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\begin_body

\begin_layout Title
Math Review #3 Supplement (Math Camp 2007)
\end_layout

\begin_layout Author
John Morrow
\end_layout

\begin_layout Date
8/26/07
\end_layout

\begin_layout Standard
What follows is a rough sketch of material for Day 3.
\end_layout

\begin_layout Section
Day 3
\end_layout

\begin_layout Enumerate
Optimization Primer
\end_layout

\begin_deeper
\begin_layout Enumerate
Local/Global Maxima and Minima.
\end_layout

\begin_layout Enumerate
Uniqueness of maxima/minima for strictly concave/convex functions (Mention
 Multivariate Case).
\end_layout

\begin_layout Enumerate
Necessity of critical points for maxima/minima.
\end_layout

\begin_layout Enumerate
Critical points sufficient for maxima/minima under concavity/convexity.
\end_layout

\begin_layout Enumerate
Basic Checklist for simple problems.
\end_layout

\begin_deeper
\begin_layout Enumerate
Check continuity.
\end_layout

\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset

Compactify
\begin_inset Quotes erd
\end_inset

 Problem.
\end_layout

\begin_layout Enumerate
Check for concavity/convexity.
\end_layout

\begin_layout Enumerate
Find critical points.
\end_layout

\begin_layout Enumerate
Be creative and use your intuition!
\end_layout

\end_deeper
\begin_layout Enumerate
Examples: Monopolist Profit Maximization
\end_layout

\begin_deeper
\begin_layout Enumerate
Linear Demand, constant costs:
\begin_inset Formula \begin{alignat*}{3}
P(q) & \equiv a-bq & \; & \; & C(q) & \equiv c\end{alignat*}

\end_inset


\end_layout

\begin_layout Enumerate
Linear Demand, constant marginal costs:
\begin_inset Formula \begin{alignat*}{3}
P(q) & \equiv a-bq & \; & \; & C(q) & \equiv cq\end{alignat*}

\end_inset


\end_layout

\begin_layout Enumerate
Fancy Demand, constant marginal costs:
\begin_inset Formula \begin{alignat*}{3}
P(q) & \equiv a-bq^{\alpha} & \; & \; & C(q) & \equiv cq\end{alignat*}

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
Examples: Cost minimization
\end_layout

\begin_deeper
\begin_layout Enumerate
Need to produce 
\begin_inset Formula $q$
\end_inset

 units, have two technologies:
\begin_inset Formula \begin{alignat*}{3}
c_{1}(q) & \equiv cq & \; & \; & c_{2}(q) & \equiv q^{\frac{1}{2}}+k\end{alignat*}

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Enumerate
Some Miscellaneous Items
\end_layout

\begin_deeper
\begin_layout Enumerate
Exp and Log properties, change of base.
\end_layout

\begin_layout Enumerate
Elasticity, 
\begin_inset Formula $\mathcal{E}_{q,p}$
\end_inset

 def, as %change and as log deriv.
\end_layout

\begin_layout Enumerate
Homogeneous functions.
\end_layout

\end_deeper
\begin_layout Enumerate
More Derivatives
\end_layout

\begin_deeper
\begin_layout Enumerate
Partial derivative notations, 
\begin_inset Formula $dx$
\end_inset

 and 
\begin_inset Formula $Df(x)$
\end_inset

.
\end_layout

\begin_layout Enumerate
Derivative as a linear operator
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\begin_inset Formula \begin{alignat*}{3}
f(x) & =\sum_{i=1}^{N}i\cdot x^{i} & \; & \; & f(x) & =\sum_{i=1}^{N}e^{i\beta x}\end{alignat*}

\end_inset


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\end_layout

\begin_layout Enumerate
Partial derivatives 
\begin_inset Formula \begin{alignat*}{3}
f(x,y) & =x^{\alpha}y^{\beta} & \; & \; & f(x_{1},\ldots,x_{N}) & =(\sum\alpha_{i}x_{i}^{\rho})^{\frac{1}{\rho}}\end{alignat*}

\end_inset


\end_layout

\begin_layout Enumerate
Definition of derivative for 
\begin_inset Formula $\mathbb{R}^{N}\rightarrow\mathbb{R}$
\end_inset

, how it looks, Hessian, 
\begin_inset Quotes eld
\end_inset

Young's Theorem
\begin_inset Quotes erd
\end_inset

.
\end_layout

\begin_layout Enumerate
Hessian and Strict Concavity/Convexity.
\end_layout

\begin_layout Enumerate
Sufficient conditions for multivariate maxima.
\end_layout

\end_deeper
\begin_layout Enumerate
More Linear algebra (doubt we'll make it this far today).
\end_layout

\begin_deeper
\begin_layout Enumerate
Dot product, vector 
\begin_inset Quotes eld
\end_inset

multiplication
\begin_inset Quotes erd
\end_inset

 and distance.
\end_layout

\begin_layout Enumerate
Solving 
\begin_inset Formula $Ax=b$
\end_inset

, inverses.
\end_layout

\begin_layout Enumerate
Eigenvectors/Eigenvalues and Definiteness.
\end_layout

\end_body
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