Gali (2008), Chapter 3

Transcription

1 Set 4 - The Basic New Keynesian Model Gali (28), Chapter 3 Introduction There are several key elements of the baseline model that are a departure from the assumptions of the classical monetary economy. First, imperfect competition (i.e., monopolistic competition) in the goods market is introduced by assuming that each firm produces a differentiated good for which it sets the price (Dixit-Stiglite), instead of taking the price as given. Moreover, there are a continuum of goods. As a result, an individual firm can change its own price, but it has no affect on the aggregate price. Also, there is no strategic component between the firms. Second, some constraints are imposed on the price adjustment mechanism by assuming that only a fraction of firms can reset their prices in any given period. In particular, and following much of the literature, a model of staggered price adjustment due to Calvo (1983) and characterized by random price durations is adopted. The resulting framework is referred to as the basic New Keynesian model. In recent years, this model has become the workhorse for the analysis of monetary policy, fluctuations, and welfare. The introduction of differentiated goods requires that the household problem be modified slightly relative to the one considered previously. That modification is discussed before turning to the firms optimal price-setting problem and the implied inflation dynamics. Households Once again, assume a representative infinitely-lived household, seeking to maximize E { t= β t U (C t,n t ) where C t is now a consumption index given by ] ε C t C t (i) 1 1 ε di 1

2 with C t (i) representing the quantity of good i consumed by the household in period t, while C t represents the total amount of goods consumed. As stated before, assume the existence of a continuum of goods represented by the interval [,1]. The period budget constraint now takes the form (i)c t (i)di + Q t B t B t 1 +W t N t + T t ; t =,1,... where (i) is the price of good i in period t, N t denotes hours of work (or the measure of household members employed), and W t is the nominal wage. In addition, B t represents purchases of oneperiod bonds (at a price Q t ). B t > means that the household purchased bonds (i.e., it saved), and B t < means that the household sold bonds (i.e., it borrowed). Finally, T t is a lump-sum component of income (which may include, among other items, dividends from ownership of firms). The above sequence of period budget constraints is supplemented with a non-ponzi game condition of the form lim E t {B T, t. T Notice that {C t (i) i [,1], B t, and N t are the control variables of the model. We assume that U N,t <, so working causes disutility for the agent. For the consumer, ε can be thought of as the elasticity of substituion between the goods. If ε, then then consumer considers all the goods perfect substitutes. If ε 1, then the consumer considers all the goods perfect complements, i.e., he wants to consume each good in equal amounts. We can divide the problem into two steps: max {Z t,b t,n t s.t. E { where C t is the solution of max {C t (i) i [,1] s.t. t= β t U(C t,n t ) Z t + Q t B t = B t 1 +W t N t + T t C t (i) 1 1 ε di ] ε (i)c t (i)di = Z t Solving for C t First, we wish to maximize the consumption index C t for any given level of expenditures 1 (i)c t (i). The objective funciton of this maximization problem is known as the constant elasticity of substitution (CES) aggregator function. max {C t (i) i [,1] s.t. C t (i) 1 1 ε di ] ε (i)c t (i)di = Z t Page 2 of 2

3 Lets setup the Lagrangian ] ε L = C t (i) 1 1 ε di [ + λ Z t ] (i)c t (i)di Important to note, in this construction, i is just an index characterizing all the goods. Our goal is to optimize L with respect to a specific good (i.e., good j). Thus, the first-order condition of L with respect to good j is L C t ( j) = [ˆ ε 1 ] C t (i) ε 1 ε di ε 1 ε C t( j) 1 ε λpt ( j) = ] C t (i) ε di [C t ( j)] 1 ε = λ ( j); j [,1] Ct ( j) 1 ε Pt ( j) = ; k, j [,1] (1) C t (k) (k) Ct ( j) Pt ( j) log = εlog ; k, j [,1] C t (k) (k) d log Ct ( j) C t (k) ) = ε (2) d log ( Pt ( j) (k) The left hand side of equation (2) is known as the elasticity of substitution between every pair of products. As one can see, the elasticity of substitution in the Dixit-Stiglitz economy is constant, which is not very realistic, but it makes the derivations much simpler. With a large ε, the agent does not care about mixing because he only wishes to consume the goods that are relatively cheaper. As a result, demand is very responsive to a price change. With a small ε, the agent wishes to diversify his consumption independent of the price. Hence, demand is not very responsive to a price change. From equation (1), we have C t ( j) = C t (k) Pt ( j) ε ; j,k [,1] (3) (k) Let us substitute equation (3) into the constraint. ( j)c t ( j)d j = Z t Pt ( j) ε ( j) C t (k)d j = Z t (k) C t (k)[ (k)] ε [ ( j)] 1 ε d j = Z t (4) Page 3 of 2

4 Now, we define the Dixit-Stiglitx aggregate price index as ] 1 ( j) 1 ε d j 1 ε (5) Since P 1 ε t ( j) 1 ε d j, then equation (4) becomes C t (k)[ (k)] ε Pt 1 ε = Z t Pt (k) ε Z t C t (k) = (6) which is a necessary condition for maximizing the consumption index. Next, we substitute equation (6) into the consumption idex. ] ε C t (k) 1 1 ε dk = = [ (Pt ) (k) ε ] 1 1 ε Zt dk [ 1 ε ] Zt ε ε ( Pt (k) ε) ε dk ε = 1 ε Zt (k) 1 ε dk ] ε (7) Finally, from the aggregate price index, equation (7) yields 1 ε Zt Pt ε = Z t (8) To summarize our derivations, the solution to the problem max {C t (i) i [,1] s.t. C t (i) 1 1 ε di ] ε (i)c t (i)di = Z t yields the set of demand equations Pt (i) ε Z t C t (i) = ; i [,1] (9) and the maximized level of the consumption basket C t = Z t (1) Page 4 of 2

5 [ ] 1 1 where (i) 1 ε 1 ε d j optimal behavior, is the aggregate price index. Furthermore, and conditional on such (i)c t (i)di = C t (11) i.e., total consumption expenditures can be written as the product of the price index times the quantity index. Now, we can plug equations (9) and (1) it into the household s constrained utility maximization problem. The Households Problem Revisited In the previous subsection, we found the set of demand equations for each individual good, the maximized consumption index, and total consumption expenditure. With this information, the household no longer has to worry about choosing its consumption for each individual good. Instead, it chooses its consumption basket as a whole. Thus, its constrained maximization problem becomes max {C t,b t,n t s.t. E { t= βt U(C t,n t ) C t + Q t B t = B t 1 +W t N t + T t which is formally identical to the problem faced by the households in the single good economy analyzed in Set 3 of the lecture notes. After setting up the Lagrangian L = E { t= β t [U(C t,n t ) + λ t ( C t Q t B t + B t 1 +W t N t + T t )] we obtain the following first-order conditions U c,t λ t = (12) U n,t + λ t W t = (13) λ t ( Q t ) + E t {βλ t+1 = (14) Equations (12) and (13) imply Wt U n,t = U c,t (15) The LHS is the marginal disutility of labor. The RHS is the real wage times the marginal utility of consumption. So, the marignal cost of working equals the marginal benefit of working. Page 5 of 2

6 Equations (12) and (14) imply Q t = E t { β U c,t+1 Q t U c,t = E t U c,t { β U c,t (16) Note that Q t is the nominal price of a bond. So, the LHS is the real price of the bond times the marginal utility of consumption; it is what the household would receive by selling a bond. The RHS is the expected discounted marginal benefit of consumption the household gives up in t + 1 by selling the bond. Under the assumption of a period utility given by U(C t,n t ) = C1 σ t 1 σ N1+ϕ t 1 + ϕ the first-order conditions become Nt ϕ Wt = Ct σ σ Ct+1 Q t = E t {β C t +1 (17) (18) (19) Condition (18) can be interpreted as a competitive labor supply schedule, determining the quantity of labor supplied as a function of the real wage, given the marginal utility of consumption (which under the assumptions is a function of consumption only). We will return to the household s problem when we study general equilibrium. Firms A continuum of firms indexed by i [,1]. Each firm produces a differentiated good, but they all use an identical technology, represented by the production function Y t (i) = A t N t (i) where A t represents the level of technology assumed to be common to all firms and to evolve exogenously over time. All firms face an identical isoelastic demand schedule given by (9), and take the aggregate price level and aggregate consumption index C t as given. Following the formalism proposed in Calvo (1983), each firm may reset its price only with probability 1 θ in any given period, independent of the time elapsed since the last adjustment. So, θ is the probability that a firm s price will remain the same in any given period. Thus, each period a measure of 1 θ producers reset their prices, while a fraction θ keep their prices unchanged. As a result, the average duration of a price is given by (1 θ) 1. In this context, θ becomes a natural index of price stickiness. (2) Page 6 of 2

7 Optimal Price Setting Firm i reoptimizing in period t will choose its price Pt that maximizes the current market value of its profits generated while that price remains effective. Formally, it solves the problem { ( max θ k E t Q t, P Pt t Y t W ) Y A t k= subject to the sequence of demand constraints ( P ) ε Y t = t C ; k =,1,... (21) +k where σ Q t, β k C C t +k is the stochastic discount factor for nominal payoffs, Y t denotes output in period t + k for a firm that last reset its price in period t, is the Dixit-Stiglitx aggregate price index (it is the general price level faced by all the firms in period t), and Pt is the optimal price set in period t by firms reoptimizing their price in that period. Notice that, as shown above, all firms will choose the same price because they face an identical problem. The first-order condition associated with the problem above takes the form { ( θ k E t Q t, Y t + P Y t t k= Pt W ) Y t A Pt = { ( θ k E t Q t, [Y t + Pt W ) ] Y t = k= k= θ k E t { ( Q t, [Y t + k= Multiplying both sides by Pt { ( θ k E t Q t, Y t k= P t W A A P t ) ( ε) Y t { θ k E t Q t, Y t (1 ε + ε 1 P 1 t Pt W A ] ) = = 1 ε yields Pt M W ) = (22) A where W A is the nominal marginal cost in period t +k for a firm which last reset its price in period t, and M ε > 1. This is known as the optimal price setting condition. Note that in the limiting case of no price rigidities (θ = ), each firm reoptimizes every period. Then, the previous condition collapses to the familiar optimal price-setting condition under flexible prices Pt M W t = Pt = M W t A t A t which allows us to interpret M as the desired markup in the absense of constraints on the frequency of price adjustment. Henceforth, M is referred to as the desired, or frictionless, gross markup. Page 7 of 2

8 Aggregate Price Dynamics From the above environment, we can derive an equation that describes the aggregate price dynamics. = = (i) 1 ε di [ ] 1 1 ε ˆ (1 θ)(pt ) 1 ε + i S(t) 1 (i) 1 ε di ] 1 1 ε where S(t) [,1] is the set of firms that are not reoptimizing their price in period t, and Pt is the same for all the firms reoptimizing their price in period t. = = [ (1 θ)(pt ) 1 ε + θ( 1 ) 1 ε] 1 ε 1 [ θ( 1 ) 1 ε + (1 θ)(pt ) 1 ε] 1 ε 1 Pt 1 ε = θ( 1 ) 1 ε + (1 θ)(p ( P Πt 1 ε ) 1 ε = θ + (1 θ) t (23) 1 t ) 1 ε where Π t 1 is gross inflation rate between t 1 and t. Note that aggregate price dynamics are described by equation (23). Equilibrium There are two market clearing conditions: a. Market clearing in the goods market requires Y t (i) = C t (i) for all i [,1] and all t. Letting aggregate output be defined as Y t follows that Y t = C t must hold for all t. Aggregate supply equals aggregate consumption. b. Market clearing in the labor market requires N t = N t (i)di [ ] 1 Y t(i) 1 1 ε ε di for all t. The hours of work chosen by the households equals the hours of work demanded by firms. it Page 8 of 2

9 Using the optimality conditions of the household and firms, along with the market clearing conditions, the economy can be summarized by the following seven equations. N ϕ t = W t Ct σ Q t = E t {β k= ( Ct+1 C t ) σ { ( θ k E t Q t, Y t Π 1 ε t = θ + (1 θ) Y t = C t N t = N t (i)di +1 P t M W ) = ( P ) 1 ε t 1 Q t is set be the central bank A Now, we will study the dynamics of this economy by simplifying the above system of equations, performing log-linearization on the equations that remain, and then analyzing the log-linearized equations in Dynare. Simplification Using the competitive labor supply schedule (equation 18) Nt ϕ Wt = Ct σ W t = N ϕ t C σ t and the market clearing condition for the goods market, the above system of seven equations can be simplified. Q t = E t {β k= Π 1 ε ( Ct+1 C t ) σ 1 Π t+1 { [ ( θ k P ϕ t N E t Q t, Y t M Y ] )Π σ t 1, = (25) 1 A ( P ) 1 ε t = θ + (1 θ) t (26) 1 N t = N t (i)di Q t is set by the central bank (28) (24) (27) Page 9 of 2

10 where Π t 1, +k/ 1. We now have a system of 5 equations and 5 endogenous variables (Q t,y t,π t, P t 1,N t ). Benchmark Case As stated previously, a useful benchmark is the case of flexible prices (θ = ). Our focus is on a zero inflation, steady state equilibrium. It follows from (26) that in a steady state equilibrium with zero inflation (i.e., Π = 1), Π t = 1 P t 1 = 1 so Pt = 1 = for all t. Then, at k =, equation (25) implies ( Y t t 1 M Nϕ t Yt σ ) = MNt ϕ Yt σ = A t (29) Also, since all firms are symmetric, then A t Y t = = = ] ε Y t (i) 1 1 ε di = A t N t ] (A t N t (i)) 1 1 ε ε di ] (A t N t ) 1 1 ε ε di (definition) (definition) (symmetry) (3) Thus, from equations (29) and (3), we have ϕ Yt M Yt σ = A t A t Yt σ+ϕ = 1 M A1+ϕ t [ ] 1 1 ϕ+σ Y t = M A1+ϕ t Y n t (31) where Yt n is known as the natural level of output. It is the equilibrium level of output under flexible prices, and it is determined independently of monetary policy (again). This is why we will introduce price stickiness. Page 1 of 2

11 The level of output that solves the social planner s problem { max E β t Ct 1 σ 1 σ N1+ϕ t 1+ϕ {C t (i),n t (i) i [,1],t=,1,... t= s.t. C t = ˆ1 N t = C t (i) 1 1 ε di ˆ1 N t (i)di C t (i) = A t N t (i) is the socially efficient level of output. In Homework 6, it was shown that this optimal level of output is Ct 1+ϕ ϕ+σ = At Yt e Notice that when M = 1 (perfect competition), the natural level of output corresponds to the equilibrium level of ouput in the classial economy, as derived in the Set 3 notes. The presence of market power by firms has the effect of lowering that output level uniformly over time, without affecting its sensitivity to changes in technology. Thus, because of markup power, Yt n is less than the socially efficient level of output. ε The Sticky Price Model As stated previously, we have a simplified the sticky price model into a system of 5 equations (equations 24 through 28) and 5 endogenous variables (Q t,y t,π t, P t 1,N t ). After some tedious calculation, we can reduce our current system into a brand new one with only 3 equations and 3 endogenous variables (π t,ỹ t,i t ), where ỹ t y t yt n is the log deviation of output from its flexible price counterpart, and following convention, it is referred to as the output gap. Important to note, the lower case variables represent the log values of their upper case counterparts (i.e., x t logx t ). Using equation (24), we will derive the Dynamic IS Equation, and using equations 24 through 27, we will derive the New Keynesian Phillips Curve. Log-Linearization Dynamic IS Equation To begin, we rewrite equation (24) as follows σ Yt+1 1 Q t = E t {β Y t Π t+1 exp( i t ) = E t {exp[ ρ σ(y t+1 y t ) π t+1 ] Page 11 of 2

12 where i t logq t is the is nominal interest rate, π t logπ t is the inflation rate, and ρ logβ is the real interest rate in the steady state. Example. Here is a numerical example concerning the nature of the different variables. Q t =.95; logq t.513; i t logq t.513. = 1.5; 1 = 1; Π t = / 1 = 1.5; π t logπ t = Now, we log-linearize the new version of equation (24) around the steady state. î t = E t {[ σ(ŷ t+1 ŷ t ) π t+1 ] Important to note, the hat over the lower-case variables represents the deviation of the variable from its steady state (i.e., ˆx t x t x ss ). So, we have î t = i t i ss. Since there is no inflation in the steady state (i.e., π t = ), then i ss = ρ. Moreover, ŷ t+1 ŷ = y t+1 y y t + y = y t+1 y t. Thus, we can rewrite the equation above as (i t ρ) = E t { σ(y t+1 y t ) π t+1 y t = E t {y t+1 1 σ (i t E t {π t+1 ρ) y t yt n { = E t yt+1 yt+1 n 1 σ (i t E t {π t+1 ρ) yt n { + E t y n t+1 y t yt n { = E t yt+1 yt+1 n 1 [ it E t {π t+1 ρ + σ ( y n { )] t E t y n σ t+1 ỹ t = E t {ỹ t+1 1 σ [i t E t {π t+1 r n t ] (32) where ỹ t y t y n t is the output gap and r n t is the natural rate of interest given by r n t = ρ σ ( y n t E t { y n t+1 ) = ρ + σ ( E t { y n t+1 y n t ) r n t = ρ + σe t { y n t+1 (33) Henceforth, equation (32) is known as the Dynamic IS Equation (or DIS, for short). It was shown that the natural level of output, y n t, is a linear function of technology shock, so it is an exogenous variable. Hence, the natural rate of interest, r n t, is also exogenous. Remember, in a steady state with zero inflation, yt n is the equilibrium level of output under flexible prices. Because of monopoly power, the natural rate of output is strictly less than the efficient rate of output, yt e, while the actual rate of output, y t, fluctuates around yt n. Under the assumption that the effects of nominal rigidities vanish asymptotically, lim T E t {ỹ t+t =. In that case one can solve equation (32) forward to yield the expression ỹ t = 1 σ k= (r r n ) where r t i t E t {π t+1 is the expected real return on a one period bond (i.e., the real interest rate). The previous expression emphasizes the fact that the output gap is proportional to the sum of current and anticipated deviations between the real interest rate and its natural counterpart. Page 12 of 2

13 New Keynesian Phillips Curve Continuing with the derivations, we log-linearize equation (25), i.e., the optimal price-setting condition, { [ ( θ k P ϕ t N E t Q t, Y t M Y ] )Π σ t 1, = 1 A k= around the zero inflation steady state. In the zero inflation steady state, Pt / 1 = 1 and Π t 1, = 1. Furthermore, constancy of the price level implies that Pt = +k in that steady state, from which it follows that Y t = Y and N ϕ Y σ A = Nϕ Y σ A because all firms will be producing the same quantity of output. In addition, Q t, = β k must hold in that steady state. Let us focus on the expression within the expectation. We apply a first-order Taylor approximation of equation (25) around the zero inflation steady state. [ ( P ϕ t N Q t, Y t M Y ] σ [ ( N )Π t 1, β k ϕ Y σ )] Y 1 M 1 A A [ ( N ϕ Y σ )] +(first order terms of Q Y t ) 1 M A +β k Y [log P t M Nϕ Y σ ] ϕ ˆn + σŷ â + logπ t 1, 1 A Note that ( ϕ N Y ) σ A is the real marginal cost in period t + k for a firm whose price was last set in period t. Since P t = M (nominal marginal cost) Pt ( ϕ N = M t Y σ ) t A t then, in the steady state with zero inflation (i.e., Pt = for all t), we have [ ( N ϕ Y σ )] ( N ϕ Y σ ) ( N ϕ Y σ ) 1 M = M = 1 = 1 A A A M As a result, we can rewrite equation (25) as k= θ k E t {β k Y [pt p t 1 ( mc ˆ + p p t 1 )] = Page 13 of 2

14 where mc ˆ ϕ ˆn + σŷ â = mc mc denotes the log-deviation of the marginal cost from its steady state value mc = µ, and where µ logm is the log of the desired gross markup (which, for M close to one, is approximately equal to the net markup M 1). After rearranging terms, we have (βθ) k (p t p t 1 ) = (βθ) k E t { mc ˆ + (p p t 1 ) k= k= Since < βθ < 1, then k= (βθ)k = 1 (1 βθ). Thus, p t p t 1 = (1 βθ) k= (βθ) k E t { mc ˆ + (p p t 1 ) (34) If marginal cost is high, then a firm will set its price high and above marginal cost. Also, if the general price level, p p t 1, is high, then the firm will also set a higher price. In order to gain some additional intuition about the factors determining a firm s price-setting decision, it is useful to rewrite (34) as p t = µ + (1 βθ) k= (βθ) k E t {mc + p Hence, firms resetting their prices will choose a price that corresponds to the desired markup over a weighted average of their currenct and expected (nominal) marginal costs, with the weights being proportional to the probability of the price remaining effective at each horizon θ k. For a moment, we will turn our attention towards the remaining equations in the system. First, we focus on equation (26), which illustrates the aggregate price dyanmics. ( P Πt 1 ε = θ + (1 θ) t 1 ) 1 ε (1 ε)π t = (1 θ)(1 ε)(p t p t 1 ) π t = (1 θ)(p t p t 1 ) (35) Equation (35) makes clear that, in the present setup, inflation results from the fact that firms reoptimizing in any given period choose a price that differs from the economy s average price in the previous period. Moving on, we consider equation (27). N t = = = = Y t A t N t (i) di Y t (i) 1 A t A t d i Pt (i) ε Y t di Pt (i) ε di Page 14 of 2

15 where the third equality follows from equations (9) and (1) and the goods market clearing condition. Taking log on both sides, n t = y t a t + d t ] where d t log Pt (i) ε di and di is a measure of price (and, hence, output) dispersion across firms. In a neighborhood of the zero inflation steady state, d t is equal to zero up to a firstorder approximation. Pt (i) ε di = exp[ ε(p t (i) p t )]di Since = (i) 1 ε di 1 = = 1 (1 ε) = 1 ] 1 1 ε, then Pt (i) 1 ε d i [1 ε(p t (i) p t )]di [ ε exp[(1 ε)(p t (i) p t )]d i [1 + (1 ε)(p t (i) p t )]d i (p t (i) p t )di Since ε > 1, then (1 ε), so we must have (p t (i) p t )d i ] Pt (i) ε dt log di] log(1) = (p t (i) p t )di =. Hence, Thus, one can write the following approximate relation between aggregate output, employment, and technology y t = a t + n t To summarize, we have derived the following three equations p t = (1 βθ) k= π t = (1 θ)(p t p t 1 ) y t = a t + n t (βθ) k E t { mc ˆ + p (36) Page 15 of 2

16 i.e., equations (34), (35), and (36), which we use to rewrite the optimal price setting equation. p t = (1 βθ) k= (βθ) k E t { mc ˆ + p = (1 βθ)( mc ˆ t + p t ) + (1 βθ) k=1 = (1 βθ)( mc ˆ t + p t ) + (βθ)e t {(1 βθ) (βθ) k E t { mc ˆ + p k= (βθ) k E t+1 [ mc ˆ t+1+k + p t+1+k ] Important to note, that we use the power rule for expectations, i.e., E t {E t+1 [x ] = E t {x. Hence, the optimal price setting equation can be rewritten as pt { = (1 βθ)( mc ˆ t + p t ) + βθe t p t+1 is a weighted average. A firm reoptimizing in period t will set a price higher if marginal cost and/or the relative price is high. Furthermore, since the firm may not be able to reset its price again in period t + 1, then it must also take into account its expected optimal price for next period. So, if a firm expects the general price level to be higher in period t +1, then the firm will set a higher optimal price in period t as insurance since it may not be able to change its price in the next period. Subtracting p t 1 from both sides yields pt { p t 1 = (1 βθ) mc ˆ t + π t + βθe t p t+1 p t Equation (35) implies { π t πt+1 = (1 βθ) mc ˆ t + π t + βθe t (1 θ) (1 θ) π t (1 θ)π t = (1 θ)(1 βθ) mc ˆ t + βθe t {π t+1 π t = (1 θ)(1 βθ) mc ˆ t + βe t {π t+1 θ π t = λmc ˆ t + βe t {π t+1 (38) where λ (1 θ)(1 βθ) θ. This equation determines inflation dynamics. Current inflation rate depends on the current marginal cost and the discounted expected inflation rate in the next period. So, if a higher inflation rate is expected in t + 1, then current inflation will increase as a result When θ is high, the probability of firms being able to change their prices is low. Consequently, firms change their prices less often signifying that prices are stickier. Important to note, λ is strictly decreasing in the index of price stickiness θ. Thus, a large rise in marginal cost does not have a substantial impact on the inflation rate. So, in the case of θ being large, when a firm does have the opportunity to change its price, the firm will choose not to deviate greatly from the prices of its competitors because it is trying to keep its relative price ratio in line with the other firms. (37) Page 16 of 2

17 Solving (38) forward, inflation is expressed as the discounted sum of current and expected future deviations of real marginal costs from the steady state π t = λ k= β k E t { mc ˆ Equivalently, and defining the average markup in the economy as µ t = mc t, it is seen that inflation will be high when firms expect average markups to be below their steady state (i.e., desired) level µ, for in that case firms that have the opportunity to reset prices will choose a price above the economy s average price level in order to realign their markup closer to its desired level. It is worth emphasizing here that the mechanism underlying fluctuations in the aggregate price level and inflation as laid out above have little in common with the mechanism at work in the classical monetary economy. Thus, in the present model, inflation results from the aggregate consequences of purposeful price-setting decisions by firms, which adjust their prices in light of current and anticipated cost conditions. By contrast, in the classical monetary economy analyzed in Set 3, inflation is a consequence of the changes in the aggregate price level that, given the monetary policy rule in place, are required in order to support an equilibrium allocation that is independent of the evolution of nominal variables, with no account given of the mechanism (other than an invisible hand) that will bring about those price level changes. Next, we hypothesize that the output gap is related to marginal cost. If the output is overheated above the reference level, i.e., the natural output level, then marginal cost will be higher than its reference point. mc ˆ t = ϕ ˆn t + σŷ t â t = ϕn t + σy t a t (ϕn + σy a) where n, y, and a are the log steady state values, so (ϕn + σy a) is the log steady state marginal cost. Since the steady state value of marginal cost is mc = µ, then the above equation becomes mc ˆ t = ϕn t + σy t a t + µ where µ logm. Recall that in a flexible price equilibrium, (i) 1 = M (Nn t ) ϕ (Y n t ) σ A t = ϕn n t + σy n t a t + µ µ = ϕn n t σy n t + a t We plug this expression for µ into the equation for marginal cost. mc ˆ t = ϕn t + σy t a t (ϕnt n + σyt n a t ) = ϕ(n t n n t ) + σ(y t y n t ) = 1, and thus, Page 17 of 2

18 Since y t = a t + n t and y n t = a t + n n t, then we obtain mc ˆ t = ϕ(y t a t yt n + a t ) + σỹ t = ϕỹ t + σỹ t mc ˆ = (ϕ + σ)ỹ t (39) i.e., the log deviation of real marginal cost from its steady state is proportional to the output gap. When the economy is overheated relative to the adjusted benchmark (i.e., the natural rate of output, which accounts for the technology level), then the marginal cost will be high, relative to some benchmark. By combining equation (39) with equation (38), one can obtain an equation relating inflation to its one period ahead forecast and the output gap π t = βe t {π t+1 + κỹ t (4) where κ λ(σ + ϕ). Equation (4) is ofter referred to as the New Keynesian Phillips curve (or NKPC, for short), and constitutes one of the key building blocks of the basic New Keynesian model. Summary of Results The Dynamic IS Equation and the New Keynesian Phillips curve together with an equilibrium process for the natural rate rt n (which in general will depend on all the real exogenous forces in the model, i.e., the technology shocks) constitute the non-policy block of the basic New Keynesian model. That block has a simple recursive structure: The NKPC determines inflation given a path for the output gap, whereas the DIS equation determines the output gap given a path for the (exogenous) natural rate and the actual real rate. In order to close the model, supplement those two equations with one or more equations determining how the nominal interest rate i t evolves over time, i.e., a description of how monetary policy is conducted. One such equation is known as the Taylor Rule, which is given by i t = ρ + φ π π t + φ y ỹ t + ν t (41) where ν t has been included to account for monetary policy shocks. It is one interest rate rule that the central bank can follow to determine the nominal interest rate. Assume φ π and φ y are nonnegative coefficients, chosen by the monetary authority. Note that the choice of the intercept ρ makes the rule consistent with a zero inflation steady state. Now, we have a system of 3 equations, 3 endogenous variables (ỹ t,π t,i t ), and 2 exogenous shocks (rt n,ν t ). All of the remaining endogenous variables in the model can be explained by relating them to one or more of the three equations and endogenous variables above. Thus, and in contrast with the classical model analyzed in Set 3, when prices are sticky the equilibrium path of real variables cannot be determined independently of monetary policy. In other words, monetary policy is non-neutral. Page 18 of 2

19 Economic Intuition Dynamic IS curve ỹ t = E t {ỹ t+1 1 σ (i t E t {π t+1 r n t ) If real interest is higher than some benchmark, then the households will decrease consumption in period t, and instead reallocate the resources to purchase more bonds (i.e., households will save ). This is essentially another version of the Euler equation. New Keynesian Phillips Curve π t = κỹ t + βe t {π t+1 When a firm sets its price, it considers the relative price (the price it sets compared to its peers, i.e., the price index) and marginal cost. κ captures how long the firm must keep the price it sets before it can change it again. If production is higher than the benchmark (higher than the current technology dictates), then households must supply more labor and suffer more disutility. To compensate the households, the firms must increase wages, and as a result, marginal cost will be higher, which will lead to higher prices and inflation. Taylor Rule i t = ρ + φ π π t + φ y ỹ t + ν t Under the assumption of non-negative coefficients (φ π,φ y ), it can be shown that a necessary and sufficient condition for uniqueness is given by κ(φ π 1) + (1 β)φ y > (42) [Bullard and Mitra (22)], which is assumed to hold, unless stated otherwise. Hence, the economy is stable around the steady state (i.e., the natural level of the endogenous variables), if and only if condition (42) is satisfied. To interpret this condition, let us ignore (1 β)φ y because β 1. Then, we get φ π 1 > (43) which is known as the Taylor Principle. Thus, if the central bank follows this recommendation, then a 1% increase in inflation must lead to a more than 1% increase in nominal interest rate. Consequently, real interest rate will rise, which, in turn, prevents the emergence of multiple equilibria. Under the Taylor Rule, the model dictates that the central bank must fight inflation so as to keep the economy stable and prevent multiple equilibria. Page 19 of 2

20 Clarida, Gali, and Gertler (CGG) The paper starts with 2 observations 1. Both inflation and output gap become stable. Great Moderation. Standard Deviation of Chairman of the Fed Inflation Output Gap Pre-Volker Volker - Greenspan (79:3-96:4) Post CGG policy rule estimate r t = βe t {π t+1 + γχ t r t = ρr t 1 + (1 ρ)r t where ρ is known as inflation rate inertia, χ t is the output gap, r t is the nominal interest rate, and r t is the target rate. They estimate the parameters by using the data associated with the previous table. Important to note, the central bank changes its behaviors by altering the following parameter values. Parameter Values Chariman of the Fed β γ ρ Pre-Volker Volker-Greenspan CGG s Model π t = δe t {π t+1 + λ(y t z t ) y t = E t {y t+1 1 σ (r t E t {π t+1 ) + g t rt = βe t {π t+1 + γχ t r t = ρr t 1 + (1 ρ)rt where δ is the discount rate, z t is the natural rate of output, and g t exogenous demand shock. The first equation is the New Keynesian Phillips Curve and the second is Dynamic IS Curve, while the third and fourth polic rule equations are policy rules. These parameter values do not satisfy the Taylor Principle, so the existence of a unique equilibrium is not guaranteed. Page 2 of 2