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Real Business Cycles Real Business Cycles JesusFernandez-Villaverde University of Pennsylvania 1 Business Cycle U.S. economy uctuates over time. How can we build models to think about it? Do we need di erent models than before to do so? Traditionally the answer was yes. Nowadays the answer is no. We will focus on equilibrium models of the cycle. 2 Business Cycles and Economic Growth How di erent are long-run growth and the business cycle? Changes in Output per Worker Secular Growth Business Cycle Due to changes in capital 1/3 0 Due to changes in labor 0 2/3 Due to changes in productivity 2/3 1/3 We want to use the same models with a slightly di erent focus. 3 Stochastic Neoclassical Growth Model Cass (1965) and Koopmans (1965). Brock and Mirman (1972). Kydland and Prescott (1982). Hansen (1985). King, Plosser, and Rebelo (1988a,b). 4 References King, Plosser, and Rebelo (1988a,b). Chapter by Cooley and Prescott in Cooley's Frontier of Business Cycle Research (in fact, you want to read the whole book). Chapter by King and Rebelo (Resurrection Real Business Cycle Mod- els) in Handbook of Macroeconomics. Chapter 12 in Ljungqvist and Sargent. 5 Preferences Preferences: 1 t t t t E0 (1 + n) u ct s ; lt s tX=0 t t for ct s 0; lt s (0; 1) 2 where n is population growth. Standard technical assumptions (continuity, di erentiability, Inada con- ditions, etc...). However, those still leave many degrees of freedom. Restrictions imposed by economic theory and empirical observation. 6 Restrictions on Preferences Three observations: 1. Risk premium relatively constant CRRA utility function. ) 2. Consumption grows at a roughly constant rate. 3. Stationary hours after the SWW Marginal rate of substitution be- ) tween labor and consumption must be linear in consumption. uc t = wt s u ) l t t t ct s f lt s = wt s ) t t t c0f lt s = w0 Explanation: income and substitution e ect cancel out. 7 Parametric Family Only parametric that satisfy conditions (King, Plosser, and Rebelo, 1988a,b): (cv (l))1 1 if > 0, = 1 1 6 log c + log v (l) if = 1 Restrictions on v (l): 1. v C2 2 2. Depending on : (a) If = 1, log v (l) must be increasing and concave. 1 (b) If < 1, v must be increasing and concave. 1 (c) If > 1, v must be decreasing and convex. 3. v (l) v (l) > (1 2 ) v (l) 2 to ensure overall concavity of u: 00 0 8 Three Useful Examples 1. CRRA-Cobb Douglass: 1 t t 1 ct s 1 lt s 1 1 t t E0 (1 + n) 1 tX=0 2. Log-log (limit as 1): ! 1 t t t t E0 (1 + n) log ct s + log 1 lt s tX=0 n o 3. Log-CRRA t 1+ lt s 1 t (1 + n)t log c st E0 8 t 9 > 1 + > tX=0 < = > > : 9 ; Household Problem Let me pick log-log for simplicity: 1 t t t t E0 (1 + n) log ct s + log 1 lt s tX=0 n o Budget constraint: t t t t t t 1 ct s + xt s = wt s lt s + rt s kt s , t > 0 8 Complete markets and Arrow securities. We can price any security. 10 Problem of the Firm I Neoclassical production function in per capita terms: 1 t zt t 1 t t yt s = e kt s (1 + ) lt s Note: labor-augmenting technological change (Phelps, 1966). We are setting up a model where the rm rents the capital from the household. However, we could also have a model where rms own the capital and the households own shares of the rms. Both environments are equivalent with complete markets. 11 Problem of the Firm II By pro t maximization: 1 1 zt t 1 t t t e kt s (1 + ) lt s = rt s z t 1 t t t (1 ) e tkt s (1 + ) lt s = wt s Investment xt induces a law of motion for capital: t t 1 t (1 + n) k s = (1 ) kt s + xt s t+1 12 Evolution of the technology st = zt zt changes over time. It follows the AR(1) process: zt = zt 1 + "t "t (0; 1) N Interpretation of and : 13 Arrow-Debreu Equilibrium . t t t A Arrow-Debreu equilibrium are prices pt(s ); wt(s ); rt(s ) and f gt1=0;st St t t t 2 allocations c^t(s ); lt(s ); kt(s ) t1=0;st St such that: f g 2 b b b b b t t t t 1. Given p^t(s ) t1=0;st St; c^t(s ); lt(s ); kt(s ) t1=0;st St solves f g 2 f g 2 b b 1 t t t t max E0 (1 + n) log ct s + log 1 lt s c^ (st);l (st);k (st) t t t t1=0;st St tX=0 n o f g 2 b b 1 t t t s.t. p^t(s ) ct(s ) + (1 + n) kt+1 s t=0 st St X X2 1 t t t t t pt(s ) wt(s )lt(s ) + rt(s ) + 1 kt+1 s t t tX=0 sXS 2 t b b ct(s ) 0 forb all t 14 t t 2. Firms pick lt(s ); kt(s ) t1=0;st St to minimize costs: f g 2 1 1 bzt bt 1 t t t e kt s (1 + ) lt s = rt s z t 1 t t t (1 )be tkt s (1 + )blt s = wt s b 3. Markets clear: b b b t t ct(s ) + (1 + n) kt+1 s = z t 1 t t 1 t 1 e tkt s (1 + ) lt s b + (1 ) kt s b for all t;allst St b b 2 b 15 Sequential Markets Equilibrium I. We introduce Arrow securities. t t t t 1 Household problem: ct(s ); lt(s ); kt(s ); at+1(s ; st+1) st+1 S t t 2 t=0;s S solve n o 2 1 t t t t max E0 (1 + n) log ct s + log 1 lt s tX=0 n o i t t t t s.t. ct(s ) + (1 + n) kt+1 s + Qt(s ; st+1)at+1(s ; st+1) t st+1 s Xj t t t b t t wt(s )lt(s ) + rt(s ) + 1 k s + at(s ) t+1 t t t ct(s ) 0 for all t; s S b b 2 a (st; s ) A (st+1) for all t; st St t+1 t+1 t+1 2 Role of A (st+1): t+1 16 Sequential Markets Equilibrium II t A SM equilibrium is prices for Arrow securities Qt(s ; st+1) t1=0;st St;s S, f g 2 t+12 i t t t t 1 allocations c^ (s ); lt(s ); kt(s ); a^t+1. (s ; st+1)b and in- t s S t t t+12 t=0;s S t t n o 2 put prices wt(s ); rbt(s ) t1b=0;st St; such that: f g 2 1. Given Q (st; s ) and w (st); r (st) b t bt+1 t1=0;st St;s S t t t1=0;st St f g 2 t+12 f g 2 b t t t t b 1 b c^t(s ); lt(s ); kt(s ); a^t+1(s ; st+1) solve the prob- st+1 S t=0;st St n o 2 2 lem of theb household.b t t 2. Firms pick lt(s ); kt(s ) t1=0;st St to minimize costs: f g 2 1 1 bzt bt 1 t t t e kt s (1 + ) lt s = rt s z t 1 t t t (1 )be tkt s (1 + )blt s = wt s b 3. Markets clear for all t; all st St b 2 b b t t z t 1 t t 1 t 1 ct(s )+(1 + n) k s = e tkt s (1 + ) lt s +(1 ) kt s t+1 b b b b b 17 Recursive Competitive Equilibrium Often, it is convenient to use a. third alternative competitive equilib- rium concept: Recursive Competitive Equilibrium (RCE). Developed by Mehra and Prescott (1980). RCE emphasizes the idea of de ning an equilibrium as a set of func- tions that depend on the state of the model. Two interpretation for states: 1. Pay-o relevant states: capital, productivity, ..... 2. Other states: promised utility, reputation, .... Recursive notation: x and x . 0 18 Value Function for the Household . Individual state: k: Aggregate states: K and z: Recursive problem: v (k; K; z) = max log c + log (1 l) + (1 + n) Ev k0;K0; z0 z c;x;l j n o s:t: c + x = r (K; z) k + w (K; z) l (1 + n) k = (1 ) k + x 0 (1 + n) K = (1 ) K + X (K; z) 0 z0 = z + "0 19 De nition of Recursive Competitive Equilibrium A RCE for our economy is a value function v (k; K; z), households policy functions, c (k; K; z) ; x (k; K; z) ; and. l (k; K; z), aggregate policy func- tions C (K; z) ;X (K; z), and L (K; z), and price functions r (K; z) and w (K; z) such that those functions satisfy: 1.
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