# Another example

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"Another example"

```					          Another example
• Want to find equilibrium level of income &
interest rate for following 2 equations:
– IS: 0.3Y + 100i –252 =0
– LM: 0.25Y – 200i –176 =0
• Need to convert equations to reduced form
– 0.3Y +100i =252
– 0.25Y – 200i = 176

1
Define Matrices

 0.3 100 
A                Y 
X  
0.25  200      i 

252
B    
176 
2
Solution
• Det A = -85
• A-1B =  800 
0.12
    
• Hence Y = \$800 and i = 0.12 = 12%
• This the level of income at which the IS
curve cuts the LM curve

3
Input-Output Analysis
– This models the economy as a number of interrelated
industrial sectors
• Interrelationship => output from a sector can be used in the
same sector or as intermediate input in another sector or by
final users.
– Used by companies or Planning Authorities to
forecast impact of changing demand & use of inputs
• E.g. find the production levels for each industry which, given
a set of prices, are just sufficient to supply the total demands
from industry and consumers

4
I/O analysis
• Total demand x for product i will be sum of all
intermediate demand for product + the final
demand by all types of users
– final demand can be from:
• Consumers, investors, government or exporters
• For any I/O table total demand = Gross
production
– All production used as intermediate inputs or for final
demand
– can express in matrix form
– See following example with 4 production sectors
5
Example:an interindustry transaction
demand table or I/O table, in \$mills
Sector of Destination

Sector of    Steel   Coal       Iron         Auto    Final    Total
Origin                                               Demand   demand

Steel           80      20         110         230     160      600
Coal           200      50          90         120     140      600
Iron           220     110          30         40       0       400
Auto            60     140         160         240     400     1000
Value           40     280          10         370
Gross          600     600         400        1000
Production

6
Notes:

• Value added = labour, profits, rents
• Can be expressed in matrix form as:
– AX + B = X, where
•   A = matrix of technical coefficients
•   X = matrix of Total demand
•   B = matrix of final demand
•   What order do each of these matrices need to be for this
example?

7
I/O matrix notation

• A is the matrix of technical coefficients
– Value of input i required to produce \$1 worth of
product j
– 1st row in A is:
•   a11 = 80/600 = 0.133
•   a12 = 20/600 = 0.03
•   a13 = 110/400 = 0.275
•   a14 = 230/1000 = 0.23
– Continue to complete matrix A by dividing each
amount of input in each industry by the industry’s
gross production
• Hence this will yield a 4 x 4 matrix
8
Matrix A

0.133   0.033 0.275 0.23
0.333                   
0.083 0.225 0.12
A
0.367   0.183 0.075 0.04
                        
 0 .1   0.233 0.4 0.24
9
Other matrices in I/O
• Matrix B = matrix of final demand
– It will be 4 x 1
• Matrix X = Total demand = Gross
Production
– It will be 4x 1

160             600 
140             600 
B            X      
 0              400 
                    
1000
 400                       10
How to study changes in I/O
model
• To find level of output (intermediate & final)
needed to satisfy total demand, then solve for X
– We have X – AX = B
– => (I – A)X = B
– =>X = (I – A) -1B
• (I – A) is called the Leontief matrix
– This matrix takes into account all the direct and
indirect requirements for inputs to satisfy the given
level of total demand

11
– Note it is important to learn how to interpret
results and explain the values obtained.
• Chap 12.6 Dowling for examples on input –output
analysis

• Next topic 3: Intro to basic Statistical concepts &
hypothesis testing

12

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 views: 3 posted: 9/11/2012 language: English pages: 12