Abstract
This paper examines quantitytargeting monetary policy in a twoperiod economy with fiat money, durable goods and default. Short positions in longterm loans are backed by collateral, the value of which depends on monetary policy. The Quantity Theory of Money turns out to be compatible with longrun nonneutrality 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 debtdeflation 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 debtdeflation theory as being distinct from Keynes’ liquidity trap. It sheds some light on the pros and contrast of nonconventional monetary policies.
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Notes
See, e.g., Dubey and Geanakoplos (2003b).
This contrasts with, e.g., Kehoe and Levine (1993), where the default penalties constrain borrowing in such a way that there is no equilibrium default.
Real and nominal securities, as well as derivatives, are added in the previous version of this paper, see Giraud and Pottier (2012).
Although there is no short sale constraint, trades of financial assets would be bounded as well because of the scarcity of collateral.
Existence of such a robust liquidity trap had been already shown by Dubey and Geanakoplos (2006b) in a model without default. Here, however, the liquidity trap is but one out of three possible scenarios that completely characterize the equilibrium set.
Equivalently, if margin requirements are high.
Which can be easily verified by checking whether shortrun interest rates are strictly positive.
Throughout this paper, for a vector x of a real vector space \({\mathbb {R}}^n\), we denote \(x>0\) if x has nonnegative components and at least one positive, and \(x\gg 0\) if all its components are positive (we then write \(x\in {\mathbb {R}}_{++}^n\)). \(\delta _{\ell }\) is the vector \((0,\ldots ,0,1,0,\ldots ,0)\) of \({\mathbb {R}}^n_+\) where 1 stands in the \(\ell \)th coordinate.
Each type of agent is thought of as represented by an interval, [0, 1], of identical clones, with the Lebesgue measure. Hence, each agent takes macrovariables (prices and interest rates) as given. Throughout the paper, we focus on typesymmetric equilibria.
We make this assumption because collateral for assets enter the storage function. If we had excluded collateral from the storage function, or used a specific linear storage function for collateral, we could have used a general concave storage function.
Lin et al. (2010a) provide a particular instance of this interpretation.
Thus, in the parlance of Woodford (1994), when \(\overline{m}_s>0\) for some state s, we are considering nonRicardian monetary policies. Since our purpose is not to study the optimal public policy, we take these transfers as exogenously given. As we shall see, equilibria exist whatever being the size of these transfers—even when they are zero.
This is in conformity with current observation. On the Repo market, for instance, there is virtually no default, and even in crisis periods (such as 1994, 1998 or 2007–2010), the rate of default remained hardly significant.
To keep the anonymity of markets, all transactions on the monetary markets pass through the Bank.
In other words, there is no private banking system in this paper.
That is, there is netting on loan.
Such constraints are standard in strategic market games, cf. Giraud (2003).
Remark that, when cast as a (typesymmetric) Nash equilibrium of the underlying strategic market game, this definition rests on the implicit assumption that players cannot condition their actions in period 1 on the actions observed from period 0. This is consistent with the anonymity property of large markets (see Giraud and Stahn 2003 for the impact of allowing for nontrivial monitoring in strategic market games with incomplete security markets). Prices are the unique signal on which players coordinate.
As already said, several alternative interpretations of outside money are conceivable: When viewed as cash inherited from some unmodelled past default, the conclusions of this subsection fail.
Given the collateral constraints, markets are (endogenously) incomplete even though financial assets’ returns would span the complete space of uncertainty.
\({\varDelta }(5)\) refers to the amount of cash saved by agent h at the end of period 0 (see the previous section for details).
Recall that \(K_s\) is defined by (15). Of course, whenever the longrun monetary market is closed, \(K_s=0\) in every scenario.
See, e.g., Geanakoplos and Zame (2014).
For this it is necessary that \(\lim _{M_0\rightarrow \infty }M_s(M_0)= 0\).
This confirms the remark already made for oneshot economies by Dubey and Geanakoplos (2003a).
Although it is unbounded, this inflation should be distinguished from hyperinflation phenomena, where the price level skyrockets to infinity for a finite increase of inside money. Here, thanks to the quantity theory of money, for every finite amount of inside money, the level of commodity prices arbitrarily remains finite.
It is somehow reminiscent of the Keynesian interpretation of Walrasian indeterminacy in incomplete markets provided by Geanakoplos and Polemarchakis (1986). The difference, however, is that we do not rely on any indeterminacy of equilibria for a given economy. Our results point toward viewing financial regulation and monetary policy as consisting in driving some key parameters of the economy, whose change induces a bifurcation in the type of equilibrium that is going to prevail—even though this equilibrium might be unique.
Only above a certain threshold of injected money do we recover the dichotomy between the real and the nominal spheres, due to the boundedness of physical trades.
Note that money created by default on longterm loan plays the role of outside money. This old idea is exploited, e.g., in Lin et al. (2010b).
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Acknowledgments
We wish to thank Christian Hellwig, Thomas Mariotti, Udara Peiris, Herakles Polemarchakis, Dimitrios Tsomocos, Alexandros Vardoulakis and Myrna Wooders for fruitful discussions, as well as participants of seminars at Paris1 university, Paris School of Economics, Bielefeld, Naples, Vigo, Toulouse School of Economics, Exeter (EWGET12), Taipei (PET 12) and the Central Bank of Austria. The usual caveat applies.
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Appendix: Proofs of properties of CME
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 righthand 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 longterm 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. Righthand side of (8) is decreased by \(\epsilon \), but the lefthand 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.^{Footnote 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 secondperiod 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 \({\varDelta }((1+r_0r_{\overline{0}})1/(1+r_s))\) (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 secondperiod state, and some agent can still increase her utility.

(iv)
Summing over h the liquidity constraints from (1) to (4), one gets: \(r_0M_0\le \overline{m}_0 + (1r_{\overline{0}})M_{\overline{0}}\). 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 longterm 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 longterm 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
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.
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Giraud, G., Pottier, A. Debtdeflation versus the liquidity trap: the dilemma of nonconventional monetary policy. Econ Theory 62, 383–408 (2016). https://doi.org/10.1007/s0019901509147
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DOI: https://doi.org/10.1007/s0019901509147
Keywords
 Central bank
 Liquidity trap
 Quantitative easing
 Collateral
 Default
 Debtdeflation
JEL Classification
 D50
 E40
 E44
 E50
 E52
 E58
 G38
 H50