MXVlER Journal of Monetary tconomics 35 (1995) 215 242

JOURNALOF Monetary

The macroeconomics of self-fulfilling prophecies A review essay

(Recewed October 1994: final verbion received November 19W)

Roger Farmer, The Macror~ot1otttic.s of Self-Fuljilliny Prophecies (MIT Press. Cambridge, MA).

1. Introduction

A key cor:,nonent of Keynes’ L,jneral Theory. as reflected in his reference to the ‘animal spirits’ of businessmen. is that forecasts of economic activit) can change independently of movements ,n economic fundamentals and that such a change is an important cause of business cycles. Yet, despite :hc critical role that Keynes’ accorded agents’ beliefs in determining economic behavior, this factor has not been widely incorporated into modern macroeconomic analysis. With the publication of The XII;, :‘wt*onontic~s ofS4j~Fu!filliny Prophecies (MIT Press, Cambridge, MA) Roger Earmer of L’CLA attempts to fill this void. This book is promoted as a graduate text: however, it is closer to a treatise wthich advocates the hypothesis that the belief structure of agents matters for equilib- rium behavior and that. within certain environments. this structure can be exogenously specified like preferences ar?d technology. While few economists would a.pue with the primacy of expectations. the latter idea will no doubt be troubling to those economists who view the specification of tastes and tech- nology as the singular bedrock of cconcnic models. Sirze. within macrocco- norniTs. it is real business cycle practitioners who are most aligned with this view. Farmer directly addresses the concerns of these economists by choosing to

I am indebteu to Timothy Cogley. Thomas CooIcy. Kevin Hoover. and Steven ShelTrin for insightful comments and suggestions on an earlier draft.

0304-3932;95.$09.50 t lY95 Elsevier Science B.V. All rights reserved SSDl 030439329401 I86 E

play their game he sets up B standard stochastic growth model. parameterizcs it ihrough calibration, and then analyzes the cyclical properties of the economy through a simulation exercise. The result of these efforts is a provocative piece of work that deserves the serious attention of macroeconomists; morcovcr. Farmer is in be commended for engaging in analysis that promotes dialogue between methodologically disparate camps. While the book contains chapters which discuss traditional topics such as general equilibrium theory and overlapping generations models. it is the inclusion of models with self-fulfilling behavior **hich is its distinguishing feature. Hence, this concept will be the focus of my review.

2. The modeling of beliefs

The idea of self-fulfilling expectations influencing equilibrium behavior is usually associated with asset price bubbles and sunspot equilibrium. Farmer is interested in studying stationary* stochastic rational expectations equilibrium so that his analysis is most ciosely related to sunspot models. It is distinguished, however, from typical sunspot models by the use of a representative agent stochastic growth model rather than the usual overlapping generations setup. His motivation for doing this is clear:

Perhaps the most important reason for the profession’s neglect of station- ary sunspot equilibria is that until recently there have been very few models of the economy in which one can take the idea -eriously. Most work in the area has been conducted in the context oi P two-period overlapping generations model in which the period of ;he model is difficult to reconcile with the observed transactions frequency. (p. 187)

The growth model that Farmer develops is closely related to that described in Woodford (1991) who also uses an infinitely-lived, representative agent model with imperfect competition to study the effects of self-fulfilling expectations. As noted above, though, Farmer’s analysis goes one step further than Woodford’s by conducting a calibration exercise and comparing the results to a typical RBC model.’

In order for self-fulfilling beliefs, or extrinsic uncertainty, to influence equilib- rium in the growth model studied by Farmer, it is necessary that the equilibrium dynamics implied by the model be indeterminate; that is, there exists a unique steady-state equilibrium but there exist multiple paths to the steady-state

’ Rotemberg and Woodford (1994) also calibrate a real busmess cycle model with imperfect competition.

consistent with equilibrium. In contrast. the typical real business cycle (RBC) model with intrinsic uncertainty. i.e., the technology shock, is characterized by a unique saddle path equilibrium for a given realization of the technologv shock. Therefore. in the RBC model. the exogenous distribution for the technology shock induces a probability distribution for the equilibrium characteristics of the endogenous variables with consistency imposed by the assumption of rational expectations. In the indeterminate setting, the technology shock can be eliminated entirely from the model; uncertainty is then reintroduced by assum- ing that agents have an exogenous belief structure I-epresented by a distribution over the possible multiple equilibrium paths. This distribution, as in the RBC model, will induce a stationary rational e; pectations equilibrium charac!erized by a stationary probability distribution for the endogenous variables. Two critical issues are raised: (1) What is the justification for the assumed distribu- tion characterizing agent’s beliefs? (2) What is the cause of indeterminancy of equilibrium in the g iswth model? I defer discussion of the latter issue until the following section and now turn to the modeling of beliefs.

Farmer sees no problem in treating agents’ beliefs as an unobservable, much like preferences, and using the implied equilibrium behavior to determine whether the assumed distribution for beliefs is reasonable. This thinking is reflected in the following passages:

To understand how agents behave in a world of indeterminate equilibria we must pin down their beliefs. If we are prepared to grant to beliefs the same methodological status that one usually reserves for preferences and endow- ments, there is a simple solution to the problem of indeterminacy. Beliefs pick the equilibrium and each possible belief is associated with a different possible realization of the observable variables of the model. (p. 59)

The rational expectations assumption. in this class of models, is a consist- ency principle that restricts the class of belief functions that are admissible to the modeler, and it plays the same role in models with multiple equilibria that the assumption of transittvity plays in the theory of rational choice. Since every belief function has a unique implication for the time series behavior of the data. this assumption does not pose any new philosophical problems. (p. 183)

This stance raises some interesting but knotty methodological issues. For instance, it requires a tremendous degree of coordination between agents in that, somehow, all are using the same belief function in forecasting future economic activity.’ Of course, this uniformity of expectations is an assumption common to theories such as RBC models which are driven by fundamentals; but at least in

*This problem of coordination is also raised by Silvestre (lW4)

thcsc models there is an cxogcnously specified distribution for the fundamentals which provides something of an anchor for agents’ forecasts. In a model driven entirely by extrinsic uncertainty. it is not clear how agents would come to agree on a single forecasting rule.

Provided the assumption of coordination doe5 not make one too uncomfort- able, the interaction between beliefs and equilibrium behavior alluded to in the quotations above implies some interesting possibilities. In particular, it suggests that it should be possible to use actual data to measure the belief shock once the belief function of agents has been specified. This use of theory to identify the exogenous shock process is similar to that in RBC models in which the specification of technology permits the measurement of the technology shock as the Solow residual. While use of the Solow residual as a proxy for tne technol- ogy shock has been criticized by several economists, e.g., Hall (1990) and Hartley (1994), it does permit a comparison between the model’s predictions and actual time series. In the next section. I will attempt to exploit these observable implications in order to assess the merits of the model with self-fulfilling beliefs relative to that of a typical RBC model.

3. Increasing returns and indeterminacy

I now turn to the other critical feature of the class of models Farmer studies, namely, the indc;crminacy ot equilibrium. Fzrmer studies two models, a non- monetary growth model and a cash-in-advance monetary economy. I will focus en!irely on the growth model for two reasons. First, because the model can be calibrated to US data, its equilibrium characteristics can be directly compared to those of a typical RBC model. Second, my primary criticism of Farmer’s analysis, which centers on the use of vector autoregressions (VARs) to study the implications of the artificial economies, applies equally to the monetary model.

3.1. The ndd

Farmer’s analysis draws heavily from his article co-authored with J.T. Guo (1994). In the book, he presents this model after discussing linear rational expectations models, general equilibrium theory, overlapping generations models, and an approximate solution method for solving nonlinear models. Consequently, a student by this time is fairly well prepared for the ideas discussed in this key chapter ;Chapter 7).

Farmer considers a generalized real business cycle model that does not impose constant returns to scale. Specifically, it is assumed that the aggregate production function can be expressed as

y, = z,k,“h;, (1)

where r denotes output. k is capital. and /I is labor. The technology shock, 2,. is assumed to follow a first-order autoregressive process of 2, + , = z/‘c, + , . where I: is an i.i.d. log normally distributed random variable. Farmer presents two scenarios which are consistent with the aggregate produc!ion function exhibt- ting increasing returns to scale. The first is based on a nonmarketable positive externality of production similar to that found in Romer (1986). Consequently, individual firms face a constant returns to scale production function (i.e., firms make zero profits) while in aggregate the economy exhibits increasing returns to sca!c The second scenario is one in which firms have monopoly power due to increasing returns in producing an intermediate good which is then sold in a competitive final goods sector. This final goods sector produces a single good through the aggregation of intermediate products so that the concepts of GNP. aggregate consumption and investment are well-defined. Under the assumption that the nonconvexity in the technology of producing intermediate goods is offset by the downward-sloping demand curve faced by firms, an interior solution will exist. The critical aspect of both of these explanations is that factor shares. while constant in the economy, are not equal to the exponents 1’ and p. I will further discuss these ideas when presenting Farmer’s calibration exercise.

Households own firms and both factors of production and use the proceeds of the sale of these inputs along with any profits from firms to finance consumpti nr and investment. These purchases are made in order to maximize discounted expected utility. That is, agents face the following intertemporal problem:

i fi’(lnc, + (T - 11,)) 1=0 1

subject to

c, + i, = r,k, t ,.,A, + rr,, with k,+, = k,(l - (5) + i,.

Here E denotes the expec::jtions operator, T denotes the total time endowment, R is used to denote profits while depreciation rate of capital is given by 6. Note Farmer assumes that utility is hnear in leisure which is justified by appealing to the Hansen (1985) indivisible labor model.”

Provided that the aggregate production function given in Eq. (1) is consistent with a competitive equilibrium in the factor markets (which will always be the case under constant returns to scale as well as the two scenarios given above if increasing returns are present). the agents’ necessary conditions from this opti- mization problem will characterize a competitive equilibrium. Two additional

“Typically, a parameter is included in the utility function which indicates the relative weight that leisure has in generating utility. I follow Farmer in stipulating that (the lop of) consumption and leisure have equal weight in the utility function. By the choice of T, the fraction of time .:pent in labI activities in steady-state will match that seen in the data.

equations which mus! hold in a compe ittvc equilibrium are the laws of motion for the capital stock and the tcchnolog:/ shock with the former evaluated at the market clearing quantities. A stationary rational expectations equilibrium in this setting is defined by a stationary ‘unction for consumption which has as arguments the state variables k, and 2,. That is c, = ~(k,. 2,). Given this function. the equilibrium behavior of investment is determined by the law of motion for ihe capital stock, equilibrium hours is determined by the intratemporal effici- ency condition, and equilibrium outptn is determined through the production function.

Since, in general, no closed rorm soluton for c(k,. 2,) exists, numerical approx- it..ation methods must be employed in crder to compute the equilibrium. In the case of constant returns to scale, it is possible to appeal to the second welfare theorem thereby transforming the problem from finding a competitive equilib- rium to that of computing a Pareto op!imum. Then. approximation methods such as Kydland and Prescott’s (1982) original linear-quadratic approach or a value function ircration method as described In Danthine and Donaldson (1994) can be used to compute the optimum. Wh’le it is possible to modify these approaches to include economies characterized by increasing returns (see Hansen and Prescott, 1994). Farmer employs a more direct technique which is closely related to that described in King. Plosser. and Rebel0 (1988). This involves a linearization of the equilibrium necessary conditions by taking first-order Taylor series expansions of these expressions around the steady-state values. The generated linear structure implies that the equilibrium policy func- tion for consumption will itself be linear. Somewhat surprisingly, F.:rmer fails to place this approximation method in the context of others used in the profession nor does he cite similar approaches. (The lack of references is somewhat of a problem throughout the book.) For instance, a student reading the articles by Kydland and Prescott (1982) or Hansen (1985) might be interested in how the linear-quadratic approximation method used in those papers compares with that described in Farmer’s book. Since Farmer is promoting this book as a graduate text, such contextual information seems appropriate, if not required.

Letting (?,. &,, 1,) denote the percentage deviation of these variables from their steady-state values at time t [e.g., t, = (c, - i’)/?J, a first-order Taylor series expansion around the steady-state of the conditions describing a competitive equilib-ium yields the following set of linear stochastic difference equations:

(2)

The elements of the matrices A. B, and C are functions of the parameters which characterize preferences, technology, and the shock to technology as implied by the Taylor series expansion.

Letting J = 4 - ‘B. and taking conditional expectations of Eq. (2) yields

Given the system of equations described in Eq. (3). the eigenvalues of J deter- mine the time series properties of consumption and capital and, importantly, the determinacy of equilibrium. Note that since J includes the motion of the technology shock which is assumed to be stationary, one of the eigenvalues of J will be equal to (l/p) and thus greater than unity. In a typical (i.e., constant returns to scale) real business cycle model, the remaining eigenvalues will bracket unity as is consistent with a saddle path equilibrium. In this case. Eq. (3) implies a unique solution in which consumption is a function of the two state variables, R, and 2:. If, however, the eigenvalues of J both lie outside the unit circle, the equilibrium is indeterminate and, in this situation, there are multiple equilibrium paths for consumption and capital :o the steady-state.

Turning first to the typical case. the solution can be computed by decompos- ing the matrix J:

J = Q/IQ-‘,

-Nhere Q is a matrix whose columns are the eigenvectors of J and n is a diagonal matrix whose elements are the eigenvalues of J. denoted i.i. Then premultiplying Eq. (7) by Q-l yields

Q-f ;J = U, = AQ-iE,(lt’:) = nE,(U,+,). (4)

But Eq (4) implies three independent equattons:

Ui.f = j-iEt(ui.t+ I 1. i = I, 2. 3. (5)

As stated above, if a unique saddle path equilibrium exists, than one of the eigecva!ues, say j.3, is !ess than one. Therefore, repeated iteration on Eq. (5) implies that u~,~ = 0. It is this linear restriction which generates the solution: c’, = o,E, + 022,. Given this solution, the equilibrium path of capital. labor, output, and investment (all represented as percentage deviations from the steady&ate) can be readily computed.

The technique described above is only slightly different from that in King, Plosser, and Rebel0 (1988) (in fact, the solution itself is identical); the real distinction of Farmer’s analysis is in permitting all of the eigenvalues of J to be greater than one -- the resulting indeterminacy of equilibrium means that, for a given level of capital at time r. there are an infinite number of paths for

consumption and capital cnnvcrging to the steady-state. In this case. it is possible to ~!iminate the technology shock frcpm the above system (set 5, = 0. Vr) and still generate a stationary rational cxpcctations equilibrium oy assuming that agents have ;I bclLf function or distribution over these multipic equilibrium paths. That is. replacmg the technology shock with a belief shock. denoted e,. which is assumed to bc indcpcndcntly and identically distributed. permits Eq. (3) to bc expressed as

In this setting, agents’ forecast errors. as described by the distribution for t’,. can permit a self-fulfilling rational cxpcctations equilibrium. That is. since both eigenvalues of J are assumed to be greater than one. Eq. (6) implies the following stable equilibrium law of moticrli:

(7)

The assumption of rational expectations imposes consistency between beliefs (the distribution for e,) and the equilibrium behavior of capital and consump- tion. Furthermore, since it is assumed that the belief shock has a stationary distribution, Eq. (7) implies that this will be the case for capital and consumption as well. Hence, given a time path for e,, the elements of J--l can be used to generate time series for equilibrium consumption and capital similar to the RBC model.

This discussion raises two key issues: (1) If one permits indeterminacy. what are the characteristics of equilibrium and how do they compare to a typical real business cycle mode? (2) Are the parameters which generate indeterminacy reasonable? I turn to each of these issues below.

My analysis involves several steps: Using the same parameter values as Farmer, 1 first compare the equilibrium behavior of the mode! with self-fulfilling beliefs to that of a typical RBC model. I depart, however, from traditional calibration analysis as well as that of Farmer’s by using the models to obtain an observable time series for the exogenous shock processes -- i.e., the technology shock in the RBC model and the belief shock in the self-fulfilling expectations model. Given these constructed shock series, the equilibrium in both models is computed and studied. Some of this duplicates Farmer’s analysis -. I show as Farmer does that the volatilities of consumption, investment, and labor relative to output are roughly the same in the two models and are reasonably close to that seen in the data. But my analysis departs from Farmer’s in several dimen- sions: (1) The steady-state behavior of the indeterminate model is demonstrated to be inconsistent with the long-run behavior of the US economy. (2) I am

critical of Farmer’s use of VAKs to study the dynamics of the artificial ccono- mies. (3) The forecasts of the two models are compared to actual data: the results from this exercise implies that the RBC model dominates the model with self-fulfilling beliefs. (4) Finally. I show that indeterminacy of equilibrium is fairly tenuous, i.e.. small departures from the parameter values which Farmer employs results in a determinate equilibrium.

3.2. Compnrison qf equilibrium characteristics

One oi the strengths of the real business cycle model is its ability to replicate many of the features of US business cycles when calibrated to US data. For example, the model can produce approximately the same relative volatilities of consumption, output. and investment that are observed in the data. The ques- tion is raised whether this property as well as others are observed in the increasing returns model (IR) in which self-fulfilling shocks are the sole distur- bances.

3.2. I. Steady-state hehacioj Before addressing the question of cyclical behavior. I first study the steady-

state of both models: a t\-pic which Farmer did not address in the book. This behavior is important. in a calibration exercise. for determining key parameter values. The use of iong-run behavior to pin down parameter values for prefer- ences and technology (but not, however, the time series behavior of the tech- nology shock) provides the discriminating power of the calibration analysis employed in RBC models. For this exercise, I use the same parameters described in Chapter 7 of Farmer’s book. t‘hesc are given in Table 1.

Note that in the RBC model. the exponents of the production function dre equal to th: factor shares (m is capital’s share and n denotes labor’s share) while in the increasing returns (IR) model this is not the case. Farmer justifies the particular parameter values he uses as following: First, measurement error in the National Income and Product Accounts, attributed to the inclusion of pro- prietor’s income, does not make it unreasonable to justify that labor’s share of income is 70% rather than the typically used value of 64%. The remaining 30% of GNP goes to capital of which only 23% is in the form of capital rents. The

Table I Model paramewr values

RBC model 0.64 0.64 0.36 0.36 0.95 0.99 0.025 IR model 0.7 21 0.23 0.4 NA 0.99 0.025

Tat-k 2 Model skady-WC v;~lucs

~---_--. ---.----- I; j i j Fj

RBC model IO.24 0.26 0.74 IR model 6.54 016 0.x4

remaining 70,6 is due to profits which Farmer theoretically justifies by alluding to the previously mentioned model of increasing returns in the production of imermediate goods. Farmer shows that this model. based upon the aggregation function for intermediate goods which he assumes, imphcs that m = i/c and n = ic. Furthermore, the value of i can be related to the Lerner index:

I-i.= P - hlC

P -

where P is the price of output and MC represents the marginal cost of produc- tion. Appealing to studies by Domowitz, Hubbard, and Peterson (1988) and Hall (1986), Farmer reports that estimated values for i. range from 0.9 to 0.49 - hence the value of 0.58 which he uses does not appear to be unreasonable.

Finally, with respect to the disturbances in both economies, the production shock is assumed to have an autoregressive parameter equal to 0.95. Again, this value is common to many RBC models. In the increasing returns economy, the consumption or beliefs shock is assumed to be white noise. The remaining parameters are set equal to those typically used in RBC studies: fl = 0.99 and S = 0.025. These parameter values imply steady-state ratios in each economy as given in Table 2.

The steady-state values from the RBC model indicate why these parameter values are used in most studies .- the implied capital-output ratio and invest- ment-output ratios match those in the data fairly well. In post World War II US data, these ratio are roughly 10 and 0.25 respectively.4 The IR model, in borrowing &he depreciation and discount rates from RBC studies but imposing different factor productivities, severely understates the capital-output ratio so that the flow of consumption relative to income is overstated. Since Farmer’s primary purpose was to investigate the cyclical properties of the IR model when buffeted by self-fulfilling shocks, perhaps this inconsistency between data and model should not be construed as being particularly troubling. However, since,

‘For these ratios, the capital stock is measured as plant and equipment and income is measured as a quarterly llow.

as we shall see below. the presence of indeterminacy depends critically on the parameter values chosen. the full implications of these parameters choices should not be ignored.

3.2.2. Cvclical heharior Using these steady-state values along with the parameter values in the above

model implies a J matrix as defined above with the eigenvalues shown in Table 3 (I have ignored the eigenvalue from the RBC model that corresponds to the technology shock). The RBC model’s eigenvalues bracket unity implying a saddle path equilibrium. The row of the matrix Q--l defined above that corresponds to the eigenvalue of 0.93 imposes the linear restriction between (c’,, c,, 2,) consistent with a stationary equilibrium in the model. In contrast, both roots of the IR model lie outside the unit circle so that the equilibrium is indeterminate. Hence, the equilibrium time path of consumption and capital i; determined by the matrix J-’ [defined in Eq. (7)J. Both eigenvalues of this matrix will be less than one and, importantly, will be complex numbers. This implies that the steady-state is a sink and that the transition path to the steady-state will be cyclical.

To explore the cych,al properties of both models, I propose that their observable implications be taken seriously; by this I mean that the models should be used to identify an observable time series for the respective exogenous shocks. That is, in the RBC model, the shock is defined as the production residual not accounted for by the productivity of the inputs while, in the IR model, the shock is defined as the response of consumption not predicted by the elements of P-l. Hence. the observed path for these series in conjunction with the definitions imply a time series for the shocks. Note that the two models require a different set of assumption in order to identify a time series for the exogenous shock from data. In the RBC model. the calibrated parameter values for technnlogy together with the assumption of constant returns to scale allow one to uncover the technology shock. In the IR model, the calibrated parameter values for preferences and tecnnology ad the assumed i.i.d. distribution for the consumption shock permit an identification of the realization of the consump- tion shock. The italics highlight the fact that, in contrast to the RBC model. the assumed behavior of the belief shock is embedded in the parameters of the I-’ matrix.

Table 3 Eigenvalucs ol J matrix

Root 1 r.001 2 -

RBC model I .06 0.93 IR model I .07 * 0. I I I 1.07 -O.ll/

For thz RBC ,nodcl. the technology shock is given by

IO~Z, = log jas, - 0.64 log km, - 0.36 log ht. (‘)I

where ~trs. kus. and /US are the level of per capita US output. capital stork, And iabor supply.’ The residuals from :egrcssing this constructed series against a linear time trend (consistent with constant technological progress) are defined

A as z,. For the IR model, the consumption shock is defined as

2, = cUIS, - 1.1 S~ri.s, _ , -t- 0.086&, _ , . (10)

The parameter values in Eq. (10) are taken from the mairix J’- ’ and the series (clis, klis) are the deviations from trend as given by the H-P filter.h

Both series are plotted in Kg. 1. Note that both arc centered on zero by construction but have different time series characteristics. Specifically, the tech- nologically shock exhibits much more autocorrelation than the consumption shock. ‘To more closely examine the time series proper.ies of these series. each were regressed on their lagged values. Letting r denote the autocorrelation c0efficier.t. this produced the results reported in Table 4 (r-statistics are given in parentheses). Note that the null c.)f the consumption shock being serially uncor- related cannot be rejected while the autocorreletion parameter for the technol- ogy shock is identical to that assumed in the model. Moreover. the estimated standard deviation of the innovation to the technology shock (0.007) is one that is common to many RBC models and ir Identical to that used by Farmer in his simulations. dased on these properties. the constructed shock series for both models appears to be reasonable.

Using these shock processes. both models were solved and the iesulting output was passed through the H-P tiltcr. The volatilities of consumption, hours, and investment relative to that of output from both models are reported below along with that from the actual data.’ Also. for comparison I provide in Table 5 the relative volatilities given in Table 7.2 of Farmer’s book.

’ For US data the fo!lowing quarter11 data over the period 1960.1 19Y3.1 wa; used: Consumption was measured as expenditures on nondurablcs :m.i servicer Investment includes buwws invest- ment. purchases of durables. ano govewment investment snd net exports (to be consistent with National Income Accounting), Labor is measured by manhwrh per quarter. Income IS measured as the sum of GNP and estimated Ilows of income from dura& goods and government capital. Measures for private and government capital wrc taken from Musgravc I 1992;. The methods for computing these measures are described in Cooley and Prescott (1994).

“Since the variables in Eq. (IO) are measured as deviations from the steady-state, an estimate of the steady-state values is necessary. I use the H-P trend 10 construcl these estimates.

‘As Woodlord (1991) has pomted out, the self-fulfilling equilibrium puts no restrictions on the absolute magnitude of volatility but does place strong rcstrictlons on the relative volatilirres(as well as Ihe scrlal correlation properties of the endogenous variables). 11 IS for this reason that I report the relative moments in Table 5.

-0 04. ,..:: :

;) -0 08-t.:

-0121., , .,, ., . ., .., J 60 65 70 75 80 ti5 90

!- consumpllon shock (IR model) -----technology shock @-A

Fig. 1. Constructed shuck series.

Table 4 Time series properties of shock processes

z

2, 0.14 (164)

2, 0.95 (46.75)

-.

7 2. of wgression

0.005

0.007

S.D. of shock

n.oos

OS-C4

Table 5 Relative volatilities

US economy

RBC wdel [shocks from Eq. (913 Farmi -‘h RL’C’ m del

IR model [shocks from Eq. (IO)] Farmer’s IR model ---

n.59 3.x1 0.99

0 3.1 3.10 0.75 029 3.3 0.76

0.14 5.1-I CJHI 0.24 5.12 0.83

-- -

First. note that comparing Farmer’s rest& to mine further establishes that tnc generated shock process which I use is not influencing the results, at least with regard to volatility. Secord. tht numbers reported in Tab!e 4 indicate that the relative volatilities of the components of GNP as well es that of the labor input in the two models are roughly the same and, to a reasonable degree, mimic

those observed in the data. While thltsc results do not permit one to discriminate between the models, they do provide some insights into their underlying strut tures. It is well known that the relative volatilities in the RBC model are sensitive to the autocorrelation of the technology shock through the implications for agents’ permanent income. For instance, if shocks are not autocorrelated, consumption becd ames smoother while investment becomes more volatile (see King, Plosser. and Rebelo. 1988). Hence, it is interesting that the IR model can duplicate these relative volatilities even in he absence of a nocorrelated shocks. This property reflects the inherent dynamics implied by thf complex roots of the J matrix defined above. Since the belief shock affects c msumption behavior directly, it is tempting to interpret the shock as a preference shock. However, such an interpretation in and of itself would not imply the equilibrium charac- teristics exhibited in the Farmer model in the absence of productive externali- ties8 As Farmer explains. it is these externalities. through their effect on the labor market, which create the interesting dynamics of the model. Specifically, in the IR model with the parameter values Farmer employs (specifically v > l), the aggregate demand curve for labor is upward-sloping with respect to the wage rate while the labor supply curve is horizontal (because of the linearity of leisure in the utility function). This implies that, ceteris paribus, an increase in the capital stock will cause equilibrium hours to contract resulting in self-damping dynamics.

The inherent dynamic behavior of the two models, therefore, is quite different and it is along this dimension which Farmer attempts to discriminate between the two economies. However, I do not find the evidence which he presents persuasive; in fact, I believe it provides little insight into the relative merits of the models. Before presenting Farmer’s arguments, I present some impulse response functions from the two models in Fig 2. In the grap-s, the effects of a one standard deviation shock in the exogenous proccjs of both models is shown. Note that both models imply the same relative response in the magnitudes of the endogenous variables, however in the RBC model the effects monotonically decline with time while the IR model generates a damped cyclical response particularly with respect to investment. The monotonic response in the RBC model implies that it will not be able to gener- l a ryclical and autocorrelated behavior similar to that seen in the actual data unless the technology shock itself is highly autocorrelated. This reliance of RBC models on the time series

‘In a closely related model Farmer. Baxter, and King (1991) also study an increasing returns economy. However. the parameters which they use produce a determinate equilibrium. They examine the e’fects of (autocorrelated) preference shocks and demonstrate hat, in a constant returns to scale econony. equilibrium characteristics are quite different than that observed in the data. For instance, IL contemporaneous correlation of output and investment is negative and the volatility of consumpticn is 150% that of output.

R6C MOM Shock lo T~chnolopy

Oo5Y 0 04

1 ‘.

5 10 15 2tJ 25 30 35 1 IR Model Shock to Consompoon

006.

\ \

004. \ \

\ \

002. \ ‘\

-. --__.---

5 10 15 20 25 36 35 40

Fig. 2. impulse responses IO a one standard deviation shock IO exogenous procc,,

characteristics of the exogenous shc;k process to generzie dynamic charactcr- istics consistent with the data has been criticized by several authors (e.g., Cogiey and Nason, 1994b). In contrast, the complex eigenvalues of the IR model, as discussed previously, do imply inherent cyclical dynamics.

Farmer does not study the dynamics of the models using graphs like those just discus

consumption. income. and invcstmcnt. In addition, the intratemporal efxciency condition in the RBC model implies that consumption. output. and labor (measured in deviation form) are linearly related. In order to estimate the same VARs as Farmer, I added an i.i.d. shock to each series that had mean zero and standard deviation of 0.001 (i.c.. mcasuremerlt error). Farmer does not state how he overcame this problem.

More important, however, is the interpretation of the estimates from these VARs. Farmer believes this exercise provides important implications for the dynamic behavior of the economy; this is reflected in the following quote:

Since all of the models that we are looking at are explicitly dynamic. they NilI a:so have implications for the cross moments between variabizs dated at different points in time. One of the most important tools in the applied macroeconomist’s tool kit is the intprt/sr> responsejunction, which is designed to capture exactly these correlations. (p. 146j

However, it is well known that esr.,r,ated impulse response functions rely on an arbitrary matrix decomposition in order to orthogonalize the VAR error terms and that different ordering of variables in the VAR can lead to different impulse response functions (for instance, see Cooley and LeRoy. 1985). While Farmer does provide some discussion of the normalization issue. he does not indicate the ordering of variables which he uses nor does he provide much in the way of caution when interpreting the results. Thus he states:

A simple resolution to the normalization issue seems to be to ignore the contemporaneous correlations in the data and ask how the system in the absence of shocks would return to the steady-state if it were displaced in each of the n dimensions that correspond to the coordinate axes. Strictly speaking, this object is not an impulse response function, but it gives jusr us much information about the dynamics ol' the system. (p. 147, emphasis added)

Fig. 3 presents the impulse response functions implied by the VAR estimated from the actual data and the data generated from the two artificial economies using the constructed shock series. The following ordering is used: (J: c, h, i). The impulse response functions in Fig. 3 are virtually identical to those given in Farmer’s book (Fig. 7.4) again implying that the two constructed series for the shocks do not seem to be influencing the dynamic behavior of the models.9 The

91t is possible that even though the constructed technology shock appears to be a highly autocor- related AR(I) series with a real roof and therefore consistent with the assumptions of the theory, its motion could be governed by a complex root. If this were true. the dynamics of the RBC model would bc biased toward cyclical behavior. The fact that I was able to duplicate Farmer’s impulse response functions suggests that this is not a problem. Clexly, a more thorough analysis would er,.ploy formal tests.

231

US Data

004-

RBC Model

IR Model

5 10 15 20 25 30 35 4c

- output ---. Hcurs -----' Caswnplion --- Investment

Fig. 3. hi~uke responses lo ;I one :, .lard deviation shock IO output. ordering of variables: (y.c,h.il.

fact that the impulse response functions in the US data and the IR model dIsi)iay cyclical behavior while the RBC model does not is interpreted by i-armer as strong evidence against the RBC model. Again quoting from the book:

This concerns the fact th.tt the US data favors a model with complex roots .-. the evidence for this statement is the fact that the impulse response functions in the data r!;arly cycle in their return to !he steady- state. (P. 147)

If one believes the impulse response functions estimated from the US data capture the dynamics due to a true exogenous shock, then the IR model is preferred to the RBC model. However, the evidence appropriate to support this conclusion would be the graphs in Fig. 2, not those in Fig. 3, since the shocks analyzed in Fig. 2 are indeed exogenous. By using estimated VARs, Farmer is imposing a particular causal structure on the data as well as on the models. We are not told what that causal ordering is nor why it was chosen. I present in Fig. 4 the impulse response functions from imposing an arbitrarily chosen alternative causal structure - one implied by ordering the variables in the VAR as (h, c, J, i). Again, these graphs trace the effects of a one standard deviation shobk to output. Note that with this causal ordering, the cyclical behavior of all variables other than investment in the actual data is much less pronounced. Also, the RBC model does display some cyclical behavior but clearly this does not match that in the data. The IR model continues to display cyclical behavior but now the responses are opposite to what is seen in the data. Should one conch~de from this that the IR model is a poor one? Or, does this imply that the original causal structure was correct? As a final exercise, I study the impulse response functions implied hy a one standard deviation shock to consumption since it is consumption shocks which drive the IR model. In estimating the VARs, the following ordering was imposed: (c, 11, i, ~7). The impulse response functions are graphed in Fig. 5. Again, the cyclical behavior in the actual data is not as pronounced as in ihe IR model while the RBC model is characterized by primarily monotonically declining responses. Also, both the RBC and IR mod- els initial responses are declining while, in the data, all variables initially increase. My point in presenting these examples is to stress the fact that the interpretation of VARs is imposstblc without imposing a priori some causal structure. Therefore, a reasonable approach, it seems, would be one in which the causal structure implied by :he RBC’ and IR models is used when estimating VARs from the data. But Farmer does not discuss precisely what these causal orderings are when he does his empirical analysis so that meaningful con- clusions are difficult to make at best. As a final note, the linear structure of both models permits one to calculate analytically the covariances that are implied. Hence comparing these analytical constructions from both models might prove

RBC Model

O.Oor,m

lb 1‘5 2il 2% 3il js I IR Model

002-

-0.03-. , , .--.. , , , 5 10 15 20 25 30 35 40

- output ---. tj- ---_ _. Cmwmp(m ---. ,mn

Fig. 4. Impulse responses to a one standard deviation shock IO outpost. ordering of variables: (h.c,y.i).

US Data 0020

,’ ‘, 0015 #’ !

! ’ I

0010 i \ \

fTz_ _

i \ ooos 1 ,,;kr-

,7 *\ --.

.\ \ --

OOCO ‘\ L. /--

‘. ---.____

I. ,-+/------

/a -___-- 0005 _, , _. -p ,

5 10 is m 25 30 35 I

004- RBC Model

003.(

‘L,

002, i \

-\

IR Model

5 10 is m 25 30 35 40

- outpil ---. Hours .----. Consumptmn ---. Investmsnt

Fig. 5. Impulse responses to a one standard deviation shock IO consumption, ordering or variables: (c. Il. i. j’i.

more interesting and insightful then impulse response functions estimated from simulated data.

A side topic related to the use of VARs is that Farmer employs unfiltered data for his estimations. However. in comparing the relative volatilities of the output from the models, filtered data was employed. Farmer gives no justification for why different detrending procedures were used for rhe two analyses when a unified approach would seem to be desired. As is well known, the choice of detrenting is not an innocuous one. I present in Fig. 6 impulse response functions identical to those in Fig. 3 which were estimated using HP filtered data. Note that the RBC model now displays responses mu;h like that of the IR model and the data itself.‘(’ The fact that the H-P filter can influence the serial correlation properties of a series has been raised by several authors (e.g., Cogley and Nakon, 1994a) and is an issue that needs more careful study. While for brevity’s sake I do not report the figures, the standard deviation of consumption, investment, and hours relative to that of income do not appear to be dramati- cally altered by the H-P filter. These are admittedly difficult issues and take one far afield from the modeling issues that Farmer focuses on, but some discussion is merited.

.3.2.3. Comparing the models through a forecasting exercise While I disagree with Farmer’s use of VARs to discriminate between models,

clearly some method needs to be employed which provides more discriminating power than is entailed in the casual comparison of a set of judiciously chosen second moments. Strictly in the form of preliminary results, I explore one method which is based on the forecasting ability of two calibrated models when the exogenous shock process is set equal to actual time series. [This approach is similar to that described in Fair and Shiller (1990); recently Canova, Finn, and Pagan (1992) have used a related method to compare calibrated models. J Using each model’s implications to determine the path of the exogenous process as described above [see Eqs. (9) and (lo)], it is possible to compare the information content implied by the endogenous output of the model through a regression analysis. For instance the RBC and IR models’ forecasts for output can be compared via the regression:

where JYLS denotes US GNP, yrhc denotes output from the RBC model, and yir is output from the IR model. Throughout, all variables are deviations from the

‘” 1 also studied the impulse response functions for the RBC model when the shock was assumed to be an AR(l) with autoregressive parameter of 0.35 rather than the constructed series defined by Eq. (I 5). These impulse response functions. using H-P filtered data. also displayed cyclical behavior.

236

US Data ” - -

1

002.7

\ I

0 01. i

i _ \ .---. 000 ----- ‘\ / ______---

k..- \ -’ h \ ,/’ ,’ -0 01 ‘Y,. / 5 10 15 20 2s 30 js-

RBC Model

i 002 \

\

co1 L i i \,i ow *- 001

I

,----__ -- ._-___-

- 1 \ i P' '-3 a

\I-'

002~,.. , .,...., 5 10 15 2a 2s 24 35

IR Model

O,lST 0.10 !

\ \

0.05 '\, \

I-

o* -5 i

,' ./-., ___--- ------v

'\ -+-/--

.i

-00s '\ .I

\ / .d'

- CGpbl ---_ HJuun ------ ct?,Wl,,Dtii 1 ---. h&“Y’tnt

Fig. 6. Impulse responses IO a

OS- I :s

60 65 70 7s 60 ‘.z---z-

I-I-US _---- ,e,R --- I-Rf,CI

T-9 60 65 70 75 80 65 so

r-Yv-US __--- Y-IR --- Y-R6Cj

Fig. 7. Plots of actual an4 arriticial series

steady-state which is defined as the trend pretitcted by the H-P filter. Under the null that neither model provides information, /I, = /I2 = 0; if the RBC model provides information while the IR model does not, iht null is fir # 0, /Y2 = 0; and vice versa if the IR model contains information but the RBC model does not.

Note that since the endogenous capital stock was used in constructing output from both models, any forecast errors are compounded over time. Hence an alternative approach would be to use the actual capital stock series (kus) along with the estimated shock to generate a series of contemporaneous forecasts for output, consumption, labor, and investment. Determining the relative merits of these approaches is left to future research. The behavior of each generated series relative to the data are presented in Fig. 7. Both models track all the serves reasonably well, while the IR model exhibits greater volatility than the data for all series other than consumption. This behavior. I believe, reflects merely a scaling issue and does not carry serious implications for the validity of the model.

Table 6 Forecast comparison of RHV ;md IR models

.___ .~

Output Consumptron Labor Investment

11, (RBC model) 0.457 O.‘6H O.?4Y 0.564 (8.31 I (,1.34) (2.6Oj (X.44)

/II (JR model) 0.008 0.5Y8 -- 0.057 0.0 I 5 (0.35) (16.39) ( - 1.57) (0.8771

--

LJsing the output from the two models and running the regression,

where s, = (y,, c,, Ir,, i,) generated the results given in Table 6 (t-statistics are in parentheses).

Note that the RBC model provides statistically significant information (or the behavior of all series while the IR model does so only in the case of consump- tion. But. given the construction of i, as defined in Eq. (lo), the degree of correlation between the model and data along this dimension is not particularly surprising nor terribly informative in a discriminating sense.’ r Overall, it ap- pears that the RBC mode) is superior to the IR model -. while it is important to note this, it is equally important to state that this comparison is clearly unfair to the latter model. ‘I hat is, Farmer’s motivation for constructing the ;R model was to show that a model in which self-fulfilling shocks are the sole source of disturbance has the ability to generate equilibrium characteristics much like a typical RBC mode). In doing this. I do not believe he was making the case that technology shocks are irrelevant. The purpose of my comparison exercise is primarily to advocate (in the spirit of Farmer’s book) that more work needs to be done on developing a methodology which discriminates between calibrated, rather than estimated, models. Additionally, it would seem to me that a neces- sary conaition for such a discrimirldiing methodology would be one that is based upon observable implications.

3.3. Empirical evidence ftir the IR model

While the above comparison suggests that the determinate RBC model is superior to the indeterminate IR model, this does not imply that the evidence tar increasing returns is weak. To bolster the case of increasing returns, Farmer

” It can be argued that output in the RBC model, since it is a linear function of the technology shock which is constructed from observed output. provides little additional information other than that already contained :n the technology shock. More research is needed on the effects of detrending, the definition of the steady-state as well as a host of other issues before such questions can be addressed.

239

i i’ \ Root I Root 2 -- --

0.58 0.400 I.21 1.07 +O.ll/ 1.07 - 0.1 I I

0.60 0.383 I.17 1.09 + 0.131 1.09 - 0.131 0.62 0.371 1.13 I.13 + 0.15 I.13 - 0.15/

0.64 0.359 1.09 1.24 + 0.161 1.24 - 0.161 0.66 0.348 I .06 3.12 1.24 0.68 0.338 I .03 1.17 0.22 0.75 0.307 0.933 I I2 0.78

0.85 0.271 0.823 I.1 I 0.86

references the study by Caballero and Lyons (1992) which finds empirical evidence for positive external effects of aggregate output on individual firm productivity. Additional evidence for increaiing returns is reported by Rotemberg and Wood- ford (1994) in which they cite studies by Morrison (1990) and Ha!i (1988, 1990). These observations have led to the development of several business cycle models which study the effects of market power on equilibrium characteristics; see the surveys by Silvestre (1994) and Rotember& and Woodford (:994). In these models, however, equilibrium is determinate and the economy is subject to exogenous shocks, e.g., technology shocks or randomness in the markup of price over marginal cost. These models have focused on exogenous shocks because of the fact that for the IR model to exhibit indeterminacy, the degree of market power, and increasing returns must be at the upper range of estimated values. [This fact is noted by Rotemberg and Woodford (1994).3 The critical nature of these pa- rameter values is demonstrated in Table 7. There I report the eigenvalues of the J matri:. for different values of i [as defined in Eq. (IS)] which, as discussed earlier. is inversely related to the markup of price over marginal cost. (Factor shares and profit margins are the same as used previously; see Table 2.)

Note all values of 11 and v in Table 7 imply increasing returns to scale; at the

To summarize. b‘armcr has prcswrcd a pvocative model of business CYCICS driven by self-fulfillmg expectations. However. the evidence in support of his model in the form (of impulse response functions from estimated VARs is not terribly persuasive and. in a preliminary forccacting comparison analysis. the RBC model dominates the IR model. In spite of thcsc negative marks, I main- tain that the Ideas prcss;‘nted in this boc>k deserve serious consideration by macroeconomists;.

4. Overall merits as 9 text

I have used The ~fllc~~oec,ont,,,l,c,s oJ‘Sd$ Fdfilhq Prophecic~s as a text in the second half of the first->ear macroeconomics sequence at the University of California. Davis and, hence. can report some first-hand observations. I (as well as the students) found the primary strengt!) of the book to be its coherent and accessible presentation of ideas. III particular, the development of the approxi- mate solution method is especially clear and the discussion of regular and irregular equilibrium is extremely readable. Furthermore, since most of the macroeconomic topics presented in the book are applications of general equilib- rium theory, this is given central, although at times a bit brief, treatment. These concepts arc then integrated nicely into the discussion of the equilibrium properties of representative agent and overlapping generations models. For instance, in discussing the possibility of inefficient competitive equilibrium in OLG models (with no intrirlsic uncertainty), Farmer explains that. as shown by Shell (1971), it is the double infinity of agents and commodities which leads to the failure of the first welfare theorem. The fact that macroeconomists, as opposed to general equilibrium theolists. often incorrectly attributed this break- down to the incompleteness of markets (see the examples in Marshall, Sonstelie. and Gilles, 1987) underscores the importance of introducing these general equilibrium ideas early on in mscroeconomists’ training. Finally, Farmer infuses the presentation of ideas with an enthusiasm and honesty that is refreshing and all too often lacking in texts. For example, he offers the following concluding statements:

As a reader of this book you may or may not be sympathetic to an approach that argues that equilibria may be driven by sunspots. You may or may not be sympathetic to the idea of explaining sticky prices with models in which all markets clear. And you may or may not be sympathetic to the idea that governments have a role in managing tt.e economy. But whatever side you take on a whole range of issues, both normative and positive, I hope to have persuaded you to be open to the idea that general equilibrium theory can provide us with a common language. (p. 234)

My negative thoughts on the book. in addition to those presented m the previous sectior.. b;rn aimosr entirely from Farmer’s s-Jccessful efforts to limit the *:cope (and 51~ *) o!‘ the book. As a result of these efforts. this book is not appropriate as a b:d’ld alone text (such as Blanchard and Fischer, 1989); but, then, Farmer most likely did not intend it as such. Still, one can not help but wish for a richer discussion on some topics. For instance, the necessary condi- tions associated with the consumers’ maximization problem in RBC models are merely stated without any discussion of dynamic optimization under uncertain- ty and references to this topic are not provided. Also, Farmer briefly mentions (as an alternative to I ..libration) the use of generalized method of moments estimation without providing the reader any references.

Still. in spite of these omissions. I intend to use The Macroeconomics qf SeljlFu/ji!/ing Be/i+ in my class once again. The research agenda which Farmer is promoting is a promising one and the book admirably represents to students an example of intriguing and serious economic investigation.

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