Question

Suppose that the real money demand function is(M/P)^d=800-50r+20y, where r is the interest rate in percent, y is the real income. The money supply is 2,000 and the price level is fixed at 5.

a. What is the equilibrium interest rate if real income is fixed at 10?

b. What happens to the equilibrium interest rate if the supply of money is reduced from 2,000 to 1500? How will real income change to make the equilibrium interest remain constant?

c. Suppose the equation for the IS curve is: Y=C(Y-T ̅ )+I(r)+G ̅   , where C(Y-T ̅ )=10+0.5(Y-T ̅),I(r)=3-0.25r and government spending equals to tax which are both fixed at 2. So what is the IS-LM model equilibrium point?

d. If money supply increases from 2,000 to 3,500, what will happen to the equilibrium point?

Answer 1

a) We are given the real money demand function as: (M/P)^d = 800 - 50r+20y       

With Nominal Money Supply: M = 2000 and Fixed Price: P= 5, the real money supply can be written as:

              (M/P)^s = 2000/5 = 400

For the money market equilibrium:

Demand for Real Balances = Money Supply

(M/P)^d = (M/P)^s

800 -50r + 20y = 400

800 - 50r + 20(10) = 400           {given real income as y = 10}

800 - 50r + 200 = 400

1000 – 400 = 50r

600 = 50r

600/50 = r

r = 12          

b) If nominal money supply is from 2000 to 1500, then real money supply falls to:

(M/P)^s = 1500/5 = 300

For the money market equilibrium:

Demand for Real Balances = Money Supply

800 -50r + 20y = 300

800 - 50r + 20(10) = 300           {given real income as y = 10}

800 -50r + 200 = 300

800-50r+ 200 = 300

1000 – 300 = 50r

700 = 50r

r = 700/50

r = 14

So when real money supply falls from 400 to 300, the equilibrium interest rates rises from 12 to 14.

In order to make the equilibrium interest rate constant i.e. it should remain at 12, the real income should fall. A fall in the real income would reduce the demand for money and shift the money demand curve downwards. The value of the reduced real income can be found out using the demand function by keeping the interest rate r = 12 and keeping the real money supply equal to 300.

At the new equilibrium:

800 - 50r+20y = 300

800 – 50(12) + 20y = 300

800 – 600 + 20y = 300

20y = 100

y = 100 /20

y = 5

Hence, in order to keep the equilibrium interest rate constant to 12, the real income should fall from 10 to 5 units

c) For deriving the equilibrium point in both goods market as well as in money market, we need to find the IS equation and the LM equation. Here we assume money supply is M = 2000 and P = 5, thus real money supply is 400

For the IS equation we use the below goods market equilibrium condition (Given T = G = 2):

Y=C(Y-T ̅ )+I(r)+G ̅

Y = 10+0.5(Y-T ̅) + (3-0.25r) + 2

Y = 10+0.5(Y -2) + 3 -0.25r + 2

Y = 10+0.5Y – 1 + 3 – 0.25r + 2

Y = 14 + 0.5Y – 0.25r

Y – 0.5Y = 14 – 0.25r

0.5Y = 14 – 0.25r

Y = 28 – 0.5r

0.5r = 28 –Y

r = 56 – 2Y      {IS Equation}   

For the LM equation we use the below money market equilibrium condition:

800 - 50r + 20Y = 400

400 – 50r + 20Y = 0

400 + 20Y = 50r

r = 8 + 0.4Y    {LM equation}

At the equilibrium point:

IS = LM

56 – 2Y = 8 + 0.4Y

48 = 2.4Y

Y = 20      {Equilibrium Income}

For equilibrium interest rates we use the value of equilibrium income in either IS or LM equation:

IS : r = 56 – 2(20) = 56 – 40 = 16

Or

LM : r = 8 + 0.4(20) = 8 + 8 = 16

Thus, equilibrium interest rate is r = 16 when equilibrium income is Y = 20.

d. If money supply increases from 2000 to 3500 then real money supply changes to (with prices fixed at P = 5)

(M/P)^s = 3500/5 = 700

With and unchanged money demand, the new LM curve can be derived as:

800 - 50r + 20Y = 700

800 – 700 + 20Y = 50r

100 + 20Y = 50r

2 + 0.4Y = r        {New LM equation}

At the equilibrium point:

IS = LM

56 – 2Y = 2 + 0.4Y

56 – 2 = 2Y + 0.4Y

54 = 2.4Y

Y = 54/2.4

Y = 22.5              {New equilibrium income}

For equilibrium interest rates we use the value of equilibrium income in either IS or LM equation:

IS : r = 56 – 2(22.5) = 56 – 45 = 11

Or

LM : r = 2 + 0.4(22.5) = 2 + 9 = 11

Thus, when money supply increases to 3500, equilibrium interest rate falls to r = 11 and equilibrium income rises to Y = 22.5. Graphically this can be seen as a rightward shift in the LM curve.