The path for government spending

and where T( K , Ai , t) is a temporary transition term which is defined as follows:

	for K0Ai
'MK, t)	(15.54)
to-xl t	for K = Ai

Before developing the economic interpretation of the solutions for consumption and the capital stock, as given in (15.52)-(15.53), it is useful to first look at the shape of the temporary transition term T( K , Ai , t). In Figure 15.5 we illustrate the shape of this term for a range of values of K . In this figure, the adjustment speed of the economy is set at the value implied by the calibration, i.e. )1/4.1 = 0.0646 (see the text below equation (15.50)).

We observe from Figure 15.5 that, provided 4•K is strictly positive, the transition term is a non-negative bell-shaped function of time. Furthermore, this term is zero both at the time of the shock (t = 0) and in the long run (t 00). The lower is the value of 4k, the later is the time at which the transition terms reaches its peak and the slower is the decline towards zero as time goes on. In the limiting case, with K = 0, the shock is permanent and the transition term is proportional to an adjustment term of the form A(Ai, t) Hence, for K = 0 the transition function is not bell-shaped—see the back ridge in Figure 15.5.

We are now in a position to study the intuition behind the macroeconomic effects of a temporary public spending shock. The aim is to firmly establish the link between the impulse-response diagrams contained in Figures 15.7-15.10 and the analytical results given in (15.52)-(15.53). This task is facilitated by considering the phase diagram presented in Figure 15.6.

497

The Foundation of Modern Macroeconomics

16

14

12

10 ° 8

`) 6 1=7 4 2 0

0.1		100
0.2		80
		60
0.3
		40 Time
0.4	20
	0.5

Figure 15.5. Transition term

R(0)=0

k(t)

Figure 15.6. Phase diagram for temporary shock

In Figure 15.6, CSE0 and CE0 are, respectively, the initial capital stock equilibrium and consumption equilibrium curves, and E0 is the initial equilibrium. The effect of a permanent shock, which was also studied in Figure 15.4, is to shift the CSE curve

to CSEpS . The economy adjusts by jumping from E0 to A ps at impact and by moving gradually along the saddle path, SPps , from Aps to Elm.

Next we consider what rary. It follows from the

in consumption is larger for this means that for a tempc vertical dashed line connect of shock persistence, we poi that < q, i.e. the shot,. jumps associated with the

points Al and A2. Consumption falls regar,i

lump-sum taxes make the n back on goods consumptiuthe more persistent is the sti shock by accumulating or d not provide an unambiguou economy jumps to. This is w

It follows from the first investment is given by:9

K(0) = [(p 3)(0 — 1) —

The impact effect on net i working in opposite direr LA, shock is very persistent (EK k side of (15.55) is positive a,. since consumption falls art( in labour supply), the in, crowding out of private inv path at impact is upward 15.6. The phase diagram time, the capital stock equil the early part of the trans.. direction, say from Al to B1 catches up with the then r, accumulation ceases, i.e. t.. economy returns to the old

If the labour supply is transient ( 1( high), then d (15.55) is negative and

> (p + 8)(0 — 1), and the

9 This expression is obtained bl and noting that dT/dt = 1 for t = I

498

100

K

shock

ti al capital stock equilibrium :1 equilibrium. The effect of 5.4, is to shift the CSE curve at impact and by moving

Chapter 15: Real Business Cycles

Next we consider what the adjustment path looks like when the shock is temporary. It follows from the comparison of (15.38) and (15.53) that the impact reduction in consumption is larger for a permanent than for a temporary shock. In Figure 15.6 this means that for a temporary shock the economy jumps somewhere along the vertical dashed line connecting E 0 and Aps . In order to study the qualitative effects of shock persistence, we postulate two values for K , sayK and q, and we assume that < q, i.e. the shock is relatively more persistent for K. The consumption jumps associated with the two K values are illustrated in Figure 15.6 by, respectively, points Al and A2.

Consumption falls regardless of the degree of shock persistence. The additional lump-sum taxes make the representative agent poorer as a result of which he cuts back on goods consumption and leisure. This negative human wealth effect is larger the more persistent is the shock. Next we consider whether the agents react to the shock by accumulating or decumulating assets. The diagram in and of itself does not provide an unambiguous answer because it is not a priori clear which region the economy jumps to. This is where the analytical results can provide further guidance.

It follows from the first expression in (15.52) that the impact effect on net

investment is given by: 9
K(0) = [(p + 8)(0 – 1 ) – Y*wG°	(15.55)
+

The impact effect on net investment depends on the interplay of two mechanisms working in opposite directions. If labour supply is highly elastic (0 high) and the shock is very persistent (1( low), then the term in square brackets on the right-hand side of (15.55) is positive and net investment rises at impact (R(0) > 0). Intuitively, since consumption falls and output increases strongly (because of the large boost in labour supply), the increase in government consumption does not cause any crowding out of private investment. Hence, for K < (p 8)(0 – 1), the transition path at impact is upward sloping—see the dashed line from point A l in Figure 15.6. The phase diagram can now be used to characterize the transition path. Over time, the capital stock equilibrium locus starts to shift back towards CSE0. During the early part of the transition the equilibrium trajectory runs in a north-easterly direction, say from A l to B 1 in Figure 15.6. By the time the equilibrium trajectory catches up with the then relevant capital stock equilibrium locus (CSE 1 ), net capital accumulation ceases, i.e. the trajectory is vertical at point B1. After that time, the economy returns to the old equilibrium along the trajectory from B1 to E0.

If the labour supply is not very elastic (0 close to unity) or the shock is very transient high), then the term in square brackets on the right-hand side of

(15.55) is negative and net investment falls at impact (K(0) < 0). In that case, > (p + 8)(0 – 1), and the economy jumps at impact from E0 to A2, after which it

9 This expression is obtained by differentiating the first expression in (15.52) with respect to time and noting that dT /dt = 1 for t = 0.

499

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p. 133:

The Foundation of Modern Macroeconomics

We conclude this section by briefly touching on what has been labelled by Baxter and King (1993) as one of the four classic fiscal policy experiments, namely the relationship between policy persistence and the magnitude of impact effects. By using (15.25) and (15.53) and noting that capital is predetermined at impact (k(0) = 0), we find that there exists a simple relationship between the output multiplier for permanent and temporary increases in government consumption in the impact period:

dY (0) 1	= A2	dY (0)r	> 0,	(15.56)
dG	A2 ± 4.K	dG

where [dY(0)/dG] i<=0 is given in (15.42) above. It follows from (15.56) that the impact multiplier is smaller the less persistent is the shock to government spending, i.e. the higher is 4K. We thus confirm analytically the conclusion reached on the basis of numerical simulations by Baxter and King (1993, p. 326). 11

15.5 The Lucas Research Programme

One of the lasting contributions of the rational expectations revolution of the 1970s (see Chapter 3) has been a methodological one. Throughout the 1950s and 1960s macroeconomists engaged in a huge model construction programme in which the insights of the IS-LM model and its refinements were estimated by econometric means. These macroeconometric models were quite popular in both public and private sectors because they could be used for prediction and simulation purposes. Two developments occurred in the early 1970s which led to a drastic reduction in the popularity of these models. First, a lot of the macroeconometric models then in use included a relatively poorly specified supply side and consequently were ill equipped to predict the effects of the various oil price shocks that occurred at the time. Of course, this criticism is not deadly per se as macroeconometric models can be (and indeed, have been) re-specified to better deal with shocks affecting the supply side of the economy.

A second—potentially much more lethal—criticism was raised by Lucas (1976). The so-called Lucas critique was discussed above-see Chapter 3. Loosely put, it states that macroeconometric models that are not based on a consistent set of optimizing foundations are non-structural and cannot be used for policy evaluation. The reason is that the estimated parameters of the model's equations are mixtures of structural and policy parameters and are therefore not invariant across different policy regimes (see Chapter 3 for a simple example of this point). To avoid the critique that now carries his name, Lucas (1980, 1987) argued forcefully and eloquently

11 In the classic analyses of Hall (1980) and Barro (1981), exactly the oppostite result holds, i.e. temporary spending shocks have larger effects than permanent ones. The reason for this discrepancy is that these papers do not allow for capital accumulation. See Baxter and King (1993, p. 326).

that macroeconomists shol optimizing behaviour of t posed what Christiano, Eil

Lucas (research) programn:

As Lucas (1980, p. 272) a unrealistic and artificial. T subjecting them to shocks would react. The more din economies give to simy questions". He goes on to

On this general view of the tion of assertions about the 1 instructions for building a pa A "good" model, from this pr but will provide better imiwi

In a seminal paper, KVL.. Lucas and his co-workers b agents doing as well as th ,. tic shocks. Their model .w (RBC) research programme ask themselves the folios% . tions in actual economies parameter estimates that and Prescott, 1982, mating its equations econo Kydland and Prescott ( I been rejected statistically 1 its abstract nature. Instead actual statistics characterii. to mimic the behavior of statistics ... would be gr(

The aim of this section successful in passing the t( Kydland-Prescott model is i pler RBC model based on PI model does surprisingly the end of this section we some of the possible extL

necessity, our discuss' is referred to Plosser (1989), Da and Rebelo (1999) for much m

502