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\begin{document}
\title{Final Exam}
\author[1]{Ian Anderson}%
\affil[1]{George Mason University}%
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\date{\today}
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\textbf{1.a. Find the general equilibrium price ratio $px/py$ for a two-agent $(i = 1, 2)$, two commodity
economy with the following characteristics:
Agents' utility functions are given by:
$u1 = A_1x^αy^β
u2 = A_2x^ay^b$
* Agents' initial incomes are given by:
$m_i = p_xx^e_i +p_yy_i^e$
where $x^e_i$ and $y^e_i$ are respectively an agent's initial endowment of $x$ and $y$ and $p_x$
and $p_y$ are the equilibrium prices of $x$ and $y$ respectively.}
Our goal is to maximize utility, derive demand curves and make sure there's no excess demand
-Everyone has to be maximizing utility
-There is no excess demand (all demand $=0$) which we get by a summation of all goods in the economy (for this problem, $x^*_1+x^*_2=$ sum of initial endowments, which are written as $x^e_1$ and $x^e_1$)
-There's only one line in the edger box that goes through the tangency point and the initial endowment. That's the line we want.
Agent 1 utility: $u_1=x_1^\alpha * y_1^\beta$
Initial Endowment (budget constraint, same thing): $m_1=p_xx^e_1+p_yy^e_1$
Agent 2 utility $u_2=x_2^a \cdot y_2^a$ (him giving us the parameters as Greek letters purportedly makes this easier for us)
Initial Endowment $m_2 = p_xx^e_2+p_yy^e_2$
Derive $D_x^1$ (the demand curve for Agent 1 and good x)
Max $u_1$ s.t. $m_1=p_xx_1^*+p_y y_1^*$
Max $L = x_1^\alpha y_1^\beta - \lambda (p_x x_1 + p_y y_1 - m_1)$
See photo here for partials wrt x and y = 0
and then we rewrite the budget constraint: $m_1=p_xx^e_1+p_yy^e_1$
Then we combine the partials to get $x_1$ in terms of $y_1$ or vice versa and then you plug it back in to find the demand curve for the other one.
$\lambda =\frac {\alpha x_1^{\alpha-1}y_1^\beta}{p_x} = \frac {\beta x_1^{\alpha}y_1^{\beta-1}}{p_y}$ so set those second two equal to each other.
Equations cancel when you divide by $x^{\alpha-1}$ to cancel that term on the left and leave $x_1^1$ on the right. Do the same with $y_1$ and multiply both sides by $p_x$ and that leaves us with $y_1=\tfrac{\beta p_x x_1}{\alpha p_y}$ which we plug into the budget constraint:
$m_1=p_xx_1+p_yy_1$
$m_1= p_xx_1+p_y \cdot \tfrac{\beta p_x x_1}{\alpha p_y} = p_xx_1+ \tfrac{\beta p_x x_1}{\alpha} = p_x x_1 (1+{\beta / \alpha})$
$x_1^* = m_1 / p_x \cdot \tfrac{\alpha}{\alpha+\beta}$ AND THEREFORE $x_2^* = m_2/p_x \cdot \tfrac{a}{a+b}$
Since $y_1^*$ has an identical functional form we know $y_1^*=\frac{m_1}{p_y}\cdot\frac{\beta}{\alpha+\beta}$
These equations are Agent [1 or 2]'s demand for [x or y]
Now we find the price ratio:
Set $x_1^*+x_2^*=x_1^e+x_2^e$
$m_1 / p_x \cdot \tfrac{\alpha}{\alpha+\beta} + m_2/p_x \cdot \tfrac{a}{a+b} = x_1^e+x_2^e$
Now we plug back in the values for $m_1$ and $m_2$, which are in a photo but take the form $m_1=p_xx_1^e + p_yy_1^e$ etc etc and then you do algebra to try to get either $p_y/p_x$ or $p_x/p_y$. We start with factoring out the fraction to get the third picture's bottom row of math mess
Final fraction is also in a pic
\textbf{1.b. Find the formula for the effect of a change in $x^e_1$: $\tfrac{\partial (p_x/p_y)}{\partial x^e_1}$? What is the economic interpretation of this formula?}
This is marginal change in the price ratio as a result of a marginal change in the initial endowment of good X to agent 1.
\selectlanguage{greek}\textbf{1.c. What is $p_x/p_y$ when $x^e_1 = 9$, $x^e_2 = 5$, $y^e_1 = 37$, $y^e_2 = 55$, $α = .5$, $β = .5$, $a = .6$, $b = .4$ ?}\selectlanguage{english}
\textbf{2. Let $S$ and $D$ be the supply and demand functions that describe a market for good $x$, where $D(p,t)=-.8(p+t)$ and $S(p) = .9p$, where $p$ is own-price and $t$ is a per-unit tax. Derive the formula for $\frac{dp^*}{dt^*}$}
Euler function yields $D'_t \cdot \frac{dt}{dt}+D'_p \cdot \frac{dp}{dt}=S'_p \cdot \frac{dp}{dt}$
Do algebra to get $\frac{dp}{dt}=\frac{D'_t}{S'_p-D'_p}$ and then plug in values
\textbf{3. Prove Product Exhaustion Theorem for $Y=3K^{.4}L^{.6}$}
min $p_1x_1+p_2x_2 $ s.t. $ \bar y=x_1^\alpha x_2^\beta$
Lagrangian lets us write Min $p_1x_1+p_2x_2 - \lambda(x_1^\alpha x_2^\beta-\bar y)$
FOCs:
$p_1 - \lambda \alpha x_1^{\alpha-1} x_2^\beta =0$
$p_2 - \lambda \beta x_1^{\alpha} x_2^{\beta-1} =0 $
$\bar y = x_1^\alpha x_2^\beta$
Since $f_1'=\alpha x_1^{\alpha-1} x_2^\beta$ we can rewrite our equations as $\lambda f'_i=p_i$
We multiply by $x_i$ to get $\lambda f'_ix_i=p_ix_i$ and then using the properties of equations we combine the values for 1 and 2 (capital and labor, respectively)
$\lambda f_1x_1+\lambda f_2x_2=p_1x_1+p_2x_2$
Under competitive conditions, lambda (change in cost when production increases) is equal to q, the price of the commodity produced. We use Euler's theorem: $f(x_1,x_2)=f'_1x_1+f'_2x_2$ to give us the final equation
$qf(x_1,x_2)=p_1x_1+p_2x_2$
\textbf{Set up and solve the inter-temporal optimization problem for an individual with the utility function $u_{1,2}=lnc_1+\frac{1}{1+\rho}lnc_2$ knowing that her wage is $w_1$ in $t_1$ and $w_2$ in $t_2$ and that she faces interest rate $r$}
We set the utility function as $U = \ln{c_1} + \frac{1}{1+\delta} \ln{c_2}$, with $c_1, c_2 > 0$ and $\delta$ representing the discount rate. $c_2$'s relationship to income is as follows:
$c_2 = y_2 - (1+r)(c_1-y_1)$, where if $c_1$ is greater than $y_1$, they're borrowing, and if $y_1$ is less than $c_1$, they're lending. Either way, the interest rate $r$ is applied to the remainder as interest on their loan or savings. We sub the $c_2$ equation into the utility equation to get
$U = \ln {c_1} + \frac{1}{1+\delta} * \ln{y_2-(1+r)(c_1-y_1)}$
Differentiating to find the maximization point via the first order condition, we get
$\frac{1}{c_1} - \frac{1+r}{1+ \delta} * \frac{1}{y-(1+r)(c_1-y_1)} * (r-1) = 0$
which we rewrite so that it's a single fraction by making the denominators equal:
$\frac{[1+\delta]*(y_2-(1+r)(c_1-y_1)) - (1+r)c_1)}{c_1(1+\delta)(y_2-(1+r)(c_1-y_1)) }= 0$
Several values cancel out, leaving us with our function for consumption in the first period as a function of interest, discount rate and income:
$c_1^* = \frac{(1+\delta)[(1+r)y_1 + y_2]}{(2+\delta)(1+r)}$
This equation also allows us to determine our Polonius' Point by setting $(1+\delta)y_2 = (1+r)y_1$
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