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. FEDERAL RESERVE SANK OF ST. LOUIS 49 A I ‘I II~ MARCH/APRIL 199$ 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$mt1(A,FK(K~,L,)k, here is designed to hammer the functions when we find them. model iota the mnld required later In addition to capital, there may be many fnr a sptciic anwenical sniotan +A, ~.(K,, L, )l, — i,, I — method. For trample, i~coaldeasi- financial assets with a net supply of zero, hut, ly be eliminated from the pmblem since consumers are identical and the economy is closed, these assets would be redundant. In period 0, the consumer is choosing i and osivg 3, hot that would be incnnve- 0 t neat Intec Nevertheless, the prices of these assets are o. Rewrite utility as FEDERAL RESERVR RANK OF St. LOUIR 50 o i~i~ MARCH/APRIL 199$ (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.