Abstract
In a dynamic economy, money provides liquidity as a medium of exchange. A central bank that sets the nominal rate of interest and distributes its profit to shareholders as dividends is traded in the asset market. A nominal rates of interest that tend to zero, but do not vanish, eliminate equilibrium allocations that do not converge to a Pareto optimal allocation.
We want to thank Jean-Jacques Herings, Felix Kubler and Yannis Vailakis and the Hotel of Gianicolo for hospitality.
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Notes
- 1.
Search theoretic models of monetary economies (Diamond 1984) or (Kiyotaki and Wright 1969) are, evidently, more satisfactory, but the simple cash-in-advance formulation here, as in much of the literature, offers analytical tractability and does not play an otherwise important role in the argument argument.
- 2.
- 3.
Minor changes of notation from the abstract argument that follows facilitate the exposition.
- 4.
\( x^- i \) is the negative part of x.
- 5.
The set of all real valued maps on \(\mathcal {L}\) is \(\varvec{L}=\mathbb {R}^\mathcal {L}\). An element \(\varvec{x}\) of \(\varvec{L}\) is said to be positive if \(\varvec{x}\left( l\right) \ge 0\) for every l in \(\mathcal {L}\); negligible if \(\varvec{x}\left( l\right) = 0\) for all but finitely many l in \(\mathcal {L}\). For an element \(\varvec{x}\) of \(\varvec{L}\), \(\varvec{x}^+\) and \(\varvec{x}^-\) are, respectively, its positive part and its negative part, so that \(\varvec{x}=\varvec{x}^+-\varvec{x}^-\) and \(\left| \varvec{x}\right| =\varvec{x}^++\varvec{x}^-\). The positive cone, \(\varvec{L}^+\), of \(\varvec{L}\) consists of all positive elements of \(\varvec{L}\). Also, \(\varvec{L}_0\) is the vector space consisting of all negligible elements of \(\varvec{L}\). Finally, for every element \(\varvec{x}\) of \(\varvec{L}\),
$$\varvec{L}\left( \varvec{x}\right) =\left\{ \varvec{v}\in \varvec{L}: \left| \varvec{v}\right| \le \lambda \left| \varvec{x}\right| , \text{ for } \text{ some } \lambda >0\right\} $$is a principal ideal of \(\varvec{L}\). Unless otherwise stated, every topological property on \(\varvec{L}\) refers to the traditional product topology. We remark that, throughout the paper, the term ‘positive’ is used to mean ‘greater than or equal to zero’.
- 6.
As the allocation is Malinvaud efficient, it is Pareto efficient within every generation. As the allocation is individually rational and preferences are strictly monotone, if positive net trades vanish within a generation, so do negative net trades. Thus, using the fact that all generations are identical, \(\epsilon >0\) above does not exist only if no-trade is a Pareto efficient allocation within a typical generation. This does not occur generically in preferences and endowments.
- 7.
As far as individuals and commodities are concerned, notation is as in Sect. 15.3. In particular, an element \(\varvec{x}\) of \(\varvec{L}=\mathbb {R}^{\mathcal {T}\times \mathcal {N}}\) decomposes, across periods of trade, as
$$\begin{aligned} \varvec{x}=\left( x_0,\ldots ,x_{t-1},x_t,x_{t+1},\ldots \right) , \end{aligned}$$where each \(x_t\) is an element of \(\mathbb {R}^{\mathcal {N}}\); an element \(\varvec{x}\) of \(\varvec{E}=\mathbb {R}^{\mathcal {T}}\) decomposes, across periods of trade, as
$$\begin{aligned} \varvec{x}=\left( x_0,\ldots ,x_{t-1},x_t,x_{t+1},\ldots \right) , \end{aligned}$$where each \(x_t\) is an element of \(\mathbb {R}\).
- 8.
The discussion here is only suggestive, so that we avoid details on conditions for well-defined, though not finite, limits.
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Bloise, G., Polemarchakis, H. (2017). An Argument for Positive Nominal Interest. In: Nishimura, K., Venditti, A., Yannelis, N. (eds) Sunspots and Non-Linear Dynamics. Studies in Economic Theory, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-319-44076-7_15
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