Real Business Cycle Theory

Real Business Cycle Theory Barbara Annicchiarico Università degli Studi di Roma "Tor Vergata" April 202

General Features I Theory of uctuations (persistence, output does not show a strong tendency to return to its long-run trend) I Business cycles driven by technology shocks (real shocks) I Focus on the real side of the economy I I I Quantities (aggregate production, consumption, hours, investments etc.) Relative prices (real wage, real interest rate) Classical dichotomy: money is a veil (nominal variables do not a ect real variables)

General Features I Absence of frictions or imperfections and symmetry I I I I I I I Perfect competition in all markets All prices adjust instantaneously Rational expectations No asymmetric information The competitive equilibrium is Pareto optimal Firms are identical and price takers In nitely lived identical price-taking households I Seminal papers: Kydland and Prescott (982), Nelson and Plosser (982), Long and Plosser (983), Prescott (986).

Firms and Technology The production function is a Cobb-Douglas y t = A t k α t h t α y t production (real) A t stochastic technology process (technology innovation is Hicks neutral) k t capital stock h t hours of work A > 0, 0 < ρ z <, 0 < α < A t = A exp z t z t = ρ z z t + ε z,t ε z,t iid.n(0, σ 2 z )

Firms and Technology Firms hire workers at wage w t and rent capital from the households at a rental rate r t Problem of the typical rm (max pro ts) (N.B. This is a static problem!) max fk t,h t g By pro t maximization: αa t kt α ht α {z } PMK 0 B @A t kt α ht α C w {z } t h t r t k t A y t = r t ( α) A t kt α ht α = w t {z } PML Firms hire labour and capital until the (real) wage rate is equal to marginal product of labour and (real) rental rate of capital is equal to marginal product of capital. Factor payments exhaust all output: y t = w t h t + r t k t

Households and Preferences The representative household maximizes max E 0 fc t,l t,k t+ g t=0 β t [u(c t ) + V (l t )] c t consumption (real) l t leisure! l t = h t β is the time discount factor Households earn a wage w t and own the rms, so they receive rents r t k t. The budget constraint is c t + k t+ = w t h t + r t k t + ( δ)k t 8t > 0 δ rate of capital depreciation Remarks: k t is a predetermined endogenous variable; w t and r t are taken as given (they depend on the tech process A t )

Households and Preferences Solve the household intertemporal problem. De ne the Lagrangian function: L 0 = E 0 β t [u(c t ) + V ( h t )]+ +λ t=0 t [w t h t + r t k t + ( δ)k t c t k t+ ] λ t is the Lagrange multiplier. At the optimum u 0 (c t ) = λ t V 0 ( h t ) = λ t w t λ t = βe t λ t+ (r t+ + δ)

Households and Preferences Labour supply: V 0 ( h t ) u 0 (c t ) = w t The marginal rate of substitution between consumption and leisure is equal to the wage. The time path of consumption is described by the stochastic Euler equation: u 0 (c t ) = βe t u 0 (c t+ ) (r t+ + δ) Household equates the cost from saving one additional unit of today s consumption to the bene t of obtaining more consumption tomorrow! consumption smoothing Consumption depends upon expected future wealth as opposed to current income.

Households and Preferences The transversality condition lim s! βs u 0 (c t+s ) k t+s = 0 It provides an extra optimality condition for the consumer s intertemporal optimization problem: if the time horizon were t + s, then it would not be optimal to have any capital left at time t + s (if consumed it would give a discounted utility of β s u 0 (c t+s ) k t+s at time t).

Functional Forms We assume u(c t ) = log c t V ( h t ) = ψ log( h t ) where ψ > 0 is a preference parameter.

Equilibrium Given an initial level of capital k 0 and the exogenous process governing A t, a competitive equilibrium is characterized by the sequence fc t, h t, y t, i t, k t+, w t, r t gt=0 satisfying the following conditions. The consumption Euler equation from the household s problem The labour supply equation = βe t (r t+ + δ) c t c t+ ψc t = w t ( h t ) The labour demand and the capital return ( α) A t k α t h α t = ( α) y t h t = w t αa t k α t h α t = α y t k t = r t

Equilibrium The production function y t = A t k α t h t α The capital accumulation equation i t = k t+ ( δ)k t The aggregate resource constraint of the economy y t = c t + i t Tech stochastic processes A t = A exp z t z t = ρ z z t + ε z,t

Steady State. c t = βe t c t+ (r t+ + δ) ) = β (r + δ) 2. ψc t = w t ( h t ) ) ψc = w ( h) 3. ( α) y t h t = w t ) α = wh/y 4. α y t k t = r t ) α = rk/y 5. y t = A t kt α ht α ) y = Ak α h α 6. i t = k t+ ( δ)k t ) i = δk 7. y t = c t + i t ) y = c + i 8. A t = A The steady state solution solves the above system of equations

Solution strategy The model does not have a paper and pencil solution. I What to do:

Solution strategy The model does not have a paper and pencil solution. I What to do:. Find the necessary equations characterizing the RE equilibrium - OK

Solution strategy The model does not have a paper and pencil solution. I What to do:. Find the necessary equations characterizing the RE equilibrium - OK 2. Solve for the deterministic steady state - OK

Solution strategy The model does not have a paper and pencil solution. I What to do:. Find the necessary equations characterizing the RE equilibrium - OK 2. Solve for the deterministic steady state - OK 3. a) Loglinearize the intra and intratemporal optimality conditions around the deterministic steady state (apply linear methods); b) use the non-linear model (apply non-linear methods perturbation methods based on second-order approx or projection methods)...

Solution strategy The model does not have a paper and pencil solution. I What to do:. Find the necessary equations characterizing the RE equilibrium - OK 2. Solve for the deterministic steady state - OK 3. a) Loglinearize the intra and intratemporal optimality conditions around the deterministic steady state (apply linear methods); b) use the non-linear model (apply non-linear methods perturbation methods based on second-order approx or projection methods)... 4. Calibrate parameter values (α, β, δ, ρ z ) and critical ratios

Solution strategy The model does not have a paper and pencil solution. I What to do:. Find the necessary equations characterizing the RE equilibrium - OK 2. Solve for the deterministic steady state - OK 3. a) Loglinearize the intra and intratemporal optimality conditions around the deterministic steady state (apply linear methods); b) use the non-linear model (apply non-linear methods perturbation methods based on second-order approx or projection methods)... 4. Calibrate parameter values (α, β, δ, ρ z ) and critical ratios 5. Compute the policy functions

Solution strategy The model does not have a paper and pencil solution. I What to do:. Find the necessary equations characterizing the RE equilibrium - OK 2. Solve for the deterministic steady state - OK 3. a) Loglinearize the intra and intratemporal optimality conditions around the deterministic steady state (apply linear methods); b) use the non-linear model (apply non-linear methods perturbation methods based on second-order approx or projection methods)... 4. Calibrate parameter values (α, β, δ, ρ z ) and critical ratios 5. Compute the policy functions 6. Analyze the model plotting impulse responses, computing moments and running stochastic simulations.

Log-linearized model. ĉ t = ĉ t+ ˆr t+

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt 3. ŷ t bh t = ŵ t

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt 3. ŷ t bh t = ŵ t 4. ŷ t ˆk t = ˆr t

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt 3. ŷ t bh t = ŵ t 4. ŷ t ˆk t = ˆr t 5. ŷ t = Â t + α ˆk t + ( α)ĥ t

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt 3. ŷ t bh t = ŵ t 4. ŷ t ˆk t = ˆr t 5. ŷ t = Â t + α ˆk t + ( α)ĥ t 6. î t = ˆk δ δ t+ δ ˆk t

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt 3. ŷ t bh t = ŵ t 4. ŷ t ˆk t = ˆr t 5. ŷ t = Â t + α ˆk t + ( α)ĥ t 6. î t = ˆk δ δ t+ δ ˆk t 7. ŷ t = c y ĉt + i y ît

Log-linearized model. ĉ t = ĉ t+ ˆr t+ 2. ĉ t = ŵ t h h ĥt 3. ŷ t bh t = ŵ t 4. ŷ t ˆk t = ˆr t 5. ŷ t =  t + α ˆk t + ( α)ĥ t 6. î t = ˆk δ δ t+ δ ˆk t 7. ŷ t = c y ĉt + i y ît 8.  t = ρ z  t + ε z,t

Log-linearized model I By substitution, reduce the problem dimension so as to obtain a system which can be written as 0 0 ĉ t+ ĉ t @ ˆk t+ A = B @ ˆk t A + B 2 ε z,t+ Â t+ Â t I B is 3 3 matrix, B 2 is a 3 vector I forward looking variable: ĉ t I backward variables: Â t, ˆk t

Blanchard - Kahn (980) conditions on matrix B Rational expectation equilibrium is determinate (equilibrium uniqueness) if and only if: I the number of eigenvalues jλ i j > is equal to the number of forward looking variables ( in this case) I the number of eigenvalues jλ i j < is equal to the number of backward looking / predetermined variables (2 in this case)

I ) BK condition holds: in this case there is one solution, the equilibrium path is unique and the system has saddle path stability I 2) The number of jλ i j > is larger than the number of forward looking variables. Too many unstable roots: no solution, paths are explosive, transversality condition violated. I 3) The number of jλ i j < is less than the number of forward looking variables. Too many stable roots: multiple equilibria

Solution strategy I Here we use Dynare, a pre-processor and a collection of MATLAB R (and GNU Octave) routines which solve non-linear models with forward looking variables using perturbation methods. For details see http://www.dynare.org and Collard and Juillard (200a, 200b). I Basic idea of perturbation methods: formulate a general problem (our model), nd a particular case that has a known solution (the deterministic steady state), use that particular case and its solution as a starting point for computing approximate solutions to the nearby problems (methods relying on Taylor series expansions, implicit function theorem see Judd 998, Schmitt-Grohé and Uribe 2004). I Dynare uses a Taylor approximation, up to third order, of the expectation functions. I Here we undertake a second order approximation of the model. I No need to loglinearize the model if you use Dynare.

Calibration α = 0.33 β = 0.99 δ = 0.023 ρ Z = 0.95 y = h = 0.3 capital share discount factor depreciation rate persistence of tech shock output hours (NB: we use the label "Labour" in the gures) The scale paratmeters A, ψ are set so as to obtain y = and h = 0.3.

E ects of a Technology Shock (RBC) 0.75 Consumption, c.6 Output, y 0.8 Labour, l 0.7 0.65.4.2 0.6 0.6 0.4 0.55 0.5 0.45 0.8 0.6 0.4 0.2 0 0.4 0.2 0.2 Capital, k 6 Investments, i 0.8 Wage, w 0.8 5 4 0.7 0.6 3 0.6 0.4 0.2 2 0 0.5 0.4 0

E ects of a Positive Technology Shock (RBC) Propagation of the shock I Productivity "!MPL "!wage " I I I Substitution e ect increases labour supply (prevails) Income e ect decreases labour supply However a transitory productivity shock, which temporarily raises the real wage rate, increases labour supply today (agents work more today to be able to consume more in the future when the wage is expected to be lower) I Productivity "!MPK "!rental rate of capital " I I I I I I Substitution e ect increases savings (prevails) Income e ect decreases savings Consumption increases gradually (consumption smoothing) Investments increases on impact (the volatile component) Capital accumulates gradually and then slowly returns to its initial level As a result y increases more than proportionally.

E ects of a Technology Shock (RBC) with Low Persistence ρ Z = 0. (basic RBC in red) 0.8 Consumption, c 2 Output, y.2 Labour, l 0.6.5 0.8 0.4 0.6 0.4 0.2 0.5 0.2 0 0 0 0.2 Capital, k 8 Investments, i 0.8 Wage, w 0.8 6 0.6 0.6 0.4 4 2 0.4 0.2 0 0.2 0 2 0

Adding Frictions to the Basic RBC Model I External Habit I Adjustment Costs on Investments I Labour Adjustment Costs I Other common frictions not considered here: e.g. indivisible labour à la Hansen 985 (it is assumed that hours of work are xed by rms and individuals simply decide whether or not to participate in the labour force), variable capacity utilization, taxes on capital and/or consumption.

RBC Model with External Habit As above... but now the utility derived from consumption of the representative household i is u(c i t hab e c t ) I Habit persistence (the period utility function depends on a quasi-di erence of consumption) I hab e 2 (0, ) = the intensity of habit formation. I Here habits are external to the consumer: the stock of habit depends on the past value of aggregate consumption, c t ( catching up with the Joneses, see Abel 990). I Under external habit an increase in the value of aggregate consumption will increase the marginal utility of consumption of each consumer i in the next period, inducing her to consume more. I Motivation: in the data the response of consumption to expansionary shocks is hump-shaped (persistence).

RBC Model with External Habit Under log speci cation of the utility of consumption and leisure the Euler equation is = βe t (r t+ + δ) c t hab e c t c t+ hab e c t the labour supply is now Set hab e = 0.4 ψ h t = c t hab e c t w t

RBC Model with External Habit. NEW: c t hab e c t = βe t c t+ hab e c t (r t+ + δ) 2. NEW: ψ h t = c t hab e c t w t 3. ( α) y t h t = w t 4. α y t k t = r t 5. y t = A t kt α ht α 6. i t = k t+ ( δ)k t 7. y t = c t + i t 8. A t = A exp z t z t = ρ z z t + ε z,t

E ects of a Technology Shock (RBC with External Habit & RBC in red) 0.9 Consumption, c 2 Output, y 0.8 Labour, l 0.8 0.7.5 0.6 0.6 0.5 0.4 0.2 0.4 0.3 0.5 0 0.2 0 0.2.4 Capital, k 8 Investments, i 0.8 Wage, w.2 6 0.7 0.8 4 0.6 0.6 0.4 0.2 2 0 0.5 0.4 0 2

RBC Model with Adjust. Costs on Investments I Motivation: Simulations of standard RBC models produce too volatile investments. Convex capital installation costs make investment less volatile! more gradual dynamic adjustment in response to shocks. I Why do we care? From the perspective of stabilizing policies and the impact of monetary policy investment is a key variable in the transmission mechanism. I Many scal policy instruments a ect the economy performance through the in uence they have on capital accumulation. I Here we assume that capital accumulation is costly (installation costs, disruption of productive activities, the need to retrain workers, the need to change the production process etc...).

RBC Model with Adjust. Costs on Investments The new budget constraint of the household is c t + k t+ = w t h t + r t k t + ( δ)k adj(i t, k t ) where adj(i t, k t ) = φ i 2 it k t 2 δ k t Remark: in steady state adj(i, k) = 0 since i = δk. The new Lagrangian is 8 >< L 0 = E 0 β t t=0 >: log c t + ψ log( h t ) +λ t w t h t + r t k t i t c t φ i 2 it k t +ξ t [i t + ( δ)k t k t+ ] 2 δ kt 9 >= >;

RBC Model with Adjust. Costs on Investments At the optimum (from the FOCs wrt to i t and k t+ ): it φ i δ = q t k t q t λ {z} t = βe t λ t+ r t+ + β ( δ) E t q t+ λ {z + } ξ t βe t λ t+ adj(i t+, k t+ ) k t+ where q t = ξ t λ t is the Tobin s marginal q (it measures the expected marginal gains of more capital) λ t is the marginal bene t in terms of the utility of sacri cing a unit of current consumption in order to have an extra unit of investment, and so extra capital tomorrow ξ t is the marginal bene t in terms of utility of an extra unity of investments ξ t+

RBC Model with Adjust. Costs on Investments. New: q t ct = βe t ct+ r t+ + β ( δ) E t q t+ c+ βe t ct+ adj(i t+,k t+ ) k t+ 2. ψc t = w t ( h t ) 3. ( α) y t h t = w t 4. α y t k t = r t 5. y t = A t kt α ht α 6. i t = k t+ ( δ)k t 7. y t = c t + i t + φ i 2 it k t δ 2 kt 8. A t = A exp z t z t = ρ z z t + ε z,t 9. NEW NEW: φ it i k t δ = q t

E ects of a Technology Shock (RBC with Adjust. Costs on I & RBC in red) 0.8 Consumption, c.6 Output, y 0.8 Labour, l 0.7.4.2 0.6 0.6 0.4 0.5 0.4 0.8 0.6 0.4 0.2 0 0.2 0.2 Capital, k 6 Investments, i Wage, w 0.8 5 4 0.9 0.8 0.6 3 0.7 0.4 0.2 2 0 0.6 0.5 0.4 0

RBC Model with Adjust. Costs on Labour I Motivation: models of business cycle depend crucially on the operation of labor markets; attempts to forecast macroeconomic conditions often resort to consideration of observed movements in hours and employment to infer the state of economic activity; policy interventions in the labour market are numerous. I Firms do face adjustment costs when changing the level of employment. Hiring and ring workers entail additional costs for rms (search and recruiting; training; explicit ring costs; variations in complementary activities; reorganization of production activities; capital accumulation etc.). I As a result: the impact of adjustment costs on labour demand is to moderate changes in employment across the business cycle.

RBC Model with Adjust. Costs on Labour The representative rm s optimization problem is now intertemporal. The pro t function is 2 Π t = E 0 β t λ t t=0 λ 0 6 4A t kt α (h t ) α 7 r t k t w t h t adj(h t ) 5 {z } y t where adj(h t ) = φ 2 h ht 2 h t. convex cost of adjusting employment (ad hoc assumption: no quit rate, no discontinuities, no xed costs). The new demand of labour is ( α) A t k α t h α φ h βe t λ t+ λ t t ht+ h t h t h t + ht+ φ h ht h 2 t = w t 3

RBC Model with Adjust. Costs on Labour. c t = βe t c t+ (r t+ + δ) 2. ψc t = w t ( h t ) 3. NEW: ( α) A t kt α ht α φ ht h h t h t + φ h βe t c t ht+ c t+ h t ht+ ht 2 4. α y t k t = r t 5. y t = A t kt α ht α 6. i t = k t+ ( δ)k t = w t 2 7. y t = c t + i t + φ h ht 2 h t 8. A t = A exp z t z t = ρ z z t + ε z,t

E ects of a Technology Shock (RBC with Adjust. Costs on L & RBC) 0.75 Consumption, c.6 Output, y 0.8 Labour, l 0.7 0.65.4.2 0.6 0.6 0.4 0.55 0.5 0.45 0.4 0.8 0.6 0.4 0.2 0.2 0 0.2 Capital, k 6 Investments, i 0.8 Wage, w 0.8 5 4 0.7 0.6 3 0.6 0.4 0.2 2 0 0.5 0.4 0

RBC with All Frictions. λ t = c t hab e c t 2. q t λ t = adj(i βe t λ t+ r t+ + β ( δ) E t q t+ λ t+ βe t λ t+,k t+ ) t+ k t+ 3. ψ = λ t w t ( h t ) φ ht h h t h t + c t ht+ c t+ h t ht+ = w ht 2 t 4. ( α) A t k α t h α t φ h βe t 5. α y t k t = r t 6. y t = A t kt α ht α 7. i t = k t+ ( δ)k t 8. y t = c t + i t + φ h 2 ht h t 2 + φ i 9. A t = A exp z t z t = ρ z z t + ε z,t 0. φ it i k t δ = q t 2 it k t δ 2 kt

E ects of a Technology Shock (RBC with All Frictions & RBC) 0.8 Consumption, c.6 Output, y 0.8 Labour, l 0.7.4.2 0.6 0.6 0.4 0.5 0.4 0.8 0.6 0.4 0.2 0 0.2 0.2 Capital, k 6 Investments, i 0.8 Wage, w 0.8 5 4 0.7 0.6 3 0.6 0.4 0.2 2 0 0.5 0.4 0

Conclusions and problems I The bulk of economic uctuations could be interpreted as an equilibrium outcome resulting from the economy s response to real shocks. Cyclical uctuations do not necessarily re ect ine ciencies (stabilization policies may not be desirable). I The leading role of technology shocks as a source of economic uctuations and of the persistence of output deviations from its trend (alternatively: model growth as endogenous!) I Labor supply elasticities with respect to the wage are implausibly high. Empirical estimates are all below. But if we assume an elasticity of in the RBC model with do not get much action in employment relative to data. (It indicates that the competitive spot market characterization of the labor market has to be replaced by something else)

Conclusions and problems of the basic RBC (con t) I The notion that there would be sharp movements in the production frontier from quarter to quarter, highly correlated accross sectors, is not plausible. The di usion of technology is steady and breaktrhoughs are rare. Shocks to technology as measured by uctuations in TFP are unlikely to be good measurs of underlying technical change. I Changes in capital utilization and labor e ort do have a role. I They cannot explain the degree of autocorrelation in output growth observed in the data. I Strong evidence that monetary shocks have important real e ects. I Money is a only veil (classical dichotomy). RBC models are not suited for studying in ation, nominal interest rates and monetary policy. One-to-one relationship between prices and money aggregates.

References Abel, A. B. (990), Asset Prices under Habit Formation and Catching Up with the Joneses, American Economic Review, 80(2). Adda, J., Cooper, R. (2003), Dynamic Economics, The MIT Press. Collard, F., Juillard, M. (200a), Accuracy of stochastic perturbation methods: The case of asset pricing models, Journal of Economic Dynamics and Control, 25(6-7). Collard, F., Juillard, M. (200b), A Higher-Order Taylor Expansion Approach to Simulation of Stochastic Forward-Looking Models with an Application to a Nonlinear Phillips Curve Model, Computational Economics, 7(2-3). Galí, J. (2008), Monetary Policy, In ation and the Business Cycle, Princeton University Press. Judd, K. L. (998), Numerical Methods in Economics, The MIT Press. Kydland, F. E., Prescott, E. C., (982), Time to Build and Aggregate Fluctuations,Econometrica, 50(6). Long, J., Plosser, C., (983), Real business cycles, Journal of Political Economy, 9.

References McCandless, G. (2008), The ABCs of RBCs, Harvard University Press. Nelson, C.R. Plosser, C.I., (982),Trends and random walks in macroeconomic time series, Journal of Monetary Economics 0. Prescott, E.C., (986), Theory Ahead of Business.Cycle Measurement, Carnegie-Rochester Conference Series on Public Policy, 25. Romer, D. (2006), Advanced Macroeconomics, McGraw-Hill. Schmitt-Grohe, S., Uribe, M.(,2004), Solving dynamic general equilibrium models using a second-order approximation to the policy function, Journal of Economic Dynamics and Control, 28(4). Wickens, M. (2008), Macroeconomic Theory, Princeton University Press. For an introduction to log-linearizations see: http://www.vwl.unibe.ch/studies/3076_e/linearisation_slides.pdf