Monday, June 22, 2015

Warm Up on Computable General Equilibrium Theory (Before We Start Coding)

Before we start a basic CGE, please be comfortable with:
1. Input Output Table in quantities and notation
2. Input Output Table in Dollar Values and notation
3. The namig of IO table (DVCOM, DVFAC, DVCOMIN, DVFACIN and HCONS)
3. The consumer's and producers problem, their optimum solution and linearized solution
4. Commodity and factor market clearing condition and their linearized solution
5. The concept of household income (We have assumed all the income will be spent in consumption).
6. Walrus Law
7. Why we set Price $P$ as $1$.
8. The properties of all the linearized solutions.

Here is the glimpse of problem, optimum solution and linearization.But prior to that one should understand the logic of Input. Output table. Here is the link (here).

Problems
Optimum Solutions
Linearized Solution
Consumer’s Problem
$\begin{align}
  & {{U}_{\max }}=\prod\limits_{i=1}^{2}{X_{i0}^{{{\alpha }_{i0}}}} \\
 & s.t. \\
 & Y=\sum\limits_{i=1}^{2}{{{P}_{i}}{{X}_{i0}}} \\
\end{align}$
Where, $\sum\limits_{i=1}^{2}{{{\alpha }_{i0}}}=1$ and ${{\alpha }_{i0}}>0$.
\[{{X}_{i0}}=\frac{{{\alpha }_{i0}}Y}{{{P}_{i}}}\]

\[{{x}_{i0}}=y-{{p}_{i}}\]


Here is the link for solution (here).
Producer’s Problem
$\begin{align}
  & \min {{C}_{j}}=\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{ij}}} \\
 & s.t. \\
 & {{X}_{j}}={{A}_{j}}\prod\limits_{i=1}^{4}{X_{ij}^{{{\alpha }_{ij}}}} \\
\end{align}$
for j=1 and 2 and $\sum\limits_{i=1}^{4}{{{\alpha }_{i1}}=1;}\sum\limits_{i=1}^{4}{{{\alpha }_{i2}}=1;}{{\alpha }_{ij}}>0$


\[\begin{align}
  & {{X}_{ij}}={{\alpha }_{ij}}{{X}_{j}}Q\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}}/{{P}_{i}} \\
 & where, \\
 & Q=\left( \frac{1}{{{A}_{j}}}\prod\limits_{i=1}^{4}{\alpha _{ij}^{-{{\alpha }_{ij}}}} \right) \\
\end{align}\]
Here is the link for solution (here).
\[{{x}_{ij}}={{x}_{j}}+\sum\limits_{i=1}^{4}{{{\alpha }_{ij}}{{p}_{i}}}-{{p}_{i}}\]


Here is the link for solution (here).
Price Formation

\[{{P}_{i}}=\lambda =Q\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}}\]
${{p}_{i}}=\sum\limits_{i=1}^{4}{{{\alpha }_{ij}}{{p}_{i}}}$

Here is the link for solution (here).
Commodity Market Clearance

$\sum\limits_{j=0}^{2}{{{X}_{ij}}={{X}_{i}}}$
for i=1 and 2
${{x}_{i}}=\sum\limits_{j=0}^{2}{{{\beta }_{ij}}{{x}_{ij}}}$

Here is the link for solution (here).
Factor Market Clearance

$\sum\limits_{j=1}^{2}{{{X}_{ij}}={{X}_{i}}}$
for i=2 and 3
${{x}_{i}}=\sum\limits_{j=1}^{2}{{{\beta }_{ij}}{{x}_{ij}}}$

Here is the link for solution (here).
The Household Income
$Y={{P}_{3}}{{X}_{3}}+{{P}_{4}}{{X}_{4}}$
Drop the equation using walrus law

Discussed below.
Price
${{P}_{1}}=1$
${{p}_{1}}=0$

Discussed below.


Here is a quick video to get you started:




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