Find the Pareto Efficient allocations and Competitive Equilibrium when both agents have funky functions

Question

I'm trying to solve this General Equilibrium excercise which I find quite challenging as both agents have funky utility functions.

Find the Pareto Efficient allocations and Competitive Equilibra for this pure exchange economy:

$U_A(x,y)=\max(5,\min(x,y))$

$U_B(x,y)=\min(x,\frac{3}{2}y)$

$\omega_A = (20,0)$

$\omega_B = (0,15)$

Here I have my Edgeworth box where

Red: Indifference curve shape for A (L shape above and to the right of the kink)

Blue: Indifference curve shape for B (L shape below and to the left of the kink)

Green: Budget line for $\frac{p_1}{p_2}=\frac{1}{3}$

Orange: Leontief expansion path for A: $x=y>5$

Pink: Expansion path for B: $y=\frac{2}{3}x$

Is the Pareto efficient set the portion of the blue indifference curve that intersects the y axis?

Now for the competitive equilibria: The set of optimal bundles for B is the entire pink curve.

For A, the optimal bundles are: the set below the budget line for $p=\frac{1}{3}$, which is supported by prices $\frac{p_1}{p_2}\leq \frac{1}{3}$ union the orange curve which is supported by prices $\frac{p_1}{p_2} > \frac{1}{3}$.

So would the competitive equilibria allocations be the interesction of the optimal points for A and B, which would be the portion of the pink segment below the budget line for $\frac{p_1}{p_2} = \frac{1}{3}$ ?

If this is correct, how would I write the Competitive Equilibria set in the form $(p,x)$ where $p$ is the price vector and $x$ is the allocation vector, i.e. the quantities each consumer gets of each good.

Also, are there any equilibria for $p_1 = 0$ or $p_2 = 0$ ?

Answer 1

Given the pure-exchange economy:

• Utility functions: $u_A=\max(5,\min(x_A,y_A))$, $u_B=\min(x_B,\frac{3}{2}y_B)$
• Endowments: $\omega_A=(20,0)$, $\omega_B=(0,15)$

Set of Feasible allocations is given by:

$\mathcal{F}=\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=20 \ \wedge \ y_A+y_B=15\}$

Set of Pareto optimal Allocations is given by:

$\mathcal{PE}=\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|x_A=0 \ \wedge 0\leq y_A\leq \frac{5}{3}\} \ \cup \ \{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|5<\frac{5}{3}+\frac{2}{3}x_A\leq y_A \leq x_A\} $

Set of competitive equilibrium allocations is given by

$\mathcal{CE}=\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|y_B=\frac{2}{3}x_B\geq 10\}$

Each allocation in the set above corresponds to exactly one equilibrium price ratio $\frac{p_X}{p_Y}$ in the set $[\frac{1}{12},\frac{1}{3}]$. To be specific, for each $x_B\in[15,20]$, the equilibrium allocation is $y_B=\frac{2}{3}x_B$, $x_A=20-x_B$, $y_A=15-\frac{2}{3}x_B$ and the corresponding price ratio is $\frac{p_X}{p_Y}=\frac{45-2x_B}{3x_B}$.

There are no equilibria with either $p_X=0$ or $p_Y=0$ because at $p_X=0$, consumer 2 will demand at least $22.5$ units of $X$ leading to excess demand for it, and at $p_Y=0$, consumer 1 will demand at least $20$ units of $Y$ causing exceed demand for $Y$.