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 ...
Michael Greinecker's user avatar
5 votes
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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 ...
BakerStreet's user avatar
  • 3,697
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, ...
Amit's user avatar
  • 8,466
4 votes
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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 ...
Amit's user avatar
  • 8,466
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 ...
Amit's user avatar
  • 8,466
4 votes
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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 $\text{MRS}^{A}_{v,w} = \text{MRS}^{B}_{v,w} = \text{MRS}^{C}_{v,w}$ ...
Ricardo's user avatar
  • 56
4 votes

Does the contract curve always have to connect the initial points on an edgeworth box? Why or why not?

The contract curve is the locus of Pareto optimal points in an Edgeworth box. What we get from that: To be P.O., an allocation must be feasible. So, the contract curve does not extend beyond the ...
123's user avatar
  • 2,911
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*}...
Amit's user avatar
  • 8,466
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 ...
Giskard's user avatar
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2 votes
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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 ...
Michael Greinecker's user avatar
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}^...
Ricardo's user avatar
  • 23
2 votes
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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 $...
Regio's user avatar
  • 4,188
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 ...
Nicolas Torres's user avatar
2 votes
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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,...
Amit's user avatar
  • 8,466
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}) & , \...
mynameparv's user avatar
2 votes
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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 ...
Giskard's user avatar
  • 29.5k
2 votes
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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 ...
Amit's user avatar
  • 8,466
2 votes
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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 $\...
Amit's user avatar
  • 8,466
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}...
Amit's user avatar
  • 8,466
1 vote
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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 ...
VARulle's user avatar
  • 6,805
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 ...
Giskard's user avatar
  • 29.5k
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 ...
mynameparv's user avatar
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 ...
Jovan Jezdic's user avatar
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 ...
Michael Greinecker's user avatar
1 vote
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 red curve. Just do the slope of ICs comparisons at the boundaries and you will get that.
Amit's user avatar
  • 8,466
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 ...
Amit's user avatar
  • 8,466
1 vote
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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 $(...
Art's user avatar
  • 2,784
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 ...
Erel Segal-Halevi's user avatar

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