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Problem Set 3 Solutions

ECON 3 − Principles of Macroeconomics

University of California San Diego

Christopher Gibson

Monday, April 20th

1. (a) Discuss the six sources of economic growth and give an example of each.

• See lectures slides.

(b) If population grows at a rate of 1% per year and the share of the population with

jobs levels out to 45%, then the population in year 2 with respect to year 1 is

POP2 = POP1 · (1.01)

i. If GDP remains constant, then Y2 = Y1. Then the growth rate of GDP per

capita from year 1 to year 2 is( Y2

POP2 − Y1

POP1 Y1

POP1

) · 100% =

( Y1

POP1(1.01) − Y1

POP1 Y1

POP1

) · 100%

= POP1 Y1

· (

Y1 POP1(1.01)

− Y1

POP1

) · 100% =

1 − (1.01) 1.01

· 100%

= − (

0.1

1.01

) · 100% = −0.99%

ii. The growth rate g of GDP that would leave GDP per capita constant satisfies

Y2 POP2

= Y1

POP1 ⇐⇒

Y1(1 + g)

POP1(1.01) =

Y1 POP1

=⇒ 1 + g = 1.01 =⇒ g = 0.01

so that in percentage terms GDP growth is 1%, exactly the rate of population.

iii. The growth rate g of GDP that would grow GDP per capita by 1% each year

satisfies( Y2

POP2 − Y1

POP1 Y1

POP1

) · 100% = 1% ⇐⇒

( Y1(1+g)

POP1(1.01) − Y1

POP1 Y1

POP1

) = 0.01

⇐⇒ POP1 Y1

· (

Y1(1 + g)

POP1(1.01) −

Y1 POP1

) = 0.01 ⇐⇒

(1 + g) − (1.01) 1.01

= 0.01

⇐⇒ 1 + g = (1.01) · 0.01 + 1.01 = (1.01)2 ⇐⇒ g = (1.01)2 − 1 = 0.0201

so that in percentage terms, GDP would need to grow by 2.01% per year for

GDP per capita to grow by 1% per year.

2. (a) Using the “years to double” shortcut, the number of years it would take one dollar

to double is

i. 72 2

= 36 years

ii. 72 4

= 18 years

iii. 72 7 ≈ 10.29 years

iv. 72 9

= 8 years

(b) Using the exact compound interest calculation, the multiple by which each dollar

actually increases in the number of years implied by the shortcut is

i. (1.02)36 ≈ 2.040

ii. (1.04)18 ≈ 2.026

iii. (1.07)72/7 ≈ 2.006

iv. (1.02)9 ≈ 1.993

The shortcut provides a good estimate, and in fact a near-perfect estimate for an

interest rate of 7%.

with numbers higher than 72. For example if you tried 100/i for i = 2%, this would

give you 50 years, and checking this would give (1.02)50 ≈ 2.69, so you would need more years to triple. Eventually you would find that some numerator in the range

of 110-115 would provide a decent estimate, giving you an estimate of 110 i

years to

triple.

If you wanted to solve for this analytically, you could solve the equation (1 + i)n = 3

for n. Taking the natural log of each side gives

n ln(1 + i) = ln(3) =⇒ n = ln(1 + i) = ln(3)

ln(1 + i)

For i = 0.02 this reduces to n ≈ 55.5 years. Then taking 111 i

is a pretty good estimate

for i = 2. Notice that the estimates get worse much more quickly than in our years

to double estimate.

3. Suppose country A has grown its GDP at a rate of 2% over the past 30 years. Country

B, on the other hand, has only grown at a rate of 0.5% over the past 30 years.

(a) Country B may have a slower rate of growth because of poor business structure,

inadequate incentives for innovation, turbulent political climate, etcetera.

(b) If the countries start with the same GDP, then after 30 years country A would have

GDP Y0 ·(1.02)30 ≈ 1.81·Y0 and country B would have GDP Y0 ·(1.005)30 ≈ 1.16·Y0. Then as a percentage

1.81 ·Y0 − 1.16 ·Y0 1.16 ·Y0

· 100% ≈ 55.96%

the GDP of country A would be about 55.96% higher than that of country B.

of growing 1.16 ·Y0 at rate g for 30 years, so GDP would be Y0 · (1.005)30 · (1 + g)30. For this to equal the GDP of country A which has grown steadily at 2% for 60 years,

Y0 · (1.005)30 · (1 + g)30 = Y0 · (1.02)60 ⇐⇒ (1.005)(1 + g) = (1.02)2

=⇒ g = (1.02)2

1.005 − 1 ≈ 0.03522

so country B must grow at a rate of about 3.52%.

(d) In 60 years countries again have the same GDP, call it Y60. If these growth rates

continue then after another 30 years country A would have GDP Y60·(1.02)30 ≈ 1.81· Y60 ≈ 5.94·Y0, and country B would have GDP Y60·(1.03522)30 ≈ 2.82·Y60 ≈ 9.27·Y0. Then as a percentage, after another 30 years

2.82 ·Y60 − 1.81 ·Y60 1.81 ·Y60

· 100% ≈ 55.96%

the GDP of country B would be about 55.96% higher than that of country A. Notice

that this is exactly the percentage difference country A was ahead after only 30

years.

4. Suppose Robert expects that for each hour he works, he will earn taxi fares

1st hr 2nd hr 3rd hr 4th hr 5th hr 6th hr 7th hr 8th hr 9th hr 10th hr

$19 $23 $21 $20 $19 $18 $17 $15 $13 $12

If Robert must pay for one gallon of gas per hour at a price of $3.00 per gallon, for each

hour Robert is left with profit

1st hr 2nd hr 3rd hr 4th hr 5th hr 6th hr 7th hr 8th hr 9th hr 10th hr

$16 $20 $18 $17 $16 $15 $14 $12 $10 $9

(a) If Robert decides to buy a car, he will work as long as his profit exceeds his value of

leisure time and he will thus work 7 hours per day. His daily profit will be

$(16 + 20 + 18 + 17 + 16 + 15 + 14) = $116

(b) If Robert works 5 hours per week then each week he can make $116 · 5 = $580 per week. If Robert faces a time-frame of 52 weeks, Robert expects his total revenue

from driving will be $580 · 52 = $30, 160. At the end of the year he can sell his car for the full amount, but must pay back the loan of $5,000 with interest. Since he

paid $5,000 of his own money, we must account for that as well. Then his net profit

− $5, 000 + $10, 000 − $5, 000(1 + 0.05) + $30, 160 = $30, 160 − $5, 000 · 0.05

= $29, 910

If, on the other hand, Robert keeps his current job, he will make $540·52 = $28, 080 in wages for the year, but he can also lend his $5,000 at the interest rate of 5%.

Then his total earnings will be $28, 080 + $5, 000 · 0.05 = $28, 330. Thus his outside option is less profitable than investing in his own business ($29, 910 > $28, 330) so

Robert will buy the car.

the end of the year would be

− $5, 000 + $10, 000(1 − 0.1) − $5, 000(1 + 0.05) + $30, 160

= $30, 160 − $5, 000 · 0.05 − $10, 000 · 0.1

= $28, 910

which is still greater than his outside option of $28,330 so he would still invest in

the car.

(d) Robert would be indifferent between investing and staying at his current job if de-

preciation δ satisfies

− $5, 000 + $10, 000(1 − δ) − $5, 000(1 + 0.05) + $30, 160 = $28, 330

⇐⇒ $29, 910 − $10, 000 · δ = $28, 330 =⇒ δ = 0.158

So if the car depreciates at a rate of 15.8%, Robert would be indifferent between

starting his own business and continuing his current job.

(e) If depreciation is 10% and the interest rate is also 10%, by investing Robert would

make profit

− $5, 000 + $10, 000(1 − 0.1) − $5, 000(1 + 0.1) + $30, 160

= $30, 160 − $5, 000 · 0.1 − $10, 000 · 0.1

= $28, 660

If Robert keeps his current job he will make $28,080 in wages and by lending his

$5,000 at a rate of 10% he would make $500, so his outside option is worth $28,580.

His outside option is still less than his payoff from investing in the car, so Robert

will invest.

(f) With depreciation of 10%, Robert would be indifferent between investing and staying

at his current job if interest i satisfies

− $5, 000 + $10, 000(1 − 0.1) − $5, 000(1 + i) + $30, 160 = $28, 080 + $5, 000 · i

⇐⇒ $30, 160 − $5, 000 · i− $10, 000 · 0.1 = $28, 080 + $5, 000 · i

⇐⇒ $10, 000 · i = $1, 080 =⇒ i = 0.108

So if the rate of interest is 10.8% Robert would be indifferent between starting his

own business and continuing his current job.

(g) If Robert were to buy a car and gas prices rose to $4.50 per gallon, how many hours

per day would he drive and what would be his daily profit? If gas prices were instead

$4.50 per gallon, for each hour Robert earns net profit

1st hr 2nd hr 3rd hr 4th hr 5th hr 6th hr 7th hr 8th hr 9th hr 10th hr

$14.50 $18.50 $16.50 $15.50 $14.50 $13.50 $12.50 $11.50 $10.50 $8.50

So if Robert values his leisure time at $13 per hour, he will only work 6 hours per

day.