Pure Exchange Economy

Question

I need help drawing the Pareto Set for an Edgeworth economy. I know how to find the contract curve given an allocation, and I think that ends up being the competitive equilibria, but drawing the Pareto line is much harder than anticipated.

We are given 2 goods and 2 agents, with log utility $(\alpha_i > 0)$:

$$u_i(x_i) = \alpha_i \ln x^1_i + \ln x^2_i$$

We get the tangency condition:

$$\alpha_1 \frac{x_1^2}{x_1^1} = \alpha_2 \frac{x_2^2}{x_2^1}$$

And combine with the resource constraints:

$$x_1^1 + x_2^1 = r^1$$ $$x_1^2 + x_2^2 = r^2$$

A lot of algebra:

$$\alpha_1 \frac{x_1^2}{x_1^1} = \alpha_2 \frac{r^2 - x^2_1}{r^1 - x^1_1}$$

$$\implies \alpha_1 (x^2_1 r^1 - x^2_1 x^1_1) = \alpha_2 (x^1_1 r^2 - x^1_1 x^2_1)$$

$$\implies \alpha_1 x^2_1 r^1 - \alpha_2 x^1_1 r^2 = (\alpha_1 - \alpha_2) x^1_1 x^2_1$$

$$(\alpha_1 \cdot \frac{1}{x^1_1} \cdot r^1) - (\alpha_2 \cdot \frac{1}{x^2_1} \cdot r^2) = \alpha_1 - \alpha_2$$

Which gets us:

$$\boxed{x_1^1 = \frac{\alpha_1 r^1 x_1^2}{\alpha_1 x_1^2 - \alpha_2 x_1^2 + \alpha_2 r^2}}$$

For our Pareto Set. (You could also solve in terms of $x_1^1$.)

As denesp notes, if $\alpha_1 = \alpha_2 - r^1 = r^2$, then $x_1^1 = x_1^2$.

The question is, how would I draw this for different values of $\alpha$? What is the intuition behind the slope of it?

Answer 1

As you note in your question the (inner points of the) Pareto set are defined by $$ x_1^1 = \frac{\alpha_1 r^1 x_1^2}{\alpha_1 x_1^2 - \alpha_2 x_1^2 + \alpha_2 r^2} $$ In order to better examine this curve, let us treat it as a function. Let $$ f(x) = \frac{\alpha_1 r^1 x}{\alpha_1 x - \alpha_2 x + \alpha_2 r^2}. $$ As I stated in my comment $$ f(0) = 0 \mbox{ and } f(r^2) = r^1, $$ so the set indeed goes from one corner to the other. In between you have $$ \frac{d \ f(x)}{d x} = \frac{\alpha_1\alpha_2r^1r^2}{\left(\alpha_1 x - \alpha_2 x + \alpha_2 r^2\right)^2} $$ and $$ \frac{d^2 f(x)}{dx^2} = 2 \cdot \frac{(\alpha_1\alpha_2r^1r^2) \cdot (\alpha_2-\alpha_1)}{\left(\alpha_1 x - \alpha_2 x + \alpha_2 r^2\right)^3}. $$ The first derivative is always positive because $r^2 > x$ and all other parameters are positive. For the same reason, the sign of the second derivative depends on the sign of $\alpha_2-\alpha_1$.
Thus the Pareto set will be a strictly increasing curve, which is convex if agent 2 values good 1 'relatively more', concave if agent 1 values good 1 'relatively more', and a straight line if they value it 'relatively equally'.