1 the Household's Problem

Professor Mark Gertler1 New York University Macroeconomic Theory I Fall 2002

Lecture 8 A Dynamic New Keynesian Model of the Business Cycle with Capital

Basic Setup Here we develop a small scale macroeconomic model useful for business cycle analysis and for the evaluation of monetary policy. The model has the descriptive richness of a traditional IS/LM model, but is optimization-based. The model is a dynamic general equilibrium framework with monopolistic competition and nominal price rigidities. It includes households, ﬁrms and a government sector:

Households consume, supply labor, hold real money balances, capital (which they rent • to ﬁrms), and save.

There are three types of firms. Final goods producers are competitors that produce • output using intermediate goods. Intermediate goods firms are monopolistic competitors that each produce a differentiated product using capital and labor. These firms set nominal prices on a staggered basis. Finally, capital goods producers are competitors that produce new capital using raw final goods output. Adjustment costs lead to a variable price of capital.

The government conducts ﬁscal and monetary policy. • 1 The Household’s Problem

The economy is composed of a continuum of inﬁnitely-lived individuals, whose total is nor- malized to unity. Each of them consumes a ﬁnal good, Ct, and supplies labor, Nt.Saving

1 with Simon Gilchrist.

1 can be held in the form of real money balances, Mt ,bonds,B , and physical capital, K . Pt t t The instantaneous utility function is

1 γ − m 1 1 γ am Mt an 1+γn Ut = Ct − + Nt . (1) 1 γ 1 γ Pt − 1+γ − − m µ ¶ n

Mt+i Bt+i ∞ The representative household chooses Ct+i,Nt+i, P , P ,Kt+i to maximize t+i t+i i=0 n o 1 γm ∞ i 1 1 γ am Mt − an 1+γ E β C − + N n . (2) t 1 γ t+i 1 γ P 1+γ t+i ( i=0 " m t − n #) X − − µ ¶ subject to

1 W M M ( Rn )Bt Bt 1 t t t 1 t+1 − − Ct = Nt+ZtKt 1+Πt+TRt − − Qt [Kt (1 δ)Kt 1] , (3) Pt − − Pt − Pt − − − − with γ,γ < 1 and with γ > 0. In (3), Wt is the real wage, Z is the real rental cost of m n Pt t capital, δ is the depreciation rate, Πt are profits from the firms, TRt are transfers rebated n lump-sum, Rt+1 is the nominal interest rate, and Qt is the price of installed capital. Define the Bellman equation:

Mt 1 Bt 1 Mt Bt v − , − ,Kt 1 =max Ut + Etβv , ,Kt , (4) P P − P P µ t t ¶ · µ t+1 t+1 ¶¸ subject to (3). Substituting for the budget constraint in the objective function, and diﬀer- entiating with respect to N , Mt , Bt ,andK ,wegetthefollowingFOCs: t Pt Pt t

Wt γ γn Ct− = anNt (5) Pt

γm γ Pt γ Mt − C− = E βC− + a (6) t t P t+1 m P ½ t+1 ¾ µ t ¶ γ γ Ct− = Et Rt+1βCt−+1 (7)

γ [Zt+1 +(1© δ)Qt+1ª] γ C− = Et − βC− , (8) t Q t+1 ½ t ¾ where Rt+1is the real interest rate, deﬁned as

n Pt Rt+1 R Et . (9) ≡ t+1 P µ t+1 ¶ 2 Using (7) in (6) and (8) the household FOCs may be written as

Wt γn γ = anNt /Ct− (10) Pt

1 1 γ M 1 − γm 1 − γm t = 1 C γm (11) P a − Rn t t µ m ¶ µ t+1 ¶ P 1=E Rn E t β(C /C ) γ (12) t t+1 t P t+1 t − ½ µ t+1 ¶ ¾ Zt+1 +(1 δ)Qt+1 γ 1=Et − .β(Ct+1/Ct)− (13) { Qt } 2Firms

The Final-Goods Sector

Firms in the ﬁnal good sector produce a homogeneous good, Yt, using intermediate goods, Yt(z). There are a continuum of intermediate goods a measure unity. The production functionsthattransformsintermediategoodsintoﬁnal output is given by

ε 1 ε 1 ε 1 − − Yt = Yt(z) ε dz , (14) ·Z0 ¸ where ε>1. Note that this production function exhibits constant returns to scale, diminishing marginal product, and constant elasticity of substitution.

The representative ﬁrm chooses inputs Yt(z) to max

Et Yt (15) − Pt subject to equation (14) and to 1 Et = Pt(z)Yt(z)dz, (16a) Z0 where Pt(z) is the price of the intermediate good Yt(z). We assume that final goods are competitive; they accordingly take Pt(z) as given. Given constant returns to scale and competition, the size of any final goods firm is inde- terminate. We can, however, determine input demands from the following cost minimization problem: choose Yt(z) to min 1 Pt(z)Yt(z)dz (17) Z0 3 subject to ε 1 ε 1 ε 1 − − Yt(z) ε dz Yt (18) ≥ ·Z0 ¸ It is straightforward to show that solution yields the following constant price elasticity demand function for good z that is homogenous of degree one in total final output:

ε P (z) − Y (z)= t Y . (19) t P t · t ¸ where >1 is the price elasticity of demand. Combining this demand function with the production function (14) yields the following price index for intermediate goods:

1 1 1 1 − Pt = Pt(z) − (20) ·Z0 ¸ The Wholesale Sector In the wholesale sector there is a continuum of monopolistically competitive ﬁrms owned by consumers, indexed by z [0, 1].Eachwholesaleﬁrm z faces the demand curve (19) for its ∈ product. In addition it uses both labor Nt(z) and capital Kt(z) to produce output according to the following constant returns technology2:

α 1 α Yt(z)=AtNt(z) Kt(z) − (21) where At is a technology parameter. Capital is freely mobile across firms. Firms rent capital from households in a competitive market on a period by period basis. Wholesale firms set nominal prices on a staggered basis. Following Calvo [1], we assume that each period a firmadjustsitspricewithprobability1 θ and keeps it price fixed with − probability θ. The adjustment probability is independent across time and across firms (i.e., it does not depend on how long a firm’s price has been fixed.) The average time a price i 1 i 1 remains fixed is given by (1 θ) i∞=0 θ − i = i∞=0 θ = 1 θ . − · − Firms that are able to adjustP price, chooseP price optimally. These firms maximize expected discounted profits given the production technology, (21), the demand curve, (19), and constraint on the frequency of price adjustment. Firms that do not adjust price, adjust

2 We use the following convention for notation: Since capital is not predetermined at the firm level, we use a t subscript to denote capital use by firm z at time t. Capital at the aggregate level is predetermined; 1 hence we use a t 1 subscript for total capital available at t,whereKt 1 = Kt(z)dz. − − 0 R 4 output to meet demand. Both types of firms choose inputs to minimize costs, given output demand. It is first useful to characterize the input choice problem, conditional on output. After- wards, we proceed to characterize the optimal price adjustment decision. Input Demand and Marginal Cost

Firm z chooses Nt(z) and Kt(z) to minimize total cost, given by

Wt Nt(z)+ZtKt(z) (22) Pt subject to α 1 α AtNt(z) Kt(z) − Yt 0. (23) − ≥

Let MCt denote the Lagrange multiplier with respect to the constraint. Note that MCt is the ﬁrm’s real marginal cost (it is the derivative of total cost with respect to Yt.)

The ﬁrst order conditions with respect to Nt(z) and Kt(z) are given by:

Wt/Pt = MCt (24) αYt(z)/Nt(z)

Zt = MCt, (25) (1 α)Yt(z)/Kt(z) − The ﬁrst order conditions imply that inputs adjust to equalize marginal cost across diﬀerent factors, where the marginal cost of a factor is the ratio of the factor price to the marginal product.

By combining equations (23), (24) and (25) we may express MCt as a function of wages, rents and technology, as follows:

α 1 α 1 Wt Zt − MCt = . (26) At α (1 α) µ ¶ · − ¸ Since the ﬁrm takes Wt,andZt as given, real marginal cost is constant at the ﬁrm level. Constant marginal cost is an outcome of constant returns to scale and perfect factor mobility.

Finally, it is instructive to note the link between marginal cost and the net markup µt. Rearranging (24) and (25) yields

Yt(z) Wt α =(1+µt) (27) Nt(z) Pt

5 Yt(z) (1 α) =(1+µt) Zt (28) − Kt(z) with 1 1+µt ≡ MCt At the optimum, the firm adjusts each input to the point where the marginal product equals the gross markup µt times the factor price. By definition, the markup equals the inverseofrealmarginalcost. Optimal Price Setting n n Let MCt denote the firm’s nominal marginal cost at time t (i.e., MCt = Pt MCt)and γ · Ct+i − let Λt,i be the ratio of marginal utility of consumption at t+i to marginal utility ≡ Ct at t. A firm³ that´ is allowed to change its price at time t chooses Pt(z) to max

n ∞ Pt(z) MC (θβ)i E Λ − t+i Y (z) (29) t t,i P t,t+i i=0 t+i X · ¸ subject to ε P (z) − Y (z)= t Y , (30) t,t+i P t+i · t+i ¸ i where β Λt,i is rate at which the ﬁrm discounts earnings at t + i. The ﬁrmtakesasgiventhe n paths of MCt ,Pt and Yt.

Using the demand curve to eliminate Yt,t+i(z) in the objective function implies that the

ﬁrm chooses Pt(z) to max

n ε ∞ Pt(z) MC P (z) − (θβ)i E Λ − t+i t Y (31) t t,i P P t+i i=0 ( t+i t+i ) X · ¸ Since all firms adjusting price at t face the same decision problem (i.e., there are no firm specific state variables), each chooses same optimum Pt∗. The first order necessary condition is given by

n 1 ∞ 1 1 P ∗ MC P − E (θβ)i Λ (1 ε) Y (z) ε t − t+i Y (z) t∗ =0. (32) t t,i P t,t+i P P t,t+i P i=0 ( − t+i − t+i t+i t+i ) X · ¸ This equation sets expected present value of future marginal revenue equal to the expected present value of future marginal cost. Rearranging terms yields the following expression for

6 theoptimalpriceattimet:

i n 1 Et ∞ (θβ) Λt,iMC Yt,t+i(z) i=0 t+i Pt+i Pt∗ =(1+µ) . (33) h i 1 i EPt ∞ (θβ) Λt,iYt,t+i(z) i=0 Pt+i h i 1 P where 1+µ = 1 1/ε is the steady state gross markup. − Combining equations (30) and 33 yields:

∞ n Pt∗ =(1+µ) ϕt,iMCt+i (34) i=0 X where 1 ε i 1 − Et (θβ) Λt,iYt+i Pt+i ϕ = · ¸ (35) t,i ³ ´ 1 ε i 1 − Et ∞ (θβ) Λt,iYt+i i=0 Pt+i · ³ ´ ¸ In general, the optimal price equalsP the steady state markup times a weighted average of expected future nominal marginal cost. The weights depends on how the firm discounts future cash flowsineachperiodt + i (taking into account that the price remains fixed in t + i), and also the relative proportion of revenues expected each period. Note:

1. for the limiting case of perfect price ﬂexibility (θ =0), price is just a constant markup over nominal marginal cost: n Pt∗ =(1+µ) MCt . (36)

2. In general, the optimal price depends on future expected values of aggregate variables n (Yt+i,Pt+i) as well as future marginal costs MCt+i.

Finally, given that (i) all firmsthatadjustint choose the same price and (ii) the average price of firms that do not adjust is simply last period’s price level Pt 1 (since firms adjusting − are a random draw from the total population), we may express the price index as,

1 1 ε 1 ε 1 ε Pt = θPt −1 +(1 θ)Pt∗ − − (37) − − Note that equation (37) implies a di£ﬀerence equation for the¤ price level.

7 Capital Producers After production of ﬁnal output each period, competitive capital producers make new capital goods. Capital producer j purchases raw output to use as materials input It(j) and capital Kt(j) as factor of production. They rent capital after it has been used to produce

ﬁnal output within the period. They sell new capital produced at the market price Qt. k The production function for new capital Yt (j) is given by : I (j) Y k(j)=φ t K (j). (38) t K (j) t · t ¸ with I I φ0( ) > 0,φ00 ( ) < 0,φ(0) = 0,φ( )= · · K K I where K is the steady state ratio of investment to capital.

Notice that the production function exhibits constant returns to scale in It(j) and Kt(j).

There is diminishing marginal product to It(j), holding Kt(j) constant, due to the concavity of φ( ). This concavity captures convex costs of adjusting investment. The problem of the · representative ﬁrm is to choose It(j) and Kt(j) to max

It(j) k Qtφ K(j)t It(j) Z Kt(j), (39) K (j) − − t · t ¸ k where Zt is the rental price of capital (used for producing new capital.

The ﬁrst order condition for It(j) is given by:

It Qtφ0 1=0 (40) Kt 1 − µ − ¶ 1 1 where It = Kt(j)dz and Kt 1 = Kt(j)dz. Notice that equation (40) implies that 0 − 0 (i) all capitalR producers will choose theR same investment/capital ratio (hence we drop the index j) and that (ii) there is a monotonic relation between the capital price and the investment/capital ratio. Rearranging (40) yields

It Qt = Φ( ). (41) Kt 1 − with I 1 Φ( t ) (42) K It t 1 ≡ φ0 − Kt 1 − ³ ´ 8 Equations (41) and (42) imply that Q is increasing in It ,asinstandardQ investment t Kt 1 − theory.

The ﬁrst order condition for Kt(j) is given by

It(j) It(j) It(j) k Qt φ( ) φ0( ) = Zt (43) { Kt(j) − Kt(j) Kt }

I I I Notice that in the steady state, since, φ( K )= K , accordingly φ0( K )=1. It follows that the steady state value of the price of capital, Q, is unity and that the steady state rental value of capital for producing new capital, Zk is zero, i.e.,

I I I k Q φ( ) φ0( ) = Z (44) K − K K · ¸ I I = =0 {K − K } k Accordingly, so long as we operate in a small neighborhood of the steady state, Zt is approximately zero, and can be ignored.

3 Aggregation, Resource Constraints and Government Policy

Aggregate Production We first aggregate production across individual intermediate goods producing firms in order to derive a relation between total final goods output and total factors inputs. The CES production function given by equation (14) makes exact aggregation difficult. In particular, combining equations (21) and (14) yields

ε 1 ε 1 ε 1 − α 1 α −ε Yt = AtNt(z) Kt(z) − dz . (45) ½Z0 ¾ £ ¤ Since relative prices differ across firms (due to staggered price setting) input usage will differ, implying that it is not possible to simply equation (45).

Consider instead the aggregator Yt0, which is the simple sum of intermediate goods output across ﬁrms: 1 1 α 1 α Yt0 = Yt(z)dz = AtNt(z) Kt(z) − (46) Z0 Z0 9 Since all intermediate goods ﬁrms choose the same capital/labor ratio, it is possible to derive a relation for Yt0 in terms of aggregate factor inputs:

α 1 α Yt0 = AtNt Kt 1 − , (47) − with Nt and Kt deﬁned as 1 Kt 1 = Kt(z)dz, (48) − Z0 and 1 Nt = Nt(z)dz. (49) Z0 0 In general, Yt and Yt diﬀer. However, within a local region of the steady state they are approximately equal. In particular, within a local region of the steady the elasticity of Yt

0 and Yt with respect to output of any intermediate goods producer are the same:

0 ∂Yt Yt(z) ∂Yt Yt(z) ( ) ss =( ) ss =1 ∂Yt(z) Yt | ∂Yt(z) Yt0 |

0 Given that the dynamics of Yt and Yt are similar within a local region of the steady, we can use the following approximate aggregate production function for local analysis:

α 1 α Yt = AtNt Kt 1 − , (50) − Resource Constraints The economy resource constraint for the ﬁnalgoodsisgivenby

Yt = Ct + It + Gt. (51) where Gt is government consumption. The capital accumulation equation is

It Kt = φ Kt +(1 δ)Kt 1. (52) Kt 1 − − µ − ¶ The Government Thegovernmentbudgetconstraintisgivenby:

Mt Mt 1 − − = Gt + TRt. (53) Pt

We assume Gt is exogenous.

10 Monetary policy determines Mt. We will consider simple interest rate rules of the general form: Pt Yt r n n γπ γy εt Rt+1 = R ( ) ( ) e (54) Pt 1 Yt∗ − n R is the steady state real rate of interest, Yt∗ is the natural level of output (the level that r would obtain under ﬂexible prices), εt is an i.i.d. monetary policy shock, and where γπ > 1 and γy > 0. Under this formulation, the target net rate of inﬂationiszero(andunityfor the gross rate). Note also that even though the nominal interest rate is the instrument of monetary policy, the feedback rule indirectly determines Mt since the central bank must 3 n adjust the money supply to satisfy money demand , given the choice of Rt+1.

4 Equilibrium

Equilibrium We define an equilibrium with monopolistic competition as follows: a vector n of quantities (Yt,Ct,It,Nt,Kt) an a price vector (Zt,Wt,Pt,Rt+1,Qt,MCt) such that all agents are maximizing subject to the respective constraints they face, supply equals demand in each market, and all resource constraints are satisfied, given the values of the predetermined variables, Kt 1 and Pt 1, and the exogenous variables Gt and At. − − In practice it is convenient to express the equilibrium as the vector (Yt,Ct,It,Nt,Kt, n Pt,Rt+1,Qt,MCt) that satisfies the following system of equations, given Kt 1 and Pt 1. − − Aggregate Demand

Yt = Ct + It + Gt (55)

n γ 1 Ct = Et R βC− − γ (56) { t+1 t+1} Yt+1 MCt+1(1 α) +(1 δ)Qt+1 Kt+ n Pt Et − − Λt,1 = Et Rt+1 Λt,1 (57) { Qt } { Pt+1 }

It 1 = Φ− (Qt); (58) Kt 1 − Aggregate Supply α 1 α Yt = AtNt Kt 1 − (59) −

Yt 1 γn 1 γ α = anNt − /Ct− (60) Nt MCt 3 see equation (11)

11 1 1 ε − 1 ε ∞ 1 ε Pt = θPt −1 +(1 θ)((1 + µ)Et ϕt,iPt+i MCt+i ) − (61) − " − { i=0 · } # X Evolution of Capital It Kt = φ Kt +(1 δ)Kt 1. (62) Kt 1 − − µ − ¶ Monetary Policy Rule

Pt Yt r n γπ γy εt Rt+1 = R( ) ( ) e (63) Pt 1 Yt∗ − 5 The Log-Linearized Model

Thestrategyistolookforanapproximate analytical solution by transforming the model into a system of log-linear diﬀerence equations. Let Xt be a stationary random variable with a constant unconditional mean X.Wedenotexˆt the logarithmic deviation of a variable Xt from its steady state value X :

xˆt =logXt log X (64) − and we make use of the fact that

xˆt Xt = Xe X(1 +x ˆt). (65) ' We also, when necessary, make use of the fact that 1 Et log Xt log EtXt var log Xt (66) ' − 2 and assume that within a local region of the steady state that var log Xt is a constant. Finally, we observe that

f(Xt) f(X)+f 0(X)(Xt X) (67) ' − f(X)+f 0(X)X(Xt/X 1) ' − f(X)+f(X)η(1 +x ˆt 1) ' − f(X)(1 + ηxˆt) ' where η ∂f(X) X . ≡ ∂X f(X) 12 In what follows, we consider a loglinear approximation of the model around a zero inﬂation, zero growth steady state. Aggregate Demand

(i) The economy resource constraint

Rewrite the resource constraint as C I G 1= t + t + t Yt Yt Yt Using (65) we can write this equation as C I G 1= (1 +c ˆt yˆt)+ I(1 + ˆıt yˆt)+ (1 +g ˆt yˆt). Y − Y − Y − where ˆıt is investment. Since at the steady state

Y = C + I + G, we can rearrange to obtain

C I G yˆ = cˆ + ˆı + gˆ . (68) t Y t Y t Y t

(ii) The Euler equation

Taking logs of each side of the consumption Euler equation (56) yields

P 0=logE Rn t β(C /C ) γ (69) t t+1 P t+1 t − ½ µ t+1 ¶ ¾ n Pt γ Et log R β(Ct+1/Ct)− (70) ' t+1 P ½ µ t+1 ¶ ¾ 1 n Pt γ + vart log Rt+1 β(Ct+1/Ct)− (71) 2 Pt+1 ½ n µ ¶ ¾ γ Et log R(1 +r ˆ )(1 +p ˆt+1 pˆt)β(1 +c ˆt+1 cˆt)− + (72) ' { t+1 − − } 1 + vart pˆt pˆt+1 γ(ˆct+1 cˆt) (73) 2 { − − − } n Et rˆ +(ˆpt+1 pˆt) γ(ˆct+1 cˆt) (74) ' { t+1 − − − }

13 given that in the steady state:

n 1 log R =logR = log β vart pˆt pˆt+1 γ(ˆct+1 cˆt) − − 2 { − − − } Rearrange to obtain, n cˆt = σ[ˆr Et(ˆpt+1 pˆt)] + Etcˆt+1 (75) − t+1 − − where σ =1/γ is the intertemporal elasticity of substitution.

(iii) Therateofreturnfromcapital

>From (57) we get:

Yt+1 n Pt Et MCt+1(1 α) +(1 δ)Qt+1Λt,1 = Et QtRt+1 Λt,1 { − Kt − } { Pt+1 }

Let mct denote the log-deviation of real marginal cost from its steady state value. According to (65), we can write

Y ˆ n Et (1 α)MC (1+mct+1 +ˆyt+1 kt)+Q(1 δ)(1+qt+1) = Et QR(1+qt+ˆr ) +constant { − K − − } { t+1 } where the constant is essentially the equity premium. Making use of the steady state rela- tionships: Q =1 Y (1 α)MC +(1 δ)=R + constant − K − 1 MC = , µ we can simplify to get

ˆ n Et (1 υ) mcˆt+1 +ˆyt+1 kt + υqt+1 qt = Et rˆ +ˆpt+1 pˆt (76) { − − − } { t+1 − } ³ ´ with (1 δ) υ = − . (1 α)MC Y +(1 δ) − K − (iv) Investment Demand: The link between the price of capital goods and investment

14 >From equation (58) we can write

It φ0 Qt =1 Kt 1 µ − ¶ which implies we may write

I ˆ φ0 Q[1 + qt η(ˆıt kt 1)] = 1 K − − − µ ¶ where I I φ00 K K η = I − φ0 ¡ K¢ I Since φ0 K Q =1, ¡ ¢ ˆ qt = η(ˆıt kt 1), (77) ¡ ¢ − − Aggregate Supply

(v) The aggregate production function

Since the aggregate production function is linear in logs, we can simply write

ˆ yˆt = at + αnt +(1 α)kt 1. (78) − − (vi) Labor market equilibrium

The labor market equilibrium (60) may be expressed as

Yt γ γn 1 MCtα Ct− = anNt − Nt which is log-linear. Accordingly,

yˆt nˆt + mcˆt γcˆt =(γ 1)ˆnt. (79) − − n − (vii) The Phillips curve

>From equation (61) we can write

P ∞ ∞ MCn t∗ E (θβ)i Λ Y P ε 1 =(1+µ) E (θβ)i Λ t+i Y P ε 1 . P t t,i t+i t+−i t t,i P t+i t+−i t ( i=0 ) ( i=0 t ) X X 15 Consider ﬁrst the left-hand side (LHS) of the above equation. Making use of (65) and dropping all terms involving a product of two or more variables in log-deviation from the steady state, we get

ε ∞ i ε ∞ i ˆ LHS (P Y ) (θβ) (ˆpt∗ pˆt)+(P Y ) (θβ) yˆt+i +(ε 1)ˆpt+i + λt,i . ' " i=0 # − i=0 − X X h i Consider now the right-hand side (RHS). Applying the same logic4:

ε ∞ i n ˆ RHS (P Y ) (θβ) (mcˆt+i pˆt)+ˆyt+i +(ε 1)ˆpt+i + λt,i . ' i=0 − − X h i Combining LHS and RHS, simplifying, and dividing through by P εY we get

∞ i n pˆt∗ =(1 θβ)Et (θβ) mcˆt+i , − ( i=0 ) X ¡ ¢ which gives the log-deviation of the newly set price as a discounted stream of the log-deviation of the nominal marginal cost. Loglinearizing the price index yields

pˆt = θpˆt 1 +(1 θ)ˆpt∗ − − Combine these last two equations to obtain:

πˆt = λmcˆt + βEtπˆt+1, (80) with (1 θ)(1 βθ) λ − − . ≡ θ and where πˆt =ˆpt pˆt 1. − − (viii) The law of motion of capital

We can rewrite the law of motion of capital as K I t = φ( t )+1 δ. Kt 1 Kt 1 − − − 4 Note that we switch to real marginal cost by dividing and multiplying by Pt+i.

16 which implies φ I I ˆ ˆ I 0 K K ˆ 1+kt kt 1 = φ( )(1 + (ˆıt kt 1)) + 1 δ − − K φ I − − − ¡ K¢ I I Given that φ K = K = δ, ¡ ¢ ˆ ˆ ¡ ¢ kt = δˆıt +(1 δ)kt 1. (81) − − Monetary Policy Rule The loglinear version of the monetary policy rule is simply

n r rˆ = γ πˆt + γ (ˆyt yˆ∗)+ε (82) t+1 π y − t t where yˆt∗ is the percent deviation of the natural level of output from the steady state, and is obtained by solving the system with mcˆt =0. Shock Processes It is assumed that the exogenous disturbances to government spending and technology obey autoregressive processes: g gˆt = ρggˆt 1 + εt (83) − a aˆt = ρaaˆt 1 + εt . (84) − 6 Investment Delays

Disturbances to the economy typically appear to generate a delayed and hump-shaped re- sponse of output. Assume therefore that investment expenditures are determined in advance. In particular, suppose that investment expenditures are chosen j periods in advance. The FOC relating the price of capital to investment (58) is modiﬁed to

1 It+j − Et Qt+j φ0 =0 (85) − Kt+j 1 ( · µ − ¶¸ ) The link between asset prices and investment now holds in expectations. Therefore, shocks to the economy have an immediate eﬀect on asset prices but a delayed eﬀect on investment and output. The log-linearized version of (85) is

ˆ Et qt+j η ˆıt+j kt+j 1 =0. (86) − − − h ³ ´i 17 Allowing, for example, for one period investment delay, we obtain the log-linear investment demand curve (86) as ˆ Etqt+1 = η ¯ıt kt , (87) − ³ ´ where investment expenditures made at time t+1 are determined by investment plans made at time t, ¯ıt:

¯ıt =ˆıt+1 (88) which implies that investment is pre-determined.

7 No investment or capital

ˆ If we set ˆıt and kt equal to zero, the model simpliﬁes greatly, but still retains the quali- ˆ tative properties of the larger system. In particular, setting ˆıt and kt equal to zero in the loglinearized model yields the following simple three equation model: IS curve:

Y G n yˆt yˆ∗ = ( − )σ[ˆr Et(ˆπt+1)] + Et[ˆyt+1 yˆ∗ ]+dt (89) − t − Y t+1 − − t+1 Phillips Curve:

πˆt = λκ(ˆyt yˆ∗)+βEtπˆt+1, (90) − t Interest Rate Rule: n i rˆt+1 = γππˆt + γyyˆt + εt (91) with G dt = [ˆgt Etgˆt+1] [ˆy∗ Etyˆ∗ ] Y − − t − t+1

mcˆt = κ(ˆyt yˆ∗) − t γ γG yˆ = κ 1[ n aˆ + gˆ ] t∗ − α t Y G t − πˆt =ˆpt pˆt 1 − − Y with κ =(γn + γ Y G α)/α > 0 and where yˆt∗ is obtained by setting mcˆt =0. − −

18 References

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