Getting to page 31 in Galí (2008)

Getting to page 31 in Galí 2008) H J Department of Economics University of Copenhagen December 4 2012 Abstract This note shows in detail how to compute the solutions for output inflation and the nominal interest rate in the classical model with money in the utility function presented in Galí 2008 p. 28 31). It utilizes the method of undetermined coeffi cients to derive the rational-expectations solution. 1 Introduction These notes present detailed computations leading to the solutions for output inflation and the nominal interest rate in a classical monetary model with money in the utility function in Galí 2008 Chapter 2 p. 31). All notation follows Galí 2008) and will not be explained unless needed. Equation numbers in this document are unique and do not correspond to the similar equations in Galí 2008). 2 Deriving the relevant optimality conditions Time is discrete and in any period t the representative household seeks to maximize a utility function { E t β t i U C i ) } i N i. i This is done while satisfying the following budget constraint: it C t + Q t B t + t B t 1 + t 1 + W t N t T t. 1) Let total financial wealth at the end of period t be defined as A t 1 B t 1 + t 1. The budget constraint 1) can then be written compactly as C t + Q t A t + 1 Q t ) t A t 1 + W t N t T t. 2) c 2012 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. 1

Written like 2) one readily sees the opportunity cost of investing resources in money rather than bonds. The yield on bonds are 1 Q t ) /Q t which approximately equals i t log Q t where i t is the nominal interest rate. We therefore have exp i t ) Q 1 t and therefore 1 Q t 1 exp i t ). 3) It thus follows that whenever i t > 0 the opportunity cost of holding money is positive. We find the necessary optimality conditions by setting up the Lagrangian: { L t E t β U t i it C i ) i N i λ i i C i + Q i A i + 1 Q i ) i A i 1 W i N i + T i )] } i where λ t is the multiplier on 2). 1 The necessary first-order conditions at any t are L t C t 0 : U ct λ t 4) L t U mt 0 : λ t 1 Q i ) t 5) L t N t 0 : U nt λ t W t 6) L t A t 0 : λ t Q t βe t {λ t+1 } 7) L t λ t 0 : C t + Q t A t + 1 Q t ) t A t 1 + W t N t T t λ t 0 and the complementary slackness condition λ t C t + Q t A t + 1 Q t ) t A t 1 W t N t + T t ) 0. 8) Note that since it is always assumed that the marginal utility of consumption is positive U ct > 0 we have from 4) that λ t > 0. Hence from 8) we have that 2) always binds. Normally one ignores this step and just state the budget constraint as an equality from the beginning. Since the focus in this version of the classical model is the inclusion of money in the utility function we start by characterizing optimal money demand. Combining 4) and 5) we readily get U mt U ct 1 Q t ) 9) which implicitly characterizes the optimal money demand as the quantity that equates the marginal rate of substitution between money and consumption to the marginal rate of transformation which here is the opportunity cost of holding money. Expressed as a function of the nominal interest rate we get U mt U ct 1 exp i t ). 10) 1 One can also write the Langrangian without having the discount factor being multiplied on Lagrange multipliers. This does not affect the results but the current formulation is the conventional and readily gives λ t the interpretation as the marginal utility of income at t. 2

From 10) we get a micro foundation for conventional money demand functions: For given consumption and labor a higher nominal interest rate reduces money demand whenever U mmt < 0 i.e. for standard concave utility. From the first-order conditions we also recover the standard optimality conditions for labor supply combine 4) and 6)] and savings combine 4) and 7)]: U nt U ct W t 11) }. 12) Q t βe t { Uct+1 U ct The model s supply side is represented by competitive firms who produce output with labor input though the production function Y t A t Nt 1 α. rofit-maximizing labor demand is characterized by 1 α) A t N α t +1 W t. 13) The description of the model is complete and the equilibrium values for five unknowns C t Y t N t W t / and / conditional on a monetary policy i t and thus Q t ) can be determined from 10) 11) 12) and 13) along with the goods market clearing condition Y t C t. 2 To facilitate a solution utility is assumed to have the following form U C i ) i N i X1 σ t i 1 σ N 1+ϕ t 1 + ϕ 14) where X t 1 ) C 1 t C 1 t + ) 1 ] 1 1 > 0 1 15) ) 1 2 Some may rightfully wonder what happened to the households budget constraint. It is not ignored. In the background of the simple representation given here the government pays out transfers T t which are financed by printing money. I.e. the government budget constraint reads T t t t 1. oreover since households are identical there will be nobody holding positive positions in bonds. I.e. in equilibrium B t 0 all t. In equilibrium the household s budget constraint is therefore C t + t t 1 + W t N t T t. Together with the public budget constraint one recovers C t W t N t. Finally since firms are competitive profits Y t W t N t are zero i.e. W t N t Y t which inserted above gives C t Y t. Hence C t Y t as claimed in the main text. So what may immediately seems too simple actually has solid foundations. In this model one is just not interested in following the evolution of net asset holdings profits or transfers the first two variables are zero all t in any case and transfers are just a residual of money creation whose value must accrue to somebody in the economy). 3

is an index aggregating utility from consumption and real money holdings. functional form we get With this U ct 1 ) Xt σ Ct ) U mt Xt σ and can therefore rewrite 10) 11) and 12) respectively as ) 1 t C t 1 exp i t )] 1 16) 1 N ϕ t Xt σ Ct 1 ) 1 W t 17) { Ct+1 ) ) } σ Xt+1 Q t βe t. 18) From this system one sees that monetary policy can have real effects whenever σ. Only when σ the system reduces to the basic classical model where monetary policy is irrelevant for the determination of e.g. C t and N t. Otherwise as changes in the nominal interest rate have impact on real money holdings through 16)] the associated change in X t will affect the labor supply decision; cf. 17). In the simulations presented in the slides associated with Chapter 2 the case of > σ was considered. 3 A shock driving up the nominal interest rate resulted in a drop in output. Although the simulations which came from Walsh 2010) were for a somewhat richer model with physical capital formation the economic transmission mechanism is the same here. A higher i t leads to a lower t /. This lowers X t which lowers the marginal utility of consumption when > σ. For a given real wage labor supply goes down cf. 17) and output will fall. 3 Deriving the linearized system The system cannot be solved analytically so one log-linearizes the relevant equations around a zero-growth steady-state this is an innocent assumption as the model does not contain any engine of sustainable growth). In particular a zero-inflation steady state is considered. This is an assumption made not only for analytical tractability but also because it makes results readily comparable to the zero-inflation steady states in the New- Keynesian models. 4 3 This appears to be the empirically relevant case when one translate the parameters to their real world counterparts. The coeffi cient of relative risk aversion σ is usually found to be around 1 5. As is the inverse elasticity of substitution between money and consumption estimates of the semi-elasticity of money with respect to the nominal interest rate give guidance to the appropriate value and an upper bound normally). As these elasticities often are in the order of 0.1 the inverse is well above the 1 5 range so as to make > σ the relevant case. 4 There the assumption is not innocent as it rules out dynamics of price dispersion as these will be of only second-order importance around a zero-inflation steady state. Also it facilitates welfare analyses based on second-order Taylor approximations of utility functions because zero inflation is welfare optimal implying that costs of inflation fluctuations will only be of second order. Thereby first-order approximations of the model equations suffi ce as the omitted second-order terms only have third- or fourth-order welfare effects. See Chapter 4 of Galí 2008) for more details. C t X t +1 4

ost of the log-linearizations are straightforward but those involving money demand and the composite consumption-real money balances index are a bit involved so we will devote some attention to these. Now under the assumptions about the steady state the consumption-euler equation 18) becomes Q β. 19) Throughout a variable without time index will be a steady-state value.) This has straightforward economic intuition. When the price of a bond today is exactly equal to the utility weight you attach to its return which is one) you have no incentive to save or dissave so as to let your marginal utility of consumption differ across periods. Using 3) in the money demand function 16) this is in steady state characterized by / C ) 1 1 Q) 1 1 ) 1 1 β) 1 1 ] 1 20) 1 β) 1 ) k m where the second line uses 19). We can now log linearize 16) around the steady state. Letting lower-case letters denote log deviations from steady state e.g. c t log C t /C) we readily get by taking logs of 16) and 20) and differencing: m t p t c t 1 log 1 exp i t)) log 1 Q)] The term in square brackets can be approximated to first order as log 1 exp i t )) log 1 Q) log 1 exp i)) log 1 Q) 1 + 1 exp i) exp i) i t i) 1 exp i) 1 i t i) where we have used 3). Note that as i t log Q t we have i log β ρ. Collecting these results we get the log-linear money demand expression: with η m t p t c t ηi t 21) 1 exp i) 1] β 1 β) 22) and where the steady-state term ηρ has been ignored. 5 The second equality in 22) follows from 3) and 19).] 5 In a linear model this has no implications for the analyses of the effects of various shocks. Alternatively one could simply work with a new variable i t ρ i.e the nominal interest rate s deviation from steady state. This is just a matter of preference for presentation. 5

We then proceed by deriving the labor supply schedule in log-deviations from steady state. From 17) we readily get w t p t σc t + ϕn t + σ) c t x t ) 23) To eliminate x t we first have to derive it. Recall that the composite consumption-real money balances index is defined as X t 1 ) C 1 t + ) 1 ] 1 1. A first-order Taylor approximation of X t around the steady state leads to X t X + + 1 ) C 1 + 1 ) C 1 + ) ] 1 1 1 1 1 ) C C t C) ) ] 1 1 1 1 ) ) where variables without time subscripts are steady-state values. Subtracting by X on both sides and rearranging this is readily rewritten as X t X X + 1 ) C 1 + 1 ) C 1 + ) ] 1 1 1 1 1 ) C C C t C) X C ) ] 1 1 1 1 ) / X t / Again letting lower-case letters denote log-deviations from steady state and acknowledging that log X t /X) X t X) /X is valid in a first-order approximation we then find x t + 1 ) C 1 + 1 ) C 1 + 1 ) C 1 + ) ] 1 1 1 1 1 ) C C X c t ) ] 1 1 1 1 ) ] 1 1 1 ) C 1 c t ) / X m t p t ) ) ] 1 1 ) 1 + 1 ) C 1 + m t p t ) ) 1 1 ) C 1 ) 1 c t + ) 1 m t p t ). 24) 1 ) C 1 + 1 ) C 1 + 6 ).

We then use 24) in 23) to obtain w t p t σc t + ϕn t + σ) c t w t p t σc t + ϕn t + σ) and thus with ) 1 1 ) C 1 ) 1 c t 1 ) C 1 + 1 ) C 1 + ) 1 1 ) C 1 + ) 1 c t m t p t )] w t p t σc t + ϕn t + χ σ) c t m t p t )] 25) χ ) 1 1 ) C 1 + ) 1 ) 1 /C ) 1. 26) 1 + /C ) 1 m t p t ) We can insert the steady-state value of /C from 20) to obtain χ 1 β) 1 ) 1 + ] 1 1 β) 1 ) 1 1 1 β) 1 )] ] 1 1 + 1 1 1 β) 1 )] 1 1 1 β) 1 ) 1 1 1 + 1 β) 27) which is the definition of χ first used in the text on page 29). Alternatively one can through 26) express χ directly as a function of the steady-state ratio of real money balances to consumption k m : χ k 1 m 1 + k 1 m k m 1 k m + k m. 7

Using 20) this can be written as χ k m 1 1 β) 1 ) + k m 1 β) k m 28) 1 + 1 β) k m which is the second variation of χ used on page 29. We now use the money demand function 21) to substitute out c t m t p t ) in 25): ore compactly this is written as where w t p t σc t + ϕn t + χη σ) i t. w t p t σc t + ϕn t + ωi t 29) ω χη σ) 1 β) k m β σ) 1 + 1 β) k m 1 β) βk m 1 σ ). 1 + 1 β) k m We here clearly see how the nominal interest rate affect labor supply as long as σ. { Ct+1 ) ) σ Xt+1 Q t βe t C t X t +1 { Ct+1 ) ) } σ Xt+1 1 βe t C t X t 1 + π t+1 We then linearize 18) around the steady state. A first-order Taylor approximation yields which becomes Q t Q β 1 C E t {C t+1 } C) + β 1 C C t C) +β σ) 1 X E t {X t+1 } X) β σ) 1 X X t X) βe t π t+1 Q t Q Q E t {c t+1 } c t ) + σ) E t {x t+1 } x t ) E t π t+1. We can write the left-hand side as Hence we get 1 Q t 1 Q) Q i t + ρ. i t + ρ E t {c t+1 } c t ) + σ) E t {x t+1 } x t ) E t π t+1 8 }

which can be re-written as i t + ρ σ E t {c t+1 } c t ) + σ) E t {x t+1 } E t {c t+1 } x t + c t ) E t π t+1 and thereby c t E t {c t+1 } 1 σ i t E t π t+1 ρ σ) E t {c t+1 x t+1 c t x t )}). 30) Again we see that for the case of σ we have the standard case with no money in the utility function. Like in the derivation of the labor supply function we can eliminate x t in 30) through 24) so as to get: c t E t {c t+1 } 1 σ i t E t π t+1 ρ χ σ) E t {c t+1 c t m t+1 p t+1 ) m t p t )]}). The log-linearized money demand function 21) is then used to eliminate c t+1 c t m t+1 p t+1 ) m t p t )] to yield c t E t {c t+1 } 1 σ i t E t π t+1 ρ ωe t {i t+1 i i }). 31) Imposing y t c t 31) readily becomes the dynamic IS curve for this classical moneyin-the utility function model: y t E t {y t+1 } σ 1 i t E t {π t+1 } ρ ωe t { i t+1 }). 32) To obtain labor market equilibrium we log linearize the labor demand equation 13) to get a t αn t w t p t which in combination with 29) gives σy t + ϕn t + ωi t a t αn t where y t c t has been used. Since the production function in logs is y t a t + 1 α) n t we use n t 1 α) 1 y t a t ) to substitute out n t : σy t + ϕ 1 α) 1 y t a t ) + ωi t a t α 1 α) 1 y t a t ) σ 1 α) + ϕ + α y t 1 + ϕ 1 α 1 α a t ωi t 1 + ϕ y t σ 1 α) + ϕ + α a ω 1 α) t σ 1 α) + ϕ + α i t 33) ψ ya a t ψ yi i t. 4 Solving for the rational-expectations equilibrium With expressions 32) and 33) we are now ready to solve for output and inflation given a specification for interest-rate policy. It is assumed that the nominal interest rate is determined according to the rule i t ρ + φ π π t + v t φ π > 1. 34) Furthermore the exogenous disturbances follow AR1) processes: a t ρ a a t 1 + ε a t v t ρ v v t 1 + ε v t. 9

4.1 Finding the relevant difference equation The solution proceeds by inserting 34) into 32) and 33): y t E t {y t+i } σ 1 φ π π t + v t E t {π t+1 } ωe t {φ π π t+1 + v t+1 }) y t ψ ya a t ψ yi ρ + φ π π t + v t ) Using the latter in the former yields ψ ya a t ψ yi ρ + φ π π t + v t ) E t { ψya a t+1 ψ yi ρ + φ π π t+1 + v t+1 ) } σ 1 φ π π t + v t E t {π t+1 } ωe t {φ π π t+1 + v t+1 }). Solving out expectations terms using the properties of the shock processes gives Rearranging: Collecting terms: Note that with Hence we can write ψ ya a t ψ yi ρ + φ π π t + v t ) ψ ya ρ a a t ψ yi ρ + φ π E t {π t+1 } + ρ v v t ) σ 1 φ π π t + v t E t {π t+1 } ωφ π E t { π t+1 } ωφ π ρ v 1) v t ]. σψ ya a t + σψ yi ρ + φ π π t + v t ) σψ ya ρ a a t + σψ yi ρ + φ π E t {π t+1 } + ρ v v t ) +φ π π t + v t E t {π t+1 } ωφ π E t {π t+1 } + ωφ π π t + ωφ π 1 ρ v ) v t. σψ ya 1 ρ a ) a t 1 + 1 ρ v ) ω σψ yi )) vt φ π 1 + ω σψyi ) πt 1 + φ π ω σψyi )) Et {π t+1 }. ω 1 α) ω σψ yi ω σ σ 1 α) + ϕ + α ) σ 1 α) ω 1 σ 1 α) + ϕ + α ϕ + α ω σ 1 α) + ϕ + α ωψ ψ ϕ + α σ 1 α) + ϕ + α. σψ ya 1 ρ a ) a t 1 + 1 ρ v ) ωψ) v t φ π 1 + ωψ) π t 1 + φ π ωψ) E t {π t+1 } leading to a first-order rational expectations difference equation in π t : π t 1 + φ πωψ φ π 1 + ωψ) E t {π t+1 } σψ ya 1 ρ a ) φ π 1 + ωψ) a t 1 + 1 ρ v) ωψ v t φ π 1 + ωψ) π t ΘE t {π t+1 } σψ ya 1 ρ a ) φ π 1 + ωψ) a t 1 + 1 ρ v) ωψ v t 35) φ π 1 + ωψ) 10

where Θ This has a unique stationary solution iff 1 + φ πωψ φ π 1 + ωψ). 1 < Θ < 1. 36) In the case where ω > 0 this is clearly satisfied as φ π > 1. However in the case where ω < 0 we cannot be sure that it holds for ωψ 1 Θ ). We assume that 36) holds in the following. 4.2 Solving for π t by the method of undetermined coeffi cients We can now solve 35) for π t by forward substitution. This is cumbersome however so instead we will solve 35) by the method of undetermined coeffi cients. This method involves two simple steps. In the first one makes a conjecture about the form of the solution as a function of unknown coeffi cients. In the second step one uses the conjecture together with the difference equation to verify the validity of the conjecture and to identify the unknown coeffi cients which then implies that a solution is obtained). In this case and in all related cases of linear rational expectations models) it is natural to conjecture that inflation is a linear function of the shocks a t and v t. I.e. π t Aa t Bv t 37) where A and B are the undetermined coeffi cients to be identified. Forward 37) one period and take expectations: E t {π t+1 } AE t {a t+1 } BE t {v t+1 } E t {π t+1 } Aρ a a t Bρ v v t. 38) We now combine 38) with 35) to see whether the conjectured form of the solution is correct: π t Θ Aρ a a t Bρ v v t ] 39) σψ ya 1 ρ a ) φ π 1 + ωψ) a t 1 + 1 ρ v) ωψ v. φ π 1 + ωψ) Wee see that it is. The conjecture is consistent with the difference equation; i.e. inflation is a linear function of the shocks. We can then identify A and B by using 37) together with 39): Aa t Bv t Θ Aρ a a t Bρ v v t ] σψ ya 1 ρ a ) φ π 1 + ωψ) a t 1 + 1 ρ v) ωψ v φ π 1 + ωψ) and note that this must hold for all values of a t and v t. Hence the following equations apply: A ΘAρ a σψ ya 1 ρ a ) φ π 1 + ωψ) 40) B ΘBρ v 1 + 1 ρ v) ωψ. 41) φ π 1 + ωψ) 11

From 40) and 41) we get A B σψ ya 1 ρ a ) φ π 1 + ωψ) 1 Θρ a ) 1 + 1 ρ v ) ωψ φ π 1 + ωψ) 1 Θρ v ) respectively giving us the solution for inflation as σψ ya 1 ρ a ) π t φ π 1 + ωψ) 1 Θρ a ) a 1 + 1 ρ t v ) ωψ φ π 1 + ωψ) 1 Θρ v ) v t 42) which is the expression in Galí 2008) p. 31. The solution for i t follows immediately by inserting 42) into 34) Galí ignores the constant ρ on p. 31; cf. the discussion in Footnote 5 on this note). I.e. we get ) σψ ya 1 ρ a ) i t ρ φ π φ π 1 + ωψ) 1 Θρ a ) a 1 + 1 ρ t + v ) ωψ φ π 1 + ωψ) 1 Θρ v ) v t + v t σψ ya 1 ρ a ) ρ 1 + ωψ) 1 Θρ a ) a t + 1 1 + 1 ρ ) v) ωψ v t 1 + ωψ) 1 Θρ v ) σψ ya 1 ρ a ) ρ 1 + ωψ) 1 Θρ a ) a t + 1 + ωψ) 1 Θρ v) 1 1 ρ v ) ωψ v t 1 + ωψ) 1 Θρ v ) σψ ya 1 ρ a ) ρ 1 + ωψ) 1 Θρ a ) a t + 1 + ωψ) Θρ v + ρ v ωψ v t 1 + ωψ) 1 Θρ v ) ρ σψ ya 1 ρ a ) v 1 + ωψ) 1 + φ ] πωψ ρ 1 + ωψ) 1 Θρ a ) a φ t + π 1 + ωψ) ωψ v t 1 + ωψ) 1 Θρ v ) σψ ya 1 ρ a ) ρ 1 + ωψ) 1 Θρ a ) a ρ t v φ π 1 + ωψ) 1 Θρ v ) v t 43) where the fifth line makes use of the definition of Θ. Finally the solution for y t follows by inserting 43) into 33): σψya 1 ρ a ) y t ψ ya a t + ψ yi 1 + ωψ) 1 Θρ a ) a t + ) ψ ya 1 + a t + References σψ yi 1 ρ a ) 1 + ωψ) 1 Θρ a ) ) ρ v φ π 1 + ωψ) 1 Θρ v ) v t ρ v ψ yi φ π 1 + ωψ) 1 Θρ v ) v t. 44) 1] Galí J. 2008 onetary olicy Inflation and the Business Cycle. rinceton University ress rinceton NJ. 2] Walsh C. E. 2010 onetary Theory and olicy Third Edition The IT ress Boston A. 12