Money in the utility model

Transcription

1 Money in the utility model Monetary Economics Michaª Brzoza-Brzezina Warsaw School of Economics 1 / 59

2 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 2 / 59

3 Introduction The neoclassical growth model by Ramsey (1928) and Solow (1956) provides the basic framework for modern macroeconomics. But these are models of nonmonetary economies. In order to explain monetary policy we have to introduce a monetary policy instrument. Sidrauski (1967) introduced money into the neoclassical growth model. At this time money was considered the monetary policy instrument. The model derives from Ramsey (1928): utility maximization by the representative agent 3 / 59

4 Introduction (cont'd) How can we introduce money? - nd explicit role for money (e.g. money is necessary to make transactions - Cash in Advancec (CIA) approach, Clower (1967)) - assume money yields utility (Money in the Utility Function (MIU) - Sidrauski (1967))- MIU is simple though not very appealing. Can be rationalized by transaction dedmand for money. The model allows us studying: - impact of money (i.e. monetary policy) on the real economy, - impact of money on prices, - optimal rate of ination. Derivation follows (approximately) Walsh (2003) 4 / 59

5 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 5 / 59

6 Household's problem - objective Suppose the utility function of the representative household is: U t = u(c t, m t ) (1) where m t = M t P t and utility is increasing and strictly concave in both arguments. The representative household seeks to maximize lifetime utility: U = E 0 t=0 β t u(c t, m t ) (2) 6 / 59

7 Constraint subject to the budget constraint: y t + τ t + (1 δ)k t 1 + i t 1B t 1 P t and the production function: c t + k t + M t P t + B t P t y t = f (k t 1 ) + M t 1 P t = (3) This means that households' income can be spent on consumption, invested as capital, saved as bonds or kept as money. 7 / 59

8 Constraint We can substitute the production function and rewrite the budget constraint: f (k t 1 ) + τ t + (1 δ)k t 1 + i t 1b t 1 + m t 1 π t = c t + k t + m t + b t (4) where π t P t /P t 1. 8 / 59

9 Lagrangian Lagrangian. max E 0 c,m,k,b t=0 β t u(c t, m t ) + λ t [f (k t 1 ) +τ t + (1 δ)k t 1 + i t 1b t 1 + m t 1 π t c t k t m t b t ] (5) 9 / 59

10 First order conditions First order conditions for this problem are: c t : β t u (c t ) = λ t (6) k t : m t : b t : λ t + E t λ t+1 [f (k t ) + (1 δ)] = 0 (7) β t u (m t ) λ t + E t λ t+1 1 π t+1 = 0 (8) i t λ t + E t λ t+1 = 0 (9) π t+1 10 / 59

11 Equilibrium conditions - consumption Euler equation FOCs can be transformed into formulae describing allocation choices in equilibrium. Substitute (6) into (9): i t u (c t ) = βe t u (c t+1 ) (10) π t+1 This is the Euler equation - key intertemporal condition in general equilibrium models. It determines allocation over time. Utility lost from consumption today equals utility from consuming tomorrow adjusted for the (real) gain from keeping bonds. 11 / 59

12 Equilibrium conditions - capital Substitute (9) into (7): E t u (c t+1 )[f (k t ) + (1 δ)] = E t u (c t+1 ) (11) π t+1 Dene real interest rate (Fisher equation): i t r t E t (12) π t+1 The marginal product of capital (net of depreciation) equals the real interest rate (up to rst order). i t 12 / 59

13 Equilibrium conditions - Opportunity cost of money Merge (6), (9) and (8): Divide by β t and rearrange: β t u (m t ) β t u (c t ) + βt u (c t ) i t = 0 u (m t ) u (c t ) = 1 1 i t (13) This equates the marginal rate of substitution between money and consumption to their relative price 13 / 59

14 Technology and utility In order to solve the model we have to assume a production and utility function. Production function: where is a stochastic TFP process. Utility function: y t = e z t k α t 1 (14) z t = ρ z z t 1 + ε z,t (15) U t = ln c t + ln m t (16) 14 / 59

15 Monetary policy Monetary policy is very simple. We assume that money follows a stochastic process: which yields: M t = e θ t M t 1 where: m t = m t 1 π t e θ t (17) Is a stochastic money supply process. θ t = ρ θ θ t 1 + ε θ,t (18) 15 / 59

16 Government and bonds Bonds are in zero net supply b = 0 and the Government budget is balanced every period in real terms τ t P t = M t M t 1 τ t = m t m t 1 π t Combine these with the household budget constraint (4): y t + (1 δ)k t 1 = c t + k t (19) 16 / 59

17 The model We now have 9 equations: (10), (11), (12), (13), (14), (15), (17), (18), (19) to solve for 9 endogeneous variables: y t, k t, m t, c t, i t, r t, π t, z t, θ t. This will be done in two steps: 1 Find the model's steady state, 2 Derive log-linear approximation arround it. 17 / 59

18 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 18 / 59

19 The steady state The steady state is where the model economy converges to (stabilizes) in the absence of shocks. In our model there is no population or productivity growth, so in the steady state output, consumption etc. will be constant. How do we solve for the steady state? 1 assume shocks are zero, 2 drop time indices (x t 1 = x t = x ss ). 19 / 59

20 The steady state - capital The steady state is where the model economy converges to (stabilizes) in the absence of shocks. In our model there is no population or productivity growth, so in the steady state output, consumption etc. will be constant. From (11), (12) and (14) we have: or α (k ss ) α 1 = 1 β 1 + δ k ss = [ ( )] α β 1 + δ α 1 (20) This determines capital (and output) in the SS. 20 / 59

21 The steady state - consumption Take (19) and (14). This yields: c ss = (k ss ) α δk ss (21) Since k ss is given by (20), this determines steady state consumption. 21 / 59

22 The steady state - money Take (13) and (10). The former yields: The latter i ss m ss = c ss i ss 1 i ss = π ss /β so that: π ss m ss = c ss π ss β (22) 22 / 59

23 Neutrality of money Denition of (long run) neutrality: in the long run real variables do not depend on money nor ination. This is true for all variables but real money holdings. But the latter are usualy excluded from the denition. So, in the MIU money is neutral in the long run. 23 / 59

24 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 24 / 59

25 Short run dynamics Steady state tells us about the long run But most interesting in most applications are short run dynamics We have a set of dierence equations that describe it But they are non-linear: dicult to solve, to estimate etc. The traditional way of analyzing the dynamics of a model is to assume that the economy is always in the viscinity of the steady state Then we take a log-linear approximation around the steady state which is relatively simple to handle The procedure of log-linearization is explained in Walsh (p ) 25 / 59

26 Log-linearization Log-linear approximation. Dene ˆx t ln ( x t x ss ) as log-deviation from steady state. It follows that: x t = x ss x t x ss = x ss e ˆ x t x ss e 0 + x ss e 0 ( xˆ t 0) = x ss (1 + xˆ t ) This can be used to derive the approximation. Some useful tricks: x t y t = x ss y ss (1 + ˆx t + ŷ t ) (23) x t = x ss y t y ss (1 + ˆx t ŷ t ) (24) xt a = (x ss ) a (1 + a ˆx t ) (25) ln x t = ln x + ˆx t (26) 26 / 59

27 LL Euler Steady state; c 1 t = βe t r t c t+1 (c ss ) 1 (1 ˆ c t ) = β r ss c ss E t(1 + ˆr t ĉ t+1 ) Divide: (c ss ) 1 = β r ss c ss (27) cˆ t = E t (ĉ t+1 ˆr t ) (28) 27 / 59

28 LL budget constraint y t + (1 δ)k t 1 = c t + k t y ss (1 + ŷ t ) + (1 δ)k ss (1 + ˆk t 1 ) = c ss (1 + ĉ t ) + k ss (1 + ˆk t ) Steady state: Substract: y ss + (1 δ)k ss = c ss + k ss Rearrange: y ss ŷ t + (1 δ)k ss ˆk t 1 = c ss ĉ t + k ss ˆk t ˆk t = (1 δ) ˆk t 1 + y ss k ss ŷt css kss ĉt 28 / 59

29 LL production function y t = e z t k α t 1 ln y t = z t + α ln k t 1 ln y ss + ŷ t = z t + α ln k ss + α ˆk t 1 y ss = (k ss ) α ŷ t = α ˆk t 1 + z t (29) 29 / 59

30 LL Fisher r t E t i t π t+1 ˆr t î t E t ˆπ t+1 (30) 30 / 59

31 LL money rule (money supply) m t = m t 1 e θ t π t ln m ss + ˆm t = ln m ss + ˆm t 1 ln π ss ˆπ t + θ t ˆm t = ˆm t 1 ˆπ t + θ t (31) 31 / 59

32 LL opportunity cost (money demand) c t m t = 1 1 i t Steady state: c ss m ss (1 + ĉ t ˆm t ) = 1 1 i ss (1 î t ) c ss m ss = i ss 1 i ss ĉ t ˆm t = ît i ss 1 32 / 59

33 LL equilibrium condition for capital [ ] [ ] 1 E t [αe z t+1 kt α (1 δ)] = E t r t c t+1 c t+1 Denote x t αe z t kt 1 α 1 + (1 δ) = α y t k t δ ] ] E t [ 1 c t+1 x t+1 x ss c ss E t(1 ĉ t+1 + ˆx t+1 ) = r ss Divide by steady state xss c ss = r ss c ss E t ˆx t+1 = ˆr t = E t [ 1 c t+1 r t c ss E t [1 ĉ t+1 + ˆr t ] 33 / 59

34 LL equilibrium condition for capital cont'd Now go back to x t α y t k t δ x ss E t (1 + ˆx t+1 ) = α y ss k ss E t(1 + ŷ t+1 kˆ t ) + 1 δ substitute for E t ˆx t+1 and x ss r ss E t (1 + ˆr t ) = α y ss k ss E t(1 + ŷ t+1 kˆ t ) + 1 δ substract steady state r ss = α y ss k ss + 1 δ r ss ˆr t = α y ss k ss E t(ŷ t+1 ˆk t ) 34 / 59

35 LL conclusions Again, we have 9 equations in 9 variables, We have some additional parameters (r ss, π ss,i ss ) that have to be calculated, From (27): r ss = β 1 From (17): π ss = 1 From (12): i ss = π ss r ss = β 1 Now we can solve the model e.g. in DYNARE and check its properties. 35 / 59

36 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 36 / 59

37 Simulations in DYNARE Simulations will be done in DYNARE What is DYNARE? Software (free :-)) for solving, simulating and estimating general equilibrium models But requires a computing platform: MATLAB (expensive) or Octave (free) 37 / 59

38 Parameters We have to assume parameter values. These are taken from the literature β =.99 (Note that in steady state this is the reciprocal of the (annualised) real interest rate of 4%) α =.33 (Capital share) δ =.025 (Depreciation approx. 10% on annual basis) ρ z = ρ θ =.9 (Some inertia) 38 / 59

39 IRFs to a productivity shock y 4 x 10 3 c 6 x 10 3 k x m x 10 4 r x pi z / 59

40 IRFs to a money supply shock 0 m 0.01 i pi theta / 59

41 IRFs - conclusions Productivity shocks have an impact on the real economy and drive the business cycle in the MIU model Monetary shocks have an impact on nominal variables: ination and nominal interest rate But monetary shocks aect only real money balances, but no other real variables In the MIU model money is neutral also in the short run. 41 / 59

42 MIU - some exercises Business cycle properties of the MIU model take the linearized version of the MIU model (MIU.mod) calibrate the standard deviation and autoregression of the productivity shock ε z,t to match the respective moments for GDP in US data. see what happens if productivity z t is assumed i.i.d. Compare impulse responses and moments with the calibration above. Optimal rate of ination take the nonlinear version of the MIU model (MIU_level.mod). Add a variable that calculates welfare check how steady state welfare depends on the ination rate 42 / 59

43 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 43 / 59

44 Optimal rate of ination One of the hot topics in monetary economics is the optimal rate of ination. This is not only a theoretical but also a very practical question: Central Banks have to choose ination targets What does the MIU model tell us about the optimal rate of ination? Utility depends on consumption and real money balances. The only impact ination has on utility is via real money balances. 44 / 59

45 Optimal rate of ination cont'd For easiness of exposition let's restrict attention to steady state. Look at (22) m ss = c ss π ss β = β css (1 + π ss β ) π ss We now that c ss is independent of ination. So ination reduces real money balances and, hence, utility. Optimal is minimal rate of ination. Given (12) and the requirement i 0 we have: π OPT = r So the optimal rate of ination (in the MIU model) is deation at rate r. This was postulated by M.Friedman. 45 / 59

46 Nonneutrality in the MIU model Our model is neutral, But it is possible to generate nonneutrality in the MIU model, One possibility: add labour-leisure choice and nonseparable preferences, Walsh (2003) shows the solution and simulations, Fig 2.3 and 2.4 from textbook, But this model cannot, under standard calibration, generate reaction functions and moments similar to those observed in the data. 46 / 59

47 Applications of DSGE models 1 The MIU model is very simple, has many drawbacks and is treated these days as a toy model 2 But is has the basic features of a dynamic stochastic general equilibrium model 3 So, what can such models be used for? 1 explaining the past 2 forecasting 3 generating counterfactual simulations 4 comparing alternative policies 5 designing optimal policy 6 and probably many others 47 / 59

48 Explaining the past Every endogenous variable can be decomposed into eects of past shocks To see this note, that the solution of a DSGE model is a VAR, and a VAR can be written in MA form But, in contrast to most VARs all shocks in a DSGE have an economic interpretation 48 / 59

49 Explaining the past - example Role of scal shocks during the nancial crisis (Coenen, Straub, Trabandt, 2011; AER) 49 / 59

50 Explaining the past - example cont'd 50 / 59

51 Forecasting Likewise, a VAR can be used to forecast We only have to make assumptions about future shocks Unconditional forecast - all future shocks assumed to be zero Conditional forecast - some shocks assumed for future periods The crucial assumption is whether these are expected or unexpected Expected shocks have an impact before they arrive This often gives counterintuitive results (see e.g. Laseen & Svensson, 2011; IJCB) Several central banks developped forecasting DSGE models: SIGMA (Fed), Ramses (Riksbank), Nemo (Norges Bank), NAWM (ECB), SOE.PL (NBP) 51 / 59

52 Forecasting - example GDP growth forecast - NBP (Kªos, 2014) 52 / 59

53 Counterfactual simulations DSGE models are robust (???) to the Lucas Critique So, they can be used to simulate dierent policies In general, two possible types of counterfactual simulations change some shocks in the past change policy in the past 53 / 59

54 Counterfactual simulations - shocks Role of nancial shocks during the crisis (Brzoza-Brzezina & Makarski 2011, JIMF) data & forecast counterfactual scenario difference / 59

55 Counterfactual simulations - policy Role of independent monetary policy during the crisis (Brzoza-Brzezina, Makarski, Wesoªowski 2014, EM) 55 / 59

56 Designing optimal / comparing alternative policies DSGE models have a natural metric for optimality This is houesehold welfare E.g. see our discussion of optimal ination rate More complicated experiments can be done numemericaly But, optimal policy can have a very complicated design Instead of looking for optimal policy we can compare how close to optimal implementable policies are (e.g. Taylor rules) We can look for robust policies In particular we can test new policies (with no history to use econometrics), e.g. macropru, ZLB 56 / 59

57 Comparing alternative policies - example Macroprudential policy in a monetary union (Brzoza-Brzezina, Kolasa, Makarski 2015, JIMF) Ind. monet., no macropru. Union, no macropru. Union, ind. macropru. frontier Union, common macropru. Union, aggressive monet. 2 Std. of credit in periphery Std. of output in periphery 57 / 59

58 Plan of the Presentation 1 Motivation 2 Model 3 Steady state 4 Short run dynamics 5 Simulations 6 Extensions 7 Conclusions 58 / 59

59 MIU - conclusions We solved and simulated a simple monetary dynamic stochastic general equilibrium model (DSGE), This was able to reect some desired business cycle features in reaction to productivity shocks, But monetary shocks have only nominal eects (except for impact on real money balances), This is inconsistent with stylised facts presented in Lecture 1, We have to look for other models, E.g. the sticky price new Keynesian model - next lecture. 59 / 59