1

Chapter 1

The Facts to Be Explained

Solutions to Problems

1. A ratio scale transforms absolute differences in the variable of interest toproportional differences. For instance, the GDP of country X, whose GDP is 10 timesgreater than country Y, will be the same distance apart as a country Z whose GDP is 10times smaller than country Y’s GDP, ie. the distance between X, Y, and Z will be thesame. On a common linear scale, the distance between X and Y would be 10 timesgreater than the distance between Y and Z. As a result, transforming Figure 1.1 into aratio scale will convey the absolute differences in the height of marchers intoproportional differences.

The Parade of World Income

$100

$1,000

$10,000

$100,000

0% 20% 40% 60% 80% 100%Percent of World Population

GDP

percapita,2000,ratio

scale

0 5.6 billion

2. Using the rule of 72, we know that GDP per capita will double every 72/2 years,ie. every 36 years. Therefore, if in year 0, GDP per capita is x, in year 36, GDP percapita will be 2x. Continuing with the same logic, in year 36 + 36 (= year 72), GDP percapita will be 4x, and in year 36 + 36 + 36 (= year 108), GDP per capita will be equal to8x. We now arrive at our solution: It will take approximately 108 years for GDP percapita to increase by a factor of eight.

2 � Weil Economic Growth

3. Using the rule of 72, we know that GDP per capita will double every 72/g years,where g is the annual growth rate of GDP per capita. Working backwards, if we start inthe year 1900 with a GDP per capita of $1,000, to reach $4,000 by the year 1948, GDPper capita must have doubled twice. To see this, note that after doubling once, GDP percapita would be $2,000 in some year, and doubling again, GDP per capita would be$4,000, exactly the GDP per capita in year 1948. Using the fact that GDP doubled twicewithin 48 years and assuming a constant annual growth rate, we conclude that GDP percapita doubles every 24 years. Solving for the equation, 72 / g = 24, we get g, the annualgrowth rate, to be 3% per year.

4. Between-country inequality is the inequality associated with average incomes ofdifferent countries. Country A’s average income is given by adding Alfred’s Income andDoris’s Income and then dividing by 2. This yields an average income of 2,500 forCountry A. Similar calculations reveal that Country B’s average income is 2,500.Because the average income for Country A is equal to that of Country B, there is nobetween-country inequality in this world.

Within-country inequality is the inequality associated with incomes of people in the samecountry. In Country A, Alfred earns 1,00 while Doris earns 4,000, making it an incomedisparity of 3,000. In Country B, the income disparity is 1,000. Therefore, we seewithin-country income inequality in both Country A and Country B.Because there is no between-country inequality, world inequality can be entirelyattributed to within-country inequality.

Equivalently, one could calculate the mean log deviation to attain values for within- andbetween-country inequality. Using the formula on page 19, the value for between-country inequality is 0 whereas, the value for within-country inequality for Country A andCountry B is 0.223 and 0.020, respectively. This implies the same conclusion as before.

5. We can solve for the average annual growth rate, g, by substituting theappropriate values into the equation:

(Y1900) * (1+g)100 = Y 2000 .

Letting Y1900 = $1,433, Y 2000 = $26,375, and rearranging to solve for g, we get :

g = ($ 26,375 / $ 1,433) (1 / 100) – 1,

g ≈ 0.0296.

Converting g into a percent, we conclude that the growth rate of income per capita inJapan over this period was approximately 2.96% per year.

To find the income per capita of Japan 100 years from now in 2100, we solve

(Y2000) * (1+g)100 = Y 2100 .

Chapter 1 The Facts to be Explained � 3

Letting Y 2000 = $26,375 and g = 0.0296,

($ 26,375) * (1+ 0.0296)100 = Y 2100 ,

Y 2100 = $ 485,443.60.

That is, if Japan grew at the average growth rate of 2.96% per year, we would find theincome per capita of Japan in 2100 to be about $485,443.60.

6. In order to calculate the year in which income per capita in the United States wasequal to income per capita in Sri Lanka, we need to find t, the number of years thatpassed between the year 2000 and the year US income per capita equaled that of 2000 SriLanka income per capita. Equating income per capita of Sri Lanka in year 2000 toincome per capita of the United States in year 2000–t, we now write an equation for theUnited States as

(YUS, 2000-t) * (1+g)t = Y US, 2000 .

Since YUS, 2000-t = YSri Lanka, 2000 = $3,527 , Y US, 2000 = $35,587 , and g = 0.019, we thensubstitute in these values and solve for t.

($3,527) * (1+0.019)t = $35,587.

(1+0.019)t =( $35,587 /$3,527)

Taking the Natural Log of both sides, and noting that ln(x y) = y ln (x), we get

t ln (1+0.019) =ln ( $35,587 /$3,527)

t = 122.81

That is, 122.81 years ago, the income per capita of the United States equaled that of SriLanka’s income in the year 2000. This year was roughly 2000–t, ie. the year 1877.

Solutions to Appendix Problems

1. (a) The level of GDP per capita in each country, measured in its own currency is

(CPUs per capita* Price) + (IC per capita* Price) = GDP per capita.

Therefore, Richland’s GDP per capita is 40 and Poorland’s GDP per capita is 4.

4 � Weil Economic Growth

(b) The market exchange rate is determined by the law of one price. As CPUs are theonly traded good, the price of computers should be the same. Consequently, theexchange rate must be 2 Richland dollars to 1 Poorland dollar.

(c) To find the ratio of GDP per capita between Richland and Poorland, we must firstconvert GDP denominations into the same currency. In the analysis that follows, Ichoose to convert GDP denominations into Poorland dollars, but converting to Richlanddollars is equally correct, similar, and will yield the same result. From part (a), weconvert Richland’s GDP per capita, denominated in Richland dollars, into PoorlandDollars by multiplying GDP per capita with the market exchange rate. Since from part(b), we know 2 Richland Dollars equals 1 Poorland Dollar, we multiply 1/2 to Richland’sGDP per capita, yielding 20 Poorland Dollars. Thus, the ratio of Richland GDP percapita to Poorland GDP per capita is 5:1.

(d) A natural basket to use is 3 computers and 1 ice cream. The cost of this basket inRichland is 10 Richland dollars. The cost of this basket in Poorland is 4 Poorland dollars.Equating the costs of baskets to be one price, the purchasing power parity exchange ratemust be, 10 Richland Dollars : 4 Poorland Dollars

(e) To find the ratio of GDP per capita between Richland and Poorland, we must firstconvert GDP denominations into the same currency. In the analysis that follows, Ichoose to convert GDP denominations into Poorland dollars, but converting to Richlanddollars is equally correct, similar, and will yield the same result. From part (a), weconvert Richland’s GDP per capita, denominated in Richland dollars, into PoorlandDollars by multiplying GDP per capita with the PPP exchange rate. Since from part (d),we know 10 Richland Dollars equals 4 Poorland Dollars, we multiply 4/10 to Richland’sGDP per capita, yielding 16 Poorland Dollars. Thus the ratio of Richland GDP per capitato Poorland GDP per capita is 4:1.

SOLUTIONS CHAP 3 WEIL 2nd ed

2. In the steady state, the growth rate of capital must be zero because investment incapital is exactly oÆset by depreciation in capital. (Note: there is no population growthhere). If we let the investment rate be given by ∞, then the investment level is equal to

∞y = ∞k12 . If capital depreciates at rate ±, then the steady state capital stock (kss) is given

by the following equality:

∞kss 12 = ±kss.

With ∞ = 0.5 and ± = 0.05, we have kss = 102 = 100. At 400, the present capital stock thusexceeds the steady-state stock. This means that the stock will go down over time. Indeed,we can verify this with the following:

¢k = ∞k12 ° ±k = 0.5 § (400)

12 ° 0.05 § 400 = °10 < 0.

At kt = 400, depreciation exceeds investment.

3. An example in biology is that of the deer population on an island. The quantity of deersthat can be supported by the island is limited by the food available on it. If there are very fewdeers, the food is abundant and their population will grow fast, i.e. births numbers exceeddeaths numbers. Conversely, if there are very many deers suddenly brought on the island,food availability per deer will be low and deaths numbers will exceed births; population sizegoes down. Between these two extremes, there must be a long-run equilibrium number ofdeers that can be supported indefinitely into the future as the numbers of deaths and birthsare equal. This is another instance of a steady-state equilibrium in a dynamic setting.

4. Assuming that output per capita can be represented by a Cobb-Douglas functionalform, i.e. y = AkÆ, we have, in the steady-state:

∞AkÆ = ±k.

Which yields the following steady-state capital stock:

kss =

µ∞A

±

∂ 11°Æ

.

Inserting this value in the output function, we get the following SS:

yss = AkssÆ = A1

1°Æ

≥∞

±

¥ Æ1°Æ

.

If two countries diÆer solely by their investment rate:

yssi

yssj

=A

11°Æ

°∞i

±

¢ Æ1°Æ

A1

1°Æ°∞j

±

¢ Æ1°Æ

=

µ∞i

∞j

∂ Æ1°Æ

=

µ0.05

0.2

∂ 1/31°1/3

= 0.5.

In the long run, i.e. at the steady-state, income per capita in country j will be twice that ofcountry i because the latter’s savings rate is four times lower.

1

2

But if Æ = 2/3, we have

yssi

yssj

==

µ∞i

∞j

∂ Æ1°Æ

=

µ0.05

0.2

∂ 2/31°2/3

= 0.0625 =1

16.

In the long run, income per capita in country j will now be sixteen times that of country ibecause the latter’s savings rate is four times lower.

5.a) If we follow the same procedure as that of the preceding problem, the Solow modelpredicts that:

yssT

yssB

=

µ∞T

∞B

∂ Æ1°Æ

=

µ0.303

0.099

∂ 1/31°1/3

= 1.75.

In reality, the income per capita ratios is:

14260

6912= 2.06,

which is somewhat close to the Solow model prediction.

5.b) In this case, the Solow model predicts:

yssN

yssT

=

µ∞N

∞T

∂ Æ1°Æ

=

µ0.075

0.146

∂ 1/31°1/3

= 0.717.

While in reality, the income per capita ratios is:

3648

17491= 0.209,

which is quite far from the Solow model’s predictions.

5.c) In this case, the Solow model predicts:

yssJ

yssN

=

µ∞J

∞N

∂ Æ1°Æ

=

µ0.313

0.207

∂ 1/31°1/3

= 1.23.

While in reality, the income per capita ratios is:

48389

43360= 1.116,

which is somewhat close to the Solow model’s predictions.

6. The fact that output per capita grows in country X suggests that its capital stockis now below its SS value, and conversely for country Y . According to the Solow model,income per capita and capital per capita at the SS both increase with the savings rate. Thefact that both countries now have the same income per capita suggests that the investmentrate in country X is higher than in country Y .

3

7.a) The per capita level of capital (kss) in SS must respect: ∞kss1/2 = ±kss. Hence

kss =°

∞±

¢2= 25 and ySS = 25

12 = 5.

7.b) In period 2, you should get: k = 16.2, y = 4.02, ∞y = 1.005, ±k = 0.81, ¢k = 0.195.Hence, the period 3 capital stock is k = 16.395. And so on. In period 8, you should get:k = 17.33, y = 4.16, ∞y = 1.041, ±k = 0.87, ¢k = 0.174.

7.c) The growth rate between years 1 and 2 is:

g =X2 °X1

X1=

4.02° 4

4= 0.005 = 0.5%.

7.d) The growth rate between years 7 and 8 is:

g =4.16° 4.14

4.14= 0.0048 = 0.48%.

6.e) The growth rate goes down the closer is the economy to its steady-state value.

8. (See accompanying graphic.)If y = c§, then investment is i = 0. If y > c§ then i = ∞(y ° c§) = ∞(f(k)° c§).The output and the depreciation curves are not aÆected by this. But the investment curves

shifts down as per the accompanying figure. There is a strictly positive income level belowwhich investment becomes nil. This income level is referred to as the subsistence incomelevel.

If the depreciation rate is not too high, it crosses the investment curve at two places.There are thus two possible steady-states, which we refer to as kss

0 et kss1 , with kss

0 < kss1 .

However, only kss1 denotes a stable steady-state. Indeed, any deviation around that value

will bring the economy back to it. In the case of kss0 , a deviation to the right-hand side will

send the economy over to kss1 in the long run, while a deviation to the left-hand side will lead

to an eventual disappearance of capital. Indeed, to the LHS of kss0 , depreciation is always

above investment, while the converse holds to the RHS.Finally, if the depreciation rate were too high, then there is no crossing between the

investment and the depreciation curves, the latter being always above the former. In thelong run, capital always disappears.