1. Money in the utility function (continued)

Transcription

1 Monetary Economics: Macro Aspects, 19/ Henrik Jensen Department of Economics University of Copenhagen 1. Money in the utility function (continued) a. Welfare costs of in ation b. Potential non-superneutrality of money c. Dynamics and calibration Literature: Walsh (2010, Chapter 2, pp , so check the Appendix as well; i.e., get a grip on the linearization technique) c 2013 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal use or distributed free.

2 Welfare Costs of In ation In ation a ects real money holdings by a ecting the nominal interest rate (the opportunity cost of holding money): u m (c ss ; m ss ) u c (c ss ; m ss ) = iss = I (money demand) 1 + iss i ss = r ss + ss (Fisher relationship) Within the MIU model framework with household utility as welfare measure, what are the welfare e ects of in ation? Is there an optimal rate of in ation? On the optimal long-run in ation rate Bailey/Friedman intuition: Private marginal cost of holding money is increasing in the nominal interest rate Social marginal cost of creating money is essentially zero Equating private and social marginal cost requires a zero nominal interest rate By the Fisher relationship it follows that ss = r ss < 0 is optimal I.e., the optimal rate of change in prices involves de ation equal to the real interest rate The Friedman Rule 1

3 This is formally con rmed in the model when nding the utility maximizing nominal money growth rate subject to resource constraint of economy Solve: First-order condition: So, max ss u (c ss ; m ss ) max ss u (f (k ss ) k ss ; m ss ) u c (f (k ss ) k ss ; m ss (f (kss ) k ss ss {z } = ss u m (c ss ; m ss ) = ss + u m (c ss ; m ss ss = With u m (c ss ; m ss ) u c (c ss ; m ss ) = iss 1 + i ss; this implies i ss = 0, and the condition determines what Friedman called the optimal quantity of money Note that with some nite m de ned as u m (c ss ; m) = 0, and u m (c ss ; m ss ) < 0 for m ss > m, this m is the optimal quantity of money 2

4 Potential non-superneutrality of money Is superneutrality of money a robust feature of the MIU model? In the model of previous lecture, endogenous savings behavior uniquely de nes steady-state capital Capital is accumulated or decumulated until its net marginal product (real interest rate) equals households subjective real interest rate: f k (k ss ) + 1 = 1 Hence, long-run superneutrality can only fail if the marginal product of capital is a ected by in ation 3

5 Natural extension is endogenous labor input in production Arises in MIU model if it is amended by a labor supply choice by households This is achieved by having leisure enter in utility function: u t = u (c t ; m t ; l t ) : The production function is y t = f (k t 1 ; n t ) ; n t 1 l t (n t is not population growth!) Households now face an additional decision: How much time should be devoted to work; how much to leisure? The relevant optimality condition is: u l (c t ; m t ; l t ) = u c (c t ; m t ; l t ) f n (k t 1 ; n t ) (2.43 ) Marginal gain of leisure is equated to the marginal cost, which is the utility loss from lower consumption times the marginal product of labor (the real wage) 4

6 In steady state, three relationships apply: u l (c ss ; m ss ; l ss ) = u c (c ss ; m ss ; l ss ) f n (k ss ; 1 l ss ) (l versus c choice) f k (k ss ; 1 l ss ) + 1 = 1 (constant capital) c ss = f (k ss ; 1 l ss ) k ss (national account) Note: If u l (c ss ; m ss ; l ss ) =u c (c ss ; m ss ; l ss ) is independent of m ss, these equations determine k ss, l ss and c ss. Long-run superneutrality holds! This will be the case if utility is separable in money; e.g. u = v (c; l) g (m) (u l and u c are a ected by m in the same way) Also, of course, it will be the case if u l and u c are not a ected by m at all But if u l (c ss ; m ss ; l ss ) =u c (c ss ; m ss ; l ss ) depends on m ss, long-run superneutrality will not hold 5

7 Note that if u l and u c are independent of m, then superneutrality holds in the short run as well dynamics collapse into a real Ramsey-style model (the Keynes-Ramsey rule depicting the evolution of marginal utility of consumption will not be a ected by money) Speci c functional form of utility function: u (c t ; m t ; 1 n t ) = ac1 t b + (1 a) m 1 t b 1 0 < a < 1, b > 0, > 0, > 0, > 0 (b; ; 6= 1) 1 1 b + (1 n t) 1 ; ((2.64)) 1 is coe cient of relative risk aversion b is inverse nominal interest rate elasticity of money demand cf. (2.32), p. 49 6

8 What is u l (c ss ; m ss ; l ss ) =u c (c ss ; m ss ; l ss ) with this speci cation? u l u c = l t a ac 1 t b + (1 a) m 1 t b b 1 b ct b Hence, if = b, u cm = 0, and superneutrality holds in the short and the long run If < b (empirically plausible), then u cm > 0. Higher expected in ation will reduce real money balances and decrease marginal utility of consumption Households substitute towards leisure, and labor supply decreases Superneutrality fails in the short and long run If > b (empirically less plausible), then u cm < 0 and superneutrality fails in the opposite direction Note that it is anticipated changes in in ation that cause real e ects. An unanticipated, temporary, change in t has no e ects, as it does not a ect the nominal interest rate and money demand. Only when t+1 is a ected, is the nominal interest rate a ected through the Fisher relationship, i t = t+1 + r t 7

9 Dynamics and calibration Given that superneutrality may fail in the short run due to endogenous labor choice, a relevant issue is whether the MIU model has short-run properties which match the data I.e., how is monetary shocks transmitted to the real economy, and how will monetary policy be able to play a stabilizing role? For this purpose a stochastic version of the model is formulated. Exogenous shocks bringing the economy away from steady state will be technology shocks and shocks to the growth rate of nominal money supply Model and private sector optimization. General case Production function is amended to y t = f (k t 1 ; 1 l t ; z t ) where z t is a technology shock: with e t being a mean-zero, white-noise shock z t = z z t 1 + e t ; j z j < 1; 8

10 Nominal money growth follows where u t is a shock to the growth rate t = ss + u t Assumption: with ' t being a mean-zero, white-noise shock. u t = u u t 1 + z t 1 + ' t ; 0 u < 1; Q 0 Note that there may or may not be serial correlation in the shocks to nominal money growth Note that money growth may or may not respond toward past technology shocks, and may be either procyclical ( > 0) or countercyclical ( < 0). Per-period utility function and budget constraint are u (c t ; m t ; l t ) and y t + t + (1 ) k t m t 1 = c t + k t + m t : (2.45 and 2.46 ) 1 + t (ignoring nancial assets b t as in last lecture...) As in MIU model without endogenous labor, households maximizes discounted lifetime utility subject to the budget constraint Again, dynamic programming method is used 9

11 Note, however, that since l t is a choice variable, it is inappropriate to treat available resources as the state variable at period t. Instead, state variables will therefore be k t 1 and a t t + m t 1 = (1 + t ) The optimization is then characterized by the value function V (a t ; k t 1 ) = max E t fu (c t ; m t ; l t ) + V (a t+1 ; k t )g where the maximization is over c, m, k, and l subject to the budget constraint and the de nition of the state variable a t. E t is the rational expectations operator. One substitutes the constraint and de nition so as to eliminate k t and a t+1 and get an unconstrained maximization problem 10

12 First-order condition with respect to c t : u c (c t ; m t ; l t ) = E t V k (a t+1 ; k t ) (2.47 ) t+1 =@c t = 0 by the de nition of a). Usual interpretation: Marginal gain of consumption must equal the expected marginal loss in terms of lower capital in next period First-order condition with respect to m t : u m (c t ; m t ; l t ) + E t V a (a t+1 ; k t ) t+1 = E t V k (a t+1 ; k t ) (2.50 ) Usual interpretation: Marginal gain in terms of current utility and expected next period monetary wealth must equal the expected marginal loss in terms of lower capital in next period First-order condition with respect to l t : u l (c t ; m t ; l t ) = E t V k (a t+1 ; k t ) f n (k t 1 ; 1 l t ; z t ) (2.48 ) Marginal gain of leisure is equated to the marginal cost, which is the value loss from less next-period capital, times the marginal product of labor (the real wage) 11

13 Mathematical digression not for lecturing", but for reading: Elimination of the value function We know that optimum will be characterized by optimal values of c t, m t, and l t as functions of the state variables. Call these functions c t = c (a t ; k t 1 ) ; m t = m (a t ; k t 1 ) ; l t = l (a t ; k t 1 ) : The value function is thus by de nition given as V (a t ; k t 1 ) = u (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) + E t V (a t+1 ; k t ) This holds for all a t, k t 1 so we have V a (a t ; k t 1 ) = u c (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) c a (a t ; k t 1 ) +u m (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) m a (a t ; k t 1 ) +u l (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) l a (a t ; k t 1 ) +E t V a (a t+1 ; k t t + E t V k (a t+1 ; k t t : We have t+1 1 = m a (a t ; k t 1 ) t 1 + t = 1 c a (a t ; k t 1 ) m a (a t ; k t 1 ) f n (k t 1 ; 1 l t ; z t ) l a (a t ; k t 1 ) t 12

14 Use this in the expression for V a (a t ; k t 1 ): V a (a t ; k t 1 ) = u c (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) c a (a t ; k t 1 ) +u m (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) m a (a t ; k t 1 ) +u l (c (a t ; k t 1 ) ; m (a t ; k t 1 ) ; l (a t ; k t 1 )) l a (a t ; k t 1 ) 2 3 +E t V a (a t+1 ; k t ) 1 m a (a t ; k t 1 ) 1 + t+1 6 {z } 7 t+1 2 t +E t V k (a t+1 ; k t ) 6 1 c a (a t ; k t 1 ) m a (a t ; k t 1 ) {z f n (k t 1 ; 1 l t ; z t ) l a (a t ; k t 1 } ) : 7 t t By the Envelope theorem, all the terms in front of c a (a t ; k t 1 ), m a (a t ; k t 1 ) and l a (a t ; k t 1 ) are zero. I.e., at an optimum, the marginal value of changing c, m, or l must be zero 3 Therefore: V a (a t ; k t 1 ) = E t V k (a t+1 ; k t ) 13

15 Likewise, we get V k (a t ; k t 1 ) = E t V k (a t+1 ; k t ) [f k (k t 1 ; 1 l t ; z t ) + 1 ] So, by the rst-order condition guiding c: u c (c t ; m t ; l t ) = V a (a t ; k t 1 ) We then get the corresponding relationship for guiding money choice similar to simple MIU model: u m (c t ; m t ; l t ) + E t u c (c t+1 ; m t+1 ; l t+1 ) t+1 = E t V k (a t+1 ; k t ) = u c (c t ; m t ; l t ) (*) 14

16 Also, the condition guiding consumption can be modi ed by use of V k (a t ; k t 1 ) = E t V k (a t+1 ; k t ) [f k (k t 1 ; n t ; z t ) + 1 ]: u c (c t ; m t ; l t ) = E t V k (a t+1 ; k t ) = E t E t+1 V k (a t+2 ; k t+1 ) [f k (k t ; 1 l t+1 ; z t+1 ) + 1 ] and using that V a (a t ; k t 1 ) = E t V k (a t+1 ; k t ), gives V a (a t+1 ; k t ) = E t+1 V k (a t+2 ; k t+2 ) and thus u c (c t ; m t ; l t ) = E t E t+1 V a (a t+1 ; k t ) [f k (k t ; 1 l t+1 ; z t+1 ) + 1 ] = E t (1 + r t ) u c (c t+1 ; m t+1 ; l t+1 ) (**) with r t f k (k t ; 1 l t+1 ; z t+1 ). I.e., (**) is the money-modi ed Keynes-Ramsey rule Finally, we get the condition for the choice of l t, which becomes u l (c t ; m t ; l t ) = u c (c t ; m t ; l t ) f n (k t 1 ; 1 l t ; z t ) : (***) Hence, the equations (*), (**) and (***), together with the budget constraint, provide solutions for the paths of c, m, l, and k. End of mathematical digression not for lecturing 15

17 Particular functional forms of utility and production functions Model is solved by numerical methods under assumptions about particular functional forms for utility and production function Utility (as before): u (c t ; m t ; l t ) = ac1 t b + (1 a) m 1 t b 1 Production function, Cobb-Douglas: Steady-state solution: Note that the real interest rate, becomes 1 1 b + l1 t 1 y t = k t 1n 1 t e z t ; 0 < < 1 (2.63) r t = f k (k t ; 1 l t+1 ; z t+1 ) r t = E t k 1 t (1 l t+1 ) 1 e z t+1 = E ty t+1 k t : By the steady-state condition 1 + r ss = 1=, this only determines the ratio y ss =k ss [holds for any CRS production function; cf. (2.61)] The ratio is independent of monetary factors, but the levels y ss, k ss, c ss are not when, e.g., l ss is a ected by m (with this utility function, when 6= b). Then, superneutrality fails. 16

18 Dynamic e ects of money and technology shocks To assess the quantitative e ects of money and technology shocks, the model is calibrated and simulated Calibration: Assign empirically plausible values to the parameters of the model Simulation: Perform a linearization of the model s dynamic equations (everything is expressed as percentage deviations from steady state); solve this system by numerical methods (various simulation programs are available on the internet); create arti cial time series data from the system From the arti cial data, one evaluates the properties of the model relative to data in terms of: Standard deviations of various relevant variables, and their s.d. relative to output Correlation coe cients of various variables with output Impulse response patterns of variables when shocks hit 17

19 Main results (when b > ; implying u cm > 0) Steady-state non-superneutrality is of the form of: Higher =) lower output If money shocks, ' t shocks, shall pay a role, persistence in money growth is necessary ( u > 0 is needed). Then, the shock will a ects expected next-period in ation, and thus through the Fisher equation period t nominal interest rate. Real money holdings change, and the consumption-leisure trade-o is altered The e ects of money shocks on labor and output are stronger the more persistence in money growth, but the e ects are quantitatively very small (1% increase in money stock lowers output by 0.2% at max) If technology shocks are met with procyclical money growth, output is more stable. The magnitude, however, is modest Main e ects of money shocks are on in ation and nominal interest rates Positive money shocks lead to higher nominal interest rates. In contrast with usual IS/LM story (where a liquidity e ect is present: nominal rates fall to increase money demand). Reason is exible prices in the MIU model (contrary to the sticky price IS/LM model). Prices adjust instantaneously so as to reduce real money supply, matching the fall in demand resulting from higher nominal interest rates. Wildly at odds with empirical evidence on short run correlations (corr(m; i) < 0) 18

20 Summary The MIU framework provides a setting in which the welfare costs of in ation can be assessed, and where the optimal long-run in ation rate can be determined This, in turn, is equivalent of determining the optimal quantity of money The stochastic, dynamic model without the superneutrality property can be used to assess the importance of monetary shocks for economic uctuations In the calibrated, MIU model with endogenous labor, money matters for business cycle uctuations, but not very much. Moreover, the missing liquidity e ect is a major weakness. This is an indication that ex-price models may be ill-suited for analysis of monetary phenomena in the short run, i.e., at business cycle frequency 19

21 Plan for next lectures Tuesday, February Cash-in-Advance Models Literature: Walsh (2010, Chapter 3, pp ) (NB: Material on shopping-time models only recommended) Thursday, February 28 Exercises: QUESTION 2 from June 15 exam, 2006 (CIA constraint on investment purchases) 20