Decentralizing lottery allocations in markets with indivisible commodities

Abstract

In economies with indivisible commodities, consumers tend to prefer lotteries in commodities. A potential mechanism for satisying these preferences is unrestricted purchasing and selling of lotteries in decentralized markets, as suggested in Prescott and Townsend [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. However, this paper shows in several examples that such lottery equilibria do not always exist for economies with finitely many consumers. Other conditions are needed. In the examples, equilibrium and the associated welfare gains are realized if consumptions are bounded or if lotteries are based upon a common "sunspot device" as defined by Shell [-mimeo, 1977] and Cass and Shell . The paper shows that any lottery equilibrium is either a Walrasian equilibrium or a sunspot equilibrium, but there are Walrasian and sunspot equilibria that are not lottery equilibria.

Walrasian equilibrium in an exchange economy with indivisibilities

“…One may think why the price vector p ∆ = (1.65, 1.65) is not stable for (8). At these prices, we have that Thus, all Euler polygonal arcs of (8) will move away from the prices (1.65, 1.65) and never come back to it.…”

“…When the central network announces that the process begins, each agent starts to place his market order of an object bundle he is willing to buy at the prices p(t 0 ). Based on the new information of the market orders, the central network adjusts the prices according to (8). Then the process moves to period 1 and the new adjusted prices p(t 1 ) become the new market prices and are posted on each terminal.…”

“…9 LetP denote the set of all such price functions. 8 Precisely the price vector p should be in the space R 2 n + , satisfying the nonarbitrage condition: p(e(S)) = j∈S pj for all S ∈ Σ. Since our price system is linear and satisfy p(S) = j∈S pj for all S ∈ Σ, the two are equivalent.…”

“…In an economy with a Walrasian equilibrium, there are no price vectors other than the Walrasian price vectors that have the property of global stability under the Euler price adjustment process of (8). In an economy without a Walrasian equilibrium, there are no price vectors other than the market equilibrium price vectors in W that have the property of global stability under the Euler price adjustment process of (8). In an economy that has a Walrasian equilibrium, the market price vectors are nothing but the Walrasian equilibrium price vectors.…”

Competitive Equilibria in Economies With Multiple Divisible and Indivisible Commodities and No Money

“…Modelling economies with indivisibilities is therefore meaningful and realistic. However, studying such discrete economies stands in general a daunting challenge; see for example Koopmans and Beckman [13], Debreu [6], Henry [10], Kelso and Crawford [12], Gale [7], Quinzii [18], Shapley and Scarf [22], and Scarf [19,20,21], and more recently Kaneko and Yamamoto [11], Yamamoto [24], Shell and Wright [23], Garratt [8], Garratt and Qin [9], Ma [17], Bevia et al [1], Bikhchandani and Mamer [2], van der Laan et al [15], Yang [26]. In Danilov et al [5] it was shown that discrete convex analysis is an appropriate tool to deal with indivisibles.…”