Question

An economy has full-employment output of 6000. Gov-
ernment purchases, G, are 1200. Desired consumption
and desired investment are
Cd == 3600 - 2000r + 0.10Y, and
I
d == 1200 - 4000r,
where Y is output and r is the real interest rate.
a. Find an equation relating desired national saving,
sd
, to r and Y.
b. Using both versions of the goods market equilibrium
condition, Eqs. ( 4.7) and ( 4.8), find the real interest
rate that clears the goods market. Assume that
output equals full-employment output.
c. Government purchases rise to 1440. How does
this increase change the equation describing
desired national saving? Show the change graph-
ically. What happens to the market-clearing real
interest rate?

Answer 1

a. We know $S^d = Y – C^d – G$ . Substituting in the above equations, we get:

$S^d$ = Y – (3600 – 2000r + 0.1Y) – 1200

= – 4800 + 2000r + 0.9Y

This equation relates $S^d$ to the real interest rate (r) and output (Y).

b. $Y = C^d + I^d + G$

Y = (3600 – 2000r + 0.1Y) – (1200 – 4000r) – 1200

Y = 6000 – 6000r + 0.1Y

So, 0.9Y = 6000 – 6000r

At full employment, Y = 6000.

Substituting this into the above, we get 0.9x6000 = 6000 – 6000r

Finally, solving for r, we get r = 0.10 (or 10%).

$S^d = I^d$

– 4800 + 2000r + 0.9Y = 1200 – 4000r

0.9Y = 6000 – 6000r

When Y = 6000, r = 0.10 (or 10%).

So, we can use either Eq. (4.7) or (4.8) to get to the same result.

c. When G = 1440, desired saving becomes:

$S^d = Y – C^d – G= Y – (3600 – 2000r + 0.1Y) – 1440.$

This becomes 0.9Y = 5040 – 2000r S^d is now 240 less for any given r and Y; this shows up as a shift in the$S^d$ from$\;S^1 \;to \;S^2$

What about r? Set $S^d = I^d$ , we get:

–5040 + 2000r + 0.9Y = 1200 – 4000r

6000r + 0.9Y = 6240

At Y = 6000, this is 6000r = 6240 – (0.9 x 6000) = 840, and so r = 0.14 (or 14%)

Thus, the market-clearing real interest rate increases from 10% to 14%.