Question

Consider an economy with full-employment output of 5000 and government purchases G of 1000. Desired consumption and desired investment are:

Cd=2500-1000r+0.5Y,  Id=500-2000r.

Find an equation relating desired saving Yd to r and Y.

Find the real interest rate that clears the goods market.

Government purchases rise to 1200. How does this increase change the equation describing desired national saving? What happens to the market-clearing real interest rate?

Answer 1

The full employment output is

Y = 5000

Government Spending is

G = 1000

Desired Consumption is

Cd = 2500 - 1000.r + 0.5.Y

Desired Investment is

Id = 500 - 2000.r

Interest Rate = r

Hence, desired savings (Sd) is

Sd = Y - Cd - G

or, Sd = Y - (2500 - 1000.r + 0.5Y) - 1000

or, Sd(r, Y) = -3500 + 0.5.Y + 1000.r

The equation relating desired savings to r and Y is: Sd(r, Y) = -3500 + 0.5.Y + 1000.r

Now, at goods market equilibrum, Full Employment GDP (Y) equals Aggrigate Expenditure (AE).

Y = AE

or, Y = Cd + Id + G

or, Y = 2500 - 1000.r + 0.5.Y + 500 - 2000.r +1000

or, 0.5.Y = 4000 - 3000.r

or, 0.5×5000 = 4000 - 3000.r

or, 3000.r = 1500

or, r* = 0.5 or 50%

The real interest rate that clears the goods market is 50%.

✓ Now, desired national savings is

NS = Y - C - G

or, NS = Y - (2500 - 1000.r + 0.5.Y) - G

or, NS = -2500 + 1000.r + 0.5.Y - G.........(1)

Now, initially, G = 1000

Hence, initially

NS = -2500 + 1000.r + 0.5.Y -1000

or, NS = -3500 + 1000.r + 0.5.Y

Now, G increases to 1200. Hence, G' = 1200.

Now,

NS' = -2500 + 1000.r + 0.5.Y - 1200

or, NS' = -3700 + 1000.r + 0.5.Y

This is the changed desired national savings equation.

Hence, the desired national savings falls by 200 at each level and the equation changes to

NS' = -3700 + 1000.r + 0.5.Y

Now, at equilibrum, desired national savings equals the desired investment.

Hence,

NS' = Id

or, -3700 + 1000.r + 0.5.Y = 500 - 2000.r

Putting Y = 5000, we get

-3700 + 1000.r + 0.5×5000 = 500 - 2000.r

or, 3000.r = 1700

or, r' = 0.57 or 57%

Hence, market clearing interest rate increases to 57%.

Hope the solution is clear to you my friend.