How to find the contract curve when both agents have linear utilities?

Question

I'm trying to solve the following excercise:

Find the contract curve for an exchange economy where agents' ($A$ and $B$) preferences and endowments are given by:

$u_A = x_A + y_A$

$u_B = s x_A + y_A$

$(\omega_{1A},\omega_{2A}) = (\alpha,\beta)$

$(\omega_{1B},\omega_{2B}) = (\beta,\alpha)$

Here the indifference curves for $A$ are lines with slope $-1$, while the indifference curves for $B$ are lines with slope $-s$, where $s$ is a parameter.

The contract curve can be found by solving

$\max x_A + y_A$

subject to

$s x_B + y_B = \overline{U}$

$x_A + x_B = \alpha + \beta$

$y_A + y_B = \alpha + \beta$

What I have tried is the usual $\frac{MU_{A,x}}{p_x} > \frac{MU_{A,y}}{p_y}$ approach used in partial equilibrium analysis.

Let's set $p_x = 1$.

Since $MU_{A,x} = MU_{A,y} = 1$, the statement above corresponds to $1 > \frac{1}{p_y}$, equivalently, $p_y > 1$.

This implies that if $p_y > 1$, $A$ would buy as much $x$ as they can get by selling $y$.

So we would get

• $p_y > 1$

$x_A = \alpha + \beta$

$y_A = \beta (1 - \frac{1}{p_y})$

$x_B = 0$

$y_B = \alpha + \frac{\beta}{p_y}$

Conversely, if $p_y < 1$, $A$ would buy as much $y$ as they can get by selling $x$.

So we would get

• $p_y < 1$

$x_A = 0$

$y_A = \beta + \frac{\alpha}{p_y}$

$x_B = \alpha + \beta$

$y_B = \alpha (1 - \frac{1}{p_y})$

Now, if $p_y = 1$, $A$ would be indifferent between buying or selling any amount of $x$ for the same amount of $y$.

So we would get

$x_A \in [0,\alpha + \beta]$

$y_A = \alpha + \beta - x_A$

$x_B = \alpha + \beta - x_A$

$y_B = x_A$

We have that

$y_B = x_A \iff \alpha + \beta - x_A = x_A \iff \alpha + \beta = 2 x_A \iff \frac{\alpha + \beta}{2} = x_A$

So the solution would be

• $p_y = 1$

$((x_A,y_A),(x_B,y_B)) = ((\frac{\alpha + \beta}{2},\frac{\alpha + \beta}{2}),(\frac{\alpha + \beta}{2},\frac{\alpha + \beta}{2}))$

which is the midpoint of the Edgeworth box.

I then drew an Edgeworth box containing indifference curves for $A$ and $B$, for some $s > 1$, then plotted the segments of the 'contract curve' parametrized by $p_y$, this is what I got:

Red pen: $A$'s indifference curves

Black pen: $B$'s indifference curves

Blue pen: Exchange Area

Pencil: 'Contract curve' = The $y_A$ axis without the endpoints $\cup$ The segment of the $y_B$ axis where $\alpha < y_B < \alpha + \beta$ $\cup$ The midpoint of the Edgeworth box.

What I found weird is in General Equilibrium we don't even know the relative prices of the goods, and my method uses the relative prices.

Graphically, only the $y_A$ axis seems to be part of the contract curve, as in the rest of the 'contract curve', moving upwards along $B$'s indifference curve would give $A$ a higher utility while keeping $B$'s constant (a Pareto improvement).

Moreover, the endpoints of the $y_A$ axis seem to be part of the contract curve, as in the upper endpoint, the only possible movements along $B$’s indifference curve are downwards, taking away utility from $A$; while in the lower endpoint, there is nowhere even to move along either indifference curve ($A$ and $B$’s indifference curves would consist only of that point), hence no room for Pareto improvement.

I also noted if I were to take $s < 1$, it is now the $y_B$ axis that seems to be part of the contract curve and none of the $y_A$ axis.

How do we actually find the contract curve?

Answer 1

I rewrite the problem of maximization you wrote (I omit the endowments):

$\max x_A + y_A \;\;\qquad (1)$

subject to

$s x_B + y_B = \overline{U}\qquad (2)$.

This problem can be seen as a problem of linear programming, and we can remember that in such problems the optimal points are, in general, in the corners of the feasible area. After all, this is the usual solution when we have linear utility functions.

In our problem the corners of the constrained area are the intersection of the constraint $(2)$ with the coordinate axes. This already suggests that the contract curve should be on the coordinate axes.

I represent graphically, below, a solution through the usual Edgeworth box.

I show separately (to avoid visual confusion) the optimization problem from the point of view of consumer $A$, as stated in $(1)-(2)$, in the first picture, and the symmetric problem for consumer $B$ in the second picture (even if this second maximization is not strictly necessary).

$$Fig. \;1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ $\;$

$$Fig. \;2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ $\;$

Consider a given indifference curve of consumer $B$, as $u_B= \overline U_{B1}$, ending at a point $A$ on the $x$ axis $0_AC$.

As it should be clear from Figure $1$ , the problem $(1)-(2)$ has its optimal solution at point $A$, on the $x$ axis, given a level of utility of consumer $B$, $u_B$, equal to $\overline U_{B1}$.

Changing the given level of $u_B$, so that the blue indifference curve which is the constraint of the problem shifts, the optimal point will go along the $x$ axis $0_AC$, covering the whole line as the blue indifference curve shifts continuously.

That is, the optimal points of consumer $A$, as $u_B$ changes assuming all possible values, coincide with the $x$ axis $0_AC$.

Analogously, if we have a given indifference curve of consumer $B$ ending at a point $H$ on the $y$ axis $C0_B$, the optimal point is $H$, and shifting the given indifference curve we obtain the whole $C0_B$ axis.

We can, therefore, conclude that the contract line is the set composed of the $x$ axis $0_AC$ and the $y$ axis $C0_B$., the violet set in Figure 1.

$\;$

If we want to verify that this is actually the contract line$^1$, we can re-formulate the problem from the point of view of consumer $B$, maximizing their utility for a given level of utility of consumer $A$.

This is represented in Figure $2$, where the green lines are the constraints, given by fixed indifference curves of consumer $A$. Following the same reasoning as for consumer $A$, we see that the optimal point for consumer $B$ is on the $x$ axis at point $B$, and actually it coincides with point $A$ of Figure $1$.

Changing the given level of utility of consumer $A$ we obtain again the $x$ axis $0_AC$.

And analogously, taking as constraint indifference curves as $u_A=\overline U_{A2}$ we obtain the $y$ axis $C0_B$.

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$^1$ ^{The first optimization is sufficient, but let's verify we are not mistaken.}