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Chapter 22 / The Edgeworth Box

22.2 Preferences over Allocations


When we introduced the notion of indifference curves, we motivated our discussion by grounding it in the axiom of complete preferences: that is, given any two alternatives $X$ and $Y$, we assume that an agent will say that she either prefers $X$ to $Y$, prefers $Y$ to $X$, or is indifferent between the two. Therefore, an indifference curve through bundle $X$ divides the choice space into the “preferred set” (those bundles the agent prefers to $X$) and the “dispreferred set” (those bundles the agent disprefers to $X$).

The same logic may apply to the Edgeworth Box: if we presented Alison and Bob with a range of alternatives within the Edgeworth Box, they could each tell us which one they preferred or dispreferred. In particular, for any allocation $X$, they could tell us the set of bundles they prefer to $X$, the set of bundles they disprefer, and the set between which they are indifferent. In other words, for any bundle $X$, we can draw both Alison’s indifference curve and Bob’s indifference curve through it.

If we draw the indifference curves and shade in the preferred sets, something profound reveals itself:

If we look at the shaded regions, we can see that for any bundle X, the indifference curves partition the box into four regions:

It’s this last category that’s most important to us as economists. Think about what this means: simply by reallocating goods from one person to another, we can make both of them better off. In other words: the world is improvable. There is cause for optimism.

To be more precise, we can say that if we have two bundles, $X$ and $Y$, then $Y$ Pareto dominates $X$ (or that moving from $X$ to $Y$ is a Pareto improvement) if $Y \succeq_i X$ for every agent $i$, and $Y \succ_i X$ for at least one agent $i$. That is, it makes at least one person strictly better off, and makes no agent worse off.

Generally speaking, at any allocation for which the agents’ marginal rates of substitution are not equal, there may be a set of allocations that represent potential Pareto improvements. For example, if Alison’s MRS is less than Bob’s, this would mean that Alison values her last unit of good 1 (in terms of good 2) less than Bob would value it; so there is the potential that by trading some of her good 1 for some of Bob’s good 2, they can each be made better off. Visually, there would be a “lens” of allocations which could potentially represent Pareto improvement above and to the left of the initial endowment, as shown in the diagram above. If Alison’s MRS is greater than Bob’s, on the other hand, the lens of potential Pareto improving allocations would lie below and to the right of the initial allocation. (Try dragging point $X$ to the upper-left part of the box and/or changing the parameters of the utility function to see how this works.)

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Copyright (c) Christopher Makler / econgraphs.org