Find the Pareto Efficient allocations and Competitive Equilibrium when both agents have funky functions
Find the Pareto Efficient allocations and Competitive Equilibrium when both agents have funky functions
Loading saved threads...
Nicolas Torres · External communityPost link
External question — Economics Stack Exchange
Author: Nicolas Torres
Original post: https://economics.stackexchange.com/questions/59222
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
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
$p_2 = 0$
?
Quote
Report
Amit · External communityPost link
External answer — Economics Stack Exchange
Author: Amit
Original post: https://economics.stackexchange.com/a/59223
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
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
$Y$
.
Quote
Report
Post Reply
Quoted from Forex.com.bd-Editorial External question — Economics Stack Exchange Author: Nicolas Torres Source score (net votes, not local likes): 3 Original post: https://economics.stackexchange.com/questions/59222 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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 $p_2 = 0$ ?
Checking account access…