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