Question

In the analysis of an exchange between two people, suppose both people have identical preferences. Will the contract curve be a straight line? Explain. Can you think of a counter example?

Short answer

The contract curve can be a straight line.

A counter example would be a case of perfect substitutes.

Step-by-step solution

Step 1. Explanation 

We know that contract curves for each individual intersect at the origin. Therefore, a straight-line contract curve will be a diagonal line running from one origin to another inside the Edgeworth box diagram.

If we take an example to prove a straight-line contract curve; we need to prove that , and this point lies on a straight-line contact curve.

We know that the slope of contract curves is,$\frac{X}{Y}$

, where Y and X are the total amount of goods of y and x plotted on vertical and horizontal axis respectively. The amount of goods allocated to one person is$\left(x_1,y_1\right)$ and to the second person is $\left(x_2,y_2\right)=\left(X-x_1,Y-y_1\right)$

The equation for straight-line contract curve will differ since they won’t intersect and it would be;

$y=\left[\frac{Y}{X}\right]x$.

Now to prove that $MR{S}_1=MR{S}_2$

on a linear contract curve, let us consider a utility function.

$$\begin{gathered} {\text{U = x}}_{\text{i}}^{\text{2}}\text{y,} \\ \text{MRS =}\frac{{\text{MU}}_{\text{x}}}{{\text{MU}}_{\text{y}}}\text{=}\frac{\text{2xy}}{{\text{x}}^{\text{2}}}\text{=}\frac{\text{2y}}{\text{x}} \end{gathered}$$

Now if then, this utility function would be like this;

$\left(\frac{\text{2y}}{\text{x}}\right)\text{=}\left(\frac{{\text{2y}}_{\text{2}}}{{\text{x}}_{\text{2}}}\right)$

Given the above amount allocations, you have

$2\left(\frac{y}{x}\right)=2\left(\frac{Y-y}{X-x}\right)$

$$\begin{gathered} \frac{{\text{y}}_{\text{1}}\left({\text{X -x}}_{\text{1}}\right)}{{\text{x}}_{\text{1}}}{\text{= Y -y}}_{\text{1}} \\ \frac{{\text{y}}_{\text{1}}{\text{X -y}}_{\text{1}}{\text{x}}_{\text{1}}}{{\text{x}}_{\text{1}}}{\text{= Y -y}}_{\text{1}} \end{gathered}$$

And,

$$\begin{gathered} \frac{{\text{y}}_{\text{1}}\text{X}}{{\text{x}}_{\text{1}}}{\text{-y}}_{\text{1}}{\text{= Y -y}}_{\text{1}} \\ \frac{{\text{y}}_{\text{1}}\text{X}}{{\text{x}}_{\text{1}}}\text{= Y} \\ {\text{y}}_{\text{1}}\text{=}\left(\frac{\text{Y}}{\text{X}}\right){\text{x}}_{\text{1}} \end{gathered}$$

Step 2. Counter Example

In case of perfect substitutes, the contract curve is not defined properly due to the fact that each point leads to a point of tangency between 2 ICs and thus there is no unique contract curve.