Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

lel economy II

rt , Y1 )

0.87

0.99

0.05

0.98

0.87

the model. Some of this

most RBC modellers fol- n-aluate the usefulness of

.1 match the data for an ual and model-generated ' 0 99, p. 956). Usually the ons) of output, consumpen the contemporaneous su compared. 19

Hansen (1985) for the US •he (asymptotic) standard een xt and Yt . In panel reported. The following * - 52). First, investment is

)n of investment is a (It) = ich equals a (Yt ) = 1.76.

_A (a (Ct ) = 1.29). Third,

on and output (0- (Kr) = output (a (Lt ) = 1.66).

1.18). Sixth, all variables

•is rather weak for the

•rs can be computed for the

Chapter 15: Real Business Cycles

In panel (b) of Table 15.5 the model-generated standard deviations and correlations are reported. Hansen (1985, pp. 319-320) uses the unit-elastic model to generate these results and employs the following calibration parameters: coG = 0, EL = 0.64, p = 0.01, 8 = 0.025, and cc = 1/3. These parameters imply: y* = (p + 3) 1 (1 — EL) = 0.097, wi = 8/y* = 0.257, we = 1 — (.0/ = 0.743, and (by (15.46)) WLL = 2.321. The persistence parameter and standard deviation of the technology shock (€f in (15.83)) are set at, respectively, pz = 0.95 and az = 0.00712.

A comparison of panels (a) and (b) reveals that the model captures the facts that consumption is less and investment is more volatile than the aggregate output. It also matches the output correlations of consumption, investment, capital, and employment quite well but it overpredicts the correlation between output and productivity. Given the extremely simple structure of the unit-elastic model, the match between actual and model-generated moments is quite impressive. There are, however, also a number of facts that are not well explained by the model. Following Stadler (1994, pp. 1757-1761) we focus on some stylized facts about the labour market which the model is unable to mimic.

Employment variability puzzle

In reality employment and output are almost equally variable (see Table 15.5), and employment is strongly procyclical, whilst wages are only mildly procyclical. If general productivity shocks are the source of the variability, then a positive shock should shift labour demand, and for a given upward-sloping labour supply curve, there should be a reaction in both wages and employment. Microeconomic evidence, however, suggests that the labour supply curve is almost vertical, so that the variability in wages should be high and that in employment should be low. In panel (b) of Table 15.5 we therefore observe that the unit-elastic model underpredicts the variability of employment by a significant factor, i.e. the model predicts a (Lt ) = 0.70 whereas in reality for the US o- (Lt ) = 1.66.20

Procyclical real wage

The unit-elastic model with productivity shocks as the source of fluctuations predicts a high correlation between productivity and output (in panel (b) of Table 15.5 p(Yt ILt ,Yt )= 0.87). In reality, however, this correlation is much weaker. Since technology is represented by a Cobb-Douglas production function in the unit-elastic model, the real wage is proportional to productivity (see (T4.4) above). It thus follows that the unit-elastic model generates wage fluctuations that are much more procyclical than is consistent with reality.

20 Below we discuss Hansen's (1985) approach to bringing the model outcomes closer to reality. See section 15.5.4 as well as Hansen and Wright (1994) and Stadler (1994, pp. 1757-1762).

523

The Foundation of Modern Macroeconomics

Productivity puzzle

If productivity shocks are the predominant source of fluctuations, the shifts in labour demand would imply that hours worked and productivity move closely together. If real wage changes are small, all variation in employment is due to labour demand shocks, and the correlation between productivity and both hours worked and output should be high. In reality, however, the first correlation (productivityhours) is absent or even negative and the second correlation (productivity-output) is much weaker than predicted.

Unemployment

Since there is no unemployment in the unit-elastic model, all variation in employment is explained by fluctuations in the supply of labour by the representative household. In reality, however, about two thirds of the variation in hours is due to movements into (and out of) employment and only one third is explained by variation in the number of hours worked per employed worker (Stadler, 1994,

p.1758).

15.5.4Extending the model

Stadler (1994), Hansen and Wright (1994), and King and Rebelo (1999) discuss the several model extensions that have been proposed in the RBC literature over the past two decades. Here, we focus attention on just some of the ways in which RBC modellers have responded to the various puzzles discussed above.

Employment variability puzzle

One would observe a realistic correlation between real wages and employment if the labour supply curve is relatively flat. There are several ways to get this. First, there may be strong intertemporal substitution effects in labour supply, but this is rejected by the econometric evidence to date (Card, 1994). Second, the dominant RBC solution to the employment variability puzzle is provided by Hansen (1985) who incorporated the insights of Rogerson (1988) into an RBC model. His argument makes use of the fact that in reality about two thirds of the variation of total hours worked is due to movements into and out of employment, whilst only one third is explained by variation in the number of hours worked. Hansen (1985) assumes that the length of the working week is constant: you either have a job and work for, say, 38 hours per week, or you do not work at all. This non-convexity in the form of indivisible labour (IL) ensures that workers wish to work as much as possible when wages are high. Hansen shows that even if individual agents have a zero intertemporal labour supply elasticity, the aggregate economy behaves as if the (average) "representative agent" has an infinite intertemporal labour supply elasticity. Individual households do not choose the number of working hours per

period, but rather the I a lottery. There is a coi the household must wo vides complete insura each household gets the or not in any particular Lt = mit, and each hou,,

The IL model is obtain appearing in (15.59) is li

(I)(r) cc log Cr — 1

With this modification, (15.67) but leisure drops

Wt 1 — cc

Ct EC

In the IL model, consL. equation is horizontal. II (T4.7) is replaced by Ii't of the model presenteu In panel (c) of Table 1 (1985) calibration of t:.. calibration parameters a: such that employed irk. i.e. L = 0.53 (Hansen, 1 clear from Table 15.5 tha

model-generated and does.22 By incorporating model, the model-gerk 0.70 (in panel (b)) to dispose of the employmt Indeed, the results in p...

similar correlation betwi

21 By manipulating (15.b

Since we and EL are known, the 22 Note that the actual ay

(a) and (c) of Table 15.5. Th: feature of the calibration pr such that the variation in a_ with divisible labour (given to match the observed stone

actuations, the shifts in oductivity move closely )lovment is due to labour and both hours worked :relation (productivityon (productivity-output)

all variation in employ-

.- by the representative -iation in hours is due one third is explained

' worker (Stadler, 1994,

'helo (1999) discuss the RBC literature over the

f the ways in which RBC bove.

ages and employment if LI ways to get this. First, labour supply, but this 994). Second, the dom-

!is provided by Hansen into an RBC model. His irds of the variation of mployment, whilst only worked. Hansen (1985)

.1 either have a job and all. This non-convexity sh to work as much as f individual agents have !* e economy behaves as :mporal labour supply er of working hours per

Chapter 15: Real Business Cycles

period, but rather the probability of working. Who actually works is determined by a lottery. There is a contract between the firm and a household that specifies that the household must work L hours with probability Trt in period t. The firm provides complete insurance to the worker and the lottery contract is traded, so that each household gets the same amount from the firm, regardless of whether it works or not in any particular period. Actual per capita employment in period t will be Lt = jrtL, and each household gets paid as if it worked Lt hours in period t.

The IL model is obtained by setting aL —> co in (15.2) so that the felicity function

With this modification, the consumption Euler equation continues to be given by (15.67) but leisure drops out of equation (15.66) which becomes:

In the IL model, consumption is proportional to the wage, i.e. the labour supply equation is horizontal. In terms of the loglinearized model of Table 15.4, equation (T4.7) is replaced by Wt = Ct . Hence, in formal terms, the IL model is a special case of the model presented in Table 15.4 with 0LL —> oo•

In panel (c) of Table 15.5 we show the results that were obtained by Hansen's (1985) calibration of the IL model. With the exception of EC (and thus the calibration parameters are the same as for panel (b). The parameter EC is chosen such that employed individuals spend 53 percent of their time endowment on work, i.e. L = 0.53 (Hansen, 1985, p. 320). 21 This yields the value of cc = 0.381. It is clear from Table 15.5 that the IL model provides a much better match between the model-generated and actual variability of employment than the standard model does.22 By incorporating the assumption of indivisible labour in the unit-elastic model, the model-generated standard deviation of employment rises from

0.70 (in panel (b)) to 1.35 (in panel (c)). The IL model is thus able to dispose of the employment variability puzzle but not of the procyclical real wage. Indeed, the results in panel (c) of Table 15.5 show that the IL model predicts a very similar correlation between productivity and output as the standard unit-elastic

21 By manipulating (15.105) we find that the value of Cc can be written as follows:

E C = 1+€ - 1

Lcoc

Since coy and EL are known, the value of cc is readily obtained from this expression.

22 Note that the actual and model-generated standard deviation of output are the same in panels

(a) and (c) of Table 15.5. This cannot be counted as a success for the unit-elastic model because it is a feature of the calibration procedure. The standard deviation of the productivity shock (az) is chosen such that the variation in aggregate output is perfectly matched (Hansen, 1985, p. 320). In a model with divisible labour (given in panel (b)) a larger standard deviation of the innovation term is needed to match the observed standard deviation of output.

525

ti

The Foundation of Modern Macroeconomics

model does (i.e. p(Yt114,Yt) = 0.87 in panel (c) and p(Yt /Lt , Yt) = 0.98 in panel (b)). Because it largely solves the employment variability puzzle, Hansen's approach has

nevertheless become standard practice in the RBC literature.

Productivity puzzle

RBC theorists have also found creative solutions to the productivity puzzle, which are often based on introducing shift factors in the labour supply equation. Examples that are found in the literature include the existence of nominal wage contracts, taste shocks, government spending shocks, labour hoarding by firms, and the existence of a non-market production sector that is also subject to technology shocks (see Stadler, 1994, pp. 1759-1761). In home production models, for example, households divide their labour over market and non-market activities. If market productivity rises, agents not only intertemporally substitute labour, but also shift labour intratemporally from the non-market to the market sector.

Unemployment

In recent years, a number of authors have introduced voluntary unemployment into the RBC framework by making use of the search-theoretic approach of Diamond, Mortensen, and Pissarides (see Chapter 9). 23 Andolfatto (1996, p. 113) shows that the introduction of labour market search into an RBC model leads to three major improvements. First, the model is able to predict that labour hours fluctuate more than wages. Second, the model predicts a lower correlation between labour hours and productivity. Third, the model predicts a more realistic impulse-response

function for output.

Involuntary unemployment can be built into RBC-style models as well. Danthine and Donaldson (1990), for example, use the device of efficiency wages (see Chapter 7) to explain equilibrium unemployment. In such models, the real wage does not clear the labour market but rather is used to induce high effort by the workers. Such models typically predict some kind of real wage rigidity which can help explain the low correlation between wages and employment and magnifies the impact of shocks on output. The latter effect also ensures that productivity shocks do not have to be unrealistically large in order to explain given fluctuations in output.

15.6 Punchlines

This chapter deals with the two major themes which have been developed by new classical economists over the last two decades, namely the equilibrium approach to

23 See, e.g. Andolfatto (1996), Merz (1995, 1997, 1999), and Cole and Rogerson (1999).

I

fiscal policy and real bti the insights that were obl the 1970s.

In order to discuss ti . chapter by extending the in the previous chapter. ysis the labour supply d4 version of) the extended financed increase in govt policy shocks are conside effects are characterized b

With a permanent ina causes households to because they feel poorer and the capital stock st. ers somewhat. In the loi than one for one) and of equi-proportionally due t, bly calibrated version of tl well exceed unity. Thu. Haavelmoo multiplier, t., new classical in nature. %V1 plays the vital role in the which determines the let,

When the increase in long-run effects. Consuitil rise in the impact period. 1 shock. If labour supply is household starts to accum increase in output and c the increased governmen the labour supply effect is stock falls during the ear. : In the second half of the totypical real business cy

a stochastic process for gen nal expectations. We stud computing the analytical nomic variables. Just as for technology shock exerts a functions.

For a purely transitory I mentt and output all rim

L : .Yt )= 0.98 in panel (b)). Hansen's approach has

ture.

lroductivity puzzle, which it supply equation. ExamDf nominal wage contracts, lg by firms, and the exisct to technology shocks ion models, for example, -ket activities. If market tute labour, but also shift

- -t sector.

I

ntary unemployment into tic approach of Diamond, (1996, p. 113) shows RBC model leads to three

hat labour hours fluctuate Drrelation between labour realistic impulse-response

vle models as well. Dan- :e of efficiency wages (see

ich models, the real wage duce high effort by the i wage rigidity which can Inloyment and magnifies ensures that productivity !xplain given fluctuations

e been developed by new quilibrium approach to

d Rogerson (1999).

Chapter 15: Real Business Cycles

fiscal policy and real business cycle theory. These two themes build on and extend the insights that were obtained as a result of the rational expectations revolution of the 1970s.

In order to discuss the equilibrium approach to fiscal policy, we started this chapter by extending the deterministic Ramsey model that was studied in detail in the previous chapter. To make this model more suitable for fiscal policy analysis the labour supply decision of households is endogenized. We use (a simple version of) the extended Ramsey model to study the effects of a lump-sum tax financed increase in government consumption. Both permanent and temporary policy shocks are considered. Furthermore, the impact, transitional, and long-run effects are characterized both analytically and quantitatively.

With a permanent increase in government consumption, the increase in taxes causes households to cut back goods consumption and to supply more labour because they feel poorer (the wealth effect). Output and investment both rise and the capital stock starts to increase during transition and consumption recovers somewhat. In the long run, consumption is still crowded out (though by less than one for one) and output, capital, employment, and investment all increase equi-proportionally due to the constancy of a number of "great ratios". A plausibly calibrated version of the model shows that the long-run output multiplier may well exceed unity. Though this result is superficially reminiscent of the KeynesHaavelmoo multiplier, the mechanism behind the output multiplier is distinctly new classical in nature. Whereas the marginal propensity to consume out of income plays the vital role in the former multiplier, it is the wealth effect in labour supply which determines the latter multiplier.

When the increase in government consumption is only temporary there are no long-run effects. Consumption is crowded out and labour supply and thus output rise in the impact period. These effects are stronger the more persistent is the policy shock. If labour supply is highly elastic and the shock is relatively persistent then the household starts to accumulate capital during the early part of the transition. The increase in output and decrease in consumption together more than compensate the increased government consumption and investment is crowded in at impact. If the labour supply effect is weak and the shock is highly transitory then the capital stock falls during the early phases of the transition.

In the second half of the chapter we turn the extended Ramsey model into a prototypical real business cycle model by reformulating it in discrete time, introducing a stochastic process for general productivity, and imposing the assumption of rational expectations. We study the properties of the so-called unit-elastic RBC model by computing the analytical impulse-response functions for the different macroeconomic variables. Just as for the deterministic model, the degree of persistence of the technology shock exerts a critical influence on the shape of the impulse-response functions.

For a purely transitory technology shock, consumption, employment, investment, and output all rise in the impact period. The employment response is

527

The Foundation of Modern Macroeconomics

explained not so much by the wealth effect (which is rather weak) but rather by the incentive to substitute labour supply across time. The technology shock makes it attractive to work in the current period because the current wage is high relative to future wages. After technology has returned to its initial level, capital and consumption gradually fall back over time.

With a permanent productivity shock, consumption, capital, output, investment, and the real wage all rise in the long run. In the absence of government consumption (and the concomitant lump-sum taxes), employment stays the same because the income and substitution effects of the wage change cancel out. With positive lump-sum taxes the former dominates the latter effect and employment falls. The intuition behind the long-run results is again provided by the constancy of a number of great ratios. Consumption jumps up at impact and thereafter increases further during transition.

Next we study the impulse-response functions for a "realistic" shock persistence parameter. Most RBC modellers use the so-called Solow residual to obtain an estimate for this persistence parameter. The typical finding is that productivity shocks (thus measured) are very persistent, i.e. the persistence parameter is close to (but strictly less than) unity.

An important, somewhat disappointing, feature of the unit-elastic RBC model is its lack of internal propagation. For all cases considered, the impulse-response function for output is virtually identical to the exogenous technology shock itself. The lack of propagation plagues not just the uni-elastic model but many other RBC models as well. For this reason, one of the currently active areas of research in the RBC literature concerns the development of models with stronger and more realistic internal propagation mechanisms.

It is standard practice to evaluate the quantitative performance of a given RBC model in terms of the quality of the match it provides between model-generated and actual data. Typically, the statistics of interest are the standard deviations (and correlations with aggregate output) of some key macroeconomic variables. Despite its simplicity, the unit-elastic model is able to capture quite a few features of the real world data. For example, it correctly predicts that investment is more and consumption is less volatile than aggregate output. It also matches the output correlations of consumption, investment, capital, and employment quite closely. There are also a number of empirical facts that are difficult or impossible to reconcile with the unit-elastic model. For that reason a huge literature has emerged over the last two decades which aims to improve the empirical fit of RBC models.

Perhaps the most important contribution of the RBC approach is a methodological one. Recall that in the traditional macroeconometric approach, weakly founded relationships were typically estimated with the aid of time series data. RBC modellers have largely abandoned the macroeconometric approach and have instead forged a link with micro-founded stochastic computable equilibrium models. Attention has shifted from estimation to simulation. The approach has proved to be quite flexible. RBC models now exist which include alternative market structures (on goods

I and labour markets), pri _

just technology shocks, a cations indicates that t' from classical and Kepi

Further Reading

Some of the most important (1994). For survey articles, (1989b), Plosser (1989), Li, (1994), Stadler (1994), Cr' approach are Summers (198 of calibration, see Kydlar (1996). Watson (1993) del. u (1995) and Rotemberg and of a number of standard RRI There is a huge and gro‘N .; ral substitution mechanism On family labour supply, • inal wage contracts are st.. On search unemployment, ! Rogerson (1999). Efficiency Kimball (1994), and Geor,„ Early papers on the macrc istic approach, include Fole! Barro (1981), and Aschauer (1998) (on public infrastru, Braun (1994), Jonsson and I Fisher (1998) study the emp Temporary shocks and ti.. are also studied by Judd ( P on home production are IN Hercowitz (1991). Models v man (1991) and McGrattan ( in household consumption Hercowitz, and Huffman 1 monopolistically competit, (1993), Chatterjee and Cool Head, and Lapham (1996a, 1 and price stickiness, see H,...

Rotemberg and Woodford 11 develops a method to decorn effects.

her weak) but rather by hnology shock makes Trent wage is high relaitial level, capital and

ital, output, investment, _:overnment consumpstays the same because kncel out. With positive I employment falls. The he constancy of a num-

after increases further

,tic" shock persistence sidual to obtain an esti- "at productivity shocks meter is close to (but

RBC model i, the impulse-response

•-chnology shock itself. :1 but many other RBC areas of research in the on :er and more realistic

Lance of a given RBC een model-generated tandard deviations (and comic variables. Despite 3 few features of the real

• is more and consump- e output correlations to closely. There are also e to reconcile with the terged over the last two

xiels.

.ich is a methodologi-

•roach, weakly founded :s data. RBC modellers al have instead forged a models. Attention has ved to be quite flexi- 'et structures (on goods

Chapter 15: Real Business Cycles

and labour markets), price and wage stickiness, open-economy features, more than just technology shocks, and heterogeneous households. The broad range of applications indicates that the RBC methodology has received widespread acceptance from classical and Keynesian economists alike.

Further Reading

Some of the most important early articles on the RBC approach have been collected in Miller (1994). For survey articles, see King, Plosser, and Rebelo (1987, 1988a, 1988b), McCallum (1989b), Plosser (1989), Eichenbaum (1991), Danthine and Donaldson (1993), Campbell (1994), Stadler (1994), Cooley (1995), and King and Rebelo (1999). Early critics of the approach are Summers (1986) and Mankiw (1989). For a recent discussion on the method of calibration, see Kydland and Prescott (1996), Hansen and Heckman (1996), and Sims (1996). Watson (1993) develops a measures of fit for calibrated models. Cogley and Nason (1995) and Rotemberg and Woodford (1996) document the weak propagation mechanisms of a number of standard RBC models.

There is a huge and growing literature on various labour market aspects. The intertemporal substitution mechanism is studied in detail by Hall (1991, 1997) and Mulligan (1998). On family labour supply, see Cho and Rogerson (1988) and Cho and Cooley (1994). Nominal wage contracts are studied by Cho and Cooley (1995) and Huang and Liu (1999). On search unemployment, see Andolfatto (1996), Merz (1995, 1997, 1999), and Cole and Rogerson (1999). Efficiency wage theories are used by Danthine and Donaldson (1990), Kimball (1994), and Georges (1995).

Early papers on the macroeconomic effects of government purchases, using a deterministic approach, include Foley and Sidrauski (1971), Hall (1971), Miller and Upton (1974), Barro (1981), and Aschauer (1988). Recent stochastic models include Cassou and Lansing (1998) (on public infrastructure), Christiano and Eichenbaum (1992), McGrattan (1994), Braun (1994), Jonsson and Klein (1996), and Canton (2001). Edelberg, Eichenbaum, and Fisher (1998) study the empirical effects of a shock to government purchases.

Temporary shocks and the interaction between the graphic and mathematical approaches are also studied by Judd (1985) and Bovenberg and Heijdra (forthcoming). Key articles on home production are Benhabib, Rogerson, and Wright (1991) and Greenwood and Hercowitz (1991). Models with distorting taxes are presented by Greenwood and Huffman (1991) and McGrattan (1994). Ljungqvist and Uhlig (2000) introduce habit formation in household consumption. Studies focusing on firm investment include Greenwood, Hercowitz, and Huffman (1988) and Gilchrist and Williams (2000). Models including a monopolistically competitive goods market are formulated by Benassy (1996a), Hornstein (1993), Chatterjee and Cooper (1993),Rotemberg and Woodford (1992, 1996), Devereux, Head, and Lapham (1996a, 1996b), Heijdra (1998), and Gall (1999). On models with money and price stickiness, see Hairault and Portier (1993), King and Watson (1986), Yun (1996), Rotemberg and Woodford (1999), and Chari, Kehoe, and McGrattan (2000). King (1991) develops a method to decompose impulse-response functions into wealth and substitution effects.

529

The Foundation of Modern Macroeconomics

Appendix

Phase diagram for the unit-elastic model

In this appendix we derive the phase diagram for the unit-elastic model. We drop the superfluous time index and hold the output share of government consumption, coG G/Y, constant.

Employment as a function of the state variables

By using labour demand (T1.4), labour supply (T1.7), and the production function (T1.8), we obtain an expression relating equilibrium employment to consumption and the capital stock ("LME" designates labour market equilibrium).

with f'(L) < 0 and f"(L) > 0 in the economically meaningful interval L E [0,1]. Hence, f(L) is as drawn in Figure A15.1.

Capital stock equilibrium

Using (T1.6) in (T1.1) we observe that K = 0 holds if and only if SK = (1 - WG)Y - C. By using (T1.4) and (T1.7), and assuming 0 < Ec < 1, the capital stock equilibrium (CSE) locus

= 0) can be written as:

f(L)

Figure A15.1. Labour market equilibrium

We are clearly only inter square brackets on the ri8.. lower bound for employme

L > LMIN ECEL + (1 -

By using LMIN and (T1.8

CSE: KEL = EcE. 6(1 -

Equation (A15.4) represer relating K and L. In oruL differentiate (A15.4):

Since L > LMIN > 0 the term that g'(K) > 0. It follows f - to L = 1, K rises from K = for the CSE line, i.e. both By using (A15.1) we find tt (C, K, L) = (0, KK, 1) are b( drawn in (C, K) space.

The slope of the CSE lin

written as:

I

C = (1 - wG)Zog(K) E

where L = g(K) is the in:, (A15.6) we obtain in a few s

I

dCK ) k=0

1 - (DG )Z

where 74(K) is the elasticity

I

I

It follows from (A15.8) that side of (A15.7) goes to (1 -

!antic model. We drop the t consumption, coG G/Y,

nroduction function (T1.8), nsumption and the capital

(A15.1)

rval L E [0, 1]. Hence, f (L)

Iv if 6K = (1 - (0G)Y - C. By ck equilibrium (CSE) locus

(A15.2)

1

Chapter 15: Real Business Cycles

We are clearly only interested in positive values of output and capital so that the term in square brackets on the right-hand side of (A15.2) must be non-negative. This furnishes a lower bound for employment:

ECEL

L > LMIN =0 < LMIN <

ECEL ( 1 — WG)( 1 — EC)

By using LMIN and (T1.8) we can rewrite (A15.2):

Equation (A15.4) represents an implicit function, L = g(K), over the interval L E [LMIN, 1] relating K and L. In order to compute the slope of the implicit function we totally differentiate (A15.4):

Since L > LMIN > 0 the term in square brackets on the right-hand side is strictly positive so

that g'(K) > 0. It follows from (A15.4) that K = 0 for L = LMIN, so as L rises from L = LMIN to L = 1, K rises from K = 0 to K = KK . ((1 — COG)Zo / /EL > 0. We now have two zeros

for the CSE line, i.e. both (K, L) = (0, LMIN) and (K, L) = (KK , 1) solve equation (A15.2). By using (A15.1) we find the corresponding values for C, i.e. (C,K,L) = (0, 0, LMIN) and (C, K, L) = (0, KK, 1) are both zeros for the CSE line. In Figure 15.1 these points have been drawn in (C, K) space.

The slope of the CSE line is computed as follows. We note that the CSE line can be written as:

where L = g(K) is the implicit function defined by (A15.4). By taking the derivative of (A15.6) we obtain in a few steps:

It follows from (A15.8) that 74(0) = 0 so that the term in square brackets on the right-hand side of (A15.7) goes to (1 - EL ) as K 0. But since limx->og(K)/K = +oo it follows that the

531

The Foundation of Modern Macroeconomics

CSE line is vertical near the origin (see Figure 15.1):

The golden-rule point (for which consumption is at its maximum value) is obtained by setting dC/dK = 0 in (A15.7):

and rig (K) is given in (A15.8). The golden rule occurs at point A in Figure 15.1. For points to the right of the golden-rule point, the CSE line is downward sloping.'

The capital stock dynamics follows from (T1.1) in combination with (T1.6), (T1.8), and using the implicit function

[

from which we derive akiac < 0. See the horizontal arrows in Figure 15.1.

Consumption equilibrium

The consumption equilibrium (CE) line describes combinations of C and K for which C = 0. By using (T1.2) (in steady-state format), (T1.5), and (T1.8), we can write the CE line as follows:

where y* (p 6) / (1 - EL) is the equilibrium output-capital ratio for which the rate of interest equals the rate of time preference (r = p). It follows from (A15.13) that consumption equilibrium pins down a unique capital-labour ratio, (K/L)* (Z0 /y*) 1 /EL . By substituting

It follows from (A15.14) that the CE line is linear and passes through the coordinates (C, K) (0, KO and (C, K) = (Cc, 0) in Figure 15.1:

24 For exogenous labour supply 77,(K) = 0 for all K and the formula for KGR collapses to the usual expression (1 — toG)FK = 8 or (1 — coG)(1 — EL )(Y/K)GR = 8.

Provided 0)G < EL, the CE I line.25

The consumption dyndil the following fashion: Ci C (A15.13) we find: I

aC =ELZ°

where (A15.1) shows that 8

has been indicated with

I

Derivation of (15.52)-

We wish to solve the dy: is as defined in (15.51). In 1 written in Laplace transforr

The impact jump for con ,

C'(0) = ( A2 322

6 12

The shock term (15.51) G

Tme-w , OK -

The shock occurs at time t (A15.19), the Laplace tran .

L{/10 s} = _

L{YK, - LIM A21 s - A2

25 To show this result, we nol

Kc " 8

(kj = (p+8)(1 1

Both terms in round brackets Hence, we conclude that Kc <