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\begin{flushright}
Andrew Iannaccone\\
Econ 201C\\
Ostroy\\
4/16/08
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{\sc General Equilibrium - HW \#1}
\newcommand{\pd}[2]{\partial #1/\partial #2}
\newcommand{\pdf}[2]{\frac{\partial #1}{\partial #2}}
\begin{enumerate}
\item {\bf Gainers and Losers in Equilibrium}
\begin{enumerate}
Note that preferences are identical and homothetic,
so we may use a representative agent to find PTE's.
\item In each region, the PTE allocations are symmetric:
\begin{align*}
\omega_{A_1} &= \omega_{A_2} = (2,1) \\
\omega_{B_1} &= \omega_{B_2} = (1,2)
\end{align*}
To see this in region $A$, substitute $y=2z$. This yields symmetric
preferences and endowments, $u=xz/2$ and $\omega = (4,4)$,
so the RA sets equal prices: $(p_x,p_z)=(1,1)$. Equivalently,
\begin{align*}
{\bf p} = (p_x,p_y) = (1,2)
\end{align*}
With these prices, $A_1$ and $A_2$ have equal wealth, so each consumes
the half the aggregate endowment, obtaining $u=2$. The analysis is
identical for region B, but with ${\bf p}=(2,1)$.
\item Aggregate endowment is symmetric: $\omega = (6,6)$, so
${\bf p}=(1,1)$. Type 1 agents have twice as much wealth
as Type 2 agents, so they consume twice as much of the aggregate
endowment:
\begin{align*}
\text{Type 1: } c = (2,2); u = 4 \\
\text{Type 1: } c = (1,1); u = 1
\end{align*}
\item[(c-d)]
Any allocation that gives 1/4 of the total wealth to each agent results
in a PTE with each agent consuming (1.5,1.5) and utility $u=2.25$.
\item[(e)]
The allocation in (b) is Pareto Efficient, as is the allocation in (d).
Free trade does not guarantee a ``fair'' equilibrium,
nor one that maximizes aggregate utility.
\end{enumerate}
\newpage
\item {\bf Replica Invariance}
\begin{enumerate}
\item Any utility $u_1+u_2 \le 100$ is feasible. From the given
endowment, there are no mutually advantageous trades.
\item In $\mathcal{E}^2$ a more efficient allocation is possible:
\begin{align*}
\text{Type 1: }& c = (50,50) \\
\text{Type 2: }& c_1 = (100,0), \; c_2 = (0,100)
\end{align*}
The Type 2 agents now have increased utility, $u=100$,
and the Type 1 still have $u=50$ as
in the original endowment.
\item In a PTE, each agent maximizes utility subject to his wealth
constraint. If there is a PTE in $\mathcal{E}$, then the same prices
must yield a PTE in $\mathcal{E}^2$. \\
As we'll show in (e), the allocation in (b) is a PTE with ${\bf p}=1$.
Any PTE in $\mathcal{E}$ must therefore have the same prices. We
also know that a PTE in $\mathcal{E}$ must allocation (50,50) for each
agent, as this endowment is already Pareto Optimal.\\
This allocation, however, is not an equilibrium with ${\bf p}=1$,
because the Type 2 agents can increase utility by trading.
For example, $c=(49,51)$ yields $u=51$.
It follows that there is no PTE in $\mathcal{E}$.
\item Preferences for the Type 1 agent do not exhibit local non-satiation.
For example, at both (50,50) and (50,51), his utility is 50.
Preferences for the Type 2 agent are not convex. For example, both
(100,0) and (0,100) are strictly preferred to (50,50).
\item For $\mathcal{F}^k$, give (50,50) to each Type 1 agent, and (100,0)
to half the Type 2's, (0,100) to the other half. No agent can be made
better off without increasing the gross total $x+y$ of his consumption.
Thus, we cannot improve upon this allocation without increasing the
gross aggregate endowment, $\omega_x + \omega_y$. The allocation is
therefore efficient for all $k$.
\item With ${\bf p}=1$, each agent's bugdet constraint is $x+y=100$.
The Type 1 agents maximize utility at (50,50), while the Type
2 agents maximize at the boundaries (0,100) or (100,0).
In the allocation in (e), each agent consumes one of these utility
maximizing bundles. The allocation is therefore a PTE.
\end{enumerate}
\newpage
\item {\bf Demand Theory with Quasi-linear Utility}
\begin{enumerate}
\item Denote indirect utility as $V(p,w)$. Let any $w,w'$ be given
and denote the corresponding optimal non-money consumption as $x,x'$ respectively.
Since $x$ is optimal for $w$, we have
\begin{align*}
U(x,w-px) &\ge U(x',w-px') \\
&= v(x') + w - px' \\
&= U(x',w'-px') + (w-w')
\end{align*}
Comparing the first and last lines, we have
\begin{align*}
V(p,w) - w \ge V(p,w') - w',
\end{align*}
but our choice of $w$ and $w'$ was arbitrary, so the reverse
inequality also holds. Thus, for all $w,w'$, we have
\begin{align*}
V(p,w)-w = V(p,w')-w',
\end{align*}
So $\alpha(p) = V(p,w)-w$ is independent of $w$, as desired.
\item Denote the expenditure function as $C(p,U)$. From duality theory,
we know
\begin{align*}
U = V\big(p,C(p,U)\big)
\end{align*}
Applying the form from (a), this becomes
\begin{align*}
U = \alpha(p) + C(p,U),
\end{align*}
so expenditure is simply
\begin{align*}
C(p,U) = -\alpha(p) - U
\end{align*}
\item Given any $w, w'$, we show that we achieve the same utility
with either $x$ or $x'$:
\begin{align*}
U(x',w-px') &= v(x') + w-px' \\
&= v(x') + w'-px' + (w-w') \\
&= U(x',w') + w -w' \\
&= V(p,w') - w' + w \\
&= V(p,w)-w + w \qquad \text{ (from (a)) }\\
&= V(p,w) \\
&= U(x,w-px)
\end{align*}
Thus, $x'$ is optimal for wealth $w$, so optimal non-money
consumption is independent of wealth, as desired.
\item In equilibrium, we have
\begin{align*}
\pd{v}{x_i} = p_i
\end{align*}
By the envelope theorem, differentiating with respect to $p_i$ gives
\begin{align*}
\pdf{^2v}{x_i^2}\pdf{x_i}{p_i} = 1
\end{align*}
Assuming $v(x)$ is concave, it's second differential is negative,
so $\pd{x_i}{p_i}$ is also negative, as is own price elasticity.
\end{enumerate}
\newpage
\item {\bf Demand/Supply, Inverse Demand/Suppy and Indirect Utility/Profit}
\begin{enumerate}
\item
(1 $\Rightarrow$ 2): Suppose $(z,m) \in D(p)$. Then $pz+m=0$ and
\begin{align*}
v(z)+m \ge v(z')+m'
\end{align*}
for all $(m',z')$ with $pz'+m'=0$. So we have
\begin{align*}
v(z)-pz \ge v(z')-pz'
\end{align*}
for all $z' \in Z$. By definition, $p\in \partial v(z)$. \\
(2 $\Rightarrow$ 3): If $p \in \partial v(z)$, then for all
$z' \in Z$, we have
\begin{align*}
v(z)-pz \ge v(z')-pz'.
\end{align*}
That is,
\begin{align*}
v(z)-pz &= \text{max}{\{v(z')-pz'\}} \\
&= \text{max}{\{v(z')+m' \; | \; pz'+m'=0\}}
\end{align*}
That is, $v(z)-pz$ is the maximum utility feasible with zero total
wealth. This is the definition of the indirect utility
function $v^*(p)$.\\
(3 $\Rightarrow$ 1): By definition, $v^*(p)$ is the maximum utilty
acheivable with zero net wealth. That is,
\begin{align*}
v^*(p) = max{\{v(z')+m' \; | \; pz' + m=0 \}}.
\end{align*}
Since $v(z)-pz = v^*(p)$, this implies $(z,-pz)\in D(p)$.\\
\item The revenue from selling $z$ is $r=-pz$,
which we previously denoted $m$. The firm's profit is therefore
\begin{align*}
\pi(p,z) = -c(z)-pz = v(z)+r
\end{align*}
In this notation, the conditions above become
\begin{enumerate}
\item[(1)] $(z,r) \in \text{argmax}_{z,r} \{v(z)+r \; | \; pz+r=0 \}$
\item[(2)] $-p \in \partial c(z)$
\item[(3)] $\pi^*(p) = -c(z)-pz$
\end{enumerate}
\end{enumerate}
\end{enumerate}
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