Ecn221 Economic Applications

7/28/2019 Ecn221 Economic Applications

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Lecture 3

Economic Applications of Linear Algebra

Lecture Outline:

Linear Economic ModelsoTerminology Behavioural Equations and Identities; Endogenous

and Exogenous variables.oStructural and Reduced Forms;oComparative Statics;oExamples Supply and Demand, IS-LM.


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Input-Output ModelsoInput-output in a nutshell.oSee Alan Beggs Lecture 4 if you want more detail.

Introduction to Asset PricingoLaw of One PriceoNo ArbitrageoState PricesoCompletenessoReplicating Portfolios and Pricing DerivativesoPricing a Simple Call OptionoSee Alan Beggs Lecture 6.


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1.Linear Economic ModelsConsider the following supply and demand model:

0 1 2

0 1 2

( )q p t y

q p w

= + +

= + +

where q is quantity,p is the pre-tax price, tis a lump sum tax,y isincome and w is weather and all the and parameters are positive

constants.

If all the variables apart from tare in logs and tis a proportion, howwould you interpret the slope parameters? Answer elasticities.


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0 1 2

0 1 2

( )q p t y

q p w

= + +

= + +

The first equation is a demand equation and the second equation is asupply equation. Both equations are behavioural equations as opposed

to identities.

You could write this model as a three equation model with twobehavioural equations and one identity:

0 1 2

0 1 2

( )d

s

s d

q p t y

q p w

q q

= + +

= + +=

However, you dont gain anything by doing so.


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0 1 2

0 1 2

( )t yp

q w

q

p

= + +

= + +

Quantity q and pricep are the endogenous variables, the variablessimultaneously determined within the model. The other variables (t,y

and w) are the exogenous variables, the variables which are given from

outside the model.

Treating these variables as exogenous is only valid in a microeconomic

model, because we can ignore the feedback from q,p and ttoy.

Whether we treat a variable as endogenous or exogenous depends on thepurpose of the model. However, in general, we need as many as many

equations as endogenous variables so that the model is complete.


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The structural formof the model may be written as:

0 1 21

0 21

1

1 b

t yq

wp

+ =

+

or, in general terms asAx = b whereA,x and b are as shown.

We can solve the structural form to get the reduced form of the model,which expresses the endogenous variables as a function of the

exogenous variables only (so there is no simultaneity).

In the general case, ifAx= b is the structural form, then x= A-1b isthe reduced form, assumingA is non-singular (which is almost always

the case).


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In the supply and demand example:

1 1 11

1 1

1 1 1

1 1( ) 0

1 1 1

A A A

= = + =

+

and the reduced form is:

( ) ( )( ) ( )

0 1 21 1

0 21 1

1 0 1 2 1 0 2

0 1 2 0 21 1

11 1

1

a t a yqwp

t y wqt y wp

+ = ++

+ + + = + ++

Note: the reduced form is a function of the exogenous variables only.


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Some Comparative Statics

Given the reduced form it is easy to find the effect of a changes in the

exogenous variables on the endogenous variables. In the general case

we have:

1

1

Structural Form

Reduced FormComparative Statics

x b

x A bx A b

=

=

=

From the reduced form, we have:

1 1 2 1 1 2

1 2 21 1

1 a t y wq

t y wp

+ + = + +


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1 1 2 1 1 2

1 2 21 1

1 a t a y wq

a t a y wp

+ + = + +

Do these results correspond with your intuition? Draw the supply anddemand curves and check them out.

Remember thatp is the pre-tax price. The post-tax price is 'p p t= +

so:

1 1

1 1 1 1

'a t t

p p t t

= + = + =

+ +

For example, consider the case where supply is completely inelastic

( 1 0 = ). If taxes rise by t , the change in post tax price is 0 and the

change in the pre-tax price is - t . Who bears the incidence of the tax?


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Example of a Closed Economy IS-LM Model

The structural model is:

0 1

1

0 1

0 1 2

( )

Y C I G

C c c Y T

T t Y

I i i r

P l l Y l r

= + += +

=

= = +

which may be re-written as:

( )1 1 0 0

1 2 0

1 (1 )c t Y i r c i G

l Y l r l M P

+ = + +

= +


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To save on notation, let s denote the tax adjusted savings rate

11 (1 )c t . Then the structural form is:

0 01

01 2

c i Gs i Y

l M Pl l r

+ + = +

The reduced form is:

1

0 0 0 01 2 1

0 01 2 12 1 1

1c i G c i Gs i l iY

l M P l M P l l l sr sl i l

+ + + +

= = +


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and ourcomparative static results are:

( ) ( )( )

( ) ( )( )

2 0 0 1 0

1 0 0 02 1 1

1 l c i G i l M P Y

l c i G s l M P r sl i l

+ + + + =

+ + + +

For example, if investment does not depend on the interest rate ( 2 0i = ),

there is no crowding out effect when G rises, and the government

expenditure multiplier, Y G , is just equal to one over the (taxadjusted) savings rates.


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2. Input-Output in a Nutshell

Consider the following two sector economy:

InputsSector

Gross

Output Sector 1 Sector 2

Final

Demand

1 100 10 15 75

2 50 15 10 25

This may be written as gross output = intermediate inputs + final

demands:

10 15100 50

15 10100 50

100 10 15 75 100 75

50 15 10 25 50 25Gross Intermediate Final Inputs PerUnit of Gro s Final Ouput Input Demand Gross Output Ouput Demand

= + = +


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Gross output = (inputs per unit of gross output) x (gross output) + final

demand:

100 0.10 0.30 100 75

50 0.15 0.20 50 25Gross Inputs Per Unit of Gross Final Ouput x Gross Output A Ouput x Demand y

= +

This may be compactly written as x Ax d= + , where100

50x

=

is the

vector of gross outputs,75

25d

=

is the vector of final demands and

0.10 0.30

0.15 0.20A

= is a matrix of input-output coefficients i.e. aij is the

input of sector i per unit of output of sector j.


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The x Ax d= + representation is perfectly general so far, noassumptions have been made about the production technology etc.

Assumption

The basic assumption in input output analysis is that the unit input

requirements aijare fixed i.e. inputs are used in fixed proportions

irrespective of the scale of output or changes in relative prices. This type

ofLeontief technology is very unrealistic generally.

Basic Input Output Analysis

Now suppose you want to find the gross output vectorx associated withsome different vectordof final demands (assumingA is fixed). The

required gross output vector is 1( )x I A d= .


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The proof is simple:

1

1

1

1 1

( )( ) ( ) ( )

( ) ( ) (

( )

)

x Ax d Ix Ax d x Ix

I A x dI A I A x I A d

Ix I A d I

x I A d

I A I

= + = + =

=

=

=

=

=

Thus, the new gross output vector is simply found by pre-multiply the

final demand vectordby 1( )I A . In practise, 1( )I A exists under

very general conditions, e.g. the column sum of the aijs are all less thanone.

Ecn221 Economic Applications

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