Inflation Tax and Money Essentiality

Abstract

This chapter addresses the issue of money essentiality using Brock’s (1975) model. We present an argument which casts doubt on the conclusion reached by Obstfeld and Rogoff (1983), henceforth called O–R, based on the same theoretical framework. According to O–R, speculative hyperinflation—under a pure fiat money regime—can be ruled out only when severe restrictions are placed on individual preferences, e.g., agents must have infinitely negative utility when their real balances are zero. We argue that this restriction can be tested with data from hyperinflation experiments by looking at the behavior of the inflation tax, as real balances approach zero.

(With Alexandre B. Cunha)

Originally published in Economics Letters vol 78 (2003), pp. 187–195.

Appendix

Consider the following statements:

• A: υ displays ‘high’ degree of risk aversion on a neighborhood of zero:

  $$\displaystyle{\exists \epsilon > 0: \left [m \in \left (0,\epsilon \right ) \Rightarrow \frac{-m\upsilon ''(m)} {\upsilon '(m)} \geq 1\right ]}$$

• B: The limit $\lim _{m\rightarrow 0^{+}}m\upsilon '(m)$ exists: $\lim \inf _{m\rightarrow 0^{+}}m\upsilon 'm =\lim \sup _{m\rightarrow 0^{+}}m\upsilon '(m)$

• C: Money is always valued: $\lim \inf _{m\rightarrow 0^{+}}m\upsilon '(m) > 0$

• D; Inada condition: $\lim _{m\rightarrow 0^{+}}\upsilon '(m) = \infty$

Proposition 1.

condition A implies conditions B, C and D.

Proof.

Concerning [A ⇒ B], it is enough to show that [−B ⇒ −A]. so, assume that there are two numbers l and L satisfying

$$\displaystyle{ -\infty \leq l =\lim \inf _{m\rightarrow 0^{+}}f(m) <\lim \sup _{m\rightarrow 0^{+}}f(m) = L \leq \infty }$$

(5.21)

where we use the notation: f(m) = mυ′(m).

Let ε be any positive number. It must be shown that there exists some positive number $\bar{m} <\epsilon$ such that $-\bar{m}\upsilon ''(\bar{m})/\upsilon '(\bar{m}) < 1$. From (5.21), there exist two sequences {x _n } _n = 1 ^∞ and {y _n } _n = 1 ^∞ of positive numbers that satisfy x _n  → 0, y _n  → 0, f(x _n ) → l and f(y _n ) → L. By taking N and K large enough, we can find numbers x _N+K and y _N with the properties

$$\displaystyle{ 0 < x_{N+K} < y_{N} < \epsilon \ \mbox{ and}\ f\left (x_{N+K}\right ) < f\left (y_{N}\right ) }$$

(5.22)

From the Mean Value Theorem Bartle (1976, p. 196), there exists a number $\bar{m} \in (x_{N+K},y_{N})$ satisfying

$$\displaystyle{f'(\bar{m}) = \frac{f\left (y_{N}\right ) - f\left (x_{N+K}\right )} {y_{N} - x_{N+K}} \Rightarrow f'(\bar{m}) > 0}$$

But

$$\displaystyle{f'(\bar{m}) > 0 \Rightarrow \bar{ m}\upsilon ''(\bar{m}) +\upsilon '(\bar{m}) > 0 \rightarrow 1 > \frac{-\bar{m}\upsilon ''(\bar{m})} {\upsilon '(\bar{m})},}$$

as required. To prove that [A ⇒ C] is true, observe that condition A implies that f is a non-increasing function on (0, ε]. Hence, if m belongs to (0, ε], then mυ′(m) ≥ ε υ′(ε) > 0. Thus, $\lim \inf _{m\rightarrow 0^{+}}m\upsilon '(m) \geq \epsilon \upsilon '(\epsilon ) > 0$. Finally, since A implies both B and C, A also implies that $\lim _{m\rightarrow 0^{+}}m\upsilon '(m) > 0$. Hence, as m approaches zero from the right.

It should be pointed out that neither B, C or D implies A. Consider the function υ(m) = m ^0. 5. It satisfies both B and D but it does not satisfy A. Thus, both statements [B ⇒ A] and [D ⇒ A] are false. To show that C does not imply A a more sophisticated example is required. Define υ according to

$$\displaystyle{ \upsilon (m) = \left \{\begin{array}{c} -\int _{m}^{0.5} \frac{1} {x}e^{x}dx\ \mbox{ if}\ m \leq 0.5 \\ \int _{0.5}^{m}2e^{1-x}dx\ \mbox{ if}\ m > 0.5 \end{array} \right. }$$

(5.23)

Note that υ is increasing, strictly concave and twice continuously differentiable. For m ≤ 0. 5, mυ′(m) = e ^m. Thus, $\lim \inf _{m\rightarrow 0^{+}}m\upsilon (m) = 1 > 0$. So, the function defined in (5.23) satisfies C. For all m < 0. 5, −mυ″(m)∕υ′(m) = 1 − m < 1. Therefore, this utility function does not satisfy A. It should be emphasized that the υ defined in (5.23) also satisfies conditions B and D.