Wednesday, June 10, 2015

An Introduction to Computable General Equilibrium Models via The stylised Johansen (SJ) Model

A general equilibrium models represents a model of an economy where all agents are in equilibrium. It’s an analytical approach which looks at the economy as a complete system of interdependent components (industries, households, investors, governments, importers and exporters). Further, It explicitly recognizes that economic shocks impacting on any one component can have repercussions throughout the system. While the word “Computable” means a system that can be solved numerically. 

In this Stylised Johansen – theoretical structure (The SJ model) it represents a closed economy in which there are two sector, one household and two factor (labor and capital). The household tries to maximize their utility (we assumed a Cobb Douglas utility function) under his given budget. They receive income for selling the factors of Production (FOP) labor and capital to industries. We also assume that there is no tax system or absence of government and household will spend all the income in consumption of goods produced by industries. Since it’s a closed economy they won’t either export or import. The producers tries to minimize his cost under his given production function (represented by CD function). In this economy prices clear markets for goods and factors. And the economy holds a Zero profit condition which leads to free entry and exist of firm condition.

Here is a simple 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.

I am sure now you are comfortable with the IO table and its representation. Here, in above table we drop the HH income implementing the walrus law. Walras noted the mathematically equivalent proposition that when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium (wikipedia.org). 

In the reality you are given a IO table more specifically know as social accounting matrix (SAM). This is given in value, the quantities and prices are not separated. Hence, in the reality you have to assume price as 1 with equivalent with a commodity known as numeraie. So everything will be expressed in term of that. So, there is need to express everything in changes therefor we linearize the equations and expressed everything in change.

I want to express gratitude to Dr. Hom Murti Pant, for his explanation below. Dr. Murti is is a senior economist with Australian Bureau of Agricultural and Resource Economics–Bureau of Rural Sciences, specializing in international trade and climate change policy.

If we study the lineaized solution of consumer's problem ${{x}_{i0}}=y-{{p}_{i}}$
we see
– All household expenditure elasticities = 1
– All own price elasticities = –1
– All cross price elasticities = 0

However, this is not very realistic. Eg. Expenditure elasticities for food are usually <1, while clothing and consumer durable are usually >1. However, we can assume other function like CDE; LES, etc.

If we study the lineaized solution of Producer;s problem ${{x}_{ij}}={{x}_{j}}+\sum\limits_{i=1}^{4}{{{\alpha }_{ij}}{{p}_{i}}}-{{p}_{i}}$, we can see,
– In the absence of changes in relative prices, industry j will change the volumes of all its inputs by the same percentage as its output 
– If increase in the price of input i > average increase in the input prices, then industry j will substitute away from input i. 
${{x}_{ij}}={{x}_{j}}+\sigma \left( \sum\limits_{i=1}^{4}{{{\alpha }_{ij}}{{p}_{i}}}-{{p}_{i}} \right)$ , where $\sigma =1$.
So, in other words our price-substitution term has an elasticity of 1. Ideally, we should adopt more general production functions so that the substitution terms vary according to input substitution possibilities in different industries. For example CES, CRESH production functions yield such substitution possibilities.

If we examine price formation ${{p}_{i}}=\sum\limits_{i=1}^{4}{{{\alpha }_{ij}}{{p}_{i}}}$ then
– Change in the price of of good j is a weighted average of the changes in input prices, the weights being cost shares.

If we examine the market clearing conditions ${{x}_{i}}=\sum\limits_{j=0}^{2}{{{\beta }_{ij}}{{x}_{ij}}}$ and  ${{x}_{i}}=\sum\limits_{j=1}^{2}{{{\beta }_{ij}}{{x}_{ij}}}$
– Change in the supply of commodity i is a weighted average of the percentage change in various demands for i, the weights being sales shares. – Similarly, the change in employment of factor i is a weighted average of the changes in industrial demands for i contributed by each industry.

I hope now we are ready to do some programmings in GEMPACK. See you in next blog.

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