Equilibrium and arbitrage in incomplete asset markets with fixed prices
An equilibrium existence theorem is established in a standard two-period general equilibrium model with uncertainty and incomplete asset markets. The notion of equilibrium examined is a fix-price one, which is an extension of an extant definition of fix-price equilibrium under certainty to the current framework, with endogenously determined rationing on net trades to attain market clearing. A point of interest - and an unexpected one - is that the possibility of arbitrage is not completely eliminated by the endogenous trading constraints, as shown by an example. Very weak assumptions are needed for the existence result, in particular, the asset return matrix need not have full column rank, allowing for redundant assets. The description of the model includes the consumption sets \(X^{i},\) utility functions \(u^{i},\) endowments \(e^{i}\) for each individual \(i\in I,\) and an asset return matrix \(R(p,q)\) that specifies the payoffs of assets in each state of nature \(s\) in \(\{1,\dots,S\},\) in units of account, at prices of commodities \(p\) and prices of assets \(q\). There are \(L+1\) commodities of which the last one, \(L+1,\) is the numeraire at each state \(s.\) There are \( A+1\) assets and asset number \(A+1\) is a numeraire asset which plays the role of the medium of exchange before the state of nature is known and after the state is known it is the numeraire commodity \((L+1,s)\) which performs this role. The following assumptions are made: {A1}. Each \(\;X^{i}= \mathbb{R}_{+}^{(L+1)S};\) {A2.} Each \(u^{i}\) is continuous, quasi-concave and weakly monotonically increasing in the numeraire commodity in every state of the world; {A3.} For each \(i,e^{i}\in X^{i}.\) Given a profile of prices \(\overline{p}\), for the commodities and \(\overline{ q}\), for the assets, a definition of equilibrium (with rationing), { at these prices}, is provided, the interest being in the specification of an allocation which is consistent with the prices \((\overline{p},\overline{q}),\) with optimizing behaviour of individuals and with transparent markets, i.e., where it is not possible to find a buyer and a seller in a single market who could benefit from mutual exchange against the numeraire. In markets for goods and assets, other than the respective numeraire, endogenously determined rationing on net trades serve to attain market clearing. For simplification, rationing is assumed to be uniform across individuals. In case of excess supply in a market, all suppliers will, therefore, have equal, but limited, opportunities to supply. Rationing in the supply (demand) of commodities (other than the numeraire) is denoted by \(\underline{ z}\in -\mathbb{R}_{+}^{Ls}(\overline{z}\in \mathbb{R}_{+}^{Ls})\) and for (non-numeraire) assets by \(\underline{y}\in -\mathbb{R}_{+}^{A}(\overline{y} \in \mathbb{R}_{+}^{A}).\) Denote by \(\mathcal A\) and \(\mathcal L\), respectively, the set of assets and goods, excluding their respective numeraires. Also denote the set of states of nature by \(\mathcal S\). At a rationing scheme \((\underline{z}, \overline{z},\underline{y},\overline{y})\), the budget constraint of an individual \(i\) is the set of \((x^{i},y^{i})\in X^{i}\times \mathbb{R}^{A+1}\) satisfying \[ \begin{aligned} \overline{q}y^{i} &\leq 0;\;\overline{p}_{s}(x_{s}^{i}-e_{s}^{i})\leq \overline{R}_{s} y^{i},\quad\text{ for }s\in {\mathcal S};\qquad\underline{y}_{a}\leq y_{a}^{i}\leq \overline{y}_{a},a\in {\mathcal A}, \\ \underline{z}_{l,s} &\leq x_{l,s}^{i}-e_{l,s}^{i}\leq \overline{z} _{l,s},\quad (l,s)\in {\mathcal L}{\times }{\mathcal S} \end{aligned} \] The set of utility maximizing consumption bundles and asset portfolios in the budget set is denoted by \(\delta ^{i}(\underline{z},\overline{z}, \underline{y},\overline{y}).\) An equilibrium at prices \((\overline {p}, \overline{q})\) is a pair \(((x^{\ast },y^{\ast }),(\underline{z}^{\ast }, \overline{z}^{\ast },\underline{y}^{\ast },\overline{y}^{\ast }))\) such that (1) for each \(i,(x^{i\ast },y^{i\ast })\in \delta ^{i}(\underline{z}^{\ast }, \overline{z}^{\ast },\underline{y}^{\ast },\overline{y}^{\ast });\) (2) \(\sum x^{i\ast }=\sum e^{i}\) and \(\sum y^{i\ast }=0;\) (3) for every \(l\in {\mathcal L} \), if for some \(i^{\prime },x_{l,s}^{i^{\prime }\ast }-e_{l,s}^{i^{\prime }}= \underline{z}_{l,s}^{\ast },\) then for all \(i\in I,x_{l,s}^{i\ast }-e_{l,s}^{i}<\overline{z}_{l,s}^{\ast }\), while if for some \(i^{\prime },x_{l,s}^{i^{\prime }\ast }-e_{l,s}^{i^{\prime }}=\overline{z}_{l,s}^{\ast },\) then for all \(i\in I,x_{l,s}^{i\ast }-e_{l,s}^{i}>\underline{z} _{l,s}^{\ast };\) and (4) for every \(a\in {\mathcal A}\), if for some \(i^{\prime },y_{a}^{i^{\prime }\ast }=\underline{y}_{a}^{\ast }\), then for all \( i,y_{a}^{i\ast }<\overline{y}_{a}^{\ast }\), while if for some \(i^{\prime },y_{a}^{i^{\prime }\ast }=\overline{y}_{a}^{\ast }\), then for all \( i,y_{a}^{i\ast }>\underline{y}_{a}^{\ast }.\) Conditions 3 and 4, together with the convexity of the consumption sets and the quasi-concavity of the utility functions, imply that there is no effective rationing, simultaneously, on {both} sides of a market, ensuring transparency of the markets; where, by {effective rationing} it is meant that an individual is able to increase utility if the rationing scheme is removed. Under the additional assumption {A4}: the numeraire asset satisfies \(R(\overline{p},\overline{q})_{.A+1}\geq 0,\) the existence of an equilibrium (at prices \((\overline{p},\overline{q})\)) is established. When an arbitrage opportunity exists, because all individuals try to profit from it, intuition suggests that all individuals would be on the same side of all asset markets that are used in the arbitrage and endogenous constraints on trade that should emerge ought to make no trade possible in these markets. At an equilibrium, the market for an asset \(a\) is { closed} if \(\underline{y}_{a}^{\ast }=0\) or \(\overline{y}_{a}^{\ast }=0\), otherwise it is {open}. In particular, the market for the numeraire asset is always open. Denote the set of all markets which are open by{ A}\(^{0},\) the associated effctive prices of assets, \(q^{0},\) an effective portfolio by \(y^{0},\) and the matrix of effective payoffs by \( R(.,.)^{0};\) where an {effective arbitrage portfolio} \(\widehat{y}^{0} \) is such that \(q^{0}\widehat{y}^{0}\leq 0,\) while \(R(\overline{p},\overline{ q})^{0}\widehat{y}^{0}>0.\) While it is shown that the intuition stated above, which suggests that effective arbitrage portfolios do not exist in an equilibrium, is true when at most two asset markets are open, it is also shown, by means of an example, that, surprisingly, the result does not extend to equilibria with three or more open asset markets and three individuals, where one individual may hold an arbitrage portfolio, in equilibrium, which the other two, together, supply.
scientific article; zbMATH DE number 1839104