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Economics 214
Lecture 13
Systems of Equations
Examples of System of
Equations



Demand and Supply
IS-LM
Aggregate Demand and Supply
Demand and Supply
Demand : q d    P  I  P S
Supply : q s    P  W
Equilibriu m : q d  q s  q
q  quantity
P  price
P S  price substitute good
I  income
W  wage rate
IS-LM
IS : ln( i )     ln( Y )   ln( G )
LM : ln( i )     ln( Y )   ln( M / P)
i  interest rate
Y  real GDP
G  real gov' t spending
P  price level
Solving System of Equations


Repeated Substitution
Matrix Algebra or linear Algebra
Solving System of Equations


Economic Models typically consist of a
number of equations that represent
identities, behavioral relationships, and
conditions that constitute an equilibrium.
These equations include both variables,
which are economic quantities and
parameters, which are unvarying constants.
Solving Systems of Equations


Variables in a system are exogenous if
determined outside the system or
endogenous if the are determined within
the system.
A solution to the model is a representation
of the endogenous variables as functions of
only the parameters of the model and the
exogenous variables.
Solving our Demand and
Supply model
Using the equilibriu m condition that q d  q s  q, we can
rewrite our demand and supply equations as
demand : q    P  I  P S
supply : q    P  W
Setting the two equal, we get
  P  W    P  I  P S
   P  (   )  I  P S  W
or
or
 



S
P

I
P 
W
 


Solving our Demand & Supply
Model
To find the equilibriu m quantity, we will substitute our
solution for price into the supply equation.
  




S
q   

I
P 
W   W

 
     
           
 S       


I
P 
W




     
 S 


I
P 
W




Solving our IS-LM Model
IS : ln( i )     ln( Y )   ln( G )
LM : ln( i )     ln( Y )   ln( M / P )
In equilibriu m, the interest rate in both equations is equal.
We will therefore set both equations equal and solve for
equilbrium ln( Y ).
   ln( Y )   ln( M / P)     ln( Y )   ln( G )
    ln( Y )        ln( G )   ln( M / P) or
 


ln( Y ) 

ln( G ) 
ln( M / P )
 

Solving our IS-LM Model
We can get the equilibriu m interest rate, by substituti ng our
solution for the equilibriu m income into either the IS or the
LM equation.
  



ln( i )     

ln( G ) 
ln( M / P )   ln( M / P )

     

           
      


ln( G ) 
ln( M / P )



   


ln( G ) 
ln( M / P )


