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1 Real Business Cycle Theory Marco Di Pietro Advanced () Monetary Economics and Policy 1 / 35

2 Introduction to DSGE models Dynamic Stochastic General Equilibrium (DSGE) models have become the main tool for the analysis of macroeconomic uctuations and monetary policy. The relationships between macroeconomic variables are derived by microfounding the behavior of economic agents. Rational expectations and microfoundations make such a kind of models robust to the Lucas critique. Exogenous shocks a ect the economy and are responsible of cyclical uctuations. Marco Di Pietro Advanced () Monetary Economics and Policy 2 / 35

3 Real Business Cycle (RBC) Theory A rst approach to DSGE models was designed by Kydland and Prescott (1982), who introduced the RBC theory. RBC models are characterized by the following ingredients: There is a representative agent who has rational expectations. The evolution of the aggregate variables is described by rst-order conditions of intertemporal problems solved by rms and households. Firms have no market power, i.e., perfect competition and frictionless markets. Prices are fully exible. Model evaluation relies on parameters calibration and simulation. Marco Di Pietro Advanced () Monetary Economics and Policy 3 / 35

4 RBC basic concepts Cyclical uctuations are e cient: business cycle is the results of the optimal response of the economy to exogenous shocks. Technology shock is the main driving force of the business cycle and a ects real variables. As a consequence, stabilization policies are not required and might be also counterproductive. TFP is not only the main source of long-term growth, but it now involves economic uctuations. Monetary neutrality: monetary policy is ine ective also in the short-run, i.e., no real e ects. It only a ects nominal variables (e.g., price level). A question: are RBC assumptions and outcomes supported by the empirical evidence? Marco Di Pietro Advanced () Monetary Economics and Policy 4 / 35

5 Some empirical evidence Nominal rigidities: empirical studies based on micro data nd that prices realignments are infrequent. Monetary policy non-neutrality: evidence from SVAR suggests that monetary policy a ects real variables. Liquidity e ect is observed. Marco Di Pietro Advanced () Monetary Economics and Policy 5 / 35

6 A Classical Monetary Model Basic (strong) hypotheses: prices are fully exible and there is perfect competition in all markets. Money serves as unit of account. The economy is composed by households and rms. Households solve an intertemporal optimization problem by maximizing their utility. Firms maximize pro ts taking account of the constraint given by the production function. No capital accumulation, no scal sector, closed economy. Marco Di Pietro Advanced () Monetary Economics and Policy 6 / 35

7 Households The representative households maximizes a utility function choosing how much to consume, the quantity of labor to supply and assets purchasing: E 0 β t U (C t, N t ) t=0 Assumptions on utility function: U c,t U (C t,n t ) C t > 0, U cc,t 2 U (C t,n t ) 0, marginal utility of Ct 2 consumption is positive and non-increasing. U n,t U (C t,n t ) N t 0, U nn,t 2 U (C t,n t ) 0, marginal utility of Nt 2 labor is negative and decreasing. Budget constraint (in nominal terms): P t C t + Q t B t B t 1 + W t N t T t No-Ponzi game condition is assumed in order to prevent households from excessive borrowing. Marco Di Pietro Advanced () Monetary Economics and Policy 7 / 35

8 Households problem solution The following form is assumed for the utility function:! U = C t 1 σ N 1+γ t 1 σ 1 + γ Constrained optimization problem: max U (C t, N t ) + λ t (B t 1 + W t N t T t P t C t Q t B t ) C t,n t,b t First order conditions: W t P t = U n,t U c,t =) W t P t Q t = βe t λ t+1 λ t =) Q t = βe t = Ct σ N γ t " Ct+1 C t σ P t P t+1 # P t+1 P t = Π t+1 is the gross in ation rate Marco Di Pietro Advanced () Monetary Economics and Policy 8 / 35

9 Firms The representative rm has the following production function: Y t = A t N 1 t α The rm seeks to maximize his pro t under the constraint given by the production function, taking prices and wages as given max P t Y t W t N t + ψ Y t,n t (A t Nt 1 α Y t ) t Optimality conditions (real wage equal to marginal product of labor) P t = ψ t W t = (1 α) ψ t A t Nt α =) W t = (1 α) A t Nt α P t {z } MPL t Marco Di Pietro Advanced () Monetary Economics and Policy 9 / 35

10 The log-linear economy Euler equation where i t = q t and ρ = log β. Labor supply Labor demand TIP: why i t = q t? c t = E t c t+1 1 σ (i t E t π t+1 ρ) w t p t = σc t + γn t w t p t = log (1 α) + a t αn t Marco Di Pietro Advanced () Monetary Economics and Policy 10 / 35

11 Asset prices and interest rate Let assume that a riskless asset pays a gross interest equal to I t. Therefore, investing 100$ the asset will pay you 100xI t $ after one year. Let now assume that a riskless asset pays you 1$ after one year. How much should it cost? The present value of 1$ today is 1 I t. Thus, the asset price Q t = 1 I t ; in log terms q t = i t. Marco Di Pietro Advanced () Monetary Economics and Policy 11 / 35

12 Equilibrium Market clearing condition Labor market clearing y t = c t Asset market clearing Aggregate production function σy t + γn t = log (1 α) + a t αn t y t = E t y t+1 1 σ (i t E t π t+1 ρ) y t = a t + (1 α) n t Technology shock follows an AR(1) process a t = ϱ a a t 1 + ε a t Marco Di Pietro Advanced () Monetary Economics and Policy 12 / 35

13 Model solution (1) We insert the production function into the labor market clearing, obtaining where Ψ na = σ [a t + (1 α) n t ] + γn t = log (1 α) + a t αn t + σ (1 α) n t + γn t + αn t = log (1 α) + (1 σ) a t + (1 σ) σ(1 α)+γ+α, ϑ n = n t = Ψ na a t + ϑ n log(1 α) σ(1 α)+γ+α Marco Di Pietro Advanced () Monetary Economics and Policy 13 / 35

14 Model solution (2) The production function becomes y t = a t + (1 α) (Ψ na a t + ϑ n ) + y t = + (1 α) (1 σ) + σ (1 α) + γ + α a t + (1 σ (1 α) + γ + α α) ϑ n y t = Ψ ya a t + ϑ y 1+γ where Ψ ya = σ(1 α)+γ+α and ϑ y = (1 α) ϑ n. From the Euler equation the real interest rate, r t = i t E t π t+1, becomes r t = σe t f y t+1 g + ρ + r t = σψ ya E t f a t+1 g + ρ Marco Di Pietro Advanced () Monetary Economics and Policy 14 / 35

15 Model solution (3) The equilibrium real wage, ω t = w t p t, is given by: where Ψ wa = ω t = log (1 α) + a t αn t + ω t = log (1 α) + a t α (Ψ na a t + ϑ n ) + ω t = Ψ wa a t + ϑ w σ+γ σ(1 α)+γ+α and ϑ [σ(1 α)+γ] log(1 α) w =. σ(1 α)+γ+α Marco Di Pietro Advanced () Monetary Economics and Policy 15 / 35

16 Model dynamics The equilibrium dynamics of the real variables (n t, y t, r t, ω t ) is a ected ONLY by technology, which is the real driving force of the system. Thus, n t, y t, r t, ω t exhibit uctuations that are optimal response to innovations in a t. Policy neutrality: real variables are determined independently of monetary policy. Both output and real wage always increase following a positive technology shock. The real interest rate increases (decreases) if the change in technology is permanent (transitory). E ect on employment is ambigous and depends on σ: for σ < 1 substitution e ect on labor supply prevails over the negative wealth e ect generated by a smaller marginal utility of consumption (n t "). For σ > 1 the opposite happens (n t #); for σ = 1 substitution and wealth e ect exactly cancel out and employment remain unchanged. Marco Di Pietro Advanced () Monetary Economics and Policy 16 / 35

17 Nominal variables determination In order to determine the nominal variables we need to specify how the monetary policy is conducted. We consider three possible solutions: an exogenous path for the nominal interest rate; a simple in ation-based interest rate rule an exogenous path for the money supply In all cases we make use of the Fisher equation: r t = i t E t π t+1 Marco Di Pietro Advanced () Monetary Economics and Policy 17 / 35

18 1) An exogenous path for the nominal interest rate Let suppose that nominal interest rate follows an exogenous stationary process i t, where i t has a mean value equal to ρ, consistent with zero steady state in ation. Moreover, r t is determined independently of the monetary policy and E t π t+1 = i t r t. Expected in ation is determined by the Fisher equation, while current in ation is not. Thus, any path for the price level that satis es the following relation is consistent with equilibrium: p t+1 = p t + i i r t + ξ t+1 where ξ t+1 is a sunspot shock, i.e., a shock unrelated to economic fundamentals. An equilibrium where nonfundamentals factors can generate uctuations in economic variables is de ned as indeterminate equilibrium. This framework leads to price level indeterminacy. Marco Di Pietro Advanced () Monetary Economics and Policy 18 / 35

19 Source of indeterminacy Let assume that money supply evolves as m t = p t + y t ηi t As a consequence, money supply inherits the indeterminacy of p t. In other words, the central bank xes the interest rate and let money be determined endogenously. But since we have undetermined prices, money is undetermined as well. Marco Di Pietro Advanced () Monetary Economics and Policy 19 / 35

20 2) A simple in ation-based interest rate rule The monetary authority sets nominal interest rate according to i t = ρ + π t where 0 measures the response of the nominal interest rate to in ation. This interest rate rule determines the nominal variables and tells that the Central Bank responds to in ation pressure by raising the nominal interest rate. Plugging the interest rate rule into the Fisher equation, we obtain r t ρ = br t = π t E t π t+1 + π t = E t π t+1 + br t This is a stochastic di erence equation: We have two cases: > 1 (Taylor principle) < 1 Marco Di Pietro Advanced () Monetary Economics and Policy 20 / 35

21 Forward solution (1) Actual in ation is determined by π t = E t π t+1 At time t br t At time t + 2 π t+1 = E t+1π t+2 + E tbr t+1 + π t = 1 E t π t+2 = E t+2π t+3 + E t+1br t+2 + π t = 1 E t Et+1 π t+2 + E tbr t+1 + br t 1 Et+2 π t+3 E t+1 + E t+1br t+2 + E tbr t+1 + br t Marco Di Pietro Advanced () Monetary Economics and Policy 21 / 35

22 Forward solution (2) By the Law of Iterated Expectations (LIE) E t [E t+1 (x)] = E t (x) h i π t = 1 φ E 1 t π φ E Et+2 π t+3 t+1 π φ + E t+1br t+2 π φ + E tbr t+1 π φ + br t π φ can be π rewritten as π t = 1 φ 3 E t π t+3 + E t π t+3 π φ 3 + E tbr t+1 π φ 2 + br t π Continuing the forward solutions π t = 1 φ j+1 π E t π t+j and continuing nally we get π t = k=0 j k=0 1 φ k+1 π 1 φ k+1 E t br t+k π E t br t+k Marco Di Pietro Advanced () Monetary Economics and Policy 22 / 35

23 Model solution If > 1 the di erence equation π t = E t π t+1 + br t has only one stationary solution obtained from forward solution π t = 1 k=0 φ k+1 E t br t+k π In ation and, consequently, price level are fully determined by the real interest rate, which, in turn, is a function of fundamentals. The Central Bank, by choosing the value of, can determines the degree of in ation volatility. For < 1 the stationary solution of π t = E t π t+1 + br t takes the following form π t+1 = π t br t + ξ t+1 where ξ t+1 is a sunspot shock. Thus, any process π t satisfying the previous equation is consistent with a stationary equilibrium. Price level, in ation and nominal interest rate are undetermined. Marco Di Pietro Advanced () Monetary Economics and Policy 23 / 35

24 3) An exogenous path for the money supply We suppose that the monetary authority sets an exogenous path for the money supply m t. By combining the money demand m t = p t + y t equation r t = i t E t π t+1 we get ηi t and the Fisher r t = p t + y t η m t E t π t+1 Writing E t π t+1 as E t p t+1 η p t = 1 + η y t p t and rearranging 1 E t p t η m t + u t where u t = ηr t 1+η evolves independently of monetary policy (they are real variables). Marco Di Pietro Advanced () Monetary Economics and Policy 24 / 35

25 Price level determination Assuming η > 0 and solving forward we obtain p t = η 1 + η k=0 η 1 + η k E t m t+k + u 0 t where ut 0 = η k 1+η Et u t+k does not depend on monetary policy. k=0 The previous equation can be rewritten in terms of expected growth rate of money: p t = m t + k=1 η k E t m t+k + ut η Thus, an arbitrary exogenous path for money supply always determines the price level uniquely. Marco Di Pietro Advanced () Monetary Economics and Policy 25 / 35

26 Interest rate determination Plugging price level equation into money supply we get y t ηi t = m t m t + k=1 + i t = 1 η k=1 η 1 + η η k E t m t+k + ut η k E t m t+k + y t + u 0 t η This implies that also the nominal interest rate is determined uniquely. Marco Di Pietro Advanced () Monetary Economics and Policy 26 / 35

27 An example Let assume that money growth evolves according to an AR(1) process: m t = ϱ m m t 1 + ε m t Moreover, we assume no real shock and y t = 0, r t = 0. The price level becomes ηϱ p t = m t + m 1 + η(1 ϱ m ) m t Following a monetary shock, given ϱ m > 0 as empirically observed, the price level response is greater than the money supply increase. Nominal interest rate goes up in response to a monetary shock i t = ϱ m 1 + η(1 ϱ m ) m t Shortcomings: these results contrast with empirical evidence (slow response of prices to monetary shocks and presence of liquidity e ect). Marco Di Pietro Advanced () Monetary Economics and Policy 27 / 35

28 Money in the utility function (MIU) We assume that real money provides utility to the agents (Hint: money facilitates transactions on the market). The households problem under MIU becomes E 0 β t U C t, M t, N t t=0 P t Budget constraint P t C t + Q t B t + M t B t 1 + M t 1 + W t N t T t We de ne Λ t = B t 1 + M t 1 as the total nancial wealth. Budget constraint can be rewritten as P t C t + Q t Λ t+1 + (1 Q t ) M t Λ t + W t N t T t Interpretation: (1 Q t ) = 1 exp ( i t ) ' i t =) opportunity cost of holding money. (Remeber i t = log(q t ) =) exp (i t ) = Q t ) Marco Di Pietro Advanced () Monetary Economics and Policy 28 / 35

29 MIU optimality conditions Maximizing the utility function under the budget constraint yield the following FOCs W t P t = U n,t U c,t P t U c,t+1 Q t = βe t U c,t U m,t = 1 exp( i t ) U c,t P t+1 Model properties depend on the speci cation of the utility function. Two cases: separable in real balances non-separable in real-balances Marco Di Pietro Advanced () Monetary Economics and Policy 29 / 35

30 Separable utility The utility functional form is U C t, M t, N t = C t 1 σ P t 1 σ + (M t /P t ) 1 ν 1 ν! N 1+γ t 1 + γ Given the separability, U c,t and U n,t continue to be independent of real balances. The equilibrium level for output, employment, real interest rate and real wage is the same of a cashless economy. The money demand equation is In log-linear terms M t P t = C σ ν t [1 exp ( i t )] m t p t = σ ν c t ηi t 1 ν Money demand serves only at determining the quantity of money that the Central Bank supplies. Moreover, money has only nominal e ects. Marco Di Pietro Advanced () Monetary Economics and Policy 30 / 35

31 Non-separable utility The utility functional form is U C t, M t, N t = X t 1 σ P t 1 σ! N 1+γ t 1 + γ X t is a composite index of consumption and real balances having the following form X t = " X t = C 1 ϑ t (1 ϑ) C 1 ν t Mt P t # 1 1 ν 1 ν Mt + ϑ P t ϑ for ν = 1 for ν 6= 1 where ν denotes the elasticity of substitution between consumption and real balances and ϑ the relative weight of real balances in utility. Marco Di Pietro Advanced () Monetary Economics and Policy 31 / 35

32 New optimality conditions Being the utility function non-separable the marginal utilities of consumption and real balances are U c,t = (1 ϑ) X ν σ U m,t = ϑx ν σ t The optimality conditions now become t Ct ν ν Mt W t = Nγ t X σ ν t Ct ν P t (1 ϑ) " Ct+1 ν ν σ Xt+1 P t Q t = βe t C t M t P t = C t [1 exp( i t )] P t X t 1 ν ϑ 1 ϑ 1 ν P t+1 Implications: monetary policy non-neutral as both labor supply and Euler equation are a ected by real balances. Marco Di Pietro Advanced () Monetary Economics and Policy 32 / 35 #

33 Optimal monetary policy in a MIU context The policymaker aims to maximize the utility of the representative household subject to the resource constraint C t = A t Nt 1 α. Formally, max U C t, M t, N t P t Optimality conditions: U n,t U c,t = (1 α)a t Nt α U m,t = 0 Condition U m,t = 0 equals the marginal utility of real balances with the social marginal cost of producing real balances. Remember that households choose their optimal real money balances according to U m,t U c,t = 1 exp( i t ) Marco Di Pietro Advanced () Monetary Economics and Policy 33 / 35

34 Friedman rule U m,t = 0 is satis ed only in the case i t = 0 =) Friedman rule. The opportunity cost of holding money is given by the nominal interest rate and the latter must be zero in order to equate the social cost of holding money. Policy implication: π = ρ < 0 =) moderate de ation in the long-run. Nonetheless, Friedman rule involves that price level is not determined. Central Bank can avoid undeterminacy implementing the Taylor principle in the following rule: i t = (r t 1 + π t ) Plugging the previous rule into the Fisherian equation leads to the following di erence equation i t = E t i t+1 whose only stationary solution is i t = 0. Equilibrium in ation is π t = r t. Marco Di Pietro Advanced () Monetary Economics and Policy 34 / 35

35 References - Christiano, L.J., M. Eichenbaum and C.L. Evans (1999), Monetary policy shocks: What have we learned and to what end?, Handbook of Macroeconomics, in: J. B. Taylor & M. Woodford (ed.), volume 1, chapter 2: Galí, J. (2008), Monetary Policy, In ation and the Business Cycle. An Introduction to the New Keynesian Framework, (Chapters 1 and 2). - King R. e S. Rebelo, (2000), Resuscitating Real Business Cycles, in Woodford, M. and Taylor, J., (eds.), Handbook of Macroeconomics, vol. 1B North-Holland (NBER Working Paper 7534). - Kydland, F.E. and E.C. Prescott (1982), Time to Build and Aggregate Fluctuations, Econometrica, 50(6): Marco Di Pietro Advanced () Monetary Economics and Policy 35 / 35

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