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\begin{document}
\title{Lecture 12 Notes (math econ)}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
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\section*{Total Effects}
{\label{168270}}
Total effect of a price change = substitution effect + income effect
\begin{itemize}
\tightlist
\item
Substitution effect: the result of a change in the relative prices of
two goods (i.e. if wages go up, leisure becomes more ``expensive'' via
opportunity cost, you will choose more labor)
\item
Income effect: the result of a change in the consumer's purchasing
power (i.e. if wages go up, you have more income and can purchase more
with the same hours of labor)
\end{itemize}
This is what is captured in the Slutsky equation for a change in the
demand for good~\(x_i\) in response to a change in price of
good \(x_j\):
\begin{quote}
\(\frac{\partial x_i (p, w)}{\partial p_j} = \frac{\partial h_i(p, u)}{\partial p_j} - \frac{\partial x_i (p, w)}{\partial w} d_j (p, w)\)
\end{quote}
where ~\(h(p, u)\)is the Hicksian demand (holding utility
constant) and~\(x(p, w)\) is the Marshallian demand (which holds
income constant), at the vector price levels~\(p\), wealth
(or income) level~\(w\), with a fixed utility
level~~\(u \) given by the original maximized utility
function subject to the budget constraint.~
Alternately, the Frisch elasticity of labor supply holds the marginal
utility of wealth constant. This is accomplished by capturing the
elasticity of hours worked to the wage rate :
\begin{quote}
\(\frac{\% \Delta L_S}{\% \Delta w}\)
\end{quote}
and then keeping a constant marginal utility of wealth.
\emph{* Fun aside: Keynes (in Economic Possibilities for our
Grandchildren) predicted that our living standards would quadruple (yes)
and that we'd work 15hrs/wk. (ha ha.. no). What can economic theory tell
us about this fact? Answer: Don't forget that while we work the same
amount of hours per week, we also created childhood and retirement, so
less lifetime hours potentially. Otherwise, this tells us that the
substitution effect must be much stronger than the income effect in the
elasticity of labor supply.~}
\section*{Log Utility}
{\label{772037}}
To find a neoclassical model where labor supply \textbf{doesn't} depend
on productivity, what do I need to believe?
Consider:
\begin{quote}
\(u = \frac{z L ^{1 - \theta}}{1 - \theta} - \gamma L\)
\end{quote}
where~\(z\) is a measure of
productivity,~\(\theta\) is intertemporal time preference,
and~\(\gamma\)~represents the disutility of labor. First-order
conditions yield:
\begin{quote}
\(L^* = \gamma ^{- \frac{1}{\theta}} z ^{\frac{1}{\theta} - 1}\)
\end{quote}
as we can see above, when~\(\theta = 1\), productivity has no
impact on the labor supply. In other words, the income and substitution
effects will cancel out. Another way to see this:
\begin{quote}
\(u = \ln z L - \gamma L\)
\(\frac{\partial u}{\partial L} = \frac{1}{zL} \cdot z - \gamma\)
\end{quote}
The first fraction represents the substitution effect and the second
``z'' is the income effect, where both of these will be eliminated.~~
\section*{Cases other than Log
Utility}
{\label{505748}}
Keynes's prediction meant that income effect would win. What does this
look like? Begin again with:
\begin{quote}
\(L^* = \gamma ^{- \frac{1}{\theta}} z ^{\frac{1}{\theta} - 1}\)
\(\ln L^* = \frac{1}{\theta} \ln \gamma + (\frac{1}{\theta} - 1) \ln z\)
\end{quote}
Therefore, if~\(\theta > 1\), a rise in z means the income effect
wins and there are mass vacations (Keynes was right). However, if
\(\theta < 1\), the substitution effect wins and people will work
more. This is also called quasi-linear utility.~When
considering~\(\theta\), the real question to ask is,~\emph{how
strong is diminishing marginal utility of consumption?}
\section*{CES and near-Leontief
Behavior}
{\label{245205}}
Now we will examine ``z'' as a share parameter, that will sometimes mean
a ``lifting tide raises all boats,'' and sometimes be factor-specific.
This will depend on the type of production function that we use. If we
want to use Cobb-Douglas, our solution will be the latter:
\begin{quote}
\(u = zK^\alpha L^{1 - \alpha}\)
\(\ln u = \ln z + \alpha \ln K + (1 - \alpha) \ln L\)
\end{quote}
Thus the productivity share parameter extends to all factors, and this
is a case of CES production functions where each input is a perfect
complement. However, looking at a case of constant elasticity of
substitution of the Leontif type:
\begin{quote}
\(u = (zK)^\alpha + L^{1 - \alpha}\)~
\end{quote}
Here, z represents factor-biased technological change. For example, we
would see this when machines are increasingly getting more productive
though labor remains at a stable level.
In general, if you have
\begin{itemize}
\tightlist
\item
\(\alpha = 1\): perfect substitutes, or a linear production
function
\item
\(\alpha = 0\): Cobb-Douglas production function
\item
\(\alpha = -1\): perfect complements, or a~modest Leontief
production function
\end{itemize}
\section*{Corner Solutions \& Dixit}
{\label{639986}}
When should you spend all your money on one good?~ The evidence of this
would be a corner solution. Our answer relies on the equimarginal
principle:~
\emph{If marginal utility per dollar for good X is less than marginal
utility per dollar for good Y (regardless of the amount of good Y
already bought), spend all your money on good Y. Or,~}
\begin{quote}
\emph{\(\frac{MU_X}{\$} > \frac{MU_Y}{\$} \Longrightarrow \$ \rightarrow Y\)}
\end{quote}
Again, this is a classic corner solution. Don't buy any of a certain
good if the marginal utility of that good is always lower than the rest.
Example:~
How high does ``z'' have to be for me to spend all my income on good 2?
\begin{quote}
max~\(u = c_1 + z \sqrt c_2\)
subject to~\(c_1 + c_2 = 100\)
\(\frac{MU_1}{\$} = 1\)
\(\frac{MU_2}{\$} = \frac{1}{2} z c_2 ^{- \frac{1}{2}}\)
\end{quote}
Recall that the marginal utility of a good is the partial derivative
with respect to that good of the total utility function. Optimizing will
always mean setting these marginal utilities equal to one another, and
then the question gives us the assumption that all of income is spent on
good 2:
\begin{quote}
\(1 = \frac{1}{2} z 100 ^{- \frac{1}{2}}\)
\(z = 20\)
\end{quote}
Calculus are the wrong way to solve a corner solution. If we would have
workout out the Lagrangian, the solution would have been:
\begin{quote}
\(c_1 = -300 , c_2 = 400\)
\end{quote}
This tells us that we should be buying good 1 and turning it into good
2. In other words, it's a clue that our marginal utility will never be
high enough for good 1.~
\section*{Intertemporal Choice (in discrete
time)}
{\label{501105}}
We can treat consumption now vs. consumption later as 2 different goods.
Typically, intertemporal choice problems will involve adding up these
separable utility functions. Mathematically:
\begin{quote}
\(u(c_1, c_2) = u^A (c_1) + u^B (c_2)\)
\end{quote}
Here, the various goods have no dependence on each other. A
counter-example to this is habit formation, where past decisions impact
present utility. Now, consider the most popular general case of
intertemporal choice:
\begin{quote}
max~\(u = \frac{c_1 ^{1 - \theta}}{1 - \theta} + (\frac{1}{1 + \rho}) \frac{c_2 ^{1 - \theta}}{1 - \theta}\)
subject to~\(c_2 = (y - c_1)(1 + r)\)
\end{quote}
Recall that~\(\rho\) is the rate of time preference (or
valuation place on having a good at an earlier rather than later
date),~\(r\) signifies the real interest rate,
and~\(\frac{1}{\theta}\) is the elasticity of intertemporal substitution
(or measure of responsiveness of the growth rate of consumption to the
real interest rate). The constraint reads: second period consumption is
equal to the present income, minus present consumption, multiplied by
one plus the real interest rate.~~
Substituting in the constraint and solving for the optimal proportion of
present to future consumption will yield
\begin{quote}
\(u = \frac{c_1 ^{1 - \theta}}{1 - \theta} + (\frac{1}{1 + \rho}) [\frac{(y - c_1)(1 + r)}{1 - \theta}]^{1 - \theta}\)
\(\frac{\partial u}{\partial c_1} = c_1 ^{- \theta} + \frac{1}{1 - \rho} c_2 ^{- \theta} (-1)(1 + r) = 0\)
\((\frac{c_2}{c_1})^* = (\frac{1 + r}{1 + \rho})^{\frac{1}{\theta}}\)
\end{quote}
Exponents are elasticities! See why~\(\frac{1}{\theta}\) is the
elasticity of intertemporal substitution. This final equation states
that the optimal proportion of future consumption to present consumption
depends on those three parameters. For example, if~\(\frac{1}{\theta} = \frac{1}{10}\),
then your consumption is hugely responsive to interest rates (will be
pushed to the future if the interest rate is relatively high).~
Finally, a real-world example. If our economy grows at about 3\%, and
consumption is two-thirds of GDP growth, then it grows at 2\%. What is
the discount rate if we know the interest rate is 4\% and assume
log-utility?
\begin{quote}
\(\ln 1.02 = \ln 1.04 - \ln (1 + x) \)
\(x = 0.02 \Longrightarrow \rho = 2 \%\)
\end{quote}
There are two reasons for present consumption to be equal to future
consumption:
\begin{enumerate}
\tightlist
\item
\(r = \rho\)~: the effects will cancel out
\item
\(\theta \rightarrow \infty\) (or even 4, 5): you demand perfect equality of
consumption across time, without regard to the interest or discount
rates. Mathematically, this makes their fraction matter infinitely
less.~
\end{enumerate}
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