A Detailed Analysis of Macroeconomic Models and Solutions

Verified

Added on  2022/09/02

|6
|434
|16
Homework Assignment
AI Summary
This assignment provides a detailed analysis of macroeconomic models, focusing on Keynesian economics and related concepts. The solution explores utility functions, consumer problems, and constrained optimization, including the use of Lagrange multipliers. It examines the impact of government spending and interest rates on the economy, presenting a matrix-based model and applying Cramer's rule to solve for key variables. The analysis includes a comparison of different scenarios, such as changes in government spending and the incorporation of interest rates into the consumption function. Overall, the assignment offers a comprehensive understanding of macroeconomic principles and problem-solving techniques.
Document Page
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
tabler-icon-diamond-filled.svg

Paraphrase This Document

Need a fresh take? Get an instant paraphrase of this document with our AI Paraphraser
Document Page
p1 x1
0 + p2 x2
0=I
MRS ( x1
0 , x2
0 ¿= p1 / p2
Document Page
(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=2000.8 Y
I =10002000r
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 =10002000r
Thus ,
AX = B
Document Page
1 1 1
¿ ¿ ¿ = [
G
200
10002000 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 ]
tabler-icon-diamond-filled.svg

Paraphrase This Document

Need a fresh take? Get an instant paraphrase of this document with our AI Paraphraser
Document Page
I = -2000r + 1000
C = 4G – 8000r + 5000
Y = 5G – 10000r + 6000
(c) Now G is decreasing by 50$ billon effect on Y
Multipliers = 1
1MPC = 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 +C1000 r=200
Or
0.8 Y +C=200+1000 r
I =10002000r
Formation of new matrix :
Document Page
AX = B
1 1 1
¿ ¿ ¿ = [
G
200+1000 r
10002000 r
]
Applying the cramer’s rule ,
We got ,
Y =5 G15,000 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 .
chevron_up_icon
1 out of 6
circle_padding
hide_on_mobile
zoom_out_icon
[object Object]