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3

The answer is no both to the if and to the only if parts. If counterexample: For $u_{1,2} = \ln x + y$ and aggregate quantities (4,4) it is easy to check that the allocation (1,1),(3,3) is not Pareto-optimal. Only if counterexample: For $u_{1} = 2x + y$ and $u_{2} = \min(x,y)$ and aggregate quantities (4,4) the contract curve is the diagonal.

3

There seems to be some confusion in the expression for $x^*_i$ in the question that whether $i$ is for consumer of for the good. Assuming $i$ is for consumer: Let $x^*_i = (x_1^i,x_2^i)'$ be the equilibrium bundle for consumer $i$. Since utility function is same for both, from MRS we have: \begin{align} \frac{x_1^i}{x_2^i}=\bigg(\frac{p_1}{p_2}\bigg)^{s-1} \...

2

This answer assumes you understand the defining property of homothetic functions, namely, slope of their level curves being equal for a given input ratio. First let's understand what is the diagonal; it is a line representing endowment ratio $X_1:X_2$ in the economy. Contract curve being linear is equivalent to it lying on the diagonal, since contract curves ...

1

In the picture below offer curve of individual 1 is given by lines connecting $E$ to $O_1$ and $O_1$ to $O_2$. And offer curve of individual 2 is given by lines connecting $E$ to $O_2$ and $O_2$ to $A$ and $A$ to $O_1$. Set of competitive equilibria are given by the intersection of two offer curves. $p_X = 0$ and $p_Y = 1$ supports allocation at $O_2$ as ...

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