An Outsider's Guide to Real Business Cycle Modeling

OIEYIt~ MARCH/APRIL ‘#95

Joseph A. Rifler is a senior economist at the Federal Reserve Bank of St. Louis. Heidi L Beyer provided research assistance.

(1.) Decisions of firms and consumers should An Outsider’s be derived from fully specified intertemporal r Guide to Red optimization problems with rational expecta- -4 tions. (2) The general equilibrium of the model must be fully specified. (3) Both the Business Cycle qualitative and quantitative properties of the model should be studied. Lucas argued in Modeling 1980, before work began on RBC models, that theoretical developments beginning with Hicks, Arrow and Debreu allowed modern Joseph A5 Rifler economists to beginwork which met the first two criteria. The dramatic fall in the One exhibits understanding of business cycles price of capital (computers) has made it by constructing a model in the most literal possible to meet the third criterion as well, sense: a fully articulated artificial economy allowing macroeconomists to explore their which behaves through time so as to imitate models in much greater depth (although this closely the time series behavior of actual potential is not always realized), This article economies. is concerned mostly with giving an outsider a feel for how the third requirement is met. Robert F. Lucas (1977) It proceeds by describing the theoryunderlying a standard RBC model, explaining what con- Pa uring the last decade, guided by Lucas’ stitutes an equilibrium, and then delving into ~Jjj principle, the real business cycle (RBC) the mechanics of solving a specific model I~ model has become a standard tool for a (Hansen’s landmark indivisible labor model) large share of macroeconomists. The tool has using a specific technique. I conclude with found such widespread applicability that pro- two illustrations of how the basic methodol- ponents of this approach to macroeconomic ogy can be extended to study fiscal and modeling (and those with proper sensitivity monetary policy training) now prefer a more generic label: 4” a ~“~~4a computable dynamic general equilibriuan model. Other demographic groups often nut regard the customs and rituals of RBC propo- The typical RBC model is an Arrow- nents with some degree of bafflement. The Debreu type economy specifically a one- goal of this article is to dispel some of the sector stochastic growth model. Many iden- aura of mystery that surrounds—from an out- tical consumers who live forever maximize sider’s point of view—the specification, cali- expected utility (derived from goods and bration, solution and evaluation of RBC mod- leisure) subject to an intertemporal budget ‘Outsiders would include, rmoia els.’ It is thus concerned more with the “how’s” constraint. Competitive firms purchase factors others, those who (like the nothorl of RBC modeling than with the “whyV’ (or, for in competitive markets. Uncertainty comes were educated where these models that matter, the “why nofs”).’ Broader intro- from a stochastic shock to the economy’s were not in Eater and those who ductions to real business cycle modeling can production technology (like the authorl finished school he found in Blanchard and Fischer (1989, For simplicity suppose that consumers before these models laid a lerte chapter 7), McCaIlum (1989), Plosser (1989) own capital directly and rent it to firms. Firms market share.

and Stadler (1994). The pioneering papers are buy capital and labor services from consumers 2 In this spirit, tire progwnts osed in Kydland and Prescott (1982) andLong and and use them to produce a single output which the article nra available on the Plosser (1983). can be used as either consumption or invest- ff10 electronic brtlletn hoard. far Three criteria have guided the model- ment. Output is the numeraire, The firms’ mace information, see the back building process in the RBC literature. technology is described by an aggregate con- curer of this issae.

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stant-returns-to-scale production function determined by the model; once equilibrium which includes an exogenous aggregate tech- quantities are known, they can simply be nology shock, A,: A,F(K~,L,),where Kr and L, substituted into the Euler equation for each are the aggregate levels of capitaland labor.° asset to determine its price. The A, are usually taken to be serially corre- Since consumers and firms are identical, lated, but the exact specification can be post- this artificial economy is mathematically poned. These are simple competitive firms identical to a representative agent economy which will purchase labor and capital services in which one price-taking consumer sells labor at wage Wand rental price R, up to the point and capital services to a single price-taking where their marginal products equal W and firm. On the surface, finding an equilibrium R,, respectively: appears to be a very daunting task. Even though we have reduced the number of con- (1) W = A,F,(K,,L,), R, = A,FK(K,,L,). sumers and firms to one each, we still have Let! be a consumer’s time endowment, an infinite number of goods: consumption I, the amount of labor she supplies, c, her and leisure in various states of the world at consumption, hr her holdings of capital and i, dates from 0 to infinity However, a greatdeal her rate of gross investment. (Upper case will is known about the theory underlying this be reserved for aggregate variables.) Thecon- type of economy (Stokey, Lucas and Prescott, sumer takes prices W, and R, as given. Given 1989), and this theory provides important a starting value k , she chooses paths, that is, 0 tools that allowsimulations to be constructed. contingency plans, for i, and I, to maximize A 5iflU’flON IN P*IiMCtali$ )} For the representative consumer, the state of this economy at the beginning f t is 0 subject to a budget flow constraint summarized by the individual’s capital stock Ia,, the aggregate capital stock K, and the state (2) + i, R,hr +W,l, of technology A,. Thus, the maximum lifetime and a description of how capital accumulates utility attainable by the consumer will he a and depreciates: function V of Ia,, K, and A,. VU,, K,, A,) is the value function for the consumer’s utility max- (3) hr =~‘ S)k,.,+ i,, ooi o For a long time, thu technalngy imization problem. shack A, was the driving force in For present purposes, it is more useful to The core of the problem is to find V. To mast RRC models (hence, the frame the solution in terms of decision rules start, substitute the budget constraint (2) into ‘real’). lnth proponents and which prescribe i, and I, as functions of current the consumer’s utility function, then substitute opponents recognized this as the state variables, k,, K, and Ar: marginal products for W, and R, as described by Achilles heel of this line af research. equation I. The latter substitution implicitly i, = i(h,, K,, A,), I, = l(h,, K,, A,). One response has been the devel- defines the consumer’s rational expectations opment of models in which technol- These decision rules depend only on thestate f factor prices in terms of present and future 0 ogy is nat the only snnrce of uncer- variables which fully describe the position of values of aggregate labor and capital. In other tainty (the last sactiar contains twa the economy at the beginning oft and which, words, the consumer does not care about K, eoamplesi, though the citicism therefore, contain all of the inforanation needed goes deeper than simply claiming and 1., per se; they merely contain the same thnt there are other kinds of shocks to decide optimal levels of i, and I,. A great information as W, and R,.’ We now have (see Stadlec 1994, section IV.A.). deal of information about how the economy works—about the structure of the model, ‘The exact sequence of sobstitatons in other words—will he embedded in these E4E$mt1(A,FK(K~,L,)k, here is designed to hammer the functions when we find them. model iota the mnld required later In addition to capital, there may be many fnr a sptciic anwenical sniotan +A, ~.(K,, L, )l, — i,, I — method. For trample, i~coaldeasi- financial assets with a net supply of zero, hut, ly be eliminated from the pmblem since consumers are identical and the economy is closed, these assets would be redundant. In period 0, the consumer is choosing i and osivg 3, hot that would be incnnve- 0 t neat Intec Nevertheless, the prices of these assets are o. Rewrite utility as

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(4) te(A,F,, (K , I_Ø )h (6) V~,,K,,A, 0 0 +AOFL (K,,, L )l~—i , I — f ) = u(i(k,. K,, A,), l(k,. K,, A,). Ia,, 4 0 0 K,,L(K,,A,), A,) +$E, {ts’u(A,FK (K,, L,)k, +/3E~VUç,.K,,, A,, ,)~A,} +A,F~(K,, L,) 1,—i,, i—i,) } with (7) L(K,,A,) = l(K,,K,,A,) Examination of the second term in 4 reveals and I(K,, A,) = i(K,, K,, A,). an ianportant feature of the optimization problem; apart from the values of state Condition 6 says that, given expectations L(O, variables, the consumer will solve exactly the decision rules ie) and l(’) are optimal for same problem in period Iasin period 0. For consumers. The equations in 7 say that an optimal plan, this recursion is sumnna- expectations of aggregate labor supply and rized in the Bellman equation for the con- investment are consistent with individual sunner’s problem at I: decisions. Only a small number of examples can he solved analytically (for example, Long (5) VU,,K,,A,) and Plosser, 1983), so we must now turn to the computer = max {uo,, I,, Ia,, K,, L,,A,)

+ JIEV{UZ,,,.,, K,~,, A SO:LU.TIOI1 lay :flflyiflyfl To compare the t.inne-series behavior of the model’s equilibrium with the time-series The anaximization on the right is subject to behavior of actual economies—to evaluate the constraint (3) that connects investment us quantitative implications—requires that we and capital. Embedded in 5 is Richard simulate the model using specific functional Bellman’s deep insighn that if you know the forms and parameter values. The process of value of your prohlenn next period for the choosing parameter values, calibration, is variotas values of state variables, it is a rela- deferred to the next section, The next few tively simple matter—a static maximniza- sections illustrate solution procedures by don—to figure out the optimal action now. fully specifying, calibrating and solving Gary There are four kinds of variables in 5: Hansen’s (1985) indivisible labor mode!, an individual decision variables (i,, I,); an mdi- early landunark in the RBC literature. Hansen’s vidual state variable (In,); economy-wide state relatively sianple nnodel provides a clear illtas- variables (K,, A,); andan economy-wide variable tration, hut should not he taken as the state of loosen nod Erescnto 11 9951 would determined in t (b,).’ The staae variables are the art. The solution follows oneof the popular call I, an ‘nggnngatu decision nun’ cleternnined at the start of t or inherited fronn linear-quadratic approximation methods.’ able.’ Other sniution wathuds woold solve En, pricing rules that —1. The contemporaneously denermined specify II.nod R,nsforctnasnf econonny-wide variable L, appears because state variables rather thav fun on (for mathematical reasons evident below) nggragate decision role’ for I,. we have suhstiauted out Vv~and R,. These Subsequently the production function is in either case, the functions cnptare market-clearing prices \vould otherwise sum- assucned to he Cobb-Douglas: the sohstnnce of the nssowpton nI anarize the information contained in K, and rntnonl enpectntinns. L, that is relevant to the consumer’s decision. A,F(K,, L,) = c~’K~L~°. ‘Taylor ard llnlig 119901 camyann a Our task is to find a recursive competitive rawber of apprunches On solving equilibrium, that is, decision rules i(k,, K,, A,) The technology shock e” is driven by an stnchostic grnwth mrdtls. Other and I(k,, K,, A,) for the household, functions AR(I) process: papers in the sawn issue of the I(K,, A,) and L(K,, A,) detercnining aggregate Journal of Business nod Ecnnnmic (8) i =y)i,,,,+s, <1., investment and labor, and a value function Stutistics provide short descriptnns VU,, K,, A,) such that where the c~are independent and idcnticalb’ of the uunious salutian methods.

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distributed normal random variables that are the early RBC literature, would work like this: independent of all t —1 variables.’ It is easier Since agents are identical in this model, a to work with A, rather than A, as a state vari- Pareto optimum can be found by maximizing able from this point forward, utility (9) subject to society’s production possi- The utility function is specified in a bilities, ignoring market structure. Production somewhat unusual way that incorporates possibilities are described by the production indivisibility of labor inputs, the contribu- function, the process generating technology tion of Hansen’s paper, shocks, and thecapital accumulation equation. This is a much simpler problem. Since the

(9) U(c,, I— I,) = loge, + Bi— I,), model has no distortions, the Second Welfare Theorem applies: The Pareto optimal alloca- where B is a constant. The Bellman equation tion can be supported as a competitive equi- 5 becomes librium. Thus, the solution to the social planner’s problem replicates the outcomes (5’) V(k,,K,,A,) of a decentralized competitive system. Rather than taking the shortcut of solving = max{log(eFuK(K,,L,)k, the social planner’s problem, this section follows the more general method described in + e~ (K,, L, )l,— i,) Hansen and Prescott (1995) that also applies + B(l —1) to models with distortions.’ Two such models are briefly described in the section titled + PE{V(k,,,, K,,,, “Extending the Basic Model.” There are two keys to finding the value l-Tansen showed that using a representative function V using functional equation 5 or 5’. consumer with this utility function produces The first is approximation, described shortly the same competitive allocations as individual The second is the Contraction Mapping consuaners described in the following way Theorem, a fixed point theorem, which guar- Each consumer works either l~hours or not antees that for certain problems the following at all, but gets paid in either case. The prob- steps will converge to the value function V. ability of working, chosen by the consuaner, The theore,n does not actually apply to many is a,. Labor supply is determined indirectly RBC models, so there is no guarantee in gen- by choosing a, rather than directly by choos- eral, but this approach usually converges ing I,. Total labor time in the economy is anyway finding the correct value function. thus 1,,= a,!,. If the utility function of indi- 1. Choose an arbitrary function vidual consumers is ~ A,,). 2. The problem on the right-hand side of 5 (10) U(c, a,) = loge, +Aa, log(—l3, is now a static optianization. Solve it to

get decision rules i, = i(k,, K,, A,) and with A>0, then the representative consuaner !,= Ida,, K,, A,). Substitute these into the will have utility 9. By cnaking these anodifi- right-hand side to produce a new func- cations, 1-lansen hoped no improve on the tion, V’(k,, K,, A,). rather poor performance of the basic cnodel This specthcntinn dillens slighiy from 3. Replace to on the right-hand side of 5 (with divisible labor) in matching facts about Hansen’s, which assawes that the with V’(k,+,, K,,, A,,,), Return to step two a~elationshipsamong hours, ennployment, techrrlogy is x,F(K,, L), with e, unless \7’ and V’” are almost identical. lng’annmaliy distributed. Choice ni output and productivity nr Alll I prucnus rnnight be corsif Unfortunately, in general, step 2 will not oral punt of calibruton sirce a spec’ produce a function that can he written down thcuinn search is usuni(y port Of the Romlino to- F)nid the un/n.e Function in any compact way particularly given the nnerni process of nstimatng y. Hansen’s model could be solved using presence of the expectation in the middle of Tl,n ,nef nod. desoibad here am ir the shortcut of finding a Pareto optimum, the right-hand side of 5. This problem is the appeodis differs in detnils from than is, solving the social planner’s problem. addressed in Hansen and Prescott’s method Hansen and Prescott’s nlgrnithm, That approach, which was used extensivel in by solving a quadratic approximation of the

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model, rather than the full model. Variations one household. But the distinction between I, on the linear-quadratic approximation have and L, must be maintained because the been the most coananon method of solving household must behave as if it takes prices dynamic general equihhrium naodels, starting (W, and R,) as given, and these would be with Kydland and Prescott (1982). functions of L,, nor!,, if we had more than For simulation purposes, the model is one consumer, approximated by a Taylor series expansion So how should L, be handled? (Though it of the utility’ function (as it appears in 4 after does not appear in the model, we also need to all the substitutions) around the steady-state worry about I, for reasons that will become clear equilibrium values (K, L, A) that would occur momentarily) The mode! assuanes that house- if we set all the shocks to zero. In some models, holds have rational expectations about L, and the zero-shock equilibrium is not a steady 1,, so they recognize than equilibrium values of state, In these cases, a simple change of vari- these variables will satisfy the first-order con- ables usually produces the required steady ditions. In maxianizing the right-hand side of state. If, for example, the population were 5” at each iteration, the first-order conditions assumed to be growing, the anodel would be define a linear relationship among choice formulated in per capita terms. For Hansen’s variables I, and i,, aggregates L, and I,, and model, (5’) becomes state variables:

(5”) VU,, K,, A,) 0 = to~~+ u,,J, + u,,i, +

= max{z~Qz, + u ,I, + a,, Ia, + u ,K, + u,rA, 4 6

0 = a,, + a,,!, + u i, + to L, + PE{V(k,~n,K,,,, A,~,)~A,}} 0 51 + ce ,I, + u,,k, + ta~,K,+ ta:,A,. 4 where Z, = Ii A, Ia, K, i, I, I, L,]’. Including 1 as a state variable allows the quadratic The first-order conditions can be solved for approximation to be written as a single qua- I, and i, to get household decision rules spec- dratic form (see the appendix). ified in terms of state variables, as well as L, The beauty of the quadratic approximation and I,. However, if L, is substituted for I, and is threefold. First, one can guess (correctly) I, for i, in the first-order conditions (thus that the value function is quadratic. Second, imposing 7), the equations can be solved for it does not depend on the distribution of s, L, and I, as functions of state variables. These except for a constant that involves the covari- solutions can be interpreted as households’ ance matrix of s,. Third, this constant is not conditional expectations of aggregate labor essential for our analysis because it does not supply and investment, given the current values involve any of the state variables. Because the of state variables. Hansen and Prescott call constant is not essential, we can ignore it and these “aggregate decision rules.” The solutions the expectation along with it. For details, see for aggregate labor supply and investment Sargent (1987, section 1.8), but it is not difficult replace L, and I, in the household decision to see that the iterations described above will rules which then become functions of state always produce a quadratic if to is quadratic. variables alone.v This procedure ensures that condition 7 for a recursive competitive equilibrium is satisfied at eveny iteration. hmoacin.e Ectuiiibriinn Fond/Horn When the value function approximations Most of the pieces of a solution method converge, we have found a value function, have already beendescribed, but there is one decision rules for 1, and i,, and aggregate labor missing, namely how to handle L,. This is supply and investment functions that satisfy neither a state variable nor a decision variable equations 6 and 7byconstruction. The Con- ‘Though it appears hera formally, 1, of any agent. It is an aggregate outcome of traction Mapping Theorem does not apply to drops out of household decision households’ decisions. The aggregation hap- this particular dynamicprogramming problem, rains for this model, that is, pens to be trivial here because there is only but the algorithm does find thevalue function

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Calibration strategies are often much nnore Percentage Standard Deviations and complex than this, and thejustifications more Correlations with Output elaborate, but they always have the saane sitnple purpose, to select a plausible parauneter United States Model point at which to study the behavior of the S.D. Cart. S.D. Grr. anodel. Kydland and Prescott (1994) detail caliba-ation strategies and their own philosophy Output 1/6 100 176 LOU of calibration. Researchers often conduct ~onumption 1.29 0.85 0.51 0.86 infornnal sensitivity analyses, varying the Irnestmel 8.60 0.92 5.75 0.99 parameters whose values are uTmost uncertain. Capiaal 0.63 004 0.48 0.06 The most common such exercise seems to he to vary the risk aversion parameter in Flours 166 0/6 134 0.98 models in which utility is of the constant Produrtinity 1.18 0.42 0.51 087 relative risk aversion form.

fl55J/35 Once the value function is found and agents’ decision rules are known, it is rela- if given an appropriate to. ‘I’he results are, of tively simple to sinnulate the model. The course, subject to the caveatthat if shocks take equations of motion for A, and K, along with the economy too far from the steady state, the the aggregate decision rule for I, are a system quadratic approximation may not be accurate. of three linear difference equations in three unknowns that can easily he simulated. (Recall that K,= Ia, in equilibrium.) Starting tnni,oieiitm’tm ~ values are chosen for the state variables and There are six unknown parameters in innovations ~ are drawn randomly

the description of Hansen’s model, 6, 8, ~, B. The real and artificial data are filtered yand the variance of s. Hansen chose values hy taking logarithans and detrending with as follows, Given Cobb-Douglas technology the Hodrick-Prescott filter,” There are two o is capital’s share of outpun. He used an esti- reasons for filtering. First, the model is mate of 0 = 0.36 based on rinne series for the intended to explain phenomena at business U.S. econoany A choice of 8=0.025 (corre- cycle frequencies, and the Hodrick-Prescott sponding to an annual depreciation rate of filter highlights those frequencies. The 10 percent) was chosen as “agood compromise models are not intended to match long-run given that different types of capital depreciate growth facts, so it would be unfair to coanpare at different rates.” A steady-state annual low-frequency movements in the data with riskless real interest rate of 4 percent would those from the model. A filter that removes

be imphed by ~= 0.99. Hansen chose 8=2.85, low-frequency movements in the data and which corresponds to an apparently arbitrary model output allows the anodel to he com- value of A=2 in 10, combined with !,=0.53. pared to phenomena in the data it was The 0.53 value equated steady-state hours in designed to explain. Hansen’s divisible and indivisible labor models. Second, many macroeconoanic none The standard deviation of ewas chosen so series may not he snarionary Ifthis is true, that, for the artificial economy with indivisible their second moments do not exist. Though labor, the standard deviation of detrended it would still be possible to generate sample “This differs sighiy from ffaosen’s output wotald be about the saane as that of second moments, there would be no reason ualoe (0.0071 2i because technol- detrended GNP for the U.S. economy. A value to think that another set of observations on ogy shocks are spocthed in a differ- = 0.00717 meets this criterion.” A value of the same economy would produce sample ent watt y=0.95 was the first-order autocorrelation second moments similar to the first set. “See Kydlnnd and Prescott (1912, coefficient of the Solow residuals for the U.S. Thus, therewould be no point in trying to p. 1362). economy produce models that matched a particular set

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of sample second moments. Filtering that indivisible labor model and 1.4 for the U.S. induces stationarity removes this problem in economy He argued that the indivisible labor thesense that samples drawn from the same model showed promise because standard data-generating process would be expected to models chronically produced ratios that were produce similar sample second moments. too small. Since the real world is not charac- There are other ways to filter the data, terized by fully indivisible labor, he argued, it by taking first differences, for example. While was good that the ratio for the U.S. economy the Hodrick-Prescott filter is somewhat con- lay between the two models’ predictions. troversial (Cogley and Nason, 1995; King and A few additional tools have been used Rebelo, 1993), proponents argue that it does to evaluate model output. A fairly common a good job of emphasizing the movements in approach is to compare impulse response thedata that most macroeconomists would functions from the model with those from call business cycle movements. For example, a vector-autoregression on the data. Watson Kydland and Prescott (1990) say “...the (1993) has proposed aprocedure for evaluating implied trend path for the logarithm of real the fit of a calibrated model. A variety of new GNP is close to the one that students of busi- approaches is discussed in Pagan (1994), ness cycles and growth would draw through which is a thought-provoking overview of a time plot of this series.” Cogley and Nason calibration exercises. argue, on the other hand, that if the data are an integrated process, “the filter can generate business cycle periodicity and co-movement EXTENDING THE cc”. cam. ccc, -,m-n,,c”” even if none are present in the original data. naaaa,, ,flubpitfliii, In this respect, applying the HP filter to an The solution technique described above is integrated process is similar to detrending more powerful than needed to solve Hansen’s a random walk.” model, but makes it possible to outline how The results in Table 1 summarize 500 more sophisticated models can be handled. I’ve simulations of 115 periods each. The statistics chosen two examples from areas in which are calculated from natural logarithms of each extensive contributions have been made. These series detrended using the Hodrick-Prescott particular papers fit easily into the framework filter with a parameter of 1600. Each simula- developed above. tion chooses A, from the unconditional distribution of A,. This is more difficult for K,, so I simulate the model for 100 periods, starting FIscal Policy K at its steady-state value. The value after One obvious road to follow in generalizing 100 periods is used as K, for the simulations the basic RBC model is to add fiscal policy to reported in Table 1.” In other words, I simu- the model. Though obvious, this was not late the model for 215 periods and throw away initially an easy road because aminimally the first 100 in order to get a representative realistic model requires distorting taxes. The distribution of starting capital stocks. This first generation of solution methods that procedure allows thestatistics reported in the relied on solving the social planner’s problem table to be interpreted as averages across 500 are not appropriate for models in which the identical artificial economies with indepen- Second Welfare Theorem is not true. dent realizations of the technology shocks. A recent contribution in this area is Evaluation of RBC models’ output has McGrattan (l994a, 1994b), which developed usually been informal. Hansen, for example, a model in which agents face stochastic tax was interestedin how well this indivisible labor rates on both labor and capital income. ‘‘The velees reported in columns model performed relative to a more standard McGrattan (1994b) modified the indivisible three and four of Table 1 differ very divisible labor model (with utility given by labor model as follows. The government uses slightly from those mported by log c, — A log I,). He noted that the standard tax revenue to fund government purchases Hansen, Sampling vnninfion and a deviation of hours relative to that of produc- and lump-sum transfers. Thus, theconsumer’s slightly different pracess geaerafiag tivity in the divisible labor model was only budget constraint (2) is replaced by 2, probably account for the differ’ about 1 compared to 1.34/0.51 = 2.6 for the ences.

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c,+ i, (1—r,)R,k, money balances carried over from last period, mao, plus the lump-sum transfer of seignior- +8rka+(1~Pa)Wa!r+r~,, age revenue, gM,~. where the capital tax allows a depreciation The household budget constraint is

deduction and ~, is the lump-sum transfer. P,c, + F~i,+ rn = P,WcI, + The tax rates, r and 0, and government purchases, g~,are exogenous state variables (like ÷111,~,+ gM,,. the technology shock A,). The size of the The last term on the left represents money transfer is determined by thegovernment balances carried into t+1. budget constraint which imposes period-by- Because positive money growth results period budget balance: in inflation, it is necessary in this model to make a change of variables for the zero- = r,(R,— cS)K,+ ~WrLr~ G,. shock path to be stationary. (Recall that we need a steady state around which no form a The representative agent must form expec- quadratic approximation.) There is a steady tations about a~,assuming that her decisions state when the model is written in terms of have no influence overgovernment revenue. iii, = m,/M, and P, = P,/M,. The two con- Thus, K, and L, rather than Ia, and I, appear straints (transformed by the change of vari- in the government budget constraint. In this ables) areused to eliminate c, and I, from the model, there arefour exogenous state variables, consumer’s utility function, leaving an opti- 0 A,, ç, a and G,. McGrattan calibrated the mization over rn (money holdings carried stochastic processes for these variables by into t+1) and i,. estimating a first-order autoregression for The money supply M, is an exogenous each. She argues that her results indicate the state variable, and is added to the list of equa- addition of these fiscal shocks to the basic tions of motion. The endogenous state vari- indivisible labor model brings it into “much ables are Ia,, K, and m,,. An equation of motion better agreement with the data.” that says that this period’s purchases of money To solve the model using the procedure become next period’s money state variable is outlined above, substitute the new budget added to the list for endogenous state variables constraint into the utility function, as before.” (the B matrix in the notation of theappendix). Then substitute the right-hand side of the The aggregates that must be determined

government budget constraint for ~. Add the are an investment function I (Ar, M,,K,) and an assumed stochastic processes for tax rates and aggregate price function P0,.,, M,,K,). Since M, government purchases to the list of equations is exogenous, there is no aggregate that cor- of motion (the matrix A in the language of responds directly with the decision variable the appendix). na,. The aggregateprice level P, serves this role instead: At each iteration, I(Aa,M,,K,) and P(Ar,Ma,Kr) are chosen by setting i,=I, and Mcccv 1 (that is, rn,=M,) in the first-order con- Cooley and Hansen (1989) studied an ditions and solving for ‘a and ~aas functions RBC model with a cash-in-advance constraint. of state variables. (For Hansen’s model, by In the simpler version of their model, the contrast, we set ~a=Iaand i,=L, in the first- money supply M, grows at a constant rate, g: order conditions, then solved for I, and L,.) Details on how to handle this slight variation M, =(1+g)M,,. can be found in Cooley and Hansen (1989).

Households’ consumption decisions must satisfy the cash-in-advance constraint, CIC NCKUPIG K RBC methods have found application Paca m,_, + gM,.., in a wide spectrum of questions in business

“ Mclratean actually used e method which says that the nominal value of con- cycles, monetary economics, open economy described in Mclmttaa (1994c). sumption purchases, P,c,, must not exceed macroeconomics and finance. The linear-

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quadratic approach described in this paper is ______and ______. “Time to Build end Aggregate a popular tool that can be applied to a broad Fluctuaflons,” Econorraetrico (November1982). cross—section of the questions posed in this Lucas, Robert E. “Methods and Prnbloms in Busiaess Cycle Theory,” literature. More generally, careful study of a Journal of Money Credit cod Banking (November1980, part 2). specific method illustrates how equilibrium “Understanding Business Cycles,” Carnegie-Rochester conditions must be tied to optimization to SerIes on Public Policy (1977). find an equilibrium numerically McCallum, Bennett T. “Real Business Cycle Models” in Robert]. Barrn, ed., Modern Business Cycle Theory. Harvard University Press, 1989. ~i’fltt ncc9 III NC5 5.. McGrattan, Ellen R. “A Progress Report on Business Cycle Models,” Blanthard, Olivier Jean, and Stanley Fischer. Lectaires on Federal Reserve Bank of Minneapolis Quarterly Review (fall 1994). Macroeconomics. The MIT Ptess, 1989. ______“The Macroeconomic Effects of Distortianary Taxation,” Caglny, Timothy, and Jomes M. Nasan. “Effects of the Hodrick’Prescatt Journal of Monetary Economics (June 1994). Filter on Tread and Difference Stationary Time Series: Implications far Business Cycle Research,” Journal of Economic Dynamics and Control ______. “A Note an Connpufing Cnmpefltive Equilibria in Unear (January/February 1995). Models,” Journal of Economic Dynamicsand Con Ira! (January 1994). Coaley, Thomas F, and Gary D. Hansen. “The Inflation Tax in a Real Pagan, Adrian. “Calibration and Econometric Research: An Overview,” Business Cycle Model,” The .4meriron EconomicReview (September JournolofApplied Econometrics (December 1994, supplement). 1989). Plosser, Charles I. “Understanding Real Business Cycles,” Journalof Hansen, Gary 0. “Indivisible Labor and the Business Cycle,” Journal of Economic Perspectives (summer 1 989). Monetary Economics (November 1985). Sargent, Thomas 3. Dynamic Mocraecanomic Theory. Harvard

______and Edward C. Prescott. “Recursive Methods far Cnmputing University Press, 1987, Eqnilibria of Business Cycle Models,” chapter 2 in Thomas F. Cooley, ed., Snadler, George W. “Real Business Cycles,” Journal of Economic Frontiers of Business Cycle Research. Princeton University Press, 1995. Literature (December 1994). King, Robert G., and Sergio I Rebelo. “Lam Frequency Filtering and Stakey, Nancy L, Robert E. lucas and Edward C. Prescott. Recursive Real Business Cycles,” Journal of Economic Dynomicsond Control Methods in EconomicDynamics. Harvard University Press, 1989. (January/March 1993). Taylor, Jnhn B., and Harald Uhlig. “Salving Nonlinear Stochastic Growth Kydland, Finn E., and Edward C. Prescott. “The Computational Models: A Comparison afAlternative Solution Methods,” Journal of Experiment: An Ecanametaic Tool,” Federal Reserve Bank of Business cadEconomic Statistics (January 1990). Minneapolis Staff Report /78 (August1994). Watson, Mark W. “Measures of Fit for Calibrated Models,’ Joumol of ______and ______- “Business Cycles: Real Facts and a Pabncol Economy (December 1993). Monetary Myth,” Federal Reserve Book of Minneapolis Review (spring 1990). lang, John B., and Charles I. Plasser. “Real Business Cycles,” Journal of PoliticalEconomy (February 1983).

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FINDING THE VALUE FUNCTION

N’okttinn or, combining all three, Letp(x) be the number of rows in the vector x. Let zbethevector of exogenous state variables, s and S the vectors of individual , A 0000 s 8 andaggregate endogenous state variables,d the ~ = CZ= B B, 13, B B, S + 0 5 4 vector of decision variables, and V the vector Ba 0 B,+ B, 0 B + B, cJ 0 4 of “aggregate decision variables.” For reasons D of convenience, the first element of zis always 1. Let The matrix C is p(X) x p(Z). For Hansen’s model, z Z,=ll )La h, K i, I~i~I.]’, 5 xS A_[b0Lor’ and Z= d 13 B=[OOC1—ö)0l0001.

Using — to denote next-period values, the ,.~ r “ “ constraints are inn (icncrw corm at me nobiern Using the notation in Hansen and Prescott (1995), the problem we wish to solve in order to find an equilibrium is

~=Az+e=[A 000 01 S +e - d (Al) V(X) = max r(Z)+$E(V(X)Jz} 13 £

subject to X=CZ+ 0 5 0

~=BZ=[Ba B, B, 134 B,] S with D13(z,S). d All nonlinear constraints have been 13 substituted into the return function r. As mentioned above, the D=D(z, 5) equation summarizes agents’ rational expectations about aggregate values of their own decision

- variables, for example, labor supply Since it S = [B~ B, B, B B,1 S does not involve the choice variables d, it is not 4 13 really a constraint, but will be used to derive 13 decision rules that depend only on X rather than both X and 13. This is why the problem z specifies “with” rather than “subject to.”

S [~~a0 B,+B, 0 B +B,] S 4 ci Thereturn function will be approximated P around a steady-state solution to the model when e=0. The steady-state solution solves

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the following set of p(Z) equations in quadratic problem. The value function for elements of Z: this kind of problem is known to be quadratic. The first function, in the sequence can be any X = CZ (excluding the s rows) quadratic function, hut convergence is some- 0= (Z) + $v~(Z) [I —f3Bj’ B, times sensitive to the choice. 1-lansen and d=D Prescott recommend V”= “qI, where w~is a small negative number. 5 = S. If yc” is the p(X) x p(X) matrix of the ~ah ‘l’he first vector equation contains quadratic value function approximation, the 5 p(z)+p(S) scalar equations.’ The second (in- lb approximation is given by vector equation contains p(cJ) scalar equations max Z’QZ +$k’v”tk that are the first-order conditions for ci. They take into account the recursive nature of Al, subject to k = C’Z than, is, the fact that ci affects every future with D=GX. period by changing ~. B, is the derivative of lIZwith respect to s, that is, the p(s) columns Theequation 13 = GXsummarizes the “aggregate t of B that correspond to the s elements of Z decision rules” implied by V ” (the derivation (the p(z)+ I through p(z)+p(s) elements). of these is described below). As explained Similarly B,a consists of the p(cL) columns of above, this equation is not a constraint. B that correspond to the ci positions in Z First eliminate the equations of motion (the p(X)+1 through p(X)+p(cd) elements). for state variables to get an equivalent problem

max Z’QZ + $Z’C’V”CZ with P = GX Tne ~‘~nrc ~ncx rz”~- no 0110 102Pm f/win or Letx be a Pr row vector and let ‘V be a maxZ’RZ with P = CX. (/n-I ) x (k+],) matrix. Partition “V so that “ra is a scalar. Then Dc*cisio Namericnl methods las salving these equatans osanily eaqaiae daopping the equoton that [ [q~ ‘v,1 11 The first-order conditions for this problem 1~II “ ‘ I l=’P,, +(W/ +P,,)x describes the avolutior of the state t are the second-to-last p(cl) rows (the p(X) + I [xJ [‘P, P,,j xj - 5 vovioble 1 Most economists have through p(X) +p(cI) rows) of little trouble solving this equotov 0=(R+R’)Z. ovnlytcnlly. inweven. Thus, any quadratic function from R°into R ‘In sonne cincumstnnces U, is close of a vector x can he written as a single qua- Write these p(cl) equations as to singolnr. which leads to cvvver- dratic form in llx’I by collecting the constant 0 = UZ gence problems. An olternotine is terms in ~ the linear terms in (‘l’/+t )x, 0 to impose the equilibrium condition and the squared terms in x’t,,x. where U is p(d) xp(Z). B”d before salving fort which U The quadratic approxianation to the return Partition to confortn with the compo- lesleod requires ill,±U,) One function is subsequently written in this way nents of Z: caveat is eeeded, however. Ibis 1 Denote the quadratic approximation to r-(Z) z vanioton leads to individual deri- by Z’QZ. The matrix involves derivatives sion sales which have the ‘wrong Q S that can be calculated either analytically or weights and and 0, though the 0=U, U, U, U, U,] S numerically sum of the weights is correct lsn ci that oggeegnte decision roles are connect). If, for some tensor, indi- D twintrt’utcno f/tnt Constnf/nb and vidual decision ‘ales one needed, the carnect meg hts can be found by 54w/Or/un fltnddkin” Solving for d yields’ choosing them so that the finn- The reanainder of this appendix describes order conditions ased in finding the the process of finding a sequence of approxi- c/=—U~’Uaz+V,s +U,S+U,P). steody stole any satisfied with B nt mations to the value function that converges its eqailibnium valves hut with to the actual value function for the linear- The aggregate decision rules are obtained d D.

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by setting d=P and s=S in the first-order The value function is updated until 11””’ and conditions. This leads to V” are sufficiently close. The (1,1) elennent

of V tends to converge slowly hut, because it zl does not enter the fmrst-order condmt,ons D=—(L~+ ELY’ [U, 0 (U, + U,)] S (UZ=0), that element can he ignored when SJ testing for convergence, unless the value

= Gx. function itself is needed, Substituting this into the previous equation gives individual decision rules that are functions of state variables only:

ci = — U~(U,z + Its + U,S + U,.GX) U, U,J+U,G)X

= HX. Substitute

[ci [H1~

LD [Gj

= JX into the objective function as follows to obtain the new value function:

Z’~=H1R X [JXJ ]X

=[x’ X7’]RH []X

= {x’ X’]’][~’ Rn, 1 [ X [R,a R,,J[.JX

= [X’Ru, + X’J’ R,m X’R , + X’]’R,,] 5 [x]

= X’Rma X + X’J’ R,aX + X’R,,JX + X’]’R,,JX

=X’IRnn +J’R,, +Rn,I+I’R,,J]X

=XtV(~5+mvX.

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