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Calibrating the Equilibrium Condition of a New Keynesian Model with Uncertainty

Tobias Kranz
From the journal Review of Economics

Abstract:

This paper presents a theoretical analysis of the simulated impact of uncertainty in a New Keynesian model. In order to incorporate uncertainty, the basic three-equation framework is modified by higher-order approximation resulting in a non-linear (dynamic) IS curve. Using impulse response analyses to examine the behavior of the model after a cost shock, I find interest rates in the version with uncertainty to be lower in contrast to the case under certainty.

JEL Classification: E12; E17; E43; E47; E52

Acknowledgements:

Thanks to Matthias Neuenkirch for his helpful comments on earlier versions of the paper. I also thank participants of the 11th Workshop for Macroeconomics and Business Cycles at ifo Dresden, particularly Stefan Homburg and Christian Scharrer, for helpful comments. The usual disclaimer applies.

Appendix

A.1 Consumers – calculation steps

L/Cτ can be obtained by using the chain rule:

(A1.1)Pτλεε101Cξε1εdξεε11ε1εCτε1ε1derivative of sub-function=0
(A1.2)Pτλ01Cξε1εdξ1ε1Cτ1ε=0.

First, exponentiate the integral with ε and 1/ε for rearranging the first-order condition. Then insert C from the constraint. It follows that

(A2.1)Pτ=λCτ1εC1εPτ=λCCτ1ε
(A2.2)Pτλ=CτC1εPτλε=CτC.

To obtain Equation (5), solve Equation (4) for Cτ and insert the result for all firms in the constraint, Equation (1):

(A3.1)C=01PξPεCε1εdξεε1C=1PεC01Pξ1εdξεε1
(A3.2)Pε=01Pξ1εdξεε1P=01Pξ1εdξ11ε.

A.2 Firms – calculation steps

Equation (6) can be written in more detail. Using Equation (4) with Y and rearranging leads to

(A4)maxPτPτP1εYKPτPεY.

The first-order condition is now straightforward, using the chain rule:

(A5)Pτ=(1ε)PτPεYPK(Yτ)(ε)PτPε1YP=0.

Simplifying and denoting the optimal price with Pτ yields

(A6.1)(ε1)PτPε=K(Yτ)εPτPε1
(A6.2)1=εε1K(Yτ)PτP1
(A6.3)Pτ=εε1K(Yτ)P.

However, perfect substitutes let the monopolistic structure vanish and show the typical polypolistic result:

(A7)limεεε1K(Yτ)P=K(Yτ)P=Pτ.

Now, with a cost function in real terms of quantities Yτ defined as

(A8)K(Yτ)=cvarψ+1Yτψ+1+cfix,

where cfix are the fix costs, cvar is a measure for the variable costs and ψ represents the elasticity of marginal costs, Equation (7) becomes a micro-funded AS curve that takes the form of a power function:

(A9)Pτ=εε1cvarYτψP.

A.3 Log-linearization

It is convenient to use log-linearized variables instead of level variables in order to solve the model analytically. Also, some interpretations of the results, in terms of elasticity and growth rates, become quite useful. So both Equation (4) and Equation (A9) can be approximated through log-linearization around the steady state. Thus, the approximation becomes more precise with small growth rates. However, some preparation is necessary. Let Z be a state variable that can change over time and Zss its long-term value. When defining

(A10)zlnZlnZss,
z becomes a good approximation of zˆ, the growth rate around the steady state. Also, a first-order Taylor approximation “in reverse" shows the relationship between z and zˆ:
(A11)zˆln(1+zˆ)=ln1+ZZssZss=lnZlnZss.

Furthermore, in the steady state, long-term values for individual variables are by definition the same as for those on aggregated level, thus Zτss=Zss. The state would otherwise include endogenous forces. And finally, the long-run marginal costs equal the multiplicative inverse of the firms’ mark-up:[42]

(A12)cvarYssψ=ε1ε.

An explanation is the long-run version of Equation (A9) and hence Pτss=Pss. Now this can be applied to the previous results. First, Equation (4), the AD curve will be log-linearized. Taking logs, expanding with the log long-term values, and using (A10) gives

(A13.1)lnYτ=lnY+ε(lnPlnPτ)
(A13.2)lnYτlnY=ε(lnPτlnP)
(A13.3)lnYτlnYss(lnYlnYss)=ε(lnPτlnPss(lnPlnPss))
(A13.4)yτy=ε(pτp)
(A13.5)yτ=εpτ+εp+y,

a linearized AD curve in terms of growth rates with the slope of 1/ε. A higher elasticity of substitution would result in a flatter curve, so a change in the firm’s price growth pτ would have a stronger effect on production growth yτ.

Next, with the use of (A12), the AS curve type Equation (A9), can be rewritten in a similar way:

(A14.1)lnPτ=lnεε1+lncvar+ψlnYτ+lnP
(A14.2)lnPτlnP=lnεε1+lncvar+ψlnYτlnYss+lnYss
(A14.3)pτp=ψyτ+lnεε1+lncvar+ψlnYss
(A14.4)pτp=ψyτ+lncvarYssψlnε1ε
(A14.5)pτp=ψyτ+lncvarYssψ(ε1)/ε=0.

The latter expression shows the assumption that the log deviations of marginal costs from their long-run trend values are linear in the amount of ψ. When the firm’s optimized price growth pτ is equal to the aggregated price growth p, then there is no growth in the firm’s production.

Having log-linearized both demand and supply side, Figure A.1 sums up.

Figure A.1: Graphical results of households’ and firms’ static optimization.

Figure A.1:

Graphical results of households’ and firms’ static optimization.

Finally, inserting (A13.5) in (A14.5) combines all the results and gives

(A15.1)pτp=ψ(εpτ+εp+y)
(A15.2)pτp=ψε(pτp)+ψy
(A15.3)(1+ψε)(pτp)=ψy
(A15.4)pτp=ψ1+ψεy.

A.4 Calvo pricing – calculation steps

Dividing the first-order condition by 2k, using the fact that xt is t-measurable, and expanding the sum gives

(A16)j=0(βϕ)jxtj=0(βϕ)jEtpt+j=0.

Excluding xt from the sum, using the formula for an infinite geometric series, and multiplying by (1βϕ) gives

(A17)xt=(1βϕ)j=0(βϕ)jEtpt+j.

Again, using t-measurability (Etpt=pt) and excluding the first summand provides a sum from j=1 to infinity that can be substituted in a subsequent step:

(A18)xt=(1βϕ)j=1(βϕ)jEtpt+j+pt.

Furthermore, Equation (A17) can be rewritten for t+1 (since firms optimize in each period),

(A19.1)Etxt+1=(1βϕ)j=1(βϕ)j1Etpt+j
(A19.2)βϕEtxt+1=(1βϕ)j=1(βϕ)jEtpt+j,

for eliminating the sum in (A18):

(A20)xt=βϕEtxt+1+(1βϕ)pt.

Inserting condition (11) leads to the expression

(A21.1)ptϕpt11ϕ=βϕEtpt+1ϕpt1ϕ+(1βϕ)pt
(A21.2)ptϕpt1=βϕ(Etpt+1ϕpt)+(1ϕ)(1βϕ)pt,

that only contains parameters and variants of the variable p. Then, with the definition of (A10) and first-order Taylor expansion, the inflation rate π can be expressed through differences of p. In the same way, the conditional expectation value for period t+1 can be expressed with

(A22)Etpt+1ptEtπt+1.

Since this approximation is sufficiently exact for small values of π, an equality sign will be used for all following calculations. Now (A21.2) can be rearranged to insert approximations π and Equation (A22):

(A23.1)ϕ(ptpt1)=βϕ(Etpt+1ϕpt)+(1ϕ)(1βϕ)pt(1ϕ)pt
(A23.2)πt=βEtπt+1+(1ϕ)(1βϕ)ϕpt1ϕϕpt+β(1ϕ)pt.

A.5 Intertemporal optimization – calculation steps

The optimization problem has the constraint

(A24)CtPt+Bt+1=Wt+(1+it1)Bt,

where W is the nominal wage and B the nominal value of bonds. The latter provides the link between two periods. Depending on the definition of the interest rate, the period can vary. Here it has been chosen in a way so that the interest from period t enters the Euler condition. Dynamic Programming uses the additively separable utility function and the envelope theorem to set up optimality conditions for two consecutive periods. The procedure can be divided into three parts. The first part is to write a value function, the Bellman equation. Under the assumption that the second term of the expanded utility

(A25)U(Ct)+Ets=t+1βst1U(Cs)

is maximized in period t, the Bellman equation is

(A26)V(Bt)maxCtU(Ct)+βV(Bt+1).

The expected value vanishes since Bt+1 is determined by variables in period t in the constraint. Differentiating with respect to Ct gives the first-order condition

(A27)ddCtU(Ct)+βddCtV(Bt+1)=U(Ct)+βV(Bt+1)dBt+1dCt=0,

which results in

(A28)U(Ct)=PtβV(Bt+1).

Equation (A28) relates the marginal utility to the marginal value in the following period, the time preference, and prices in the same period. Therefore, a higher β and Pt results in a lower Ct. In the next part, the envelope theorem is used to differentiate the value function (by inserting the optimized Ct) with respect to the costate variable Bt:

(A29.1)V(Bt)=U(Ct)+βV(Bt+1)
(A29.2)dVdBt=βV(Bt+1)dBt+1dBt
(A29.3)V(Bt)=βV(Bt+1)(1+it1).

Equation (A29.3) reveals the relationship of the marginal value functions. In a third and last step, the first-order condition (A28) can be used to replace the value functions in Equation (A29.3) with the marginal utility in both periods t and t1:

(A30.1)U(Ct1)Pt1β=βU(Ct)Ptβ(1+it1)
(A30.2)U(Ct)Pt=β(1+it)EtU(Ct+1)Pt+1.

The time shift yields the Euler condition.

A.6 Jensen’s inequality – calculation steps

f(EX)E[f(X)] holds for concave functions, i.e. the logarithm and Jensen’s inequality still holds for the conditional expected value. Since the function’s curvature is sufficiently small, the accuracy is comparable to log-linearization for small growth rates. Moreover, the exactness increases for larger values because of (ln(x))′′0 for increasing x. However, resulting values will always be underestimated.

(A31.1)lnEtZt+1Zt=lnEtexplnZt+1ZtlnEt1+lnZt+1Zt
(A31.2)=ln1+EtlnZt+1ZtEtlnZt+1Zt.

A.7 Second-order taylor approximation

The Taylor series (in R) helps in finding a polynomial to substitute a certain function f(x) (i.e. exponential, logarithm, etc.) around a point x0. The generalized formula of the degree n in the compact sigma notation is

(A32)Taylor(n)=j=0nf(j)(x0)j!(xx0)j,

where f(j) denotes the jth derivative with f(0)=f as a special case. Thereby, larger values for n give better approximations of the original function f(x). In (23.1), f(x)=ln(1+x) and n=2. Formula (A32) simplifies to

(A33)Taylor(2)=ln(1+x0)+11+x0(xx0)12(1+x0)2(xx0)2.

The result in (23.1) appears with x0=0 and yt+1˜ (πt+1 respectively) as the argument of the function:

(A34)ln(1+yt+1˜)yt+1˜12yt+1˜2.

A.8 Standard targeting rule – calculation steps

The Lagrangian has to be differentiated with respect to ytˆ, πt, and it, since the central bank sets the nominal interest rate:

(A35)L(πt,ytˆ,it)=Et[s=tβstπs2+δysˆ2χs(πsβπs+1κysˆ)φsysˆys+1ˆ+1σ(isrπs+1)+12σπs+12+12ys+1˜2].

First-order conditions:

(A36.1)Lπt=2πtχt=0
(A36.2)Lytˆ=2δytˆ+χtκφt(1+ytˆEtyt+1ˆ)=0
(A36.3)Lit=φtσ=0.

Condition (A36.2) follows with Equation (27). From condition (A36.3) follows that φt=0, hence the minimized loss will not change if the IS curve shifts, as the central bank can counteract it one by one through resetting the nominal interest rate. Combining (A36.1) and (A36.2), the standard targeting rule under discretion arises.

A.9 Optimal interest rate for positive inflation targets

When the Lagrangian attains the “leaning against the wind" condition, it is extended with π (as in (29), the loss function). Therefore, the standard targeting rule changes to

(A37)πtπ=δκytˆ,

whereby the optimal output gap,

(A38)ytˆ=βκδ+κ2Etπt+1+πκδ+κ2,

comprises an additional term. After inserting (A38) in the IS curve, the interest rule also has an additional (negative) term. This would lead to a generally lower interest level.

A.10 Equilibrium condition – calculation steps

Equation (51) and Equation (52) in more detail:

(A39.1)Et(yt+1ˆytˆ)2=Et(κθ)μet+ζt+1(κθ)et2
(A39.2)=Et(κθ)2μet+ζt+1et2
(A39.3)=(κθ)2Et(μ1)et+ζt+12
(A39.4)=(κθ)(μ1)2et2+(κθ)2Vartζt+1+Etζt+12
(A39.5)=κ2θ2(μ1)2et2+κ2θ2σe2
(A39.6)=(κθ)2(1μ)2et2+σe2

and

(A40.1)it=r+αμet12(δθ)2μ2et2+σe2σ2(κθ)2(1μ)2et2+σe2+σut
(A40.2)=r+αμet12(δθ)2μ2et2+(δθ)2σe2+σ(κθ)2(1μ)2et2+σ(κθ)2σe2+σut
(A40.3)=r+αμet12(1μ)2σκ2+μ2δ2θ2et2+σκ2+δ2θ2σe2+σut.

A.11 Parameter discussion

Equation (45) includes all parameters of the model.[43] This subsection gives a brief overview over possible values, which are used to graphically depict the equilibrium conditions.

The discount parameter β is typically close to 1. Galí (2015) and Rotemberg and Woodford (1997), 321 set β equal to 0.99 (quarterly), whereas Jensen (2002), 939 uses this under an annual interpretation. Walsh (2010), 362 also sets it to 0.99. Galí and Gertler (1999), 207 estimate a value of 0.988. To keep the framework close to the actual interest setting of the central bank, all calculations are carried out quarterly and β will be set to 0.99.

The slope of the NKPC κ takes values close to zero and usually lower than 1. Roberts (1995), 982 estimates in his original NKPC article κ0.3. On a quarterly basis, Walsh (2010), 362 sets 0.05, Galí and Gertler (1999), 13 estimate 0.02, and McCallum and Nelson (2004), 47 suggest 0.010.05. Jensen (2002) calibrates an annual value of 0.142, whereas Clarida et al. (2000), 170 set 0.3 (yearly) and give a range of 0.05 to 1.22 in the literature. In the baseline simulation, κ is set to 0.04.[44]Woodford (2003a) states that a value of 1 is customary in the RBC literature for σ, the multiplicative inverse of the IES (see, e.g. Clarida et al. (2000), Galí (2015), Yun (1996)). A slightly larger value (1.5) is set by Jensen (2002) and Smets and Wouters (2003) estimate 1.4. An insightful metadata study by Havranek et al. (2015) estimates a mean IES of 0.5 (σ=2) across all countries. However, they report that more developed countries have a higher IES (lower σ). Therefore, σ will be set to 1.

The weight on output fluctuations δ is set to 0.25 in almost all the literature (see, e.g. Walsh (2010), 939), McCallum and Nelson (2004), Jensen (2002)). The latter reports values from 0.05 to 0.33 in other papers. Thus, δ=0.25 will also be assumed for the simulation.

Walsh (2003) allows values up to 0.7 for μ, the cost shock persistence. Clarida et al. (2000), 170 set 0.27 (yearly) and Galí and Rabanal (2004) estimate 0.95. Generally, Smets and Wouters (2003), 1142–1143 estimate persistencies of 0.8 and higher, which is confirmed by Smets and Wouters (2007). Thus, μ will be treated as a variable in the range of 0.60.85. The smallest value 0.6 implies 0.1296 on an annual basis.

For the standard deviation of a cost shock, Sims (2011) sets 0.01 (σe2=0.0001), Jensen (2002) sets 0.015 (σe2=0.000225), and Galí and Rabanal (2004) estimate 0.011 (σe2=0.000121). McCallum and Nelson (2004) set an annualized standard deviation of 0.02 (σe2=0.0004). The conservative value of 0.0001 will be taken for the simulation.

Table A.1:

Symbols

LetterDescription
αSummarizing parameters (αψ,αy,απ,αμ,αe,ασ)
βDiscount factor (time preference)
δWeighting on output gap in loss function
εElasticity of substitution
ζError term of cost shock
ηError term of demand shock
θAuxiliary parameter
κSlope of NKPC
μCost shock persistence
νDemand shock persistence
πInflation
σReciprocal value of the IES
τFirm index
ϕPrice stickiness
ψParameter in cost function
cConstants in cost function (cfix,cvar) \\[-1mm] eCost shock
iNominal interest rate
kCost parameter in Calvo pricing
pLog-linearized price around the steady state
rLong-run real interest rate
uDemand shock
xCalvo price
yLog-linearized output growth rate around the steady state
yˆGrowth rate of output gap around the steady state
y˜Output growth rate
BBonds
CConsumption
K(.)Cost function
PPrice; Price level
U(.)Utility function
WWage
YOutput

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Published Online: 2017-6-16
Published in Print: 2017-9-26

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