Question

1.

If y=C+I where C= 50+.75 Y and I = 50

What is Y, C, and the multiplier?

What if instead I = 50+.05Y?

2.

If Y=C+I+G where C=C - aTo + Y(a - at) and

I= lo + by solve for Y, C, I, and the multiplier.

Given a = MPC= .75, MPT = .2, MPI = .15, C = 45, To = 40

Io = 60 and G=90

3.

What is the present value of $1,000 paid at the end of five years? Assume the

relevant interest rate is 5%.

Answer 1

Ans 1)-

Given, Y = C +I     ------------------(1)

C = 50 +0.75Y, I =50

Put C and I function in equation (1)

Y = [50 +0.75Y] + 50

Y = 100 +0.75Y

Y -0.75Y = 100

0.25Y = 100                 ------------------(2) { IS equation which is here independent of interest rate (r)}

Y = 100/0.25

[Y = 400]

Put Y=400 in consumption function (C)

C = 50 +0.75*400

[C = 350]

IS Multiplier:

Multiplier can be calculated by the following formula (this can also be obtained by differentiating IS equation w.r.t. autonomous spending)-

From equation (2) we have coefficient of Y is 0.25, so

Hence, the required Y=400, C =350 and multiplier =4.

Now, if I = 50 +0.05Y

So, put C = 50 +0.75Y and I=50+0.05Y in equation (1)

Y = [ 50 +0.75Y] + [50 +0.05Y]

Y = 100 + 0.8Y

Y – 0.8Y = 100

0.2Y = 100             ------------------(3) { New IS equation which is here independent of interest rate (r)}

Y = 100/0.2

[Y = 500]

Put Y =500 in consumption function

C = 50 +0.75*500

[C = 425]

Put Y =50 in investment function

I = 50 + 0.05*500

[I = 75]

Multiplier:

From equation (3), coefficient of Y is 0.2, so

Hence, the new required Y=500, C =425, I =75 and multiplier =5.

Ans 2)- Given,

Y = C +I +G

C=C – aT₀ + Y(a - at), I= l₀ + bY

Put C and I in Y = C +I +G equation

Y = [C – aT₀ + Y(a - at)] + [I₀ +bY] +G

Y = C -aT₀ + Ya – atY + I₀ + bY + G

Y = C – aT₀ + I₀ +G +Ya – atY +bY

Y = C -aT₀ + I₀ + G + Y(a -at +b)

Y - Y(a -at +b) = C -aT₀ + I₀ + G

Y [ 1 – (a-at +b)] = C -aT₀ + I₀ + G

Now, put the given values in IS equation i.e.

a=MPC = 0.75, t=MPT = 0.2 , b=MPI = 0.15, C= 45, T₀=40, I₀=60 and G =90

So, required, Y =660

We have consumption function-

C=C – aT₀ + Y(a - at)

Put C = 45, a = 0.75, T₀ = 40, Y =660 and t=0.2

C = 45 - 0.75*40 + 660(0.75 – 0.75*0.2)

C = 45 - 30 + 660(0.75 – 0.15)

C = 15+ 396

[C = 411]

We have, I = I₀ +bY

Put, I₀ = 60, b=0.15 and Y= 660

I = 60 +0.15*660

[I = 159]

Multiplier:

To find Multiplier differentiate IS equation w.r.t. autonomous consumption i.e. differentiate w.r.t. A, by replacing A = C -aT₀ + I₀ + G in IS equation

So,

Hence, the required value of Y = 660, C = 411, I = 159 and multiplier = 4

Ans 3)- Given an amount of $1,000 paid at the end of five years i.e.

FV at the end of 5 years = $1,000

Interest rate (r) = 5% =0.05 and n=5 years

Hence, using Present value/Future value relation –

Put, FV = $1,000 , r=0.05 and n =5

Hence, the required Present value would be $783.7.