Abstract
In the past two decades, the implications of Knightian uncertainty have been explored in several general equilibrium settings. As shown by (Econometrica 60: 197–204 (1992)) Dow and Werlang (1992), (J Econ Dyn Control 22: 357–368 (1998)) Tallon (1998), (J Math Econ 61: 953–957 (2000)) Chateauneuf et al. (2000), and (Econ Theo 23: 569–587 (2004))Dana (2004), one of the most important findings in the existing literature is that indeterminacy of equilibria can be generated in the static Arrow-Debreu economy under Knightian uncertainty when there is no aggregate uncertainty.
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Notes
- 1.
See also Billot et al. (2000).
- 2.
Epstein and Wang (1994) use similar logic toward no aggregate uncertainty to show indeterminacy of equilibria in an intertemporal general equilibrium model with an infinitely lived representative agent. Note, however, that they only demonstrated nominal indeterminacy in this setting; i.e., they demonstrated the indeterminacy of the equilibrium asset prices but not of the equilibrium allocation. In their model, the single representative agent consumes the sum of the initial endowment and the dividend at each date along the equilibrium price path, and hence, real indeterminacy does not occur by the definition of the equilibrium. Also note that Epstein and Wang (1994) assume an uncertainty-averse representative agent in Lucas’ (1978) model. Essentially, the former is different from the latter only in this respect although the former assumes that consumptions are continuous over the state space to avoid some mathematical difficulty which would arise when consumptions were only measurable. Epstein and Wang (1995) overcome this difficulty by using the concept of analyticity.
- 3.
Here, an “idiosyncratic component of the asset returns” refers to the asset payoff difference across some states over which endowments remain identical. While “no trade” can occur over some price range when endowments are universally identical according to Dow and Werlang (1992); Epstein and Wang (1994) showed that it can still occur even with idiosyncratic shocks in this restricted sense. In their framework of a representative agent, the state of no trade is equivalent to an equilibrium and hence the presence of some price width with no trade implies indeterminacy of equilibria. Mukerji and Tallon (2001) incorporate Epstein and Wang’s (1994) idea into a truly general equilibrium setting and showed a “no trade” result and indeterminacy. Mukerji and Tallon (2004b) use this idiosyncratic shock to solve a “puzzle” that the public want to denominate contracts in currency units. They extend Margil and Quinzii’s (1997) general equilibrium model where both nominal and indexed bonds are available for trade by assuming uncertainty-averse agents and then derive conditions under which there is no trade in indexed bonds in any equilibrium. Furthermore, Ozsoylev and Werner (2011) use a similar idiosyncratic shock to show emergence of illiquid rational expectations equilibria by demonstrating that there exists a range of values of the signal and random asset supply over which the arbitrageurs, the suppliers of money , cease the trade in an asset pricing model with information transmission and with uncertainty-averse agents.
- 4.
See Rinaldi (2009) for additional information. Note, however, that because the MEU preference is a special case of the variational preference axiomatized by Maccheroni et al. (2006) and because any variational preference may be approximated arbitrarily well by another smooth variational preference, Rinaldi’s (2009) result is not a confirmation that Mukerji and Tallon’s (2001) result works for the whole class of variational preferences. In a similar fashion, the robust indeterminacy in this chapter is concerned with the robustness with respect to only the initial endowments and not the robustness with respect to both the initial endowments and the preferences. That is, the preferences are supposed to be fixed. Also see Guidolin and Rinaldi (2013).
- 5.
- 6.
- 7.
Interested readers may also see Ohtaki (2015).
- 8.
Recently, Ohtaki and Ozaki (2013) extended the standard dominant root characterization of the optimal allocations to the economy under Knightian uncertainty.
- 9.
Except for the preferences, the ingredients of our model are similar to those in Labadie’s work (2004). However, her objective was to examine the financial arrangements that could result in the optimal allocations, rather than to examine the in/determinacy of stationary monetary equilibrium.
- 10.
This definition of the date-event tree is standard, and can be seen in, for example, Chattopadhyay (2001).
- 11.
That is, the formation of the belief is independent of the past history of realized states. Furthermore, the set of priors may not be common to all agents who are distinguished by the states at which they are born.
- 12.
Gilboa and Schmeidler (1989) axiomatized the MEU preferences over lottery acts and Casadesus-Masanell et al. (2000) axiomatized the MEU preferences over Savage acts. Their axiomatization does not depend on whether the state space is finite or infinite, and hence, it may be applied to our situation with a finite state space.
- 13.
Labadie (2004) considered a time-separable utility index function.
- 14.
- 15.
In (i), we assume that the budget constraints hold with equalities. We can do this for the first budget constraint without loss of generality by the strict increase of u. For the other budget constraints, we simply assume it. Also note that we exclude corner solutions by assuming that \((c^y,(c^o_{s'})_{s'\in S}) \in \mathbb {R}_{++}\times \mathbb {R}_{++}^S\) and (ii).
- 16.
The MEU preferences in the Cobb-Douglas form as in this example have been axiomatized by Faro (2013).
- 17.
To be more precise, the marginal rates of substitution for indifferent curves derived from \(\hat{U}^s\), denoted by \(\widehat{MRS}_{s}(x^o)\), is calculated using
$$\begin{aligned} \widehat{MRS}_{\alpha }(x^o) =-\frac{\hat{U}^{\alpha }_1(x^o)}{\hat{U}^{\alpha }_2(x^o)} =\left\{ \begin{array}{ll} \displaystyle \frac{[(1+\varepsilon )x^o_{\alpha } -\varepsilon \bar{\omega }_{\alpha }]x^o_{\beta }}{(1-\varepsilon )(\bar{\omega }_{\alpha }-x^o_{\alpha })x^o_{\alpha }} &{}\quad {\text {if}}\quad x^o_{\alpha }>x^o_{\beta },\\ \displaystyle \frac{[(1+\delta )x^o_{\alpha } -\delta \bar{\omega }_{\alpha }]x^o_{\beta }}{(1-\delta )(\bar{\omega }_{\alpha }-x^o_{\alpha })x^o_{\alpha }} &{}\quad {\text {if}}\quad x^o_{\alpha }<x^o_{\beta } \, , \end{array} \right. \end{aligned}$$and
$$\begin{aligned} \widehat{MRS}_{\beta }(x^o) =-\frac{\hat{U}^{\beta }_1(x^o)}{\hat{U}^{\beta }_2(x^o)} =\left\{ \begin{array}{ll} \displaystyle \frac{\varepsilon (\bar{\omega }_{\beta } -x^o_{\beta })x^o_{\beta }}{[(2-\varepsilon )x^o_{\alpha } -(1-\varepsilon )\bar{\omega }_{\beta }]x^o_{\alpha }} &{}\quad {\text {if}}\quad x^o_{\alpha }>x^o_{\beta },\\ \displaystyle \frac{\delta (\bar{\omega }_{\beta } -x^o_{\beta })x^o_{\beta }}{[(2-\delta )x^o_{\alpha } -(1-\delta )\bar{\omega }_{\beta }]x^o_{\alpha }} &{}\quad {\text {if}}\quad x^o_{\alpha }<x^o_{\beta } \end{array} \right. \end{aligned}$$if \(x^o_{\alpha }\ne x^o_{\beta }\). However, it cannot be calculated at \(x^o\) with \(x^o_{\alpha }=x^o_{\beta }\), if \(\varepsilon <\delta \).
- 18.
- 19.
The detail of the box diagram presented here is provided in Ohtaki (2014) for a case of no Knightian uncertainty.
- 20.
- 21.
To be more precise, Proposition 2 of Magill and Quinzii (2003) show the indeterminacy of the candidates for “expectation functions,” each of which constructs a nonstationary rational expectations equilibrium with circulating money.
- 22.
See also Theorem of Ohtaki (2015).
- 23.
As shown by Gottardi (1996), a “zero-th order” stationary monetary equilibrium (where money prices only depend on the current states) is locally isolated. In contrast, Spear et al. (1990) showed that the “first-order” and “second-order” stationary monetary equilibria (where money prices may depend on the past states) are indeterminate. In this chapter, the stationarity always refers to the “zero-th order.”
- 24.
The basic idea is as follows: let \(s_{1}:= \arg \min _{s'\in S}\omega ^o_{s'}+q^*_{s'}\) and let \(\mathcal {M}_{1}\) be the set of probability measures in \(\mathcal {P}_s\) that assign the largest probability to \(s_{1}\). Then, let \(s_{2}:= \arg \min _{s'\in S\backslash \{s_{1}\}}\omega ^o_{s'}+q^*_{s'}\) and let \(\mathcal {M}_{2}\) be the set of probability measures in \(\mathcal {M}_{1}\) that assign the largest probability to \(s_{2}\). Continuing this process can lead to a single probability measure in \(\mathcal {P}_s\) that is the unique element of \(\mathcal {M}_s(c^M_s(q))\). In general, however, the success of this procedure hinges upon the nature of \(\mathcal {P}_s\). In this sense, the argument of this footnote stands only heuristically. For example, if \(\mathcal {P}_s\) is characterized by the \(\varepsilon \)-contamination (Example 6.3.1), the above procedure will determine a single probability measure.
- 25.
As stated in the previous footnote, the system of equations can possibly be a system of inclusions. Because we are concerned with a sufficient condition for multiple equilibria, we can neglect such a situation for our purpose.
- 26.
Interested readers may refer to Cass et al. (1992), which also provided the existence and regularity results for the stationary monetary equilibrium in a more complicated stochastic OLG model.
- 27.
Consider \(\overline{s}\) and \(\underline{s}\) such that \(\omega _{\overline{s}}=\max _{s \in S}\omega ^o_s\) and \(\omega _{\underline{s}}=\min _{s \in S}\omega ^o_s\). Because \(\pi \gg 0\) for each \(\pi \in \mathcal {P}_s\), \(\bar{c}^{\, *o}\) should satisfy that \(1<\min _{\pi \in \mathcal {P}_{\underline{s}}} \sum _{s'\in S}(\bar{c}^{\, o*}-{\omega }^o_{s'})\pi _{s'}/(\bar{c}^{\, *o}-{\omega }^o_{\underline{s}}) \le u_1(\bar{\omega }-\bar{c}^{\, *o},\bar{c}^{\, *o})/u_2(\bar{\omega }-\bar{c}^{\, *o},\bar{c}^{\, *o}) \le \max _{\pi \in \mathcal {P}_{\overline{s}}} \sum _{s'\in S}(\bar{c}^{\, o*}-{\omega }^o_{s'})\pi _{s'}/(\bar{c}^{\, *o}-{\omega }^o_{\overline{s}})<1\). However, this is a contradiction.
- 28.
This result can be extended to slightly more general preferences. See Ohtaki (2015).
- 29.
In the framework of additively time-separable preferences, Manuelli (1990) also provides uniqueness results under an additional assumption that the states’ evolution follows an i.i.d. process. He considers a stochastic OLG model with a general state space , and hence, it includes a finite state space case as in our model as a special case. Note that even when the set of priors, \(\mathcal {P}_s\), is independent of s, Corollary 6.5.2 still demonstrates the existence of a continuum of stationary monetary equilibria.
- 30.
Klibanoff et al. (2005) attempt to separate these two essentially distinct aspects of the preference. See footnote 35 of Chap. 1. Snow (2010) proposes a definition of “an increase in ambiguity” in the framework of Klibanoff et al. (2005). Gajdos et al. (2008) also separate them by using “objective” uncertainty.
- 31.
Note that the proof of Theorem 6.7.1 does not require continuous differentiability of u.
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Nishimura, K.G., Ozaki, H. (2017). Monetary Equilibria and Knightian Uncertainty. In: Economics of Pessimism and Optimism. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55903-0_6
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