Debt-deflation versus the liquidity trap: the dilemma of nonconventional monetary policy

Abstract

This paper examines quantity-targeting monetary policy in a two-period economy with fiat money, durable goods and default. Short positions in long-term loans are backed by collateral, the value of which depends on monetary policy. The Quantity Theory of Money turns out to be compatible with long-run non-neutrality of money. Moreover, we show that, provided it does not lead to a liquidity trap, an expansionary monetary policy reduces markets’ inefficiency. Finally, we prove that, as the quantity of Bank money injected in the economy grows to infinity, only three scenarios can asymptotically emerge: (1) either the economy enters a liquidity trap in the first period, because the monetary expansion is not credible; (2) or a credible expansionary monetary policy accompanies the orderly functioning of markets at the cost of fueling inflation on the commodity market; (3) else, the money injected by the central bank increases the leverage of indebted investors, fueling a financial bubble whose bursting may lead to debt-deflation in the next period. This dilemma of monetary policy highlights the default channel affecting trades and production and provides a rigorous foundation to Fisher’s debt-deflation theory as being distinct from Keynes’ liquidity trap. It sheds some light on the pros and contrast of non-conventional monetary policies.

Appendix: Proofs of properties of CME

This Appendix provides the proof of the properties of collateral monetary equilibria of Sect. 3.

Proof of Proposition 1

• (i) Summing (2) over h yields:

  $$\begin{aligned} \sum _h\Bigl [ \tilde{\mu }^h_0 + \tilde{\mu }^h_{\overline{0}} + \sum _\ell \tilde{q}^h_{0\ell } +{\varDelta }(2)^h \Bigr ]=\sum _hm^h_0 + \frac{1}{1 + r_0}\sum _h\mu ^h_0 +\frac{1}{1 + r_{\overline{0}}}\sum _h\mu ^h_{\overline{0}}. \end{aligned}$$

  The conclusion follows by (13) and (14).

  When $r_0>0$, we prove that the liquidity constraint (2) is binding, so ${\varDelta }(2)^h=0$ for all agents h. Suppose ${\varDelta }(2)>0$. One of the three terms ${\varDelta }(1), \frac{\mu _0^{h}}{1+r_0}$ or $\frac{\mu ^{h}_{\overline{0}}}{1+r_{\overline{0}}}$ must then be positive. If ${\varDelta }(1)>0$, individual h can increase her deposits on 0 by $\epsilon$; if $\frac{\mu _0^{h}}{1+r_0}>0$, individual h can reduce $\mu ^h_0$ by an amount of $\epsilon (1+r_0)$. In both cases, this increases ${\varDelta }(4)$ by $\epsilon r_0$, positive by assumption, and so the right-hand side of (7). This leads to a contradiction since individual h is thus able to increase her final allocation in consumption commodities in state s. So if ${\varDelta }(2)>0$, one must have ${\varDelta }(1)=0$ and $\mu _0^{h}=0$, so that the individual h only borrows long-term money to spend on commodities and asset. But the collateral constraint (3) forces her to spend all the money on commodities. Hence ${\varDelta }(2)=0$, which contradicts the hypothesis. We then have that ${\varDelta }(2)=0$ in all cases.

• (ii) The second inequality obtains similarly by summing (7) over h. The equality follows, since, as we now show, (7) must be binding when $r_s>0$.

  Suppose that $\epsilon :={\varDelta }(7)>0$. At least one of ${\varDelta }(6)$ or $\frac{\mu ^{h}_s}{1+r_s}$ is positive. So, let individual h increase her deposits $\tilde{\mu }^{h}_s$ by $\frac{\epsilon }{1+r_s}$ or reduce her short loan $\mu ^h_s$ on s by $\epsilon$. She can spend $r_s\epsilon /(1+r_s)$ (which is positive since $r_s>0$) more on final commodities. Thus ${\varDelta }(7)$ decreases by $\epsilon$ and thus, by construction of $\epsilon$, (7) is still satisfied. Right-hand side of (8) is decreased by $\epsilon$, but the left-hand side also decreases, because h has reduced her short loan or increased her deposits by this amount. So h has increased her utility in state s, a contradiction.

  This proves that (7) is binding. One then sums (7) up to (1) over h and uses (13), (14) to obtain the desired conclusion. For the third equation, one just needs to notice that ${\varDelta }(8)=0$. In fact, if h has some money left at the end of s, she could have borrowed more, consumed more, and still been able to repay her loan. One then sums (8) over h and uses (13) and ${\varDelta }(7)=0$ to obtain the desired conclusion. Alternatively, one can use (26) (which also relies on ${\varDelta }(8)=0$) to derive the third equation from the second.

Proof of Proposition 2

• (i) Otherwise, a player could improve her profile by borrowing more money, spending part of this extra cash on a commodity, and inventorying the money to pay back the extra loan.

• (ii) No agent has some money on hand at the end of period s (for she would rather spend it or borrow more), so that ${\varDelta }(8)=0$. This yields

  $$\begin{aligned} r_0 M_0 + r_s M_s + r_{\overline{0}}M_{\overline{0}} = \overline{m}_0 + \overline{m}_s + M_{\overline{0}} - K_s \end{aligned}$$

  (26)

  The result follows because $K_s\le M_{\overline{0}}$ with equality if, and only if, there is no default.²⁹

• (iii) We have $\frac{\sum _{h} \mu ^h_0}{1+ r_0} = M_0 + \sum _{h}\tilde{\mu }^h_0$. Because $M_0>0$, at least one player h is a net borrower on $M_0$. Suppose the claim is false. Let h borrow $\epsilon$ less on $M_0$ but more on $M_{\overline{0}}$. Player h can still act in period 0 as before: she buys the same goods with the $\overline{0}$ money as with the 0 money (because ${\mathbf {C}}={\mathbf {L}}$ she can pledge them as collateral), and she ends up with the same final utility (because only second-period consumption matters and storage map are identity). Because she has less to pay on her loan 0, she inventories $\epsilon (1+r_0 -r_{\overline{0}})$ into period 1. She deposits only $\epsilon /(1+r_s)$ on $M_s$. This will exactly reimburse the principal of her long loan. Now, she is endowed with the extra money ${\varDelta }((1+r_0-r_{\overline{0}})-1/(1+r_s))$ (which is positive in every state s by assumption) that she can spend to increase her utility. A contradiction.

  If the $r^{\varepsilon }_s$ are not identical and if there is equality in the formula, then the extra money in every state s is positive in some second-period state, and some agent can still increase her utility.

• (iv) Summing over h the liquidity constraints from (1) to (4), one gets: $r_0M_0\le \overline{m}_0 + (1-r_{\overline{0}})M_{\overline{0}}$. For the inequality on $r_s$, we have $r_s M_s \le r_0 M_0 + r_s M_s + r_{\overline{0}}M_{\overline{0}} = \overline{m}_0 + \overline{m}_s + M_{\overline{0}} - K_s$. This gives the desired result because $K_s\ge 0$.

• (v) Suppose that $r_{\overline{0}}\ge 1+r_0$. Instead of borrowing on the long-term monetary market, the agent h can borrow short term, buy the same goods as before and enjoys them from period 0 on, before paying back the loan with the money she would have spent on the long-term loan interest. This increases her utility. A contradiction.

Proof of Proposition 3

Note first that $z^{h+}_{s\ell }-\frac{z^{h-}_{s\ell }}{1+r_s}\le \frac{\tilde{q}^h_{s\ell }}{p^\ell _s}-\frac{q^h_{s\ell }}{1+r_s}$ and similarly $\theta ^{h+}_k-\frac{\theta ^{h-}_k}{1+r_0}\le \frac{\tilde{\alpha }^h_k}{\pi _k}-\frac{\alpha ^h_k}{1+r_0}$. We have therefore

$$\begin{aligned}&p_0\cdot z^{h+}_0+ \pi \cdot \theta ^{h+}- \frac{1}{1+r_0}\Bigl (p_0\cdot z^{h-}_0 + \pi \cdot \theta ^{h-}\Bigr )\\&\quad \le \sum _\ell \tilde{q}^h_{0\ell }+\sum _k\tilde{\alpha }^h_k-\frac{1}{1+r_0}\Bigl ( p_0\cdot q_0^h+\pi \cdot \alpha ^h\Bigr )\\&\quad \le m^h_0 +\frac{\mu ^{h}_{\overline{0}}}{1+r_{\overline{0}}}\left( 1- \frac{r_{\overline{0}}}{1+r_0}\right) - \tilde{\mu }^{h}_{\overline{0}} \left( 1-\frac{r_{\overline{0}}}{1+r_0}\right) \end{aligned}$$

For the last inequality we have subtracted (4) divided by $1+r_0$ from (2), and used the fact that $\mu ^h_0=0$ and ${\varDelta }(2)=0$. Since $1+r_0>r_{\overline{0}}, \tilde{\mu }^{h}_{\overline{0}}(r_{\overline{0}}/(1+r_0)-1)\le 0$, the inequality follows.

To prove the inequality in the second period, one proceeds in the same way, noting that (7) is binding.