7
votes
Accepted
Pure exchange economy: Given an initial endowment are multiple equilibria possible?
Yes. The Debreu version of the Sonnenschein-Mantel-Debreu theorem guarantees that excess demand has to satisfy very little restrictions if there are as many consumers as commodities.
An explicit ...
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5
votes
Perfect complement preferences in an exchange economy
Set of Pareto Efficient Allocations consists of feasible allocations $((x_J, y_J), (x_D, y_D))$ satisfying $y_J=\displaystyle\frac{x_J}{2}$.
Competitive Equilibrium is the price $(p_x, p_y=1)$ ...
- 9,842
5
votes
Accepted
How to find the contract curve when both agents have linear utilities?
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 ...
- 4,160
5
votes
Pure exchange economy: Given an initial endowment are multiple equilibria possible?
Here is another example with two consumers (A and B), two goods (X and Y):
\begin{eqnarray*} u_A(x_A, y_A) & = & \min(x_A, y_A), \ \omega_A = (1, 0) \\ u_B(x_B, y_B) & = & \min(x_B, ...
- 9,842
4
votes
Accepted
How do I find the set of pareto optimal allocations?
In the economy you provided, set of feasible allocations is
$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=2 \ \wedge \ y_1+y_2=1\}$
and is represented by points in ...
- 9,842
4
votes
Core in a replicated economy
In the economy provided in the question, competitive equilibrium allocations is equal to the set of efficient allocations. This along with the fact that the competitive equilibrium allocations always ...
- 9,842
4
votes
Accepted
Finding Pareto optimal allocations and Walrasian equilibrium allocations in the case of 3 goods
Pareto Optimality:
Since preferences are convex in your case, you can find Pareto optimality in the same way. You need to solve $\text{MRS}^{A}_{v,w} = \text{MRS}^{B}_{v,w} = \text{MRS}^{C}_{v,w}$ ...
- 56
2
votes
Fair and efficient allocation of "family goods"
Suppose there are two families: Family U has $n_u$ members, and family V has $n_v$ members. Utility function of member $i$ of family U is:
\begin{eqnarray*} u_i(x_u, y_u) = a_ix_u + y_u \end{eqnarray*}...
- 9,842
2
votes
Accepted
Utility Possibility Frontier with two consumers and 3 commodities
The utility possibility frontier (UPF) plots the maximum total combination of utilities that can be achieved, given the preferences and total resources. To fix ideas, let's suppose we are plotting $...
- 4,208
2
votes
Accepted
Find Pareto optimal allocations and the core for the following economies
In the economy described above, set of efficient allocations is given by the blue curve. Just do the slope of ICs comparisons at the boundaries and you will get that.
- 9,842
2
votes
Accepted
How to find the competitive equilibrium?
Here is a lazy tricks-based approach that requires almost no calculations: First, we know that every equilibrium allocation must be Pareto efficient by the first welfare theorem. If $B$ consumes ...
- 13.7k
2
votes
Finding Pareto optimal allocations and Walrasian equilibrium allocations in the case of 3 goods
To add to my previous answer (and to reply to your comment), the set of equations for Pareto optimality include the MRS equations and the feasibility equations.
MRS equations:
\begin{align}
\text{MRS}^...
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2
votes
Net and gross market clearing in endowment economy
The issue here is that market clearing can be defined in gross as well as in net terms.
Gross vs. net
Suppose I need 10 eggs for the next two weeks, and I don't intend to have any left over (I am ...
- 29.7k
2
votes
Exchange economy with two agents, what's the competitive equilibrium?
I’m going to change the notation to one I’m more comfortable with:
Let the goods be $x,y$, the consumers $A,B$; and the respective endowments $(w_{x_A},w_{y_A}), (w_{x_B},w_{y_B})$.
Let’s set as ...
- 2,351
2
votes
Accepted
How to find the Walrasian equilibrium for non monotonic utility functions?
Set of Pareto efficient allocations (or contract curve) is the set of all feasible allocations satisfying $1 \leq x_1=y_1\leq 3$. This is the line segment connecting points A and B in your graph.
Also,...
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2
votes
Finding Walrasian equilibria when Walrasian demands are not unique
let us first write the demand functions of individual $A$ and $B$
$$(x_A^d,y_A^d)(p_x,p_y,m_A)\in\left\{\begin{matrix}
(\frac{m_A}{p_x},0) & ,\frac{p_x}{p_y}<1\\
(0,\frac{m_A}{p_y}) & , \...
- 667
2
votes
Accepted
Is there/can we define a notion of Giffen goods in pure exchange economies?
"Giffenness" may follow only from the preferences, you have to filter out endowment income effects. (Substitute a fixed monetary income $m$, for details, see e.g.; Varian's Intermediate ...
- 29.7k
2
votes
Accepted
General equilibrium with market power
Given a pure exchange economy:
$u_A(x_A, y_A)=x_Ay_A$, $u_B(x_B, y_B)=x_By_B^2$
with endowments:
$\omega_A=(80,150)$ and $\omega_B=(210,180)$
To find the equilibrium, we first find the price offer ...
- 9,842
2
votes
Accepted
Why doesn't this exchange economy have a walrasian equilibrium?
The economy you described does have a Walrasian Equilibrium when $n$ is even. The Equilibrium price ratio is $\dfrac{p_X}{p_Y}=1$. The corresponding equilibrium allocation is any allocation in which $\...
- 9,842
1
vote
Finding the competitive equilibrium in an exchange economy with two perfect complements
Given a pure-exchange economy with
$u_A(x_A,y_A)=\min(x_A,2y_A)$, $u_B(x_B,y_B)=\min(2x_B,y_B)$
Endowment of A is $(k_X,k_Y)$ and of B is $(12-k_X,12-k_Y)$
Set of feasible allocations is $\mathcal{F}...
- 9,842
1
vote
Trying to understand Walrasian equilibrium (Brown & Matzkin 1996)
Consider consumer 1. Let $(x,y)$ be his optimal consumption bundle in the first equilibrium and let $(\bar x, \bar y)$ be his optimal consumption bundle in the second equilibrium.
Then the budget ...
- 12.8k
1
vote
Accepted
Edgeworth Box for exchange economy
Not quite. The line you have drawn turns out to be the contract curve, but from this figure it's not clear how you get this line, because the indifference curves for the two utility functions are ...
- 7,470
1
vote
General equilibrium with market power
Not sure where you get lost. $u_B$ is Cobb-Douglas type, thus given a price vector $\textbf{p}$ and initial endowment $\textbf{w}^B$ it is easy to determine $x_1^B(\textbf{p}),x_2^B(\textbf{p})$.
Once ...
- 29.7k
1
vote
Accepted
Utility Possibility Set and Utility Possibility Frontier
Given a pure-exchange economy:
$u_1(x_1,y_1)=\max(x_1,y_1)$, $\omega_1=(4,0)$
$u_2(x_2,y_2)=\min(2x_2+y_2,x_2+2y_2)$, $\omega_2=(0,3)$
Set of feasible allocations is given by
$\mathcal{F}=\{((x_1,...
- 9,842
1
vote
Exchange economy with two agents, what's the competitive equilibrium?
An allocation $((c^*_{a,1},c^*_{a,2}),(c^*_{b,1},c^*_{b,2}))$ is a competitive equilibrium allocation for the given economy supported by the price ratio $\frac{p_1^*}{p_2^*}$ if it satisfies the ...
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1
vote
Calculating the Competitive Equilibrium in a pure exchange economy with 3 comodities and 2 agents
First, there is no such thing as the competitive equilibrium price vector. If $(p_1,p_2,p_3)$ is a competitive equilibrium price vector, so is every positive multiple of this vector.
Second, to find ...
- 13.7k
1
vote
Accepted
Exchange economy find core
Core Allocations are Pareto efficient allocations that must satisfy individual rationality i.e. these allocations must yield at least as much satisfaction to the individuals as their respective ...
- 9,842
1
vote
Accepted
Why can’t we know the specific outcome of an exchange economy with initial resource endowment?
The idea of the contract curve is just restricting the Pareto set so that no one player is worse off than the initial allocation. There's no concept of "price" involved here. When your endowment is $(...
- 2,800
1
vote
Pure exchange economy: Set of multiple equilibria endowments
The following article:
TODA, A.A. and WALSH, K.J., 2017. Edgeworth box economies with multiple equilibria. Economic Theory Bulletin, 5(1), pp. 65-80.
though not focusing on the properties of the ...
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1
vote
Fair and efficient allocation of "family goods"
Suppose the preferences of all agents in all families are monotone and convex (the standard assumptions of consumer theory).
Then, a Pareto-efficient envy-free allocation always exists when there ...
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