1. Money in the utility function (start)

Transcription

1 Monetary Policy, 8/2 206 Henrik Jensen Department of Economics University of Copenhagen. Money in the utility function (start) a. The basic money-in-the-utility function model b. Optimal behavior and steady-state equilibrium properties: Long-run superneutrality of money Literature: Walsh (200, Chapter 2, pp ) c 206 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 Introductory remarks The standard model for (exogenous) economic growth is the simple Solow model featuring a xed savings rate a law of motion for physical capital accumulation When extended with optimizing savings behavior, we get the Ramsey model Models have no role for money and, hence, monetary policy Purpose of models/analyses in coming lectures is to introduce a role for money in these type of models By-product: Introduction to Dynamic Programming

3 Money is introduced in various ways: Often short cuts Short cuts are helpful for understanding simple features, and the more robust results are to a particular short cut, of course, the better Tobin model (965, Econometrica) extends the Solow model by postulating a demand for money (the short cut) Highlights the implications of in ation for the choice between investment in physical and nancial assets Higher in ation leads to a substitution away from nancial assets towards physical capital Higher in ation leads to higher steady-state capital stock and output Money-in-the-Utility-function models extend the Ramsey model by postulating that money gives utility (the short cut) Highlights the importance of micro foundations and optimizing private-sector behavior (absent in the Tobin model) 2

4 Money in the utility function (start) A standard Ramsey model with money (in nite-horizon setting with optimizing households and perfect competition in goods market) The short-cut here: real money provide utility per se One interpretation: Money facilitate transactions on the market and reduce shopping/search time; they solve the problem of double coincidence of wants Money are accepted as a means of payment by all (for example as money is backed/guaranteed by a central bank/government) Money is an otherwise useless commodity; see Walsh Section 3.2. for a formalization of the shopping-time argument The per-period utility function of households is U t = u (c t ; m t ) with u being increasing and concave in both arguments. It is real money that enters in u, i.e., M t s value relative to what it can buy at price P t ; m t M t = (P t N t ). 3

5 The representative household/producer maximizes: X W = t u (c t ; m t ) ; 0 < < : (2.) t=0 Households budget constraints are (ignoring, for simplicity, nancial assets B t used in Walsh): C t + K t + M t P t = Y t + t N t + ( ) K t + M t P t ; 0 < < (2.2 ) (m t is end-of-period money balances; debatable in itself whether it should be m t or m t that should give utility) Output is produced by a CRS production function: Y t = F (K t ; N t ) : In intensive form (see Walsh s Footnote 8, p. 37): kt y t = f ; y t Y t =N t ; k t K t =N t ; + n N t =N t + n The government makes per capita real lump-sum transfers (or withdrawals) t : t = M t M t P t N t This is the consolidated government/central bank ow budget constraint 4

6 In per-capita version, assuming no population growth in contrast to Walsh (set n = 0, and N t = ), we get c t + k t + m t = y t + t + ( ) k t + + t m t In ation t (P t =P t ), erodes initial real monetary resources Rewritten: c t + k t + m t =! t f (k t ) + t + ( ) k t + + t m t (2.4 )! t is the total available resources, taken as given at t by households. It is the relevant state variable when choosing the optimal paths of c, k, and m at date t Household s optimization problem is solved by dynamic programming 5

7 Dynamic programming involves use of the value function The value function is the maximal value of W, given constraints and the current state! t : X V (! t ) max s t u (c s ; m s ) fc s g s=t ;fm sg s=t ;fk sg s=t s=t = max fu (c t ; m t ) + u (c t+ ; m t+ ) + 2 u (c t+2 ; m t+2 ) + : : : g fc s g s=t ;fm sg s=t ;fk sg s=t = max fu (c t ; m t ) + [u (c t+ ; m t+ ) + u (c t+2 ; m t+2 ) + : : : ] g fc s g s=t ;fm sg s=t ;fk sg s=t = max c t ;m t ;k t = max c t ;m t ;k t u (c t ; m t ) + ( u (c t ; m t ) + max [u (c t+ ; m t+ ) + u (c t+2 ; m t+2 ) + : : : ] fc s g s=t+ ;fm sg s=t+ ;fk sg s=t+ ) max fc s g s=t+ ;fm sg s=t+ ;fk sg s=t+ X s=t+ s (t+) u (c s ; m s ) compactly written as V (! t ) = max c t ;m t ;k t fu (c t ; m t ) + V (! t+ )g Maximization is subject to the budget constraint and the de nition of! t+ (available resources one period ahead) 6

8 To make it simple here substitute out! t+ = f (k t ) + t+ + ( ) k t + then substitute out k t using k t =! t c t m t + t+ m t Then maximize (unconstrained) just over c t and m t : 8 0 >< and thus max c t ;m t max c t ;m t u (c t ; m t ) + V >: 8 >< u (c t ; m t ) + V >: f (k t) + t+ + ( ) k t >= m t C + t+ A >; {z }! t+ f (! t c t m t ) + t+ + ( ) (! t c t m t ) + m t {z + t+ }! t+ with k t substituted out 9 >= C A >; 7

9 First-order condition concerning choice of c t : u c (c t ; m t ) + V! (! t+ t = 0 u c (c t ; m t ) = V! (! t+ ) [f k (k t ) + ] (2.6 ) Marginal utility of period-t consumption equals its marginal loss (in terms of the future discounted marginal value of capital) First-order condition concerning choice of m t : u m (c t ; m t ) + V! (! t+ ) u m (c t ; m t ) + V! (! t+ t = 0 + t+ = V! (! t+ ) (f k (k t ) + ) (2.8 ) Marginal utility period-t real money (in terms of direct utility plus discounted marginal value of more future real money) equals marginal loss (in terms of the marginal value of less future capital) Furthermore, transversality conditions must be satis ed: lim t! t u c (c t ; m t ) k t = 0 lim t! t u c (c t ; m t ) m t = 0 (otherwise over accumulation of k and m is taking place lifetime utility could be improved through higher consumption by accumulating less) 8

10 In the rst-order conditions, V! is eliminated by use of the so-called Envelope Theorem Note: optimal period t consumption and money holding choices will be functions of! t De ne these as c t c (! t ) and m t m (! t ), respectively The value function is therefore by de nition given by As (*) holds for any value of! t, it follows that V (! t ) = u (c (! t ) ; m (! t )) + V (! t+ ) : (*) V! (! t ) = u c (c (! t ) ; m (! t )) c 0 (! t ) + u m (c (! t ) ; m (! t )) m 0 (! t ) + V! (! t+ t : (**) Now, t+ =@! t when c t = c (! t ) and m t = m (! t ) applies: Remember! t+ = f (k t ) + t+ + ( ) k t + + t+ m t = f (! t c t m t ) + t+ + ( ) (! t c t m t ) + + t+ m t One therefore t = [f k (k t ) + ] ( c 0 (! t ) m 0 (! t )) + + t+ m 0 (! t ) 9

11 Combining this with (**): V! (! t ) = u c (c (! t ) ; m 8 (! t )) c 0 (! t ) + u m (c (! t ) ; m (! t )) m 0 (! t ) < [f k (k t ) + ] ( c 0 (! t ) m 0 (! t )) +V! (! t+ ) : + m 0 (! t ) + t+ Collect the c 0 (! t ) and m 0 (! t ) terms: V! (! t ) = [u c (c (! t ) ; m (! t )) V! (! t+ ) (f k (k t ) + )] c 0 (! t ) u m (c (! t ) ; m (! t )) + V! (! t+ ) + t+ 5 m 0 (! t ) V! (! t+ ) [f k (k t ) + ] +V! (! t+ ) [f k (k t ) + ] 9 = ; :... and get a pleasant surprise: 2 3 V! (! t ) = 4u c (c (! t ) ; m (! t )) V! (! t+ ) (f k (k t ) + {z ) 5 c } (! t ) 2 = 0 by (2.6 ) 3 u m (c (! t ) ; m (! t )) + V! (! t+ ) t+ m 0 (! t ) 4 V! (! t+ ) [f k (k t ) + ] 7 {z } 5 = 0 by (2.8 ) +V! (! t+ ) [f k (k t ) + ] 0

12 All terms in front of c 0 (! t ) and m 0 (! t ) are zero by the rst-order conditions Note the terms capture the life-time utility e ects of changing! t marginally through the associated marginal changes in c t and m t As the value function is de ned as the life-time utility where c t and m t are optimally chosen given! t, their marginal e ects are zero Alternatively, use a contradiction argument: If a change in! t leads to a change in V through marginal changes in c t and m t, then c t and m t have not been optimally chosen, and V is not the highest life-time utility Hence, (**) reduces to V! (! t ) = V! (! t+ ) [f k (k t ) + ] Then use the rst-order condition for consumption choice, u c (c t ; m t ) V! (! t+ ) [f k (k t ) + ] = 0; to obtain Walsh s expression: V! (! t ) = u c (c t ; m t ) : (2.0) Marginal utility of consumption (denoted t in Walsh) equals marginal value of wealth

13 The rst-order conditions can then be rewritten as u c (c t ; m t ) = u c (c t+ ; m t+ ) [f k (k t ) + ] (a discrete-time, monetary version of the standard Keynes-Ramsey rule ), and u m (c t ; m t ) + u c (c t+ ; m t+ ) + t+ = u c (c t ; m t ) (marginal gain of m t equal to the marginal loss in terms of lower capital in period t + equal to the marginal utility of c t by the Keynes-Ramsey rule ) These conditions, together with the budget constraint characterize the optimal paths of c, k, and m Note: Only m t appears. Any change in M t is re ected proportionally in P t : Money neutrality For now, concentrate on the long-run properties of the model; i.e., a steady state with c t = k t = m t = 0 First, from Keynes-Ramsey rule it follows that in steady state or, = [f k (k ss ) + ] ; f k (k ss ) + = (2.9 ) This independently of any monetary factors de nes the steady-state capital per capita (and thus output per capita) 2

14 Strong contrast with the Tobin model mentioned in introduction The MIU model has micro-founded behavior; i.e., it captures behavioral responses to a policy change (here, change in nominal money growth) If, e.g., k t < k ss the current marginal product of capital is relatively high (as f kk < 0) =) optimal to postpone consumption to later =) capital is accumulated until f k (k ss ) + = = holds again If one imagined that a Tobin e ect was there; one would be self-contradictory: Assume higher in ation increases capital to a new, higher steady state Marginal product of capital decreases, and households would want to consume now rather than later =) they endogenously save less and capital decreases until k ss is reached again! In ation has no steady-state e ect on capital. Only possible in Tobin model, where individuals are modelled as having an exogenously xed savings rate Example of the importance of considering changes in private sector behavior when examining policy changes 3

15 Steady-state consumption in the MIU model? Budget constraint: Transfers are so in steady state: A constant m implies =, one gets f (k ss ) + ss + ( ) k ss + + mss = c ss + k ss + m ss t = (M t M t ) P t = t M t P t = ss = + mss t + t m t ; f (k ss ) mss + ( ) k ss = c ss + k ss + m ss f (k ss ) k ss = c ss (2.2) The economy s resource constraint (national account): Output less investment equals consumption Implication is that c ss is determined exclusively by k ss ; and thus independent of nominal money growth Model exhibits superneutrality of money in steady state The growth rate of nominal money has no real e ects 4

16 Do money growth and in ation not a ect anything? Yes, the opportunity cost of holding real money, and thus the steady-state value of real money balances Relative demand for consumption versus real money is given by [use rst-order condition for money holdings and divide through by u c (c t ; m t )] with r t f k (k t ) u m (c t ; m t ) u c (c t ; m t ) = = = is the real interest rate u c (c t ; m t ) u c (c t ; m t ) + t+ u c (c t+ ; m t+ ) + t+ u c (c t ; m t ) (f k (k t ) + ) ( + t+ ) ( + r t ) Note that the real interest rate is the nominal rate net of expected in ation: + r t = ( + i t ) = ( + t+ ) ; (r t i t t+ ) Hence, u m (c t ; m t ) i t = I t (2.2) u c (c t ; m t ) + i t So, as nominal interest rate is determined by the Fisher relationship, i t r t + t+, higher in ation leads to a higher nominal interest rate Consistent with long-run data 5

17 For given c t, m t is likely to fall when i t increases (as u mm < 0). A micro foundation for conventional money-demand function With u (c t ; m t ) = act b + ( a) m t b =( b), 0 < a <, b > 0 we get from (2.2) u m (c t ; m t ) u c (c t ; m t ) = = ac b t + ( a) m t b b=( b) ( a) m b t ac b t + ( a) mt b b=( b) ac b t b a ct = i t a m t + i t This gives a money demand function as m t = a a b it + i t b ct (2.3) In logs (often used for estimation purposes): log m t = const. + log c t b log it ; (2.32 ) + i t where =b is the elasticity of money demand wrt. opportunity cost (depending on money concept, empirically around ) 6

18 Will a unique steady-state value for m exist? Must solve u m (c ss ; m ss ) = i ss u + i ss c (c ss ; m ss ) u m (c ss ; m ss ) = I ss u c (c ss ; m ss ) Not necessarily unique... Stability properties? For separable utility, u (c t ; m t ) = v (c t ) + (m t ), resulting di erence equation (from rst-order condition) will imply a saddle-point stable m ss > 0 (m 0 in Figure 2., p. 45) Problems: One cannot necessarily rule out the paths with increasing m above steady state ( speculative hyperde ations) One cannot necessarily rule out the paths with falling m below steady state ( speculative hyperin ations), leading to m ss = 0 In the context of this model, we will not pursue the stability issue 7

19 Summary MIU model has one for one relationship between in ation and money growth MIU model exhibits superneutrality A model like the Tobin model is not superneutral; reason is the postulated and policy-invariant private sector behavior. This di erence highlights the importance of micro foundations to avoid Lucas critique MIU is a structural model where the reactions of the private sector to changes in policy (money growth) are taken into account Potentially more appropriate for analyzing policy changes (even though the micro foundation for money demand is a short cut) (At least for steady-state analyses.) 8

20 Plan for next lectures Monday, February 22. Money in the utility function (continued) a. Welfare costs of in ation b. Potential non-superneutrality of money c. Dynamics and calibration of stochastic version: Simulation examples Literature: Walsh (200, Chapter 2, pp , so check the Appendix as well; i.e., get a grip on (relive the) linearization techniques) Monday, February 29. A Cash-in-Advance Model Literature: Walsh (200, Chapter 3, pp. 9 5) (NB: Material on shopping-time models only recommended) 9