Question

Suppose an inflationary economy can be described by the following equations representing the goods and money markets:

C = 20 + 0.7Yd

M = 0.4Yd

I = 70 – 0.1r

T = 0.1Y

G = 100

X = 20

Ld = 389 + 0.7Y – 0.6r

Ls = 145

where G represents government expenditure, M is imports, X is exports, Y is national income, Yd is disposable income, T is government taxes (net of transfer payments), I is investment, r is the rate of interest, C is consumption, Ld is money demand, and Ls is money supply.

i) Use the inverse matrix method to solve for the equilibrium level of national income and the equilibrium rate of interest in this economy. (Note: ½ of the marks in this part are given for the correct set up of the equations. Explain what you are doing, including how equilibrium is established in each market.)

ii) Now use Cramer’s rule to find your answer.

Answer 1

Let us form the formula for IS and LM curve from the above formula. Then we will create a 2*2 matrix to resolve the value of Y and r.

We confirm that goods market is in equilibrium and in IM curve for IS curve. we confirm that the money market is in equilibrium.

Then, Let us consider the matrix form as, AX= C

So multiplying both side by A-1, we can get the solution for X

i.e., A-1. AX= A-1. C
or, X= A-1. C

this is the way we resolve the formula with the help of inverse matrix

Solution 1: Y= C + I + G + X-M

From the above information

C = 20 + 0.7 Yd

M = 0.4 Yd

I = 70-0.1 r

T = 0.1Y

G= 100

X= 20

Where Yd = disposable income .then, Yd = Y-T

Then, We can say that the IS curve formula as,

Y= 20 + 0.7 (Y-T) + 70 - 0.1r + 100 + 20 - 0.4 (Y-T)

that is Y = 210 + 0.7 (Y- 0.1Y) - 0. 1r - 0.4 (Y- 0.1 Y)

that is Y = 210 + 0.7 * 0.9Y -0.1r -0.4 * 0.9 Y

that is Y = 210 + 0. 63 Y - 0.1r - 0.36Y

that is 0.73 Y = 210- 0.1r

that is 0.73 Y + 0.1r = 210 -------------(1)

LM Curve Formula:

Ls = Ld

145 = 389 + 0.7 Y -0.6 r

that is 0.7 -0.6 r = - 244 --------------(2)

Please find the below attachment: