Calibrating the Equilibrium Condition of a New Keynesian Model with Uncertainty

A.1 Consumers – calculation steps

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

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

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

A.2 Firms – calculation steps

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

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

Simplifying and denoting the optimal price with Pτ∗ yields

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

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

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:

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

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]

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

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:

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.

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

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

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

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:

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

for eliminating the sum in (A18):

Inserting condition (11) leads to the expression

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

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):

A.5 Intertemporal optimization – calculation steps

The optimization problem has the constraint

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

is maximized in period t, the Bellman equation is

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

which results in

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:

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 t−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.

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

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

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

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:

First-order conditions:

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

whereby the optimal output gap,

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:

and

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 κ≈0.3. 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 0.01−0.05. 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 0.6−0.85. 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.