# 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.

### Figure A.1:

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

### Table A.1:

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 |

### References

Bassetto, M. (2004): Negative Nominal Interest Rates, The American Economic Review 94, 104–108.Search in Google Scholar

Bauer, C. and Neuenkirch, M. (2015): Forcasting Uncertainty and the Taylor Rule, Research Papers in Economics. 05-15, Department of Economics, University of Trier.Search in Google Scholar

Branch, W. A. (2014): Nowcasting and the Taylor Rule, Journal of Money, Credit and Banking 46, 1035–1055.Search in Google Scholar

Boneva, L. M., Braun, R. A., and Waki, Y. (2016): Some unpleasant Properties of Loglinearized Solutions when the Nominal Rate is Zero, Journal of Monetary Economics 84, 216–232.Search in Google Scholar

Calvo, G. A. (1983): Staggered Prices in a Utility-Maximizing Framework, Journal of Monetary Economics 12, 383–398.Search in Google Scholar

Clarida, R., Galí, J., and Gertler, M. (1999): The Science of Monetary Policy: A New Keynesian Perspective, Journal of Economic Literature 37, 1661–1707.Search in Google Scholar

Clarida, R., Galí, J., and Gertler, M. (2000): Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory, Quarterly Journal of Economics 105, 147–180.Search in Google Scholar

Collard, F. and Juillard, M. (2001): A Higher-Order Taylor Expansion Approach to Simulation of Stochastic Forward-Looking Models with an Application to a Nonlinear Phillips Curve Model, Computational Economics 17, 125–139.Search in Google Scholar

Dixit, A. K. and Stiglitz, J. E. (1977): Monopolistic Competition and Optimum Product Diversity, The American Economic Review 67, 297–308.Search in Google Scholar

De Paoli, B. and Zabczyk, P. (2013): Cyclical Risk Aversion, Precautionary Saving, and Monetary Policy, Journal of Money, Credit and Banking 45, 1–36.Search in Google Scholar

Dolado, J. J., María-Dolores, R., and Naveira, M. (2014): Are Monetary–Policy Reaction Functions Asymmetric? The Role of Nonlinearity in the Phillips Curve, European Economic Review 49, 485–503.Search in Google Scholar

Fernández-Villaverde, J., Guerrón-Quintana, P., Rubio-Ramírez, J. F., and Uribe, M. (2011): Risk Matters: The Real Effects of Volatility Shocks, American Economic Review 101, 2530–2561.Search in Google Scholar

Galí, J. (2015): Monetary Policy, Inflation, and the Business Cycle. Princeton University Press, Princeton, NJ, 2nd ed.Search in Google Scholar

Gal, J. and Gertler, M. (1999): Inflation Dynamics: A Structural Econometric Analysis, Journal of Monetary Economics 44, 195–222.Search in Google Scholar

Gal, J. and Rabanal, P. (2004): Technology Shocks and Aggregate Fluctuations: How Well Does the Real Business Cycle Model Fit Postwar U.S. Data?, NBER Macroeconomics Annual 19, 225–288.Search in Google Scholar

Havranek, T., Horvath, R., Irsova, Z., and Rusnak, M. (2015): Cross-Country Heterogeneity in Intertemporal Substitution, Journal of International Economics 96, 100–118.Search in Google Scholar

Jensen, H. (2002): Targeting Nominal Income Growth or Inflation?, The American Economic Review 92, 928–956.Search in Google Scholar

Kim, J., Kim, S., Schaumburg, E., and Sims, C. A. (2008): Calculating and Using Second-Order Accurate Solutions of Discrete Time Dynamic Equilibrium Models, Journal of Economic Dynamics & Control 32, 3397-3414.Search in Google Scholar

McCallum, B. T. and Nelson, E. (2004): Timeless Perspective vs. Discretionary Monetary Policy in Forward-Looking Models, Federal Reserve Bank of St. Louis Review 86, 43–56.Search in Google Scholar

Nobay, A. R. and Peel, D. A. (2003): Optimal Discretionary Monetary Policy in a Model with Asymmetric Central Bank Preferences, The Economic Journal 113, 657–665.Search in Google Scholar

Roberts, J. M. (1995): New Keynesian Economics and the Phillips Curve, Journal of Money, Credit and Banking 27, 975–984.Search in Google Scholar

Rotemberg, J. J. and Woodford, M. (1997): An Optimization-Based Econometric Framework for the Evaluation of Monetary Policy, NBER Macroeconomics Annual 12, 297–361.Search in Google Scholar

Rotemberg, J. J. and Woodford, M. (1999): Interest Rules in an Estimated Sticky Price Model, in J. B. Taylor (ed.), Monetary Policy Rules. University of Chicago Press, Chicago, 57–119.Search in Google Scholar

Schaling, E. (2004): The Nonlinear Phillips Curve and Inflation Forecast Targeting: Symmetric Versus Asymmetric Monetary Policy Rules, Journal of Money, Credit and Banking 36, 361–386.Search in Google Scholar

Schmitt-Grohé, S. and Uribe, M. (2004): Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function, Journal of Economic Dynamics & Control 28, 755–775.Search in Google Scholar

Sims, E. (2011): Notes on Medium Scale DSGE Models, Graduate Macro Theory II. University of Notre Dame.Search in Google Scholar

Smets, F. and Wouters, R. (2003): An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area, Journal of the European Economic Association 1, 1123–1175.Search in Google Scholar

Smets, F. and Wouters, R. (2007): Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach, American Economic Review 97, 586–606.Search in Google Scholar

Svensson, L. E. O. and Woodford, M. (2005): Implementing Optimal Policy through Inflation-Forecast Targeting, in B. S. Bernanke and M. Woodford (eds.) The Inflation-Targeting Debate. University of Chicago Press, Chicago, 19–83.Search in Google Scholar

Taylor, J. B. (1999): Staggered Price and Wage Setting in Macroeconomics, in J. B. Taylor and M. Woodford (eds.) Handbook of Macroeconomics. Elsevier, New York, 1009–1050.Search in Google Scholar

Walsh, C. E. (2003): Speed Limit Policies: The Output Gap and Optimal Monetary Policy, The American Economic Review 93, 265–278.Search in Google Scholar

Walsh, C. E. (2010): Monetary Policy and Theory, 3rd Edition, MIT Press, Cambridge, MA.Search in Google Scholar

Woodford, M. (2003a): Optimal Interest-Rate Smoothing, Review of Economic Studies 70, 861–886.Search in Google Scholar

Woodford, M. (2003b): Interest and Prices, Princeton University Press, Princeton, NJ.Search in Google Scholar

Wu, J. C. and Xia, F. D. (2016): Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound, Journal of Money, Credit and Banking 48, 253–291.Search in Google Scholar

Yun, T. (1996): Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles, Journal of Monetary Economics 37, 345–370.Search in Google Scholar

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

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

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