Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

s been labelled by Baxter kperiments, namely the ude of impact effects. By -mined at impact (k(0) = the output multiplier for 'cumption in the impact

(15.56)

s from (15.56) that the to government spending, •.elusion reached on the p. 326). 11

c revolution of the 1970s out the 1950s and 1960s programme in which the estimated by econometric ru1ar in both public and and simulation purposes. cl to a drastic reduction in

sonometric models then and consequently were ill ''cks that occurred at the aacroeconometric models I with shocks affecting the

I

as raised by Lucas (1976). , •;'r 3. Loosely put, it states sistent set of optimizing icy evaluation. The reason s are mixtures of struc- ant across different policy t). To avoid the critique rcefully and eloquently

the oppostite result holds, i.e. he reason for this discrepancy

King (1993, p. 326).

Chapter 15: Real Business Cycles

that macroeconomists should build structural models, i.e. models that are based on optimizing behaviour of the various agents in the economy. In doing so he proposed what Christiano, Eichenbaum, and Evans (1999) have recently labelled the

Lucas (research) programme.

As Lucas (1980, p. 272) argues, well-articulated structural models are of necessity unrealistic and artificial. They should be tested "as useful imitations of reality by subjecting them to shocks for which we are fairly certain how actual economies . would react. The more dimensions on which the model mimics the answers actual economies give to simple questions, the more we trust its answers to harder questions". He goes on to argue that:

On this general view of the nature of economic theory then, a "theory" is not a collection of assertions about the behavior of the actual economy but rather an explicit set of instructions for building a parallel or analogue system-a mechanical, imitation economy. A "good" model, from this point of view, will not be exactly more "real" than a poor one, but will provide better imitations. (1980, p. 272)

In a seminal paper, Kydland and Prescott (1982) accepted the challenge posed by Lucas and his co-workers by building a full-scale structural model with maximizing agents doing as well as they can in a world in which technology is subject to stochastic shocks. Their model can be seen as the starting point of the real business cycle (RBC) research programme (see also Prescott (1986)). As their testing procedure they ask themselves the following question: can shocks to productivity explain fluctuations in actual economies using a model that is plausibly calibrated, i.e. that uses parameter estimates that are not inconsistent with micro observations (Kydland and Prescott, 1982, p. 1359)? The performance of the model is not gleaned by estimating its equations econometrically and testing its implied restrictions. Indeed, as Kydland and Prescott (1982, p. 1360) suggest, the model would undoubtedly have been rejected statistically both because of measurement problems and because of its abstract nature. Instead, the model is tested by comparing model-generated and actual statistics characterizing fluctuations in the economy. "Failure of the theory to mimic the behavior of the post-war US economy with respect to these stable statistics ... would be ground for its rejection".

The aim of this section is to illustrate to what extent RBC models have been successful in passing the tests proposed by Kydland and Prescott (1982). Since the Kydland-Prescott model is rather complex, we start our assessment with a much simpler RBC model based on Prescott (1986). It is shown that even this relatively simple model does surprisingly well in mimicking the fluctuations in the US economy. At the end of this section we show some deficiencies of the simple model and survey some of the possible extensions that can potentially fix them. 12

12 Of necessity, our discussion of the RBC methodology is far from complete. The interested reader is referred to Plosser (1989), Danthine and Donaldson (1993), Stadler (1994), Cooley (1995), and King and Rebelo (1999) for much more extensive surveys of the literature.

503

The Foundation of Modern Macroeconomics

15.5.1 The unit-elastic RBC model

The model constructed in section 2 can be viewed as a deterministic version of an RBC model. To turn that model into a conventional RBC model we must reformulate it in discrete time, introduce a stochastic technology shock, and derive the rational expectations solution for the loglinearized version of the model.

In much of the early RBC literature attention was restricted to competitive models without distortions (like tax rates, useless government consumption, etc.) or externalities (like congestion, pollution, etc.). As Prescott (1986, p. 271) argues, the advantage of working with such models is that the competitive equilibrium is Pareto-optimal and unique. The solution algorithm can then exploit this equivalence between the decentralized market outcome and the social planning problem by solving the latter (easy) problem rather than the former (more difficult) problem. Here we do not pursue this approach because we wish to emphasize the link with the theoretical framework used throughout the book. As a result of this, we need to spell out the decentralized economy. (An additional advantage of doing so is that distortions, such as taxes, are easily introduced in and analysed with the model.)

The decentralized economy

The basic setup is as follows. The representative firm is perfectly competitive and produces homogeneous output, Yr , by renting capital, KT , and labour, LT , from the household sector. The production function is linearly homogeneous in capital and labour and features a unit elasticity of substitution:

where Z, is the state of general technology at time r, which is known to the firm at the time of its production decision. The firm thus faces the static problem of maximizing profit, 11, F(K, , LT ) - W,L, - RKKT , where WT is the wage rate and kr< is the rental charge on capital services. The first-order conditions are:

FL(Z,, KT , LT ) W, , FK(Zt , KT, LT ) = RK , (15.58)

and the linear homogeneity of the production function ensures that profits are zero (II, = 0)•

There is a large number of consumer-investor households. Each individual household is infinitely small and is a price taker on all markets in which it operates. By normalizing the population size to unity we can develop the argument on the basis of a single representativeac,agent. The representative household is infinitely lived and has an objectiveafunction based on expected lifetimee'utility. Denoting the planning period by t, expected lifetime utility, EtAt, is given by:

where Et is the expectati and leisure in period r , an ence. Equation (15.59) is the existence of uncertain tation operator, Et , indicate available at time t.

The household receive sum taxes to the governm investment purposes. Th. 1, t + 2, .. .) by:

Cr ± IT = W,L, +

where ir is gross investmei next equals gross invest' stock:

KT+ 1 = IT + (1 - OK_ .

with 0 < 3 < 1. Equatiu: L. to, respectively, (15.3) and Wt and Rit( but future rem because of the future techn borrow and lend freely 0' tion, labour supply, inves.. expected utility (15.59) sub Kt , as given.

We follow Chow (1997 used throughout this b•zx

,C,11 Et 1),_,

00

r=t (1+P

(Kr-F1 - 1

where kr is the Lagrange mt conditions for this probl,

=1

= 1

aKr+1 1+PI

...:terministic version of an Tiodel we must reformulate wk, and derive the rational e model.

- L- ted to competitive mod- ent consumption, etc.) or , tt (1986, p. 271) argues, competitive equilibrium is then exploit this equivasocial planning problem er (more difficult) problem. emphasize the link with a result of this, we need to

vintage of doing so is that

.ilysed with the model.)

perfectly competitive and C, , and labour, LT , from the omogeneous in capital and

(15.57)

which is known to the firm ces the static problem of Wr is the wage rate and RK, 'editions are:

(15.58)

isures that profits are zero

Ilds. Each individual house- , in which it operates. By the argument on the basis hold is infinitely lived and Denoting the planning

- L,11, (15.59)

Chapter 15: Real Business Cycles

where Et is the expectations operator, Cr and 1 - LT are, respectively, consumption and leisure in period t, and 1/(1 + p) is the discounting factor due to time preference. Equation (15.59) is the discrete-time analogue to (15.1)-(15.2) modified for the existence of uncertainty (and with crL = 1 imposed). The notation for the expectation operator, Et , indicates that the household bases its decisions on information available at time t.

The household receives wage and rental payments from the firm, pays lumpsum taxes to the government, and uses its after-tax income for consumption and investment purposes. The budget identity is given in discrete time (for r = t, t + 1, t + 2, ...) by:

where IT is gross investment. The capital stock carried over from one period to the next equals gross investment plus the undepreciated part of the existing capital stock:

with 0 < 8 < 1. Equations (15.60) and (15.61) are the discrete-time counterparts to, respectively, (15.3) and (15.14). In the planning period, the household knows Wt and Rf but future rental payments on labour and capital are stochastic variables because of the future technology shocks (see (15.58) and below). The household can borrow and lend freely on the capital market and chooses sequences for consumption, labour supply, investment, and capital {CT , LT , IT , KT _F i}r in order to maximize expected utility (15.59) subject to (15.60)-(15.61) and taking its initial capital stock, Kt , as given.

We follow Chow (1997) by tackling this problem with the Lagrangian methods used throughout this book. The Lagrangian expression is:

- (KT-4-1 - + - (RK, + 1 - 8)K, + Cr WT[1 - LTD],

where X, is the Lagrange multiplier for the budget identity in period r. The first-order conditions for this problem (for r = t, t + 1, t + 2, .. .) are:

The Foundation of Modern Macroeconomics

For the planning period (r = t) these first-order conditions can be combined to obtain one static and one dynamic equation:

Equation (15.66), which is obtained by combining (15.63) and (15.64) for period t, is the familiar condition calling for an equalization of the wage rate and the marginal rate of substitution between consumption and leisure. Note that the expectations operator does not feature in this expression. As Mankiw, Rotemberg, and Summers (1985, p. 231) explain, this is the case because (15.66) is a purely static condition as the household knows the wage rate at time t and simply chooses the optimal mix of consumption and leisure appropriately.

Equation (15.67) is obtained by using (15.63) twice (for periods t and t + 1) in (15.65) for period t and substituting (15.68). It is the discrete-time consumption Euler equation. Intuitively (15.67) says that along the optimal path the representative household cannot change his/her expected lifetime utility by consuming a little less and investing a little more in period t, and consuming the additional resources thus obtained in period t+1. The left-hand and right-hand sides of (15.67) represent, respectively, the (marginal) utility cost of giving up present consumption and the expected utility gain of future consumption (Mankiw, Rotemberg, and Summers, 1985, p. 231).

The remainder of the model is quite standard. The government is assumed to finance its consumption with lump-sum taxes, i.e. G, = T. Finally, the goods market clearing condition is given in each period by: 1 77 = Cr + I, + G. The last two expressions are the discrete-time counterparts to, respectively, (15.16) and (15.17).

Loglinearized model

The model consists of the capital accumulation identity (15.61), the consumption Euler equation (15.67), the factor demand equations (15.58), the definition of the real interest rate (15.68), the labour supply equation (15.66), the production function (15.57), plus the goods market clearing condition and the government budget restriction.

We follow Campbell (1994) by looking for analytical solutions to the loglinearized model. The advantage of this approach is that it allows us to study the economic mechanisms behind our simulation results in a straightforward fashion. The loglinearized model is reported in Table 15.4. As before we loglinearize the model around the steady state and use the notation jet log [xt /x], where x is the steady-state value of xt.

Table 15.4. The log-linear,.

1(1+1 — kt = 8[It - kt]

=tt

=—

Pit (P 8)[Cit - kr]

Yt = (oce.t + + (0G6t

it =wa[17vt

Yt =2t+ELit+ (1 E L)kt

Definitions: cod G IY : output

co, //Y: output share of investme between leisure and labour. it

Comparing the discreteof Table 15.2 reveals the L . technology term appearing between the two models lb time model agents are biL growth ( i(t)) appears in th the representative househ( future general technologyi features in the Euler equ,

The derivation of the (15.67) is not straightfor that (15.67)-(15.68) can bt

1 = Et (1 + rt+i

1 +P

By definition we have that eet+1 so we can rewrite (15.

1 = Et exp [(1 +

Al

= Et [1 + (1 + rt+i) -I-

0 = Et Ri ±P p)it+1

• id (15.64) for period t, is Ige rate and the marginal ±e that the expectations totemberg, and Summers purely static condition as

nooses the optimal mix

1r periods t and t + 1) in rete-time consumption imal path the representaty by consuming a little the additional resources des of (15.67) represent, t consumption and the ,temberg, and Summers,

wernment is assumed to

. Finally, the goods C, + IT + G. The last two L•Ply, (15.16) and (15.17).

between leisure and labour. it log kt /x1.

Comparing the discrete-time model of Table 15.4 to the continuous-time model of Table 15.2 reveals the close connection between the two models. Apart from the technology term appearing in (T4.8) but not in (T2.8), the only significant difference between the two models lies in the consumption Euler equation. In the continuous time model agents are blessed with perfect foresight and thus actual consumption growth ( a(t)) appears in the Euler equation. In contrast, in the discrete-time model the representative household does not know the future interest rate (rt+1 ) because future general technology (4 ± 1) is stochastic. As a result, the expectations operator features in the Euler equation (T4.2).

The derivation of the loglinearized Euler equation (T4.2) from its level counterpart (15.67) is not straightforward and warrants some further comment. First we note that (15.67)-(15.68) can be combined to:

(15.69)

rt+ii, Ct /C = et and CNA /C =

(15.70)

The Foundation of Modern Macroeconomics

In going from the first to the second line, we have used the approximation ext 1 + xt , and in going from the second to the third line, we relate 1 + rt+i to it+1. 13

As was the case for the deterministic continuous-time model of section 4.1, the stochastic discrete-time model of Table 15.4 can best be solved by first condensing it. This procedure yields a system of stochastic difference equations in the state variables Kt and Ct , of which the former is a predetermined variable and the latter is a jumping variable.

By using labour demand (T4.4), labour supply (T4.7), and the production function (T4.8), we solve for the equilibrium levels of employment i- t , wages 1717t , and output kt , conditional upon the two state variables and the existing state of general productivity 4:

where is defined in (15.26) above. Ceteris paribus consumption and capital, a higher than average level of general productivity (21- > 0) implies that labour demand is higher than average. As a result, employment, wages, and output are also higher than average.

By using (15.73) in (T4.6) and (T4.5), respectively, we obtain the relevant expressions for investment it and the interest rate it:

General productivity affects investment and the interest rate positively because, ceteris paribus, output and capital productivity are both higher than average if 2t > 0. By leading (15.75) by one period and taking expectations we obtain the following expression:

13 An alternative derivation, mentioned by Campbell (1994, p. 469) and Uhlig (1999, p. 33), is due to Hansen and Singleton (1983, p. 253). (See also Attanasio, 1999, p. 768.) Under the assumption that (Ct± i /Ct ) and (1 + rt+1) are jointly distributed lognormally with a constant variance-covariance matrix, (15.69) can be rewritten as:

G2

Et log (1 + rt+i) = Et log [Ct+i /Cd + log (1 + p) -

where 5 2 is the (constant) variance of log [(Ct/Ct+1)( 1 + rt+i)]. The Q 2 term is subsequently ignored by Campbell (1994) and Uhlig (1999).

Since investment is kno period's capital stock w tures in (15.76). Further next period's general p- explains why EtCr+ i en t( Finally, by using (15.7 expression for the (con-,

kt-Fi -Kt

where A r -1 0* is the J

and yiK and ytc are the sh

[ YK r_i [ Y* (c

Ytc

A number of things shout First we note that the d, to verify that Al equals:

Al = I A * I = - 01* Pc

Second, since the determ from (15.81) that the s\ denoted by -A i < 0, and If, in addition, the pararr the system is saddle-poi_

14 Denoting the typical el

From matrix algebra we kno, the determinant unchanged.

15 Checking saddle-point continuous-time context. With characteristic roots. In contr.:

d the approximation eat

re relate 1 + rt±i to rt+1-- . 13 model of section 4.1, the

colved by first condensing e equations in the state ned variable and the latter

I

and the production func- 'vment it , wages Wt , and he existing state of general

sumption and capital, a > 0) implies that labour nt, wages, and output are

we obtain the relevant

rate positively because, ,th higher than average if -, ectations we obtain the

I

0Et2t+1. (15.76)

0, and Uhlig (1999, p. 33), is due Under the assumption that nt variance-covariance matrix,

term is subsequently ignored by

Chapter 15: Real Business Cycles

Since investment is known in period t, the household knows exactly what next period's capital stock will be. Hence, the actual future capital stock (kt+ 1) features in (15.76). Furthermore, the household must form expectations regarding next period's general productivity level (Et2t+ 1) and labour supply. The latter effect explains why Etat+1 enters in (15.76).

Finally, by using (15.74) in (T4.1) and (15.76) in (T4.2) we obtain the following expression for the (condensed) dynamic system of stochastic difference equations:

where A F -1 A* is the Jacobian matrix and A* and F are defined, respectively, as:

and y1( and ytc are the shock terms:

A number of things should be noted about the dynamical system defined in (15.77). First we note that the determinants of A and A* are identical. 14 It is straightforward to verify that IA l equals:

lAl = I A * I = — 01* [wG(4) — + Ococe-L] < 0. (15.81)

Second, since the determinant is the product of the characteristic roots it follows from (15.81) that the system in (15.77) possesses one negative characteristic root, denoted by —Al < 0, and one positive characteristic root which we denote by A2 > 0. If, in addition, the parameters of the problem are such that Al < 1 it follows that the system is saddle-point stable. 15

14 Denoting the typical elements of A and A* by, respectively, 6q and ,S;; we find:

A . r -1 0* = [ 611 6'12

621 - Y218 1'1 622 — Y216 121 .

From matrix algebra we know that the subtraction of a multiple of any row from another row leaves the determinant unchanged, so it follows that IAI=I A*I. See Mathematical Appendix.

15 Checking saddle-point stability is thus more involved in a discrete-time setting than in a continuous-time context. With continuous time, the only thing that must be checked is the sign of the characteristic roots. In contrast, with discrete time, the magnitude of the roots matters, i.e. one must

509

The Foundation of Modern Macroeconomics

The shock process

Our description of the unit-elastic RBC model is completed once particular specifications are adopted for the exogenous variables, Zt and G. To keep things simple, we assume that government consumption is constant, so that a t = 0 for all t, and that the technology shock takes the following first-order autoregressive form:

log Zt = az + pz log Zr_i + 0<pz,1, (15.82)

where az is a constant, pz is the autoregressive parameter, and Er is a stochastic "innovation" term. The parameter pz parameterizes the persistence in the productivity term-the closer pz is to unity, the higher is the degree of persistence. It is assumed that the innovation term, Ef , is identically and independently distributed with mean zero and variance cry. In the absence of stochastic shocks, technology would settle in a steady state for which (1 - pz) log Z = az. Since, by definition, we have that Zt log [Zt /Z], equation (15.82) can be rewritten as follows:

Recall that agents must form an expectation at time t about technology in the next period (Et2t+ i) in order to forecast the interest rate featuring in their Euler equation

(Etit+i, see (15.76)). Since agents are aware of the shock process for technology (given in (15.83)) they will use this information to compute their forecast, i.e. they

will base their decisions on the forecast Et 2t+i = Pz2t (since Et Ef+1 = 0 this is the best they can do).

The model is now fully specified and consists of (15.77), (15.80), and (15.83). There exist several methods that can be used to solve for the rational expectations solution of the model. Campbell (1994, pp. 470-472), for example, uses the method of undetermined coefficients. Intuitively, this method works as follows. First, we guess a solution for consumption in terms of the state variables (Kt , Zt ) and unknown parameters (7,k, of the form Ct = 7rakt + 7„2t. Next, we use all the structural information contained in the model plus the assumption of rational expectations in order to relate the unknown coefficients to the structural parameters of the model. Another method is due to Blanchard and Kahn (1980)—see Uhlig (1999, pp. 54-56) for an example.

check whether they are inside or outside the unit circle. Note that (15.77) is conventionally written as:

[ kt+1 ]=,[ kt J_[ YtK

where a I + A has characteristic roots X 1 1 — Xi and A2 1 + A2 (see Mathematical Appendix). A stable (unstable) root satisfies IX < 1 (V,., > 1). Saddle-point stability thus obtains provided Il — X i I < 1 and Il + X2I > 1. See Azariadis (1993, pp. 39 and 62-67) for a very thorough discussion of the discretetime case. In the text we simply assume that X1 < 1 and we ensure that this assumption is satisfied in the simulations.

15.5.2 Impulse-respoi

In the appendix to this ch tion of the model in tern _ Campbell (1994). Here, wl different variables. The ad parison with the analyti- impulse-response functioi RBC model, especially th

We compute the impul of the shock at t = 0, and that technology was at use (15.83) to solve for th

time t = 0:

2t = 4E6.

By using (15.84) in (15.8 , the shock term affecting I form:

[ ytx = [

c -

Yt 04,

It follows from (15.85) tt curves. Since 0 <

yoc = 0. The innovation t and transition results are impulse-response functioi

where the impact jump i.

( + [(1 -

Co = coc

and Tt (ai , a2) is a non-neg I

at -

al tar'

I

-d once particular specifi- 6't . To keep things simple, that at = 0 for all t, and

autoregressive form:

(15.82)

,?r, and Er is a stochastic persistence in the producdegree of persistence. It is independently distributed :hastic shocks, technology rz. Since, by definition, we ten as follows:

I

(15.83)

"!t technology in the next lg in their Euler equation ck process for technology lute their forecast, i.e. they since Et Ef±i = 0 this is the

Chapter 15: Real Business Cycles

15.5.2 Impulse-response functions

In the appendix to this chapter we work out the general rational expectations solution of the model in terms of its state variables, following the approach suggested by Campbell (1994). Here, we focus directly on the impulse-response functions for the different variables. The advantage of doing so is twofold. First, it facilitates the comparison with the analytical discussion in the first half of this chapter. Second, the impulse-response functions nicely visualize the key properties of our prototypical RBC model, especially those related to the degree of persistence of the shock.

We compute the impulse-response functions as follows. We normalize the time of the shock at t = 0, and assume that 66 > 0 and ef = 0 for t = 1, 2, ... Assuming that technology was at its steady-state level in the previous period (2_1 = 0) we can use (15.83) to solve for the implied path of 2, that results from the innovation at time t = 0:

By using (15.84) in (15.80) (and recalling that Gt = 0 and Er2r+i Pz2t) we find that the shock term affecting the dynamical system takes the following, time-varying,

form:

(15.86)

(15.87)

(15.88)

511

The Foundation of Modern Macroeconomics

Although these expressions look rather complex, it turns out that quite a lot can be understood about them by first focusing on some special cases that have received a lot of attention in the literature. In doing so we are able to demonstrate the crucial role of shock persistence in the unit-elastic RBC model.

A purely temporary shock (pz = 0)

We follow King and Rebelo (1999, pp. 964-967) by first considering the effects of a purely transitory productivity shock. In terms of our model this means that the shock displays no serial correlation at all (i.e. pz = 0 in (15.83)) and we study the response of the system to a technology shock of the form Zo = 66 and 2, = 0 for t = 1, 2, ... Clearly, such a shock has no long-run effect on the macroeconomy as technology only deviates in the impact period from its steady-state level. The impact effect on consumption, and thus on the other variables, is, however, non-zero. Indeed, by setting pz = 0 in (15.87) we obtain the expression for the consumption jump with a purely transitory shock:

Intuitively, consumption rises in the impact period because the technology shock, brief though it may be, makes the agent richer. Since leisure, like consumption, is a normal good, the shock also causes a wealth effect in labour supply. In terms of Figure 15.11 the labour supply curve shifts up and to the left (from the solid to the dashed line). At the same time, however, the shock raises labour productivity and thus labour demand. Hence, even though the capital stock is predetermined in the impact period, the labour demand curve shifts up and to the right. As is clear from the diagram, the impact effect on the wage rate is unambiguously positive, but the impact effect on employment appears to be ambiguous as it depends on the relative magnitudes of the labour supply and demand effects.

By using (15.89) and Zo = E6 in (15.71)—(15.73) we obtain the following analytical expressions forCL0, V-170, and Yo :

For realistic calibrations of the model the labour-demand effect dominates the labour-supply effect, so that employment increases in the impact period as illustrated in Figure 15.11. The wage rate increases at impact regardless of the parameter values as the labour-demand and supply effects work in the same direction. Finally, despite the fact that the employment effect is ambiguous in general, the output

Wo

0

I

Figure 15.1 market

effect is unambiguously pc at impact, the immediate Finally, the impact effect

where the sign follows froi By substituting pz = 0 ii

transition paths for the cud

[kt 1 = [(1A8

t Co

In Figure 15.12 we plot ti shock, using the calibratio the shock has occurred, tt t = 1, 2, ...). It follows fr. , stock in period 1 (since k sumption also gradually . . confirm, investment and e during transition (it < 0 falls below its steady-starL

16 The sign of the output eft, (15.79), satisfies 0 < “(/) — 1) < 1