Extremes of the Edgeworth Box | Encyclopedia MDPI

Extremes of the Edgeworth Box: Comparison

Please note this is a comparison between Version 2 by Abigail Zou and Version 1 by Sergio Da Silva.

Extremes of the Edgeworth box concern corner allocations and their relationship to the contract curve in a two-good, two-agent exchange economy. In the standard pure-exchange setting with well-behaved preferences, the contract curve comprises all Pareto-efficient allocations, including interior tangencies and boundary corners, where no mutually beneficial trade remains. When money is introduced as a numéraire (a medium of exchange only), real feasibility and preferences are unchanged, so the contract curve remains the benchmark for efficiency. When money provides liquidity services (is valued for holding), agents may rationally abstain from trade even near interior tangencies; short-run outcomes can therefore include inaction at corners. This entry defines these objects, outlines the efficiency conditions at boundaries, and summarizes how monetary interpretations affect short-run behavior in general equilibrium and monetary economics. The Edgeworth geometry remains a real-exchange depiction; when we discuss money as a store of value, we use it as a short-run, reduced-form outside option that proxies intertemporal motives. This does not “fix” the box; it clarifies why no-trade at or near corners can be individually rational when liquidity is valued.

Edgeworth box
contract curve
corner solutions
Pareto efficiency
general equilibrium
indifference curves
budget constraint
numéraire
liquidity preference
monetary economy

The extremes of the Edgeworth box [1,2,3]^[1][2][3] are the two boundary allocations in a two-good, two-agent exchange economy where one agent holds all resources and the other holds none. Let agents

i \in \{A, B\}

have strictly increasing, strictly quasi-concave utilities

U_{i} (x_{i}, y_{i})

and endowments satisfying

x_{A} + x_{B} = \bar{x}

and

y_{A} + y_{B} = \bar{y}

. An allocation is Pareto-efficient if no feasible reallocation makes someone better off without making the other worse off.

For interior efficient points,

MRS 𝑥 𝑦 𝐴 = MRS 𝑥 𝑦 𝐵,

while at the extremes a non-negativity/boundary condition binds, so any feasible move that benefits the deprived agent necessarily harms the endowed agent; hence the corners are efficient boundary optima.

With strictly positive prices

(p_{x}, p_{y})

, a Walrasian equilibrium selects either a tangency on the contract curve or a boundary optimum when the budget line through the endowment supports it. Thus, the extremes are limit cases of price-supported efficient allocations in the Edgeworth geometry.

References

Edgeworth, F.Y. Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences; C. Kegan Paul & Co.: London, UK, 1881.
Hicks, J.R. Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory; Clarendon Press: Oxford, UK, 1939.
Nicholson, W.; Snyder, C.M. Microeconomic Theory: Basic Principles and Extensions, 12th ed.; Cengage Learning: Boston, MA, USA, 2021.