On the Welfare Costs of Business Cycles in the 20Th Century"

On the Welfare Costs of Business Cycles in the 20th Century

Osmani Teixeira de Carvalho Guillén João Victor Isslery Banco Central do Brasil and Ibmec-Rio Graduate School of Economics - EPGE Av. Presidente Vargas 730 Getulio Vargas Foundation Rio de Janeiro, RJ 20071-001 Praia de Botafogo 190 s. 1100 Brazil Rio de Janeiro, RJ 22253-900 Brazil Afonso Arinos de Mello-Franco-Neto Graduate School of Economics - EPGE Getulio Vargas Foundation Praia de Botafogo 190 s. 1100 Rio de Janeiro, RJ 22253-900 Brazil E-mail: [email protected], [email protected], and [email protected] JEL Codes: E32; C32; C53. Keywords: Business cycles ‡uctuations, economic growth variation, welfare costs, structural time-series model.

May, 2011.

Abstract

Lucas(1987) has shown a surprising result in business-cycle research, that the welfare cost of business cycles are relatively small. Using standard assumptions on preferences and a reasonable reduced form for consumption, we computed these welfare costs for the pre- and post-WWII era, using a trend-cycle decomposition method which allows measuring the welfare cost of business cycles and the welfare cost of economic growth. The post-WWII period is very quiet, with the welfare costs never exceeding 1% of per-capita consumption. However, for the pre-WWII era this basic result is altered, where welfare costs can

We gratefully acknowledge the comments of Larry Christiano, Wouter den Haan, Robert F. Engle, Pedro C. Ferreira, Antonio Fiorencio, Clive Granger, Rodolfo Manuelli, Samuel Pessoa and Octavio Tourinho. All errors are ours. We thank CNPq-Brazil and PRONEX for …nancial support. yCorresponding author reach more than 3% of consumption. These results highlight the relevance of the stabilization policies implemented after WWII. Apparently, these policies have had a lasting e¤ects on the behavior of business cycles and on economic growth, although the impact on the former was a bit larger.

1. Introduction

From the perspective of a representative consumer, who dislikes systematic risk, it makes sense for macroeconomic policy to try to reduce the variability of pervasive shocks a¤ecting consumption. The best known welfare-cost approach to this issue was put forth by Lucas (1987, 3), who calculates the amount of extra consumption a rational consumer would require in order to be indi¤erent between an in…nite sequence of consumption under uncertainty (aggregate consumption) and a consumption sequence with deterministic growth and no cyclical variability. Here, business-cycle shocks are the only source of variation for aggregation consumption. Thus, Lucas’measure is known as the welfare cost of business cycles. For 1983 …gures, using a reasonable parametric utility function (CES or Power utility function), and post-WWII data, the extra consumption is about $ 8.50 per person in the U.S., a surprisingly low amount. Several papers have been written just after Lucas …rst presented his results. For example, Im- rohoroglu (1989) and Atkeson and Phelan (1995) recalculated welfare costs using a model where the hypothesis of complete markets was relaxed. Van Wincoop (1994), Pemberton (1996), Dolmas (1998), and Tallarini (2000) have either changed preferences or relaxed expected utility maximiza- tion. In some of them, welfare costs of business cycles reached up to 25% of per-capita consumption. On that matter, Otrok (2001) notes that “it is trivial to make the welfare cost of business cycle as large as one wants by simply choosing an appropriate form for preferences.” Regarding the original setup, as in Zellner’s(1992) version of the KISS principle, Lucas Keeps It Sophisticatedly Simple: if only transitory shocks hit consumption, the best a macroeconomist can hope to achieve in terms of welfare improvement is to eliminate completely its cyclical variation, which is equivalent to eliminating all systematic risk. Of course, the implicit counter-factual exercise being performed is rather extreme. First, it dismisses any sources of uncertainty a¤ecting long-term growth. Indeed, Lucas recognizes that the setup could also include permanent shocks, which lead Obstfeld (1994) to compute welfare costs in this context; see also Dolmas, Tallarini, Issler, Franco, and Guillén (2008) and Reis (2009) on this issue. Second, by compensating the consumer to be indi¤erent between welfare computed using actual consumption and its deterministic version entails shutting out completely the uncertainty behind shocks to consumption in computing welfare costs, which is of limited practical importance as a counter-factual exercise. In a very interesting paper, Alvarez and Jermann (2004) generalized the setup in Lucas by proposing a di¤erent counter-factual exercise, where the representative consumer is o¤ered a convex

2 combination of consumption and its conditional mean, but not a deterministic sequence. Their setup now includes the total and the marginal welfare costs of business cycles. Total welfare costs are computed when, in the counter-factual exercise, all the weight goes to the conditional mean1. Marginal costs are obtained when we consider small changes in welfare in the neighborhood of observed consumption, which has a more practical appeal. More recently, the literature has focused on rare disasters –Barro (2009); on the e¤ects of model uncertainty on the welfare cost of business cycles –Barillas, Hansen, and Sargent (2009); on how the stochastic properties of aggregate consumption a¤ects welfare cost estimates – Issler, Franco, and Guillén, and Reis; on the distinction between individual and aggregate consumption risk in computing welfare costs –De Santis (2009); and on the di¤erence between welfare costs based on preference-parameter values that …t or not asset-pricing data –Melino (2010). In our view, despite the existence of a seemingly mature literature, there are still important issues to be discussed in it. Consider models where aggregate consumption is hit by permanent shocks (shocks a¤ecting economic growth) and transitory shocks (typical business-cycle shocks). These can arise in the real-business-cycles tradition, e.g., King, Plosser and Rebelo (1988), and King et al. (1991), or in new-keynesian tradition, e.g., Galí (1999). As we note in a previous paper (Issler, Franco, and Guillén), the welfare impacts of permanent and transitory shocks is completely di¤erent: for the former, its conditional variance increases without bound with time, whereas it is bounded for the latter. Hence, separating the e¤ects of these two type of shocks in a sensible way requires thinking deeper about the counter-factual exercise being performed. An easy solution is to lump all uncertainty together, computing the welfare costs of what we have labelled macroeconomic uncertainty. This approach is clearly limited in scope, given the very di¤erent roles that these two types of shocks play and their potentially di¤erent sources. Another important issue is the fact that all of the previous literature has computed welfare costs for the post-WWII period. Although this is interesting on its own right, it helps little in measuring the welfare bene…ts of counter-cyclical policies, for the simple reason that they were already in place during this period. Borrowing ideas from the treatment-e¤ect literature, post- WWII aggregate consumption re‡ected already the treatment from counter-cyclical policies, thus it cannot serve as a benchmark to compute the welfare bene…ts associated with them. A natural candidate to compute the latter is to use pre-WWII consumption data, which lead us to compute here “The Welfare Costs in the 20th Century.”Moreover, separating the samples in pre-WWII and post-WWII allows to measure by how much welfare costs have changed through time, something that could serve as a guide for current and future macroeconomic policy. Our paper has three original contributions. First, while the whole literature makes no e¤ort

1 To be equivalente to the exercise in Lucas, all the weight should go the unconditional mean instead. However, Alvarez and Jermann want to take into account the possibility that consumption is non-stationary. Thus, they focus on the conditional mean, which is still de…ned in this case.

3 to construct a setup that could separate the e¤ects of uncertainty stemming from business-cycle ‡uctuations and economic-growth variation, we explicitly make an e¤ort to do so. Here, permanent shocks to consumption arise from the unit-root component in its trend. There are empirical reasons for that, e.g., Hall (1978), Nelson and Plosser (1982), Engle and Granger (1987), King et al. (1991) and Issler and Vahid (2001). There are also theoretical reasons: in the consumption literature – e.g., Hall (1978) and Flavin (1983) – it is shown that consumption should follow a martingale; in the stochastic discount factor literature – e.g., Alvarez and Jermann (2005), and Hansen and Scheikman (2009) – it is shown that the limit stochastic discount factor must entail permanent shocks. Indeed, as stressed by Alvarez and Jermann, “for many cases where the pricing kernel is a function of consumption, innovations to consumption need to have permanent e¤ects.”Thus, we model the trend in consumption as martingale process to accommodate this need. The ‡uctuations about the trend (the cycle) are modelled as a stationary and ergodic zero-mean process. Trend and cyclical innovations are assumed to be independent sources of uncertainty. Independence allows the joint measurement of welfare costs of business cycles and of economic-growth variation. Second, we depart from Lucas in changing the sample from which to compute the moments of consumption: the whole of the literature chose to work with post-WWII data. However, for this period, actual consumption is already a result of counter-cyclical policies, and is potentially smoother than what it otherwise have been in their absence. It would be desirable to measure the welfare cost of business cycles observed in economies with no counter-cyclical policy. That is why we use pre-WWII data. Third, we take an econometric approach, and compute explicitly the asymptotic standard de- viation of welfare costs using the Delta Method. This allows us to compute con…dence bands for welfare costs. Indeed, we go back to the idea behind the original exercise done by Lucas, where he notes that: “It is worth re-emphasizing that these calculations rest on assumptions about preferences only, and not about any particular mechanism –equilibrium and disequilibrium –assumed to generate business cycles.” The paper is divided as follows. Section 2 provides a theoretical and statistical framework to evaluate the welfare costs of business cycles. Section 3 provides the estimates that are used in calculating them. Section 4 provides the calculations results, and Section 5 concludes. There is also an Appendix providing the econometric background necessary to implement the calculations carried out in the paper.

2. The Problem

Lucas (1987) proposed the following way to evaluate the welfare gains of cycle smoothing (or the welfare costs of business cycles). Suppose that consumption (ct) is log-Normally distributed about a deterministic trend:

4 t 1 2 ct = 0 (1 + 1) exp zt, (2.1) 2 z 2 where ln (zt) N 0; is the stationary and ergodic cyclical component of consumption. z Cycle-free consumption will be the sequence ct t1=0 , where t 1 2 f g t c =E(ct) = 0 (1 + 1) exp E(zt) = 0 (1 + 1) . Notice that c 1 is the resulting t 2 z f tgt=0 sequence when we replace the random variable ct with its unconditional mean. Hence, for any particular time period, ct represents a mean-preserving spread of ct.

An intuitive way of thinking about ct is realizing that:

t 1 2 t ct = lim ct = lim 0 (1 + 1) exp z zt = 0 (1 + 1) : 2 0 2 0 2 z ! z ! t Hence, ct is a degenerate random variable with all the mass of its distribution at 0 (1 + 1) , obviously risk free.

Of course, risk averse consumers prefer c 1 to ct 1 . Then, to evaluate the welfare costs f tgt=0 f gt=0 of business cycles, amounts to calculating , which solves the following equation2:

1 t 1 t E E0 u ((1 + ) ct) = u (ct) , (2.2) t=0 ! t=0 X X where Et ( ) = E( t) is the conditional expectation operator of a random variable, using t as j I I the information set, u ( ) is the utility function of the representative agent who discounts future utility at the rate . Then, the welfare cost is expressed as the compensation , that consumers would require at all dates and states of nature, which makes them indi¤erent between the uncertain stream ct 1 and the risk-free stream c 1 . f gt=0 f tgt=0 Notice that uncertainty here comes in the form of stochastic business cycles alone, since the trend in consumption is purely deterministic. One important limitation of this setup is that it prevents the existence of permanent shocks to consumption. Of course, at least since Nelson and Plosser (1982), macroeconomists have bene…tted from the dichotomy of having econometric models with permanent and transitory shocks, the …rst being associated with permanent factors in‡uencing economic growth - such as productivity, population, etc., and the second being associated with transient factors - such as monetary policy. Since Lucas modelled consumption trend as deterministic, eliminating all the cyclical variability in ln (ct) is equivalent to eliminating all its variability. Under di¤erence stationarity for (log) consumption, where econometric models now entail a permanent-transitory decomposition for shocks,

2 Notice that Lucas (1987) uses the unconditional mean operator instead of the conditional mean operator in (2.2). The same problem can be proposed using the conditional expectation instead. This is exactly how we proceed in this paper.

5 this equivalence is lost, since uncertainty comes both in the trend and the cyclical component of ln (ct). Moreover, E (ct) is not de…ned, since the stochastic component of ln (ct) is neither stationary nor ergodic. This led Obstfeld (1994) to use E0 ( ) in de…ning welfare costs:

1 t 1 t E0 u ((1 + ) ct) = u (E0 (ct)) : (2.3) t=0 t=0 X X Here, is the welfare cost associated with all the uncertainty in consumption, not just the uncertainty associated with the business-cycle component of consumption. Thus, it cannot be labelled the welfare cost of business cycles. Indeed, on an earlier paper (Issler, Mello-Franco-Neto, and Guillén (2008)), we have labelled it the total welfare cost of macroeconomic uncertainty as opposed to the total welfare cost of business cycles. An interesting generalization of the setup in Lucas ia due to Alvarez and Jermann (2004), who proposed o¤ering the consumer a convex combination of c 1 and ct 1 : (1 ) ct + c , where f tgt=0 f gt=0 t ct = E0 (ct). They make the welfare cost to be a function of the weight , ( ), which solves:

1 t 1 t E0 u ((1 + ( )) ct) = E0 u ((1 ) ct + ct) . (2.4) t=0 t=0 X X In their setup (0) = 0, and , as de…ned by Lucas, is obtained as = (1), replacing E0 ( ) by E ( ) in (2.4). They label (1) as the total cost of business cycles and de…ne the marginal cost of business cycles, obtained after di¤erentiating (2.4) with respect to as:

t E0 t1=0 u0 (ct) E0 (ct) 0 (0) = t 1. (2.5) E0 1 [ u0 (ct) ct] P t=0 From (2.2), (2.3), and (2.5), notice thatP the total and the marginal cost of business cycles can be computed if consumption is stationary and ergodic and also when it is not. The only di¤erence is whether we employ E( ) or Et ( ), respectively, in de…ning it. Despite that, the choice of how to model consumption is an important one for several reasons. As is well known, unless consumption has a unit root, we cannot consider the existence of shocks with a permanent e¤ect on it. As stressed by Alvarez and Jermann (2005), “for many cases where the pricing kernel is a function of consumption, innovations to consumption need to have permanent e¤ects.”A permanent-transitory decomposition of consumption shocks allows to explicitly isolate transient and permanent sources of welfare ‡uctuations, which could, in principle, be associated with the welfare costs of business cycles and the welfare costs of growth components. If one does not separate the welfare costs associated with permanent and transitory components, there is the risk of inconsistent estimation of business-cycle costs alone. As stressed in Issler and Vahid (2001), “theoretical models are rarely built in terms of permanent or transitory shocks. Rather, they are built in terms of real (e.g., productivity) or nominal (e.g.,

6 monetary) shocks.” Here, in the original spirit of Lucas, we will link transitory shocks to sources of business cycles. Permanent shocks will be linked to sources of economic growth. Moreover, we impose independence between them. To go one step further would be to link these shocks, respectively, to monetary policy and to productivity, something we refrain from doing here. Issler and Vahid point or that not all permanent shocks are productivity shocks, since there may be permanent demand shocks to taste, for example. Also, one could also think of transitory productivity shocks as well, challenging the link between “transitory”and “monetary.” To start the discussion of di¤erence-stationary consumption, we …rst assume that the utility function is in CES class, with risk aversion coe¢ cient :

1 c 1 u(c ) = t , (2.6) t 1 where u (ct) approaches ln (ct) as 1: ! As shown in Beveridge and Nelson (1981), every linear di¤erence-stationary process can be decomposed as the sum of a deterministic term, a random walk trend, and a stationary cycle (ARMA process). The analogue of (2.1) when consumption is di¤erence stationary is:

2 t t 1 !t ln (ct) = ln ( 0) + ln (1 + 1) t + "i + jt j (2.7) 2 i=1 j=0 X X t 2 t where ln 0 (1 + 1) exp !t =2 is the deterministic term, i=1 "i is the random walk compo- t 1 nent, j=0 jt j is the MA ( ) representation of the stationary part (cycle), which entails 0 = 1 P 2 1 and 1 < . The permanent shock "t and the transitory shock t are assumed to have a Pj=0 j 1 bi-variate Normal distribution as follows: P " 0 0 t i:i:d: ; 11 , (2.8) t ! N 0 ! 0 22 !! i.e., shocks are uncorrelated across time and are contemporaneously uncorrelated, which implies that t 1 2 2 they are also independent. Thus, !t = 11 t+22 j is the conditional variance of ln (ct), where j=0 it becomes clear that "t and t have two very di¤erentP roles in terms of uncertainty: the uncertainty of "t grows without bound with t, whereas that of t also increases with t but is bounded from 1 2 above by the unconditional variance 22 j . j=0 The fact that shocks to trends and cyclesP are independent allows to isolate factors behind growth of ln (ct) and of ‡uctuations about growth of ln (ct), i.e., business cycles. Here, economic growth is solely a function of "t 1 , whereas business cycles are solely a function of t 1 . A main f gt=1 f gt=1 objectives of this paper is to isolate the welfare costs of business cycles and the welfare costs of economic growth. As stressed by Issler, Mello-Franco-Neto, and Guillén (2008), a necessary condition

7 to examine the welfare cost of business cycles in a di¤erence-stationary world is independence of the shocks responsible for trend and cyclical movements in ln (ct). Otherwise, one is forced to examine the welfare cost of macroeconomic uncertainty altogether or to work with a measure of welfare cost of business cycles which includes some or all of the cost associated with economic growth factors. Some previous attempts to deal with this issue included only examining consumption ‡uctuations at business-cycle horizons; see Alvarez and Jermann (2004). In our view, this strategy is best viewed as an approximation in dealing with this issue, since some of the business-cycle variation in consumption is naturally due to permanent shocks. Indeed, one of the main features of the real-business-cycle literature was that permanent shocks could in principle generate business-cycle ‡uctuations; see, inter alia, King, Plosser and Rebelo (1987), King, Plosser, Stock and Watson (1991), and Issler and Vahid (2001). In the framework above, because of independence of shocks, it is natural to evaluate the welfare cost of business cycles using t, and to evaluate the welfare cost of economic growth using "t. To do so, consider the two processes below, where we start with (2.7) and shut out permanent and transitory shocks, respectively, as follows:

t 1 2 22 j j=0 2 P 3 2 t 1 T t 6 7 ct = 0 (1 + 1) exp6 7 jt j, and, (2.9) 4 5 j=0 X t P t ( 11 t) c = 0 (1 + 1) exp 2 "i. (2.10) t i=1 X T P From (2.7), we can think of ct and ct as limit cases:

T P lim ct = ct , and lim ct = ct : 11 0 22 0 ! ! We now propose measuring the welfare cost for the representative consumer of bearing the P uncertainty associated with t alone through the use of ct . Notice that the conditional means P f g P t of ct and ct are identical: E0 ct = E0 (ct) = 0 (1 + 1) . However, the uncertainty of the consumption stream c is larger than that of cP 1 . Thus, c is a mean-preserving spread t t1=1 t t=1 t P f g P of c . The risk averse consumer prefers the stream c 1 over ct 1 . Thus, we propose to t t t=1 f gt=1 measure the welfare cost associated with t alone using P , which solves: f g

1 t 1 t P 0 u ((1 + P ) ct) = 0 u c ; (2.11) E t=0 E t=0 t hX i hX i i.e., we can think of P as the welfare cost of bearing the risks associated with transitory shocks, i.e., associated with business cycles.

8 In order to implement the computation of P , we specialize the utility function to in the CES class as in (2.6). After straightforward, but tedious algebra, we get,

exp(22=2) 1; for = 1 6 P = 8 e ; <> exp(22=2) 1; for = 1 e :> t 1 2 where, for the sake of simplicity, we replace 22 j by its respective unconditional counterpart j=0

1 2 P (1 ) ( (1 )11=2) 22 = 22 j , and we assume that the convergence condition (1 + 1) exp < j=0 1 holds. P e Analogously, we can propose measuring the welfare cost for the representative consumer of T T bearing the uncertainty associated with "t alone through the use of ct . Again, E0 ct = E0 (ct) = t f g T 0 (1 + 1) , and ct is a mean-preserving spread of ct . Hence, we propose to measure the welfare cost associated with "t alone using T , which solves: f g

1 t 1 t T 0 u ((1 + T ) ct) = 0 u c ; (2.12) E t=0 E t=0 t hX i hX i i.e., we can think of T as the welfare cost of economic growth. Specializing the utility function to in the CES class as in (2.6), one can show that:

1 (1 ) ( (1 ) =2) (1 ) (1 (1+ 1) e 11 ) (1 ) 1; for = 1 (1 (1+ 1) ) 6 8 T = > ; > <> 11 2(1 ) exp 1; for = 1 > > :> (1 ) where we assume that the convergence condition (1 + 1) < 1, holds. Both P and T are what Alvarez and Jermann have labelled measures of total welfare costs.

Here, we also interested in measures of marginal welfare costs, i.e., P0 (0) and T0 (0). Starting from (2.4), and using (2.6), we can propose measures of marginal welfare costs of business cycles and

9 P T economic growth using ct and ct , respectively:

(2 1) 22 exp 2 1; for = 1 e 6 8 P0 (0) = , and (2.13) > > 22 < exp 2 1; for = 1 e > (1 )2 > 1 11 : 1 (1+ 1) e 2 ! 1; for = 1 8 2 1 11 6 1 (1+ 1) e 2 > T0 (0) = > ! ; (2.14) > <> 1 11 1; for = 1 > 1 exp( 2 ) > > where we assume that the usual:> convergence conditions apply in computing C0 (0) and T0 (0), and t 1 2 1 2 replace 22 j by its respective unconditional counterpart 22 = 22 j in computing C0 (0). j=0 j=0 As in the caseP of total welfare costs, we interpret (0) and (0) asP being the marginal welfare P0 e T0 costs of economic growth and of business cycles. Finally, we give some intuition behind the measures of welfare costs proposed above. One way we could think about (2.10) is:

P lim ct = ct = E [ct 0; "t t1=0] ; 22 0 j I f g ! P which shows that ct is the conditional expectation of ct when we have perfect foresight of the sequence "t 1 of permanent shocks. Thus, in computing the welfare costs of business cycles, we f gt=0 control for the existence of permanent shocks to consumption. This shows that the welfare-cost measures P and P0 (0) only take into account the uncertainty that goes beyond permanent shocks, T i.e., transitory shocks alone. Using (2.9), a similar reasoning applies to ct : T lim ct = ct = E [ct 0; t t1=0] : 11 0 j I f g !

3. Identi…cation and Estimation of Structural Parameters used in Computing T ,

P , T0 (0) and of P0 (0)

Next, we discuss the reduced form and the structural form used in estimating T , P , T0 (0) and of P0 (0). For the reduced form, we borrow heavily from the discussion in Issler, Mello-Franco-Neto and Guillén (2008). This is especially important regarding possible long-run constraints in the data. Our starting point is a vector autoregression (VAR), where possible cointegrating restrictions are used in estimation. We show how a simple identi…cation strategy can be used in this setup, although it does impose the restriction that E ("tt) = 0. For that reason, we also discuss structural time-series models based on Harvey (1985b) and Koopman et al. (1995)

10 3.1. Reduced Form and Long-Run Constraints

A full discussion of the econometric models employed here can be found in Beveridge and Nelson (1981), Stock and Watson (1988), Engle and Granger (1987), Campbell (1987), Campbell and

Deaton (1989), and Proietti (1997). Denote by yt = (ln (ct) ; ln (It))0 a 2 1 vector containing respectively the logarithms of consumption and disposable income per-capita. We assume that both series contain a unit-root and are possibly cointegrated as in [ 1; 1]0 yt because of the Permanent- Income Hypothesis (Campbell(1987)). A vector error-correction model (VECM(p 1)) is:

yt = 1 yt 1 + ::: + p 1 yt p+1 + [ 1; 1] yt p + t: (3.1) Here, long-run constraints in the VAR are imposed through the error-correction mechanism

[ 1; 1] yt p. As discussed in Vahid and Engle (1993), short-run restrictions in the form of common cycles can be imposed in (3.1). Let [ 1; 1]0 = and consider the following restrictions on the parameters of (3.1):

0 i = 0, for all i = 1; 2; p 1, and 0 = 0: Then, we can represent (3.1) as having common-cyclical-feature restrictions in a 2 1 system: e e

yt 1 . 0 . 1 0 0 0 2 . 3 yt = + t; (3.2) " 0 1 # " A1 Ap 1 # y 6 t p+1 7 e 6 7 6 0 yt p 7 6 7 4 5 1 0 where A1; ;Ap 1; represent partitions of 1, ::: p 1, , respectively. Notice that " 0 1 # is non-singular, which allows to recover the parameters in (3.1) from the ones in (3.2) as wee pre- 1 1 0 multiply the latter by . Indeed, (3.2) is just a more parsimonious representation than " 0 1 # (3.1). e

Proietti(1997) shows how to extract trends and cycles from the elements in yt using a state-space representation. If, for example, (3.1) is well described by a VECM(1), its state-space form is:

yt+1 = Zft+1 (3.3)

ft+1 = T ft + Z0t+1, where,

yt+1 1 0 ft+1 = 2 yt 3 ;T = 2 I2 0 0 3 ;Z = I2 0 0 ; 0yt 1 0 0 1 h i 6 7 6 7 4 5 4 5

11 and is the cointegrating vector. Labelling the random-walk trend and the cyclical component of yt respectively by t and t, the Beveridge and Nelson(1981) trends and cycles are:

l 1 t = lim Et [yt+i] = Z [I T ] T ft, and, l !1 i=1 X t = yt t: Issler, Franco and Guillén (2008) show how we can identify trend and cyclical innovations in this reduced-form setup: recall that the variance of a random variable is invariant regarding the addition of a constant. Apart from an irrelevant constant, the trend innovation in consumption "t is simply [1; 0] t, because the trend is a random walk. Its variance 11 equals V AR ([1; 0] t). 1 2 2 1 Notice that: ln (ct) Et 1 (ln (ct)) = [1; 0] t = "t + t; which identi…es t up to an irrelevant constant using [1; 0] (t t) = t. Thus, we can compute 22. The main problem with this approach is that it does not impose the constraint that permanent and transitory shocks to ln (ct) are orthogonal, i.e., that E ("tt) = 0, which, under Normality would imply independence of these shocks. For that reason, we now turn into the discussion of structural time-series models, where possible long- and short-run restrictions in (3.2) are still kept in a di¤erent setup.

3.2. Structural Time-Series Models with Long-Run Constraints

In this section we present a brief summary of the structural time-series model used here, following Harvey (1985b) and Koopman et al. (1995). We start the discussion using a univariate framework. There, the main objective is to decompose a single integrated series (I (1)) in a trend and a cycle, treating both as latent variables to be estimated by maximum likelihood. For a given economic series xt, we decompose it as two di¤erent components:

xt = t + 't where t is the I (1) trend, 't is the cycle. Shocks to each of these two components are orthogonal to each other. Under Normality, this implies that they are independent. The trend evolves as:

t = t 1 + + t; (3.4) 2 where t has variance given by , whereas the cyclical component evolves as a bi-variate V AR(1):

't cos sin 't 1 !t = + (3.5) " 't # " sin cos #" 't 1 # " !t # 12 where the component 't shows up by construction; see Harrison and Akran (1983). Both !t and !t 2 2 are orthogonal white noise errors with variances given by ! and ! , respectively. Harvey (1985b) 2 2 argues that very little is lost in terms in terms of …t if we impose the restriction that ! = ! , representing an advantage in terms of parsimony. Finally, some restrictions on parameter values should be observed:

0 and 0 1; where is the frequency of the cycle and is the discount factor for the amplitude. The last restriction makes the cyclical component stationary. One can also show that the cyclical component obeys:

(1 cos L) !t + ( sin L) ! ' = t t 1 2 cos L + 2L2 2 2 where L is the lag operator. Under ! = ! , we can put the last equation in an ARMA format as:

2 2 L 2 cos L + 1 't = (1 + L) !t where it becomes clear that 't follows an ARMA (2; 1), with = (sin cos ). This is a restriction into the ARMA class of models, since not every cycle of an economy series will be approximately well modelled as an ARMA (2; 1). Following the notation for the univariate class of models, in a multivariate setting, we can represent yt = (ln (ct) ; ln (It))0 as having a common trend and a common cycle, respectively, as:

ln (ct) 1 1 = t + 't; (3.6) " ln (It) # " 1 # " # where the I (1) trend component t follows (3.4) and the stationary cyclical component 't follows

(3.5). Here, the bi-variate system in yt is modelled with just a single stochastic trend and a single cycle, respectively. The trend a¤ects identically the two series in yt, whereas the cycle a¤ect them di¤erently. The vector [ 1; 1]0 removes the common trend and that the vector [ ; 1]0removes the common cycle, where there is an additional restriction that = 1. The structural time-series model 6 in (3.6) is analogous to its reduced-form counterpart (3.2), in which it imposes identical long- and short-run restrictions. Despite that, they di¤er in which (3.6) imposes independence for the shocks to t and 't, whereas (3.2) does not. As stressed by Issler and Vahid (2001), there are several theoretical reasons why consumption and income should cointegrate (Campbell (1987)) and have common cycles (King, Plosser, and Rebelo (1988) and King et al. (1991)). Despite that, one may be more willing to impose long-run restrictions than short-run restrictions, meaning that the the two variables in yt have two distinct cycles, but still a common trend as in (3.6). This can be easily accomodated by the structure in

13 1 (3.6), where 't is replaced by the 2 1 vector 't, " #

't cos sin 't 1 !t = + ; (3.7) " 't # " sin cos #" 't 1 # " !t # where now 't, ' , !t and ! are 2 1 vectors, is a 1 2 vector, and we impose the restriction t t that E (!t!t0) = E (!t!t0). We now discuss the identi…cation of the variances 11 and 22 using t and 't. The parameter 1 2 11 can be identi…ed using V AR (t), whereas 22 = 22 j can be identi…ed by computing j=0e V AR ('t). Finally, running a regression of t on a constantP identi…es ln (1 + 1), and therefore e 1 using Slutsky’s Theorem. Even if one uses the model with a common trend, but idiossincratic cycles as in (3.7), identi…cation of 22 is straightforward: simply compute V AR ([1; 0] 't).

e 3.3. Computing Con…dence Intervals for Welfare Costs T , P , T0 (0) and P0 (0)

Consider the set of parameters = ( ; ; 11; 22; 1)0. All welfare costs T , P , T0 (0) and P0 (0) can be expressed as speci…c non-linear functions of . Here, we follow the literature in treating

and as known, whereas the remaining parameterse in , stacked in = (11; 22; 1)0, are estimated consistently employing a su¢ ciently large sample of t = 1; 2; ;T observations. In this e setup, the uncertainty in estimating T , P , T0 (0) and P0 (0) will be a function of the uncertainty in estimating the components of alone, and the Delta Method can be used to compute asymptotic standard errors (and con…dence intervals) for T , P , T0 (0) and P0 (0). Suppose that a generic welfare measure relates to as = G (), where G () is a continuous and continuously di¤erentiable function. Start by assuming that a Central Limit Theorem holds for d d @G() @G() , i.e., that pT (0;V ). Then, pT 0; V . In practice, we !N !N @ @0 have to replace V with a consistent estimator, and evaluate @G() and @G() at = . Asymptotic b b c @ @0 con…dence intervals can be constructed based on on the asymptotic variance of . The application of the Delta Method to this end was proposed by Duarte, Issler, and Salvato (2004).b b 4. Empirical Results

Data for consumption of non-durables and services were obtained from DRI from 1929 through 2000. Data for consumption of perishables and services from 1901 to 1929 were obtained from Kuznets (1961) in real terms, and then chained with DRI data, resulting in a long-span series for consumption of non-durables and services from 1901-2000. Data for real GNP were also extracted from DRI from 1929 through 2000 and from Kuznets from 1901 through 1929. Data on population were extracted from Kuznets and DRI, and then chained. Figure 1 presents the data on consumption and income

14 per-capita for the whole period 1901-2000. The peculiar features are …rst the magnitude of the great depression in both consumption and income behavior, and second the fact that pre-WWII data present much more volatility than post-WWII data. We …tted a bi-variate vector autoregression for the logs of consumption and income. Lag length selection indicated that a VAR(2) with an unrestricted constant term was an appropriate description of the dynamic system. This was true not only in terms of minimizing information criteria but also because this speci…cation did not fail diagnostic testing. Table 1 presents results of the cointegration test using Johansen’s (1988, 1991) technique. The Trace Statistics for the null of no cointegration and of at most one cointegrating vector were respectively 16.43 and 0.18. At 5% signi…cance, we concluded that there is one cointegrating vector, given by ( 1:000; 1:005)0. Conditioning on the existence of one cointegrating vector, we tested the restriction that it was equal to ( 1; 1)0. Testing this hypothesis, using the likelihood-ratio test in Johansen (1991), yielded a p-value of 0.831, not rejecting the null at usual levels of signi…cance. An interesting by-product of cointegration analysis is testing the signi…cance of the error-correction term in each regression of the system. The t-statistics associated with this test are -0.07 and 3.16, for the regression involving consumption and income respectively. Hence, the error-correction term a¤ects income but not consumption, and the latter is long-run weakly exogenous in the sense of Engle, Hendry, and Richard (1983) and Johansen (1992). Given the cointegration vector found in the empirical analysis, we implemented the multivariate structural time-series model in the form suggested by Harvey (1985b) and Koopman et al. (1995). Figure 1 shows the result of this exercise. The consumption series and the trend are very close throughout the whole period, re‡ecting the fact that agents do update their beliefs about future income, and that the permanent-income theory is probably a reasonable approximation to consumption behavior; see Cochrane (1994) inter alia. It is obvious that the cyclical component of consumption varies much more in the pre-WWII era than after that period. Next, we present the results of the structural time-series model discussed in Section 3.2, where trends and cycles are estimated imposing that their innovations are independent. Table 2 displays the description of the data in terms of the parameters estimates associated with the log of consumption (2.7) under alternative periods in it. Estimates are obtained for four distinct periods: pre-Great Depression data –1901-1929, pre-WWII data –1901-1941, post-WWII data –1947-2000, and the whole sample –1901-2000. It is obvious that uncertainty in the pre-WWII period is much larger than in the post-WWII period. The variance of the permanent component is at least 70% larger in the pre-WWII era than in the pre-Great Depression era or in the post-WWII era. When we considered the whole sample, the variance of the trend innovations is larger because the adjustment of the trend is almost perfect. The variance of the cyclical innovations is more than one order of magnitude larger in the pre-WWII era than all other periods. These results, of course, will impact the …nal computations of welfare measures.

15 The estimates of the total welfare cost of business cycles in the 20th Century are presented in Tables 3 and 4. We start the discussion using the post-WWII period, keeping at hand the Lucas benchmark values for comparison. First, it is comforting to note that in almost all cases considered, the welfare cost of business cycles associated with transitory shocks is smaller than the welfare cost associated with permanent shocks. Second, the pre-WWII period (1901-1941) produces larger welfare costs than all other periods considered. We now turn our attention to the analysis of the pre-WWII period (1901-1941). For a reasonable degree of risk aversion ( = 5) we get welfare costs of about 1.0% of consumption if we considered only permanent shocks and 0.1% of consumption if we considered only transitory shocks, which translates into US$ 200.00 a year and US$ 20.00 in current value. Second, using the structural trend-cycle decomposition for post-WWII data we get welfare costs between …ve and ten times smaller than the pre-WWII period. Results for the pre-Great Depression era are between two to three times the welfare cost for post-WWII era. Results for the whole period 1901-2000 are indeed a combination of those of pre- and post- WWII eras. For reasonable preference parameter and discount values ( = 0:971; = 5) we get a compensation of about 1.11% and 0.01% of consumption. Table 5 and 6 presents estimates of the marginal welfare cost of macroeconomic uncertainty. For reasonable preference parameter and discount values ( = 0:971; = 5) they are about 2.78% and 1.24% of per-capita consumption considering only permanent shocks or transitory shocks, respectively, for the pre-WWII era. For the post-WWII era, the marginal welfare cost of macroeconomic uncertainty are 0.47% of per-capita consumption if we considered only permanent shocks and 0.12% of per-capita consumption if we considered only transitory shocks. This result can be compared to those found by Alvarez and Jermann(2004). For the 1954-97 period, they …nd about 0.20% when an 8-year low-pass …lter is used to extract cycles, about 0.30% when a one-sided …lter is used, and about 0.77% and 1.40% when a geometric and a linear …lter are used respectively. Lastly, results for the whole period 1901-2000 are indeed a combination of those of pre- and post-WWII eras. For reasonable preference parameter and discount values ( = 0:971; = 5) we get a compensation of about 2,86% if we considered only permanent shocks. If we take into account only transitory shocks we get 0.07% of per-capita consumption. From the discussion above we can conclude the following. First, the current marginal welfare cost of business cycles is small. Hence, it makes little sense to deepen current counter-cyclical policies. Second, from the point of view of a pre-WWII consumer, the marginal welfare costs of business cycles were fairly large. Indeed, for reasonable parameter values ( = 0:971; = 5) they were 1.24% of consumption in all dates and states of nature. Therefore, from her (his) point of view, it made sense to have had counter-cyclical policies implemented. Last, a comparison between the welfare costs of business cycles in the pre-WWII and post-WWII period can give some idea of the e¤ectiveness of counter-cyclical policies implemented in the latter period. Since the total

16 welfare cost of business cycles decreased from about 0.12% to about 0.01% of consumption, if this reduction can be credited to the existence of counter-cyclical policies, then the latter have been proven e¤ective in increasing the welfare of rational consumers. A similar result is observed for the reduction of total welfare costs: from 1.32% to 0.23% –a cost reduction of almost one percentage point of consumption. It is hard to …nd any type of implemented economic policy in the name of which it could be claimed such an impressive impact on welfare.

5. Conclusion

Using only standard assumptions on preferences and an econometric approach for modelling consumption we separate the e¤ects of uncertainty stemming from business-cycle ‡uctuations and economic growth variation. We model the trend in consumption as a random-walk process, while the ‡uctuations about the trend (the cycle) are a stationary and ergodic zero-mean process. Trend and cyclical innovations are assumed to be independent sources of uncertainty. This hypothesis allows the measurement of welfare costs of business cycles and also of economic growth variation. The whole of the literature chose to work with post-WWII data. However, for this period, actual consumption is already a result of counter-cyclical policies, and is potentially smoother than what it otherwise would have been in their absence. We use four distinct periods to avoid this limitation: pre-Great Depression data –1901-1929, pre-WWII data –1901-1941, post-WWII data –1947-2000, and the whole sample –1901-2000. We found that uncertainty in the pre-WWII period is much larger than in the post-WWII period. The variance of the permanent component is at least 70% larger in the pre-WWII era than in the pre-Great Depression era or in the post-WWII era. For the whole sample, the variance of the trend innovations is larger because the adjustment of the trend is almost perfect. The variance of the cyclical innovations is more than one order of magnitude larger in the pre-WWII era than all other periods. The welfare cost of business cycles associated with transitory shocks is smaller than the welfare costs associated with permanent shocks in almost all cases considered. The pre-WWII period (1901-1941) produces larger welfare costs than all other periods considered. For a reasonable degree of risk aversion ( = 5) we get welfare costs of about 1.0% of consumption if we considered only permanent shocks and 0.1% of consumption if we considered only transitory shocks, which translates into US$ 200.00 and US$ 20.00 a year in current value. Results for the whole period 1901-2000 are a combination of those of pre- and post-WWII eras. For reasonable preference parameter and discount values ( = 0:971; = 5) we get a compensation of about 1.11% and 0.01% of consumption, which translates into US$ 220.00 and US$ 2.00 a year in current value. For reasonable preference parameter and discount values ( = 0:971; = 5) the marginal welfare costs of macroeconomic uncertainty are about 2.78% and 1.24% of per-capita consumption consid-

17 ering only permanent shocks or transitory shocks, respectively, for the pre-WWII era. For the post-WWII era, the marginal welfare costs of macroeconomic uncertainty are 0.47% of per-capita consumption if we considered only permanent shocks and 0.12% of per-capita consumption if we considered only transitory shocks. We can conclude the following. First, the current marginal welfare cost of business cycles is small. Hence, it makes little sense to deepen current counter-cyclical policies. Second, from the point of view of a pre-WWII consumer, the marginal welfare costs of business cycles were fairly large. Therefore, from her (his) point of view, it made sense to have had counter-cyclical policies implemented. Last, a comparison between the welfare costs of business cycles in the pre-WWII and post-WWII period can give some idea of the e¤ectiveness of counter-cyclical policies implemented in the latter period. Since the total welfare cost of business cycles decreased from about 0.12% to about 0.01% of consumption, if this reduction can be credited to the existence of counter-cyclical policies, then the latter have been proven e¤ective in increasing the welfare of rational consumers. A similar result is observed for the reduction of total welfare cost: from 1.32% to 0.23% – a cost reduction of almost one percentage point of consumption. It is hard to …nd any type of implemented economic policy in the name of which it could be claimed such an impressive impact on welfare.

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22 Table 1: Cointegration test –Johansen (1988, 1991) Technique

Cointegrating Trace 5 % max 5 %

Vectors under H0 Eigenvalues Stat. Crit. Value Stat. Crit. Value None 0.150 16.44 15.41 16.26 14.07 At most 1 0.0018 0.18 3.76 0.18 3.76 Estimate of the cointegrating vector is: ( 1; 1:005) :

H0 : = ( 1; 1) ; conditional on r = 1, p-value = 0:831: 0

23 Figure 1 • Consumption (logs): trend and cycle components

1 0 .0 1 0 .0 LCDRIK1 LCDRIK1 T_LCDRIK1_1901_2000 T_LCDRIK1_1901_1941 C_LCDRIK1_1901_2000 9 .5 C_LCDRIK1_1901_1941 9 .5

9 .0 9 .0 .10 8 .5 8 .5 .10 .05 .05 8 .0 8 .0 .00 .00 7 .5 7 .5 •.05 •.05

•.10 •.10

•.15 •.15 10 20 30 40 50 60 70 80 90 00 05 10 15 20 25 30 35 40

1 0 .0 1 0 .0 LCDRIK1 LCDRIK1 T_LCDRIK1_1947_2000 T_LCDRIK1_1901_1929 C_LCDRIK1_1947_2000 C_LCDRIK1_1901_1929 9 .5 .10 9 .5

9 .0

.05 9 .0 .10 8 .5

.05 .00 8 .5 8 .0

.00 •.05 8 .0 7 .5 •.05

•.10 7 .5 •.10

•.15 •.15 50 55 60 65 70 75 80 85 90 95 00 1905 1910 1915 1920 1925

24 Table 2: Consumption –Parameter Estimates in Equation (2.7)

1901-2000 1901-1941 1947-2000 1901-1929

ln (1\ + 1) 0.0207 0.0149 0.0234 0.0181

11 0.00062 0.00047 0.00012 0.00027

22 0.00003 0.00049 0.00005 0.00037 c c

25 Table 3: The Total Welfare Costs of Economic Growth: Consumption Compensation (T %) for Di¤erent ( ; ) Values a) Lucas (1987) Benchmark Values Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950; 0:971; 0:985 0:008 0:042 0:08 0:17 (b) 1901-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:59 1:11 1:25 1:35 = 0:971 1:04 1:36 1:40 1:45 = 0:985 2:05 1:58 2:77 1:52 (c) 1901-1941 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:45 1:04 1:25 1:44 = 0:971 0:79 1:32 1:45 1:58 = 0:985 1:56 1:60 1:62 1:69 (d) 1947-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:11 0:19 0:20 0:19 = 0:971 0:20 0:23 0:23 0:21 = 0:985 0:39 0:27 0:24 0:21 (e) 1901-1929 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:26 0:53 0:60 0:62 = 0:971 0:46 0:66 0:68 0:67 = 0:985 0:90 0:77 0:74 0:71

26 Table 4: The Total Welfare Costs of Business Cycles: Consumption Compensation (P %) for Di¤erent ( ; ) Values a) Lucas (1987) Benchmark Values Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950; 0:971; 0:985 0:008 0:042 0:08 0:17 (b) 1901-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:00 0:01 0:01 0:03 = 0:971 0:00 0:01 0:01 0:03 = 0:985 0:00 0:01 0:01 0:03 (c) 1901-1941 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:02 0:12 0:25 0:49 = 0:971 0:02 0:12 0:25 0:49 = 0:985 0:02 0:12 0:25 0:49 (d) 1947-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:00 0:01 0:23 0:24 = 0:971 0:00 0:01 0:25 0:26 = 0:985 0:00 0:01 0:27 0:26 (e) 1901-1929 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:02 0:09 0:19 0:37 = 0:971 0:02 0:09 0:19 0:37 = 0:985 0:02 0:09 0:19 0:37

27 Table 5: The Marginal Welfare Costs of Economic Growth: Consumption Compensation (T0 (0)%) for Di¤erent ( ; ) Values a) Lucas (1987) Benchmark Values Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950; 0:971; 0:985 0:008 0:042 0:08 0:17 (b) 1901-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 1:19 2:33 2:75 3:35 = 0:971 2:11 2:86 3:12 3:65 = 0:985 4:22 3:35 3:41 3:87 (c) 1901-1941 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:90 2:16 2:74 3:58 = 0:971 1:61 2:78 3:22 4:00 = 0:985 3:20 3:39 3:62 4:32 (d) 1947-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:23 0:39 0:42 0:40 = 0:971 0:40 0:47 0:46 0:43 = 0:985 0:79 0:54 0:49 0:45 (e) 1901-1929 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:52 1:08 1:25 1:38 = 0:971 0:93 1:34 1:42 1:49 = 0:985 1:83 1:59 1:56 1:57

28 Table 6: The Marginal Welfare Costs of Business Cycles: Consumption Compensation (P0 (0)%) for Di¤erent ( ; ) Values a) Lucas (1987) Benchmark Values Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950; 0:971; 0:985 0:008 0:042 0:08 0:17 (b) 1901-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:00 0:07 0:28 1:12 = 0:971 0:00 0:07 0:28 1:12 = 0:985 0:00 0:07 0:28 1:12 (c) 1901-1941 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:05 0:24 5:06 21:82 = 0:971 0:05 0:24 5:06 21:82 = 0:985 0:05 0:24 5:06 21:82 (d) 1947-2000 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:00 0:12 0:50 2:00 = 0:971 0:00 0:12 0:50 2:00 = 0:985 0:00 0:12 0:50 2:00 (e) 1901-1929 Equivalent in a Yearly Basis = 1 = 5 = 10 = 20 = 0:950 0:04 0:93 3:79 16:03 = 0:971 0:04 0:93 3:79 16:03 = 0:985 0:04 0:93 3:79 16:03