5

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 example of multiple equilibria in a $2\times 2$-exchange economy can be found in Shapley, L. S., and M. Shubik. “An Example of a Trading Economy with Three ...


4

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 edges of the box (opposed to indifference curves, which we can draw as extending beyond the edges of the box) because points outside the Edgeworth box are not ...


3

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, y_B), \ \omega_B = (0, 1) \end{eqnarray*} In this case, every feasible allocation $((x_A, y_A), (x_B, y_B))$ satisfying $y_A = x_A$ is a competitive equilibrium,...


3

I would suggest that ask yourself the following questions (hopefully, this should help you figure out how to solve the problem) : If good $z \in (x,y)$ was free, what would be the demand for both agents? Is the conjunction of these demands feasible given the endowments? (this should allow you to rule out one of the cases) If the demands are feasible, which ...


3

No you are not. Preferences determine the equilibrium, even if they are identical, because they determine the value of the endowments. Consider two agents, one ("A") having ice cream only, and one ("B") having lava only. Case 1: Both hate lava Assume lava burns tongues and is useless (typical Economist). Then, the initial endowments of B have zero value, ...


2

The preferences of agent $A$ cannot be represented by any utility function and the prefeences of $B$ not by a differentiable utility function, so forget calculus approaches. Since $A$ has strictly monotone preferences, we must have $p_1>0$ and $p_2>0$ for every equilibrium. Also, $A$ is always willing to give up any amount of good $2$ to get more of ...


2

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*} where all $a_i$s are positive for all $i\in\{1,2,\ldots, n_u\}$, and utility function of member $j$ of family V is: \begin{eqnarray*} v_j(x_v, y_v) = b_jx_v +...


2

Right now I'm not sure about the equivalence of the relabeling, and therefore the usefulness of this anwer -- see comments below. This is the beginning of an answer and an attempt to demonstrate how strong the necessary assumptions would have to be to guarantee existence. Let's transform the problem into one that's equivalent but a bit easier to work with. ...


1

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 $(x, y)$ and you're asked whether you'd like to get another bundle, $(x', y')$ instead, you only check if your utility is higher. If it is, then you'd take it. ...


1

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 $u_1$ in the $y$ coordinate and $u_2$ in the $x$ coordinate. The easiest way to find the UPF is to start with a single agent, say agent 1 and give all the ...


1

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 are two families. However, it might not exist when there ar three or more families. Proofs and examples can be found in this working paper.


1

The goal of import quotas is to get people to buy domestic goods, which can only be bought for local currency. Therefore import quotas increase demand for local currency. But... Setting import quotas is a very undesirable economic policy. It is basically a state intrusion into a market economy. Quotas for imports are usually implemented as a desperate ...


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