Abstract
The power of an inverted yield curve to predict recessions is widely discussed in the financial press, yet most undergraduate textbooks provide little discussion of this stylized fact. This paper fills this gap by extending a 3-equation textbook model to include an accessible treatment of a term structure of interest rates formed by the one-period policy rate and a two-period rate that obeys the Fisher Equation. The Phillips curve features partially anchored adaptive expectations, while financial markets and the central bank have perfect foresight. Using this framework, we show that raising the policy rate in response to an inflation shock inverts the yield curve. Whether this inversion foreshadows a recession, however, depends on the bank’s monetary policy rule, which we illustrate using numerical examples. In particular, we show that, with anchoring and an output-gap averse central bank, inflation can stabilize and the yield curve can invert without an ensuing recession.
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Notes
See Wheelock and Wohar (2009) for a detailed survey of the empirical literature. Note that the 2019 yield curve inversion in Fig 1 correctly predicted the Covid recession by sheer chance, and thus offers very little informational content for the issues we are addressing, since no one would think financial markets predicted the pandemic.
A temporary inflation shock is equivalent to a temporary supply shock. Should instructors choose, the model can also easily accommodate a temporary demand shock. A positive demand shock would create an increase in inflation and a policy response using the original IS curve to determine the appropriate interest rates, treating a temporary demand shock as a one-period shift in the IS curve.
The quadratic loss function shows that the central bank experiences symmetric utility losses due to both positive or negative deviations from both its inflation and employment targets. To derive the monetary rule, one can set up the central bank’s loss minimization exercise, which involves minimizing losses (described by the loss function) subject to its constraints (namely, the Phillips curves, which describe the combinations of output and inflation available for any given level of expected inflation). The monetary rule comes out of the first order condition for this minimization exercise. For further details see, for example, pages 111-113 of Carlin & Soskice [2014].
To clarify, if we measured gap aversion absolutely using βπ and βy so that β=βπ/βy, this case would arise when the weight applied to the output gap is zero.
Because we assume \(\chi =0.5\), the Phillips curve will have shifted halfway back to its original position in \(t=1\).
If you expand the no-arbitrage condition and rearrange a little you get \(2i+\left(1+{i}^{2}\right)={i}^{P}+{i}_{+1}^{P}+(1+{i}^{P}{i}_{+1}^{P})\). The items placed in parenthesis very nearly cancel out.
Unconventional monetary policies like Quantitative Easing are aimed at these term premia. Conventional monetary policy works by exploiting the expectational component of the yield curve elaborated in this paper.
This interest rate (\({r}_{n}\)) goes by other names, including the natural rate, the neutral rate, or sometimes just \({r}^{*}\).
The implicit value of the slope term in the Phillips curve is \(\alpha =0.2\) and the implicit IS curve is \({y}_{+1}=100-5r\) (measuring rates using percentages rather than decimals).
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Appendix: Solving for the Policy Rate
Appendix: Solving for the Policy Rate
Now as promised, we can outline the mathematical procedure the central bank will use to set the sequence of policy rates in the soft-landing case. Using the yield curve and iterating forward from \(t=0\), we can solve for the initial policy rate:
Moreover, we know that \(i={\uppi }^{F}\) since \(r=0\), and that \({\uppi }^{F}=3/{2}^{t}\) by virtue of our assumption that \(\chi =1/2\) and the fact that \({\uppi }_{0}^{F}=3\). Substituting and simplifying gives:
Taking the limit of the right-hand side as \(T\to \infty\), recalling that the terminal \({i}^{P}\) is assumed to be zero, and recognizing that the alternating converging series sums to 2/3 gives us the solution \({i}_{0}^{P}=4\). We can then calculate the sequence \({\left\{{i}^{P}\right\}}_{1}^{T}\) by using the yield curve.
Instructors hoping to generate more general numerical examples can automate recursive methods like this using a spreadsheet. The key is recognizing that the terminal value of the policy rate is a parameter.
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Davis, L., Michl, T.R. The Inverted Yield Curve in a 3-Equation Model. Eastern Econ J 50, 195–212 (2024). https://doi.org/10.1057/s41302-024-00264-7
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DOI: https://doi.org/10.1057/s41302-024-00264-7