Monetary Economics Basic Flexible Price Models

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1 Monetary Economics Basic Flexible Price Models Nicola Viegi July 26, 207

2 Modelling Money I Cagan Model - The Price of Money I A Modern Classical Model (Without Money) I Money in Utility Function Approach I Cash in Advance Models (not really)

3 The Cagan Model I Originally a model of Hyperin ation I Main Point: Expectation about future fundamentals determine prices now I Used extensively in Exchange rate analysis, assets prices I A lot of modern models look like this

4 The Model Money Demand Fisher Equation M d t = L (Y t, i t ) ( + i t ) = ( + r t ) + log approximation and uncertainty log ( + i t ) = log ( + r t ) + E t p t+ p t

5 Assume Y t and r t xed, the money demand equation (in log): m d t p t = η (E t p t+ p t ) or m t p t = η (E t p t+ p t ) p t = m t + η E tp t+

6 Forward Solution p t = = m t + η E tp t+ m t + η E t = =... = m t + T s=0 η E tm t+ η m t+ E t+p t+2 η 2 + E t p t+2 s m t+s + η T E t p t+t ()

7 Law of Iterated Expectations E t (E t+ p t+2 ) = E t p t+2 No Bubble lim T! η T lim E t p t+t = 0 T! T s=0 η s m t+s <

8 Solution Generic Solution p t = s=0 η s m t+s ep t Prices at time t depend on all future expected money supply. Expected fundamentals (not actual)

9 Constant Money Supply p t = m t m E t m t+s = m (s 0) = = m s=0 η η +η m s m

10 Future One Time Money Increase at T>0 Consider an surprise announcement at time t = 0 of a change of policy at some time in the future T > 0 Thus m t = m m 0 > m t < T t T p t = m t < T m 0 t T How would the transition look like?

11 Solution In the period between 0 t T η s p t = m + s=0 p t = m + η T t s=0 η T t p t = m + m 0 m s=t η η s m 0 m t s m 0 m

12 Solution log price level m p0 m 0 T time I Prices "Jump" at the announcement, before the policy is implemented I Expectations drive economic dynamics

13 A Classical Monetary Model (Gali Ch. ) I Why Classical? I I Perfect Competition Fully Flexible Prices I Real Business Cycle Structure + Nominal Prices Determination I Three Agents: Households, Firms, Central Bank (setting nominal interest rate)

14 Households Representative household solves subject to max E 0 t=0 β t U (C t, N t ) C t + Q t B t B t for t = 0,, 2,... plus solvency constrains Optimality Conditions + W t N t + D t U n,t + W t U c,t = 0 Q t U c,t = βe t U c,t+ +

15 Households Speci cation of Utility U (C t, N t ) = C t σ σ N +ϕ t + ϕ which gives the log-linear optimality conditions (aggregate variables) w t p t = σc t + ϕn t c t = E t fc t+ g σ (i t E t fπ t+ g ρ) where i t = log Q t is the nominal interest rate, ρ = log β is the discount raate, and π t+ = p t+ p t is the in ation rate

16 Firms Pro t Maximization max Y t W t N t subject to Optimality Conditions Y t = A t N t α W t = ( α) A t Nt α log linear version (ignoring constants) w t p t = a t αn t

17 Equlibrium Equlibrium Value of Real Varialbes y t = c t Labour Market Asset market Clearing σc t + ϕn t = a t αn t b t = 0 y t = E t fy t+ g Aggregate Production relationship σ (i t E t fπ t+ g ρ) y t = a t + ( α) n t

18 Equlibrium Equilibrium Values of Real Variables σ n t = σ ( α) + ϕ + α a + ϕ t; y t = σ ( α) + ϕ + α a t σ + ϕ w t p t = σ ( α) + ϕ + α a t σ ( + ϕ) ( ρ r t i t E t fπ t+ g = ρ a ) σ ( α) + ϕ + α a t I Real varialbes determined independently of monetary policy (neutrality) I Optimal Policy Undetermined I Speci cation of Monetary Policy Needed to determine nominal variables

19 Monetary Policy and Price Level Determination A Simple Interest Rate Rule i t = ρ + φ π π t combined with the de nition of real rate r t i t E t fπ t+ g, φ π π t E t fπ t+ g + br t if φ π >, unique stationary solution π t φ (k+) π fbr t+k g k=0 if φ π <, any process π t satisfying π t+ φ π π t br t + ξ t+ where E t fξ t+ g = 0 for all t is consistent with a stationary equlibrium (Price Level Indeterminacy) I Illustration of the "Taylor Principle"

20 Monetary Policy and Price Level Determination Exogenous Path of Money Supply (Cagan Again) m t p t = y t ηi t Combining Money Demand and Fisher Equation r t i t E t fπ t+ g, η p t = +η where u t = Assuming η > and solving forward where u 0 t = k=0 E t fp t+ g + m t + u t (ηr t y t ) evolves independently of m. p t = k=0 η η +η k Et fu t+k g k E t fm t+k g + u 0 t

21 Monetary Policy and Price Level Determination Exogenous Path of Money Supply (Cagan Again) Rewrite in term of expected future money growth rates p t = m t + k= η k E t f m t+k g + u 0 t which implies the following nominal interest rate: i t = η [y t (m t p t )] i t = η η k= k E t f m t+k g + υ t i where υ t η hy t + u 0 t is independent of policy

22 Money in The Utility Function Money in the Utility function is just a modelling trick to determine money demand inside the private sector optimization The household s optimization problem (with xed labour supply) max fc t,m t,k t+ β i u C t, M t gt=0 t=0 subject to the real budget constraint K t+ + M t = ( + r t ) K t + M t + w t C t T t Production is given by a neoclassical production function with constant return to scale Y t = F (K t, L t )

23 Equilibrium Conditions u c C t, M t u c C t, M t = β ( + r t+ ) u c C t+, M t+ = u M /P C t, M t + + βu c C t+, M t+ + Pt + (2) (3) Substituting for βu c (C t+, M t+ /+ ) from (2) in (3) and rearrange, we get: u c C t, M t ( + r t+ ) + = u M /P C t, M t (4)

24 Equilibrium Conditions Remember the Fisher Equation that relates real and nominal interest rates ( + i t ) = E t ( + r t+ ) + under perfect foresight, (4) can than be written as: i t = u + i M /P C t, M t /u c C t, M t t LM Equation (5)

25 General Equilibrium in MIU Need to Specify: Government Bahaviour Factor prices T s = M s M s P s F (K, ) r t = K F (K, ) w t = F (K, ) K t K where the second relation uses the property of constant return to scale (F = L F / L + K F / K )

26 Steady State To be consistent with the data a model including money should have the following I Neutrality of Money: the real equilibrium is independent of the money stock I Superneutrality of money: the real equilibrium is independent of the money growth rate Nominal Equilibrium M t /M t = + σ / = + π C t /C t = M t / /M t / = π = σ

27 Real Equlibrium from u c C t, M t In steady state u c C, M P = β ( + r t+ ) u c C t+, M t+ + = β ( + r t+ ) u c C, M P Combining with r t = stock must solve ( + r) = /β F (K,) K,, gives that the steady state capital F K (K, ) = r = /β which depends only on the technology and the real discount rate, not on the money stock or growth. Because capital stock is constant, so consumption in steady state is uniquely determined by the real side of the economy

28 Monetary Equilibrium In steady state, in this model, money is neutral and superneutral With a value for the steady state consumption, we can solve for a value of the steady state real money balances by combining the Fisher equation and ( + i) = E t ( + r) + / ( + i) = E t ( + r) ( + π) i + i = u M /P C, M /u c C, M P P would give and equation in steady state real money and of known parameters

29 Optimal Rate of In ation Friedman (969) Social Marginal cost of producing money = 0 Private marginal cost of money = i De ation PMC = SMC i = 0 ( + i) = E t ( + r) + i = r + π = 0 π = r

30 Cost Of In ation Money Demand Function i + i = u M /P C, M P /u c C, M P

31 Conclusions I Basic Flexible Price Models Highlight: I I I I Problems: I I The Role of Expectations The Importance of Policy Design for Price Determination Cost of In ation (even when we have neutrality of money) Money has no e ect on real variables (only welfare) Friedman (968) not realistic (De ation is costly, but why?)

ECON 4325 Monetary Policy and Business Fluctuations

ECON 4325 Monetary Policy and Business Fluctuations Tommy Sveen Norges Bank January 28, 2009 TS (NB) ECON 4325 January 28, 2009 / 35 Introduction A simple model of a classical monetary economy. Perfect