Monetary Economics. Money in Utility. Seyed Ali Madanizadeh. February Sharif University of Technology

Transcription

1 Monetary Economics Money in Utility Seyed Ali Madanizadeh Sharif University of Technology February 2014

2 Introduction

3 MIU setup FOCs Interpretations and implications Neutrality and superneutrality Equilibrium steady state Simple Example Log-Linearization Short run implications A short run shock to money

4 To use the general equilibrium framework to analyze monetary issues, a role for money should be specified so that agents are willing ot hold positive quantity of money. Here, we assume that money yields direct utility by incorporating real money balances into the utility function (Sidrauski 1967) We can think of it as saving the individual s time from barter. Money does not earn interest.

5 A basic MIU model: subject to max E 0 {c t,l t,m t,b t β t U (c t, l t, m t ) } t=0 P t c t + B t + P t k t+1 + M t = W t l t + V t k t + (1 δ) P t k t + (1 + i t ) B t where m t = M t P t is the real money holding.

6 In real terms c t + B t P t + k t+1 + M t P t redefining variables = W t l t + V t k t + (1 δ) k t + (1 + i t ) B t 1 P t P t P t P t 1 P c t + b t + k t+1 + m t = w t l t + (v t + 1 δ) k t i t 1 + π t b t π t m

7 FOCs: [c t ] : β t U ct = λ t [l t ] : β t U lt = w t λ t [ ] 1 + it+1 [b t ] : λ t = E t λ t π t+1 [k t+1 ] : λ t = E t [λ t+1 (v t δ)] [ ] 1 [m t ] : U mt + βe t λ t+1 = λ t 1 + π t+1 mc and mb analysis. In Equilibrium, return on bond, money (and capital) should be equalized to prevent arbtrage. Why people holds money? Neither M nor P matter but their ratio m. Money Neutrality

8 Euler Equation [ ] 1 + it+1 U ct = βe t U ct π t+1 Note that these terms depend on m (Real money balances)

9 Interpretation U ct = U mt + βe t [ Uct π t+1 ]

10 Asset pricing Equation of money = λ t P t }{{} Unit value of C I could purchase with my 1$ U mt P t }{{} Utility value of money balanaces I buy instead [ ] 1 + βe t λ t+1 P t+1 }{{} Discounted utility value of consumtion if I spend the 1$ next period Value of Stock today = It s dividend payment today + It s expected future value.

11 Using stock price analogy, we get (one out of many stationary solution) [ 1 = 1 ( ) ] Um(t+i) E t β i P t λ t i=0 P t+i It moves like a stock price. But in reality we do t see such movements in the aggregate price index. Thus we need some price rigidities!

12 Perfect forsight: 1 + r t+1 v t δ = 1 + i t π t+1

13 Using Euler equation, FOCs for money and bond, we find that: U mt U ct = i t i t+1 Interpretation: MRS=relative price, since 1 + i t+1 = (1 + π t+1 ) (1 + r t+1 ) We get moey demand equation: As i, m Consumer surplus! and welfare cost of inflation

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15 How does the money demand curve look like? ln (m) = ln (A) ηi i 0 ln (m) ln (A) ln (m) = ln (B) η ln (i) i 0 ln (m)

16 Lucas 1994 used 1900 to 1985 Us data to estmate money demand equation Ireland 2009 used more recent data and showed semi-log money demand curve has better fit. Estimates of the welfare cost of inflation varies from 0.85% to 3% of GDP (Gillman 1995)

17 Equilibrium: HHs maximizes utility Firms only hire labor optimally. Centrl Bank supplies money (constant growth rate µ, for example) M s t = (1 + µ) M s t 1 V t = µm s t 1 Goods, labor, bond and money markets clear B t = 0 M d t = M s t

18 Neutrality Changes in level of money doesn t affect the real terms. Super-neutrality: Changes in the growth rate of money supply π i m If separable utilites: No change in real terms Super Neutral. Inflation only induces a welfare cost. Non-Seperable: Not super Neutral

19 Steady States: Separable case m constant growth of P t = graowth of M t π = µ We had (Fisher Equation): Euler Equation (EE) given β ( or ρ), i tracks π. 1 + r t = 1 + i t 1 + π t 1 = β 1 + i 1 + π 1 + ρ = 1 β = 1 + i 1 + π = 1 + r r = ρ i = π + r = µ + ρ

20 Inflation results in welfare loss due to lower money demand. Is there an optimal rate of π that maximizes the welfare? The private opportunity cost of the private market depends on the nomina interest rate. The social marginal cost of producing moey is essentially zero. If i > 0 then there is a wedge between private an social cost ineffi ciency So the optimal rate is i = 0.

21 Optimal inflation i = 0 µ = π = ρ < 0 Called "The Fiedman Rule" (Friedman 1969, Bailey 1956) Some economists do not like this rule! Phelps 1973 argues that Friedman rule holds only in environements where gov can rais lump sum taxes.

22 A Simple Example U = log (c) + α log (1 l) + γ log (M)

23 Short Run analysis Log-linearization Dynare

24 More References Chari, Christiano, Kehoe (1991, 1996) Coerria, Teles (1996,1999) Mulligan, Salai-Martin (1997)