Public Debt, Taylor Rules, and Inflation Dynamics in an Overlapping Generations Model

Abstract

This chapter employs the discipline of the overlapping generations monetary framework originally developed by Marini and van der Ploeg (The Economic Journal, Vol. 98, September 1988, pp. 772–786) in order to study the dynamic consequences of fiscal policy regimes targeting real government liabilities relative to the size of the economy, via gradual adjustment rules, which interact with interest-rate policy regimes targeting inflation, via Taylor rules. The main result is that a fiscal easing enlarging the target stock of government liabilities causes inflation to increase both in the short run and in the long run, even under an aggressive interest-rate policy and an intertemporally balanced budgetary policy stance. Differently from the fiscal theory of the price level, fiscal regimes need not be “non-Ricardian”—i.e., disregarding the public solvency constraint—to reveal the debt–inflation relationship. The view that the rules-based approach to monetary policy is not sufficient to anchor inflation at the target rate without an appropriate commitment by the government over the long-run debt position and the pace of debt reduction—even under “Ricardian” fiscal regimes—has therefore additional analytical foundations. The chapter’s results suggest that inflation is neither only a monetary nor only a fiscal phenomenon. Rather, it appears to be a monetary–fiscal phenomenon.

[T]here is a chance that macroeconomic stimulus on a scale closer to World War II levels than normal recession levels will set off inflationary pressures of a kind we have not seen in a generation. [...] Stimulus measures of the magnitude contemplated are steps into the unknown.

Lawrence H. Summers (2021)

I am profoundly grateful to Giancarlo Marini, who was a major influence on my decision to pursue study and research in the field of fiscal and monetary policy. My deepest thanks to Giancarlo for the endless discussions and the invaluable encouragement.

Appendices

Appendix A

Using the definition of total consumption (5) and the optimal intratemporal condition (7), the instantaneous utility function can be expressed as

$$\displaystyle \begin{aligned} \log \Upsilon \left( \overline{c}(s,t),\overline{m}(s,t)\right) =\log q(t)+\log \overline{x}(s,t), {} \end{aligned} $$

(A.1)

where $q(t)\equiv \Upsilon \left ( \frac {\Omega (R(t))}{\Omega (R(t))+R(t)}, \frac {1}{\Omega (R(t))+R(t)}\right )$ is identical across all generations and can be interpreted as the utility-based cost of living index of the basket of physical goods and real balances. Hence, the intertemporal optimization problem can be formulated in the following terms:

$$\displaystyle \begin{aligned} \underset{\{ \overline{x}(s,t)\}}{\max }\int_{0}^{\infty }\log q(t)+\log \overline{x}(s,t)e^{-\left( \mu +\rho \right) t}dt, {} \end{aligned} $$

(A.2)

subject to

$$\displaystyle \begin{aligned} \dot{\overline{a}}(s,t)=\left( R(t)-\pi (t)+\mu \right) \overline{a}(s,t)+ \overline{y}(s,t)-\overline{\tau }(s,t)-\overline{x}(s,t), {} \end{aligned} $$

(A.3)

the no-Ponzi game condition (4), and given $\overline {a}(s,0)$.

Optimality yields the Euler equation in terms of total consumption (8) and the transversality condition (9). Integrating forward (A.3) and using both the transversality condition (9) and the law of motion of total consumption (8), the optimal level of total consumption turns to be a linear function of total wealth, according to equation (10). From (7), one has

$$\displaystyle \begin{aligned} \overline{x}(s,t)=\Lambda (R(t))\overline{c}(s,t), {} \end{aligned} $$

(A.4)

where $\Lambda (R(t))\equiv 1+R(t)/\Omega \left ( R(t)\right )$. Time differentiating (A.4) yields

$$\displaystyle \begin{aligned} \dot{\overline{x}}(s,t)=\Lambda ^{\prime }(R(t))\overline{c}(s,t)\dot{R} (t)+\Lambda (R(t))\dot{\overline{c}}(s,t). {} \end{aligned} $$

(A.5)

Substituting (A.4) and (A.5) into (8), one obtains the law of motion for individual consumption (13).

Appendix B

Aggregate wealth in per capita terms is, by definition, given by

$$\displaystyle \begin{aligned} \overline{a}(t)=\beta \int_{-\infty }^{t}\overline{a}(s,t)e^{\beta \left( s-t\right) }ds. {} \end{aligned} $$

(B.1)

Differentiating with respect to time yields

$$\displaystyle \begin{aligned} \dot{\overline{a}}(t)=\beta \overline{a}(t,t)-\beta \overline{a}(t)+\beta \int_{-\infty }^{t}\dot{\overline{a}}(s,t)e^{\beta \left( s-t\right) }ds. {} \end{aligned} $$

(B.2)

Since $\overline {a}(t,t)$ is equal to zero, by assumption, using (3) into (B.2) yields

$$\displaystyle \begin{aligned} \begin{array}{c} \dot{\overline{a}}(t)=-\beta \overline{a}(t)+\mu \overline{a}(t)+\left( R(t)-\pi (t)\right) \overline{a}(t)+\overline{y}(t)-\overline{\tau }(t)- \overline{c}(t)-R(t)\overline{m}(t) \\ =\left( R(t)-\pi (t)-n\right) \overline{a}(t)+\overline{y}(t)-\overline{\tau }(t)-\overline{c}(t)-R(t)\overline{m}(t). \end{array} {} \end{aligned} $$

(B.3)

From (12), the per capita aggregate consumption is given by

$$\displaystyle \begin{aligned} \overline{c}(t)=\frac{(\mu +\rho )}{\Lambda (R(t))}\left( \overline{a}(t)+ \overline{h}(t)\right) . {} \end{aligned} $$

(B.4)

Differentiating with respect to time the definition of per capita aggregate consumption, one obtains

$$\displaystyle \begin{aligned} \dot{\overline{c}}(t)=\beta \overline{c}(t,t)-\beta \overline{c}(t)+\beta \int_{-\infty }^{t}\dot{\overline{c}}(s,t)e^{\beta \left( s-t\right) }ds, {} \end{aligned} $$

(B.5)

where $\overline {c}(t,t)$ represents consumption of the newborn generation. Because $\overline {a}(t,t)=0$ and $\overline {h}(t,t)=\overline {h}(t)$, (12) implies

$$\displaystyle \begin{aligned} \overline{c}(t,t)=\frac{(\mu +\rho )}{\Lambda (R(t))}\overline{h}(t). {} \end{aligned} $$

(B.6)

Substituting (13), (B.4), and (B.6) into (B.5), one obtains the time path of per capita aggregate consumption, expressed by (17).