It is your second answer. From MWG: $$\frac{\partial u(x^*)/\partial x_\ell}{\partial u(x^*)/\partial_k}=\frac{p_\ell}{p_k}\tag{3.D.5}$$ The expression on the left of $\text{(3.D.5)}$ is the ...

This example comes from Miller (1980) (the paper that introduced the uncovered set definition). Suppose that the majority's preference is defined by: \begin{aligned} x \succ y \\ x \succ z \\ y \...

I don't know what is the definition of the "law of DMRS", but I think what the author meant is summarized in these lines of MWG: Proposition 3.G.2: suppose that $u(\cdot)$ is a continuous ...

There is a proposition that says: If there is a function $u: X \rightarrow \mathbb{R}$ that represents the preference relation $\succsim$, then $\succsim$ is rational. So, the contrapositive form ...

First of all, an utility function $u: X \rightarrow \mathbb{R}$ represents the preference relation $\succsim$ if: $$\forall a, b \in X, \; u(a) \geq u(b) \iff a \succsim b.$$ Well, if another ...

The definition of elasticity of demand with respect to price is: $\varepsilon_{q,p} = \frac{dq}{dp} \cdot \frac{p}{q}$. So in your demand function we have: $$q = kp^{-\epsilon}$$ \frac{dq}{dp} = -\...

About your second question, Gérard Debreu, in his book Theory of Value, says in the notes of chapter 1 that the $\succsim$ notation is due to Herstein and Milnor. So, probably, this is the origin of ...

Proof of "the SPNE of a sequential game might not necessarily be Pareto Optimal"? I don't get it, your example is a proof of this statement. So what else do you need? If you need another example, ...

As afreelunch only answered in the comment section, I'll answer it here. The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call $S$ the set of all ...

Sketch your answer for the demand function $x = -p_x/p_y$.