22.3 Pareto Efficiency and The Contract Curve

We just showed that for an arbitrary allocation, there may be a set of alternative allocations which are Pareto improvements over that allocation: that is, it might be possible to make at least one person better off without making many anyone worse off.

However, there are some allocations for which there is no overlap between the set of bundles that Alison prefers, and the set that Bob prefers. Such an allocation is called Pareto efficient, meaning that you cannot make one person strictly better off without making at least one other person strictly worse off.

Try dragging the allocation in the following diagram around until the is no overlap in the set of allocations Alison prefers to X, and the set of allocations Bob does. Once you find such a point, try changing their preferences using the sliders, and try to figure out how the location of these kinds of points changes:

What makes a bundle Pareto efficient? For a case like, this, in which we have “well behaved” preferences (smooth, monotonic, strictly convex, along with some other technical conditions), a Pareto efficient allocation is characterized by the condition that the marginal rates of substitution are equal: \(MRS^A = MRS^B\) In the diagrams in this sections, we’ve been looking at two agents with Cobb-Douglas preferences. You might have noticed that you can change the coefficients on these preferences: Alison’s preferences have been given by \(u^A(x_1^A,x_2^A) = a \ln x_1^A + (1-a) \ln x_2^A \Rightarrow MRS^A = {a \over 1-a} \times {x_2^A \over x_1^A}\) while Bob’s preferences have been given by \(u^B(x_1^B,x_2^B) = b \ln x_1^B + (1-b) \ln x_2^B \Rightarrow MRS^B = {b \over 1-b} \times {x_2^B \over x_1^B}\) For a point $X$ to be Pareto efficient, therefore, we would need their MRS’s to be equal, or \({a \over 1-a} \times {x_2^A \over x_1^A} = {b \over 1-b} \times {x_2^B \over x_1^B}\) What can we glean from this? If both Alison and Bob have the same utility function (that is, if $a = b$), then we have \({x_2^A \over x_1^A} = {x_2^B \over x_1^B}\) That is, along the contract curve, they will each have the same ratio of good 2 to good 1, so the contract curve is a straight line connecting their two origins. However, if $a > b$, then the contract curve bends down and to the right: that is, Alison likes good 1 relatively more (and good 2 relatively less) than Bob does, so Pareto efficient allocations will generally give her more good 1 and Bob more good 2: \({x_2^A \over x_1^A} < {x_2^B \over x_1^B}\) On the other hand, if $a < b$, then the opposite is true, and the contract curve bends up and to the left, meaning \({x_2^A \over x_1^A} > {x_2^B \over x_1^B}\) Regardless, at every point along the contract (you can try dragging bundle $X$ left and right), there is no overlap Alison and Bob’s preferred sets:

Note that the locus of the contract curve is based solely on the dimensions of the Edgeworth Box and the preferences of the agents. In other words, it is a characteristic of every possible allocation to be Pareto efficient or not. It is not affected by whatever the initial allocation (“endowment”) might be.

The “Contract Curve” for non-well-behaved preferences

If preferences are not well-behaved, the set of Pareto efficient points may not be characterized by a point of tangency. For example, suppose both Alison and Bob viewed these goods as perfect substitutes, but Alison’s MRS was greater than Bob’s. For example, their utility functions might be \(u^A(x_1^A,x_2^A) = 2x_1^A + x_2^A\) \(u^B(x_1^B,x_2^B) = x_1^B + 2x_2^B\) The following diagram shows the set of bundles preferred to each allocation. Try to find the Pareto efficient points (i.e. the “contract curve”). Then check the box to reveal where it is:

What’s going on here? At every point in the Edgeworth Box, Alison likes good 1 relatively more than good 2, and Bob feels the opposite way. So there is always an opportunity for an improvement if Alison exchanges some of her good 2 for some of Bob’s good 1…as long as she has some good 2, and he has some good 1! Hence the set of points from which there is no potential improvement are the allocations in which Alison only has good 1 and/or Bob only has good 2.