## 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.

## Acknowledgements:

Thanks to Matthias Neuenkirch for his helpful comments on earlier versions of the paper. I also thank participants of the 11

## Appendix

### A.1 Consumers – calculation steps

First, exponentiate the integral with

To obtain Equation (5), solve Equation (4) for

### A.2 Firms – calculation steps

Equation (6) can be written in more detail. Using Equation (4) with

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

Simplifying and denoting the optimal price with

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

Now, with a cost function in real terms of quantities

where

### 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

Furthermore, in the steady state, long-term values for individual variables are by definition the same as for those on aggregated level, thus ^{[42]}

An explanation is the long-run version of Equation (A9) and hence

a linearized AD curve in terms of growth rates with the slope of

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

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

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

Excluding

Again, using

Furthermore, Equation (A17) can be rewritten for

for eliminating the sum in (A18):

Inserting condition (11) leads to the expression

that only contains parameters and variants of the variable

Since this approximation is sufficiently exact for small values of

### A.5 Intertemporal optimization – calculation steps

The optimization problem has the constraint

where

is maximized in period

The expected value vanishes since

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

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

The time shift yields the Euler condition.

### A.6 Jensen’s inequality – calculation steps

### A.7 Second-order taylor approximation

The Taylor series (in

where

The result in (23.1) appears with

### A.8 Standard targeting rule – calculation steps

The Lagrangian has to be differentiated with respect to

First-order conditions:

Condition (A36.2) follows with Equation (27). From condition (A36.3) follows that

### A.9 Optimal interest rate for positive inflation targets

When the Lagrangian attains the “leaning against the wind" condition, it is extended with

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

The slope of the NKPC ^{[44]}Woodford (2003a) states that a value of

The weight on output fluctuations

Walsh (2003) allows values up to

For the standard deviation of a cost shock, Sims (2011) sets

Letter | Description | |
---|---|---|

Summarizing parameters ( | ||

Discount factor (time preference) | ||

Weighting on output gap in loss function | ||

Elasticity of substitution | ||

Error term of cost shock | ||

Error term of demand shock | ||

Auxiliary parameter | ||

Slope of NKPC | ||

Cost shock persistence | ||

Demand shock persistence | ||

Inflation | ||

Reciprocal value of the IES | ||

Firm index | ||

Price stickiness | ||

Parameter in cost function | ||

Constants in cost function ( | Cost shock | |

Nominal interest rate | ||

Cost parameter in Calvo pricing | ||

Log-linearized price around the steady state | ||

Long-run real interest rate | ||

Demand shock | ||

Calvo price | ||

Log-linearized output growth rate around the steady state | ||

Growth rate of output gap around the steady state | ||

Output growth rate | ||

Bonds | ||

Consumption | ||

Cost function | ||

Price; Price level | ||

Utility function | ||

Wage | ||

Output |

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**Published Online:**2017-6-16

**Published in Print:**2017-9-26

© 2017 Oldenbourg Wissenschaftsverlag GmbH, Published by De Gruyter Oldenbourg, Berlin/Boston