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Chapter 9:
Problem 4

4. The purpose of this problem is to study the sacrifice ra- tio. Suppose that initially actual and natural real GDP both equal 11,000 and that the rate of inflation is 3.5 percent. Natural real GDPgrows by 3 percent per year over the next five years. Actual real GDPdecreases by 2 percent in the first year, but then grows by 4 percent in the second year, 5.5 percent in the third year, 4.2 per- cent in the fourth year, and 3.5 percent in the fifth year. Inflation in years 1–5 equals 3.1 percent, 2.2 percent, 1.6 percent, 1.3 percent, and 1.1 percent, respectively.

(a)Calculate natural real GDPfor years 1–5.

The natural real GDP increase by 3% for next 5 years
So 1st year Natural real GDP=11000*1.03=11,330
2nd year Natural real GDP=11330*1.03=11,669.9
3rd year Natural real GDP=11669.9*1.03=12,020
4th year Natural real GDP=12020*1.03=12,380.6
5th year Natural real GDP=12380.6*1.03=12,752

(b)Calculate actual real GDPfor years 1–5.

Actual real GDP decreases by 2 percent in the first year, but then grows by 4 percent in the second year, 5.5 percent in the third year, 4.2 per- cent in the fourth year, and 3.5 percent in the fifth year.
So 1st year 11000*0.98=10,780
2nd year 10780*1.04=11,211.2
3rd year 11211.2*1.055=11,827.8
4th year 11827.8*1.042=12,324.6
5th year 12324.6*1.035=12,756

(c)Calculate the output ratio for years 1–5.

The output ratio equals (10,780/11,330)(100) = 95.1 in the first year.
2nd year is (11,211.2/ 11,669.9)(100) = 96.1
It equals (11,827.8/12,020.0)(100) = 98.4 in the 3rd year.
It equals (12,324.6/12,380.6)(100) = 99.5 in the 4th year,
And it equals (12,756/12,752)(100) = 100.0 in the 5th year.

(d) The cumulative loss of output equals

The first year loss was 100-95.1=4.9
Second year was 100-96.1=3.9
Third year was 100-98.4=1.6
Fourth year was 100-99.5=0.5
Fifth year was 100-100=0

4.9 + 3.9 + 1.6 + 0.5 + 0 = 10.9 percent.

(e) The reduction in the inflation rate equals 3.5 − 1.1 = 2.4 percentage points. Therefore, the sacrifice ratio equals 10.9/2.4 = 4.5.

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