← 5. Where the Interest Rate Is Decided

Exercises

Problems to check your understanding of this chapter. The solutions are folded away — try them yourself first.

Medium

1Medium

Households raise base saving from S0=40S_0 = 40 to 5050 at the same time as the government runs a deficit D=20D = 20.

Compared with the starting state (S0=40S_0 = 40, D=0D = 0), what happens to the interest rate and investment?

Saving 5050, deficit 2020 — rate 7%7\%, investment 5858.
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  1. The two forces pull opposite ways.

    The saving rise pushes the rate down by (50−40)/(s+b)=10/10=1(50 - 40) / (s + b) = 10 / 10 = 1 point; the deficit pushes it up by D/(s+b)=20/10=2D / (s + b) = 20 / 10 = 2 points.

    Together: r∗=(I0−S0+D)/(s+b)=(100−50+20)/10=7r^* = (I_0 - S_0 + D) / (s + b) = (100 - 50 + 20) / 10 = 7 — one point above the starting 66.

  2. Investment is I=100−6⋅7=58I = 100 - 6 \cdot 7 = 58, down 66 from the starting 6464. Saving rose yet investment fell because the deficit's pull was the stronger one — the winner is decided by comparing the two changes, each scaled by s+bs + b.

Hard

2Hard

While a boom lifts income from Y=200Y = 200 to 220220, the central bank also raises the real money supply from M/P=90M/P = 90 to 105105.

(1) Where does the interest rate go from its starting 5%5\%?

(2) What happens to investment?

Money 105105, income 220220 — rate 4.5%4.5\%, investment 7373.
Show solution

  1. (1) The income rise lifts the money-demand intercept to L0+kY=40+0.5⋅220=150L_0 + k Y = 40 + 0.5 \cdot 220 = 150, pushing the rate up by kΔY/h=0.5⋅20/10=+1k \Delta Y / h = 0.5 \cdot 20 / 10 = +1 point; the money increase pulls it down by Δr=−Δ(M/P)/h=−15/10=−1.5\Delta r = -\Delta (M/P) / h = -15 / 10 = -1.5 points. Together: r∗=(150−105)/10=4.5r^* = (150 - 105) / 10 = 4.5 — a net move of −0.5-0.5 points.

  2. (2) The rate set in the money market rides the loanable-funds investment curve: I=100−6⋅4.5=73I = 100 - 6 \cdot 4.5 = 73, up 33 from the starting 7070. The Keynesian chain — money to rate to investment to aggregate demand — has turned once: the central bank's downward pull beat the boom's upward pressure, by a little.