Problem Set 1: Trading from an Endowment

Improvements and Efficiency

Amal has 6 units of good 1 and 2 units of good 2, and the utility function \(u^A(x_1,x_2) = x_1x_2\)

Beatriz has 4 units of good 1 and 4 units of good 2, and the utility function \(u^B(x_1,x_2) = x_1^2x_2\)

Amal proposes that he exchange 1 unit of good 1 for 2 units of Beatriz’s good 2. Would this trade represent a Pareto improvement relative to the initial allocation? Would the resulting allocation lie along the contract curve?

Beatriz counter-offers, suggesting Amal give her 2 units of good 1 for her 2 units of good 2. Would this trade represent a Pareto improvement relative to the initial allocation? Would the resulting allocation be Pareto efficient?

Illustrate your answers to (a) and (b) in an Edgeworth Box diagram showing the initial endowment as point $E$, Amal’s proposed trade as point $A$, and Beatriz’s counter-offer as point $B$. Sketch each person’s indifference curves through each point, and shade the area representing the set of trades which would represent a Pareto improvement relative to $E$. Finally, sketch the contract curve. (Note: neither the indifference curves nor the contract curve need to be mathematically precise, but they should go through certain points and have certain shapes…)

Only smooth sometimes

Consider an Edgeworth Box diagram in which there is more good 1 than good 2. For each of the following situations, draw the “contract curve” (or more precisely, the set of Pareto efficient points); and for a representative point $X$ on the contract curve, draw the set of bundles preferred by each person to $X$. For example, if you were to do this for case we saw in lecture, it would look like this:

Hint: for these examples, the “curve” isn’t really a curve at all…it might be a good idea to try with a fairly small Edgeworth Box, for example one in which there is a total of 6 units of good 1, and 4 units of good 2…

Both people have the utility function $u(x_1,x_2) = x_1 + x_2$.

A’s utility function is $u^A(x_1,x_2) = x_1 + x_2$, and B’s is $u^B(x_1,x_2) = x_1 + 2x_2$.

A’s utility function is $u^A(x_1,x_2) = x_1 + x_2$, and B’s is $u^B(x_1,x_2) = x_1x_2$

A’s utility function is $u^A(x_1,x_2) = \ln x_1 + x_2$, and B’s is $u^B(x_1,x_2) = 2 \ln x_1 + x_2$