Question

2.      Assume the following IS-LM model:

         expenditure sector:                                                       money sector:

         AD = C + I + G + NX         I      = 300 - 20i                  M_s   = 700

         C    = 100 + (4/5)YD           G    = 120                          P    = 2

         YD = Y - TA                       NX = -20                           m_d = (1/3)Y + 200 - 10i

         TA = (1/4)Y                       

        

   a.   Derive the equilibrium values of consumption (C) and money demand (m_d).

   b.   How much investment (I) will be crowded out if the government increases its purchases by DG = 160 and nominal money supply (M) remains unchanged?

   c.   By how much will the equilibrium level of income (Y) and the interest rate (i) change, if the Fed responds to this increase in government purchases by increasing nominal money supply to Ms = 1,100

Answer 1

We have the following information

Consumption: C = 100 + 0.8YD; where YD is disposal income

Investment: I = 300 – 20i; where i is the interest rate

Government spending: G = 120

Net exports: NX = – 20

Taxes: T = 0.25Y

Money supply: Ms = 700

Price level: P = 2

Money demand: Md = 0.33Y + 200 – 10i

First we will derive the IS equation

Y = C + I + G + NX

Y = 100 + 0.8YD + 300 – 20i + 120 – 20

Y = 500 + 0.8(Y – T) – 20i

Y = 500 + 0.8(Y – 0.25Y) – 20i

Y = 500 + 0.6Y – 20i

0.4Y = 500 – 20i

Y = 1250 – 50i (IS equation)

Now we will derive the LM equation

Ms/P = Md

700/2 = 0.33Y + 200 – 10i

350 = 0.33Y + 200 – 10i

0.33Y = 150 + 10i

Y = 454.54 + 30.30i (LM equation)

Equating IS and LM equations

1250 – 50i = 454.54 + 30.30i

795.46 = 80.30i

Equilibrium interest rate (i) = 9.9%

Y = 1250 – 50i

Y = 1250 – 495.30

Equilibrium income (Y) = 754.7

Consumption = 100 + 0.8YD

Consumption = 100 + 0.8(Y – T)

Consumption = 100 + 0.8(Y – 0.25Y)

Consumption = 100 + 0.6Y

Consumption = 100 + 0.6(754.7)

Consumption = 552.82

Md = 0.33Y + 200 – 10i

Md = 249.05 + 200 – 99

Money demand = 350.05

Investment = 300 – 20i

Investment = 300 – 198

Investment = 102

Now it is given that the government spending has increased by 160. New G = 280

New IS equation

Y = C + I + G + NX

Y = 100 + 0.8YD + 300 – 20i + 280 – 20

Y = 660 + 0.8(Y – T) – 20i

Y = 660 + 0.8(Y – 0.25Y) – 20i

Y = 660 + 0.6Y – 20i

0.4Y = 660 – 20i

Y = 1650 – 50i (New IS equation)

The LM equation will remain the same. Equating IS and LM equations

1650 – 50i = 454.54 + 30.30i

1195.46 = 80.30i

New equilibrium interest rate (i) = 14.89%

Y = 1650 – 50i

Y = 1650 – 744.37

New equilibrium income (Y) = 905.6

New Investment = 300 – 20i

New Investment = 300 – 297.8

Investment = 2.2

Change in investment = 2.2 – 102 = – 99.8. So, investment has declined by 99.8 due to increased government spending.

Now it is given that the Fed has increased the nominal money supply (Ms) to 1100 in response to increased government spending.

Y = 1650 – 50i (New IS equation)

New LM equation

Ms/P = Md

1100/2 = 0.33Y + 200 – 10i

550 = 0.33Y + 200 – 10i

0.33Y = 350 + 10i

Y = 1060.6 + 30.30i (New LM equation)

Equating IS and LM equations

1650 – 50i = 1060.6 + 30.30i

589.4 = 80.30i

New equilibrium interest rate (i) = 7.3%

Y = 1650 – 50i

Y = 1650 – 365

New equilibrium income (Y) = 1285