A Qualitative Game of Interest Rate Adjustments with a Nuisance Agent
Abstract
:1. Introduction
2. What Are Viability Problems and Their Solutions?
2.1. The Meaning of Viability
2.2. Mathematical Formulation
2.3. A Qualitative Game
3. A Macroeconomic Model
3.1. A Viability Theory Problem
3.2. The Central Bank’s Problem
- I.
- Output gap is the log deviation of actual output from “natural” output. Since this is a log deviation, it is interpreted as a fraction of natural output.
- II.
- Inflation is defined as the CPI inflation rate. The symbol denotes the deviation of CPI inflation from a reference value of inflation.
- III.
- Interest rate is the short-term nominal interest rate that is used by the central bank as the policy instrument. We denote the deviation of the nominal interest rate from its reference value by .Both rates are expressed as fractions rather than percentages. We also follow the convention in this literature by considering annualised rates. The reference values can be steady-state values (if available) or some typical long-term averages. In this paper, we assume that the reference values of inflation and nominal interest rate are 0.02 and 0.04, respectively. Hence, the level inflation and interest rates will be and , respectively. We will use I to denote the level interest rate.
- IV.
- Exchange rate is the log ratio ofIt can be viewed as an aggregate measure of the strength of a country’s currency. If the local currency weakens, then increases. That is, a larger value of implies real depreciation: the domestic goods become relatively cheaper when is large. Conversely, if decreases, the local currency strengthens, hence the domestic goods become relatively more expensive.
3.3. How Can the Viability Kernel Be Used by the Domestic Central Bank?
4. Calibrated Qualitative Monetary Policy NA-Game
5. Viability Analysis: Parameter-Specific Solutions
5.1. A Method for the Determination of Viability Kernels
5.2. Analysing Monetary Policy in Four Dimensions
5.3. How to Interpret a 3D Projection of a 4D Kernel
6. Impact of Foreign Real Interest Rate on the Domestic Economy
6.1. Neutral Exchange Rate
- the bank should keep the domestic interest rate close to the upper limit when the foreign real interest rate is negative;
- the bank should keep the domestic interest rate close to the lower limit when the foreign real interest rate is positive.
6.2. Undervalued and Overvalued Currency
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Viable Areas
Appendix B. Exchange Rate Responsiveness to Interest Rate Adjustments
Appendix C. VIKAASA
1 | This paper draws from [1]. |
2 | In the two-person game context, we cannot use the gender-neutral pronoun their to describe a singular player action. Instead, we will use he and his to refer to a single genderless agent. |
3 | |
4 | Viability is normally defined for an infinite time horizon, but it is also possible to define , and consider viability in finite-time. |
5 | The parameter notation is chosen to reflect some compatibility of our model with Batini-Haldane’s discrete-time model in [39]. |
6 | This is the reason we refer to our game as a nuisance agent game. |
7 | This algorithm (called the inclusion algorithm, see [9]) employed by VIKAASA will miss any viable points that cannot reach a steady state; e.g., because they form (large) orbits. However, experimenting with our monetary policy models, which consisted of using different discretisation grids and trying various controls did not lead to the discovery of points like that. |
8 | The total number of rows of the array for the selected discretisation (see Appendix C) is 683. |
9 | The penultimate subplot is the economy’s velocity norm, i.e., the sum of absolute values of the right-hand sides of (20)–(23). We consider this number a measure of an aggregate system’s velocity and call it velocity in the figure. If this velocity is close to zero, the economy has approached a steady state. If the steady-state is inside and no state variable has ever breached the bounds of K then the evolution starting point is viable; refer to Appendix C. |
10 | See [37] for a discussion on building systemic resilience in the context of viability theory. |
11 | Let denotes the set of proximal normals to D at x i.e., the set of such that the distance of to D is equal to |
12 | If D were a disc, then a contingent cone at any point of the circumference would be a half-space. When is an interior point of D, then the contingent cone for this point is the whole space. |
13 | VIKAASA stands for Viability Kernel Approximation, Analysis and Simulation Application = VIKAASA, which happens to be a Sanskrit word that means “progress” or “development”. |
14 | VIKAASA is also compatible with GNU Octave, but without the GUI. See the manual for more information. |
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0.01 | 0.02 | 0.023 | 0.04 | 0.056 | 0.07 | |
0.01 | 0.03 | −0.034 | 0.028 | 0.035 | 0.07 |
0.01 | 0.03 | −0.016 | 0.032 | 0.035 | 0.056 | |
0.01 | 0.03 | −0.032 | 0.008 | 0.021 | 0.042 |
0.01 | 0.03 | −0.029 | 0.029 | 0.007 | 0.042 | |
0.022 | 0.03 | −0.04 | −0.023 | 0.014 | 0.021 |
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Krawczyk, J.B.; Petkov, V.P. A Qualitative Game of Interest Rate Adjustments with a Nuisance Agent. Games 2022, 13, 58. https://doi.org/10.3390/g13050058
Krawczyk JB, Petkov VP. A Qualitative Game of Interest Rate Adjustments with a Nuisance Agent. Games. 2022; 13(5):58. https://doi.org/10.3390/g13050058
Chicago/Turabian StyleKrawczyk, Jacek B., and Vladimir P. Petkov. 2022. "A Qualitative Game of Interest Rate Adjustments with a Nuisance Agent" Games 13, no. 5: 58. https://doi.org/10.3390/g13050058
APA StyleKrawczyk, J. B., & Petkov, V. P. (2022). A Qualitative Game of Interest Rate Adjustments with a Nuisance Agent. Games, 13(5), 58. https://doi.org/10.3390/g13050058