Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

out that quite a lot can be tl cases that have received a to demonstrate the crucial

t considering the effects of model this means that the 15.83)) and we study the - m ko = E6 and Zt = 0 for

t on the macroeconomy as

.i.dy-state level. The impact is, however, non-zero.

.ssion for the consumption

(15.89)

'use the technology shock, :ire, like consumption, is labour supply. In terms of e left (from the solid to the es labour productivity and ock is predetermined in the ) the right. As is clear from biguously positive, but the s it depends on the relative 0

in the following analytical

(15.90)

(15.91)

(15.92)

and effect dominates the impact period as illus4ardless of the parameter he same direction. Finally,

• in general, the output

Chapter 15: Real Business Cycles

Figure 15.11. A shock to technology and the labour market

effect is unambiguously positive. 16 Since output rises and capital is predetermined at impact, the immediate effect on the interest rate is positive (see e.g. (T4.5)). Finally, the impact effect on investment is obtained by using (15.89) and setting Zo = Eo and at = 0 in (15.74). After some manipulation we obtain:

By substituting pz = 0 into (15.86) and (15.88) (and noting (15.93)) we find the

In Figure 15.12 we plot the impulse-response functions for the purely transitory shock, using the calibration values discussed above (see page 493). One period after the shock has occurred, technology is back to its steady-state level (as kt = 0 for t = 1, 2, ...). It follows from (15.94) that the economy has a slightly higher capital stock in period 1 (since K1 = Sio > 0) which is gradually run down over time. Consumption also gradually returns to its initial steady-state value. As the simulations confirm, investment and employment fall below their respective steady-state levels during transition (it < 0 and it < 0 for t = 1, 2, ...). The real interest rate also falls below its steady-state level in period t = 1 after which it gradually returns to

16 The sign of the output effect follows in a straightforward fashion from the fact that defined in (15.79), satisfies 0 < ((p. — 1) < 1.

513

The Foundation of Modern Macroeconomics

Figure 15.12. Purely transitory productivity shock

this level. Since it < 0 downward-sloping consul

A permanent shock (pz =

The second special case shocks are permanent, i (pz = 1). The impact eft

- [4) X2 — (1)ci

Co =

X2 [WC + (/) —

Consumption rises at in the representative agent obtain the impact effects

EL

0 = X2 + — 1 )

LX2 [WC ± — 1

As for the purely temp( but positive for realistic output. Finally, the impa setting 20 = €.6 and at =1

By setting pz = 1 in (1:: transition paths of the c_.

Kt

where Co is given in (15.5

I

Roc, = (1(1)c(0Y

As equation (15.100) sha written as the weighted ■

17 King and Rebelo (1999 period. Since output rises an.. interest rate rises at impact

Investment

Consumption rises at impact because the permanent technology shock makes the representative agent wealthier. By substituting (15.95) into (15.71)—(15.73) we obtain the impact effects for employment, the wage, and output:

1 Employment

Real interest rate

As for the purely temporary shock, the employment effect is ambiguous in general but positive for realistic calibrations. The wage rate rises unambiguously as does output. Finally, the impact effect on investment is obtained by using (15.95) and setting 20 = E6 and Ot = 0 in (15.74):

By setting pz = 1 in (15.86) and (15.88) we obtain analytical expressions for the transition paths of the capital stock and consumption:

As equation (15.100) shows, Kt and Ct (and thus all other variables also) can be written as the weighted average of the relevant impact and long-run effects. The

17 King and Rebelo (1999, pp. 966-967) incorrectly argue that the interest rate falls in the impact period. Since output rises and the capital stock is unchanged at impact, it must be the case that the interest rate rises at impact also.

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The Foundation of Modern Macroeconomics

transition speed of the economy, (1-A.1), determines the time-varying weights. With a permanent productivity shock both consumption and the capital stock increase in the long run—see (15.101). The intuition behind this result follows readily from the steady-state constancy of the great ratios (see also above). Imposing the steady state in equations (T4.1)-(T4.2) (and ignoring the expectations operator) we find ioo = koo and i.„„ = 0. But this implies, by (T4.5), that YO° = Roo , and by (T4.4) and (T4.8) that Roo - Loo = Woo = (1/EL )2„ where 40 = e6. With constant government spending ( ä- t = 0), the steady-state versions of (T4.6) and (15.73) can be solved for e0,0 and Yoo :

and (T4.7) can be solved for L oo :

In the long run a permanent productivity improvement makes the representative agent wealthier which prompts him to increase consumption. The investmentcapital ratio and the output-capital ratio are unchanged but the capital-labour ratio rises as does the real wage. In the absence of government consumption (wG = 0) the income and substitution effects in labour supply exactly cancel out and employment is unchanged (see (15.103)). With positive government consumption the income effect dominates the substitution effect and labour supply goes down (i.e. the household consumes more leisure).

In Figure 15.13 we present the impulse-response functions for the permanent shock, again using the calibration values discussed above (see page 493). Following their initial jumps, consumption and the wage both gradually increase further during transition. Investment and employment both overshoot their respective long-run levels. Though the impact effect on employment is positive, employment falls in the long run because the calibration is based on a positive share of government consumption (see (15.103)). The real interest jumps up at impact and gradually returns to its initial level. This explains why the time profile of consumption is upward sloping.

A realistic shock

Now that we have discussed the impulse-response functions for purely transitory and permanent technology shock, we can proceed and study the reaction of the economy to realistic productivity shocks. The seminal work by Solow (1957) has been used by RBC proponents to estimate the nature of technological change. Solow (1957) tried to determine how much of economic growth can be accounted for by fluctuations in the production factors capital and labour. He found that the

0.01 -

Figure 15.13. Permanent p

The Foundation of Modern Macroeconomics

1.4

1.2

0.8

K

0.6

0.4

0.2

0

100

Figure 15.14. Capital stock

unexplained part of output growth (later termed the Solow residual in his honour) accounted for approximately half of the growth of output in the US since the 1870s (Stadler, 1994, p. 1753). It was shown by Prescott (1986) that data on the Solow residual can be used to recover an estimate for the persistence parameter (pz ) and the standard deviation of the innovation term (denoted by az). King and Rebelo (1999, pp. 952-953) explain in detail how this can be done. 18 They use quarterly data for the US and obtain the following estimates for these parameters: pz = 0.979 and o-z = 0.0072. The key thing to note is that the technology shock displays a very high degree of persistence.

In Figures 15.14-15.20 we present impulse-response functions for all macroeconomic variables using the calibration values discussed above (see page 493). Instead of focusing on one particular estimate for the persistence parameter, we show these impulse-response functions for a range of values of pz which includes both King and Rebelo's estimate and the unit-root case (i.e. 0.5 < pz < 1 in these figures).

18 In the context of our (simple) model the procedure would work as follows. First, we take logarithms of (15.57) to derive the estimate for the Solow residual:

log SRt log 17, — EL log /4 — (1 — EL) log Kt = log Zt.

Hence, in our model the Solow residual is equal to the general productivity index Zt . By using this result in (15.82) one obtains an equation which can be estimated empirically:

log SR, = az + pz log SR, + Er .

The procedure of King and Rebelo (1999) is a little more complicated because they also allow for labour-augmenting technological change.

1.5

C

0.5

0

100

Tin g

Figure 15.15. Cc

Figure 15.16. OtI

As King and Rebelo (19 function is very sensiti\ Hence, impulse respon se: very different in shape. does not show up in Fi b, Az = 0.7 look very much case is due to the fact ti:.. transitory shocks (no mat

O

U,

3 co

O

rD

Ui

The Foundation of Modern Macroeconomics

Table 15.5. The unit-elastic RBC model

ways to improve the internal propagation mechanism of the model. Some of this literature will be discussed briefly below.

15.5.3 Correlations

As was pointed out in the introduction to this section, most RBC modellers follow the suggestion by Kydland and Prescott (1982) and evaluate the usefulness of their model by judging how well the model-generated data match the data for an actual economy. The typical approach is to compute actual and model-generated moments for a number of key variables (King and Rebelo, 1999, p. 956). Usually the moments of interest are the variances (or standard deviations) of output, consumption, investment, capital, labour, and productivity. Often the contemporaneous correlations between output and the other variables are also compared. 19

In Table 15.5 we show the results that were computed by Hansen (1985) for the US economy. In this table, 0- (xt ) and p(xt ,Yt ) are, respectively, the (asymptotic) standard deviation of xt and the contemporaneous correlation between xt and Yt . In panel

(a) of Table 15.5 the indicators for the US economy are reported. The following regularities can be distinguished (Stadler, 1994, pp. 1751-1752). First, investment is much more volatile than output, i.e. the standard deviation of investment is

8.60 which far exceeds the standard deviation of output which equals Second, consumption is somewhat less volatile than output

the capital stock is much less volatile than both consumption and output 0.63). Fourth, employment is approximately as volatile as output

Fifth, productivity is less volatile than output 1.18). Sixth, all variables are positively correlated with output, although the correlation is rather weak for the capital stock.

19 In the appendix to this chapter we show how these various indicators can be computed for the theoretical model without having to use statistical simulation methods.

In panel (b) of Table lations are reported. Ha generate these results ar EL = 0.64, p = 0.01,

Y* = (A + 8)I( 1 - EL) =

(by (15.46)) cou, = 2.321 technology shock (Ef in A comparison of panel consumption is less ai It also matches the outp employment quite well t ductivity. Given the extn between actual and m( ever, also a number of f Stadler (1994, pp. 1757market which the mod

Employment variability j

In reality employment al employment is strongly general productivity sho should shift labour der:.. there should be a reactii dence, however, sug, . the variability in wages s panel (b) of Table 15.5 dicts the variability of L 0- (Lt) = 0.70 whereas in

Procyclical real wage

The unit-elastic model mi dicts a high correlation b

P( 17tIlit,171-)= 0.87). In re,

nology is represented model, the real wage is p lows that the unit-ela n procyclical than is corb.,

2° Below we discuss Hansu - section 15.5.4 as well as Hansa