Applying The Cramer’s Rule

Solution 1
a) Let U =f ( x1 , x2) be utility function
Let V =f (x1 , x2) = F(U )
Where F’(U ) > 0 ( V is a monotonic transformation of U)
(i) We know that
MRS (x ¿¿ 1 , x2)¿ = −M U1
M U2
=−M V 1
M V 2
Hence ,
V1/V2 = U1/U2
(ii) i = 1and j = 2
V12/V21 = U12/U21
No, It is not necessary Vij and Uij always have the same sign .Because it will vary as per
constraint and utility function .
b ) Considering a standard Constrained utility maximization problem (i.e consumer’s problem) or cost
minimization problem (i.e firm’s problem)
(i) Optimization problem and corresponding Lagrangian :
Considering a consumer problem
max
x1 ,x2
u( x1 ¿, x2)¿
s.t p1 x1 + p2 x2=I
(ii) Writing a first order conditions

p1 x1
0 + p2 x2
0=I
MRS ( x1
0 , x2
0 ¿= p1 / p2

(iii) Lagrange Multiplier : It is basically applied to identify optimum situation and sometime
used to find out the maximum and minimum coordinates of the tangent points between
objective function .
(iv) Envelope theorem : This theorem says the direct effect of changes in exogenous variable
which is based considered , even though the exogenous variable might enter the maximum
value function indirectly as the part of solution to the endogenous choice variables.
The problem then becomes Maximize
U = f( x , y , α)
Subject to g( x , y , α)=0
The Lagrangian for this problem is Z = f(x, y, α) + λg(x, y, α)
Solution 4
According to Keynesian Macroeconomic model ,
Y =C + I+ G
C=200−0.8 Y
I =1000−2000r
G and r  exogenous variable
Y, C and I endogenous variable
(a) Set up of model in matrix form :
Y −C + I =G
−0.8 Y +C=200
I =1000−2000r
Thus ,
AX = B

1 −1 −1
¿ ¿ ¿ = [
G
200
1000−2000 r
]
(b) Now finding A-1 of matrix
A = [ 1 −1 −1
−0.8 1 0
0 0 1 ]
A-1 = adj A
¿ A∨¿ ¿
| A|=1+1 ( −0.8 ) −1 ( 0 ) =0.2
Writing Matrix of Minor ,B =
[1 0.8 0
1 1 0
1 0.8 0.2 ]
Now Adj A = BT¿ [ 1 1 1
0.8 1 0.8
0 0 0.2 ]
Now A-1 = BT
0.2 = [ 5 5 5
4 5 4
0 0 1 ]

I = -2000r + 1000
C = 4G – 8000r + 5000
Y = 5G – 10000r + 6000
(c) Now G is decreasing by 50$ billon effect on Y
Multipliers = 1
1−MPC = 1
0.2 =5
So dY
dG =Multiplier=5
Thus dY = 5 dG
dG = - 50
d Y = (-50) x 5 = -250
Hence income is decreased by 250 $ Billon .
(d) Now considering the consumption depends upon the interest rate as well income.
Replacing the second equation :
Y −C + I =G
−0.8 Y +C−1000 r=200
Or
−0.8 Y +C=200+1000 r
I =1000−2000r
Formation of new matrix :

AX = B
1 −1 −1
¿ ¿ ¿ = [
G
200+1000 r
1000−2000 r
]
Applying the cramer’s rule ,
We got ,
Y =5 G−15,000 r+ 6000
(e) On comparing The income we got in b part and d part and indicated that economy will have to pay
more interest for borrowing in part C . Y will be less in Part C as compare to part b if interest is same .