经济代写|ECON1110 Microeconomics

Consider a WE $\left(\boldsymbol{p}, \boldsymbol{x}^\right)$ in which $\boldsymbol{x}^{ k i} \neq \boldsymbol{x}^{* k j}$ for $i$ th and $j$ th consumers of some type $k$. Let’s consider $\hat{\boldsymbol{x}}^k \equiv \frac{1}{2} \boldsymbol{x}^{* k i}+\frac{1}{2} \boldsymbol{x}^{* k j}$, where $\boldsymbol{p} \hat{\boldsymbol{x}}^k \leq \boldsymbol{p} \boldsymbol{e}^k$. Now, since the preferences are strictly convex, we must have either $\boldsymbol{p} \hat{\boldsymbol{x}}^k \succ^k \boldsymbol{x}^{* k i}$ or $\boldsymbol{p} \hat{\boldsymbol{x}}^k \succ^k \boldsymbol{x}^{* k j}$ (or both). This contradicts to utility maximization.

(a) Suppose that all pairs are matched. In this case, by exchanging between $R$ and $L$, mismatches are caused, and so mismatched consumers are worse off. When exchanging between $R$ and $R$ or between $L$ and $L$, consumers’ utilities remains unchanged. When giving some paired $R$ to a consumer who has only $L$, there must be a consumer who is worse off.

Conversely, if some pair is not matched, we can find superior allocation by exchanging mismatched $R$ and $L$.
(b) The allocation $\boldsymbol{x}$ is the core allocation if and only if each $R$ owner receives pair of shoes. (This allocation is Pareto efficient.)
Proof
[If each $R$ owner receives pair of shoes in the allocation $\boldsymbol{x}$, this allocation is the core allocation.]

Suppose that there exists coalition $S$ which blocks $\boldsymbol{x}$. Let $J$ be the number of $R$ owners in $S$. In order to block $\boldsymbol{x}, S$ must receive at least $J$ pairs and some consumer receives more. But this is not feasible.
[If $\boldsymbol{x}$ is the core allocation, each $R$ owner receives pair of shoes in $\boldsymbol{x}$.]
Suppose not. Then some $R$ owner receives no pair of shoes (zero utility). Since there exists $I+1 L$ owners, some $L$ owner receives no $R$ (zero utility). Consider coalition $S$ which consists of $R$ owner without $L$ and $L$ owner without $R$. Then, at least one of them can receive a pair of shoes, and so $S$ blocks $\boldsymbol{x}$.
(c)
[Claim: If $(\boldsymbol{x}, \boldsymbol{p})$ is WE, then each $R$ owner receives a pair of shoes at $\boldsymbol{x}$ and $\left(p_R, p_L\right)=\left(p_R, 0\right)$, where $\left.p_R>0.\right]$

Proof
Let $p_K$ be a price of $K$ and $x_K$ be a consumption of $K$ where $K=R, L$. Then, $R$ owners’ budget constraint becomes $p_R x_R+p_L x_L \leq p_R$, and that of $L$ owners’ is $p_R x_R+p_L x_L \leq p_L$.
First, suppose that $\left(p_R, p_L\right)=(0,0)$. Then, there are excess demands of shoes, and so the market cannot clear. Next, suppose that $\left(p_R, p_L\right) \gg(0,0)$. Then, each consumer can demand at most one shoe, and total demand becomes at most 2I. Thus, at least one shoe is left unsold, which contradicts to market clearing condition.

Now suppose that $p_R=0$ and $p_L>0$. Then, every $L$ owner demands $R$, but this cannot satisfy feasibility.
Thus, on the WE $(\boldsymbol{x}, \boldsymbol{p}), p_R>0$ and $p_L=0$ are satisfied, and each $R$ owner demands one $L$.
[Claim: If price vector is $\left(p_R, p_L\right)=\left(p_R, 0\right)$, where $p_R>0$, and every $R$ owner receives a pair of shoes, then this constitutes WE.]
Proof
With this price vector, no $L$ owner can demand shoes, and each $R$ owner demands at most one shoe. Then, each $R$ owner demands one $L$, and at equilibrium, every $R$ owner receives a pair of shoes.
(d) The core allocation and the WE allocation are identical.

(i) and (ii) is obvious. For (iii), consider the sequence converging to $\left(p_1, p_2\right)=(1,0)$. In this case, associated sequence of excess demand converges to $(-1, \infty)$, and so it is not unbounded above.
Since $z_1\left(p_1, p_2\right)=-1$ for any $\left(p_1, p_2\right) \in \mathbb{R}_{++}, \boldsymbol{z}\left(p_1, p_2\right) \neq \mathbf{0}$ always.
If we allow the free goods and define the Warlasian eqm by $\boldsymbol{z}\left(p_1^, p_2^\right) \leq \mathbf{0}$, then $\left(p_1^, p_2^\right)=\left(0, p_2 *\right)$ is the Warlasian eqm price.