Endowments, Allocations, Feasibility

An exchange economy has a finite set of agents $I = \{1, \ldots, n\}$, a finite set of goods $L = \{1, \ldots, \ell\}$, preferences for each agent over consumption bundles in $\mathbb{R}^L_+$, and an initial endowment that specifies how much of each good each agent starts with. No production technology appears anywhere in the model. The total supply of every good is therefore fixed at the sum of individual endowments.

The rest of this module works in the canonical $I = 2$, $L = 2$ case: two agents — call them $A$ and $B$ — trading two goods. This is the smallest economy in which there is any trade to study. Everything that follows generalises to $I > 2$ agents and $L > 2$ goods, but the generalisation loses the Edgeworth box diagram and replaces it with higher-dimensional geometry. The two-by-two case retains the picture, and the picture carries almost all the intuition.

The Edgeworth box is a single rectangle that displays every non-wasteful feasible allocation at once. Its width is $\Omega_1 = \omega^A_1 + \omega^B_1$ — the total endowment of good 1 — and its height is $\Omega_2 = \omega^A_2 + \omega^B_2$. Every point strictly inside the box (and on its boundary) is exactly one non-wasteful allocation. There are no leftovers: whatever $A$ does not consume, $B$ consumes.

The construction uses two origins. Agent $A$'s consumption bundle $(x^A_1, x^A_2)$ is measured from the lower-left corner, moving rightward for more of good 1 and upward for more of good 2. Agent $B$'s bundle $(x^B_1, x^B_2)$ is measured from the upper-right corner, moving leftward for more of good 1 and downward for more of good 2. Because the box dimensions equal the total endowments, the two readings always sum to the totals: $x^A_1 + x^B_1 = \Omega_1$ and $x^A_2 + x^B_2 = \Omega_2$.

The initial endowment point $\omega = (\omega^A, \omega^B)$ is one particular point inside the box — the allocation the economy starts at before any trade occurs. It is no more or less feasible than any other interior point. Its special status comes from outside the feasibility criterion: it is the fallback each agent can guarantee themselves by refusing to trade, which matters for individual rationality and for the boundary conditions of the Walrasian budget set.

The box below is live. Adjust the endowment controls and watch the rectangle resize: its width tracks $\Omega_1 = \omega^A_1 + \omega^B_1$ and its height tracks $\Omega_2 = \omega^A_2 + \omega^B_2$. The dot is the endowment $\omega$, measured as $A$'s bundle from the lower-left origin $O_A$; the same point read from the upper-right origin $O_B$ is $B$'s bundle. There are no preferences or prices here yet — every point in the box is simply a non-wasteful feasible allocation.