Chapter 1
The Facts to Be Explained Solutions to Problems 1. A ratio ratio scale scale transfor transforms ms absolu absolute te differen differences ces in the variabl variablee of intere interest st to proportional proportional differences. differences. For instance, instance, the GDP of country X , whose GDP is 10 times greater than country Y , will be the same distance apart as a country Z whose whose GDP is 10 times smaller than country Y ’s ’s GDP, ie. the distance between X, Y , and Z will be the same. On a common linear linear scale, the distance distance between between X and and Y would be 10 times greater than the distance distance between between Y and Z . As a result, transformi transforming ng Figure 1.1 into into a ratio scale will convey the absolute differences in the height of marchers into proportional differences.
The Parade of World Income
$100,000
e l a c s o i t a r , 0 0 0 2 , a t i p a c r e p P D G
$10,000
$1,000
$100 0% 0
20%
40%
60%
Percent of World Population
80%
100% 5.6 billion
2. Using Using the rule of 72, we know that that GDP GDP per capita capita will will doubl doublee every every 72/2 72/2 years, years, ie. every 36 years. years. Therefore, Therefore, if in year 0, GDP per capita capita is x, in year 36, GDP per capita will be 2 x. Continuing Continuing with with the same logic, logic, in year 36 + 36 (= year 72), GDP per capita will be 4 x, and in year 36 + 36 + 36 (= year 108), GDP per capita will be equal to 8 x. We now arrive at at our solution: solution: It will will take approximat approximately ely 108 years years for GDP per capita to increase by a factor of eight.
1
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 in the year 1900 with a GDP per capita of $1,000, to reach $4,000 by the year 1948, GDP per capita must have doubled twice. To see this, note that after doubling once, GDP per capita 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 twice within 48 years and assuming a constant annual growth rate, we conclude that GDP per capita doubles every 24 years. Solving for the equation, 72 / g = 24, we get g , the annual growth rate, to be 3% per year. 4. Between-country inequality is the inequality associated with average incomes of different countries. Country A’s average income is given by adding Alfred’s Income and Doris’s Income and then dividing by 2. This yields an average income of 2,500 for Country 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 no between-country inequality in this world. Within-country inequality is the inequality associated with incomes of people in the same country. In Country A, Alfred earns 1,00 while Doris earns 4,000, making it an income disparity of 3,000. In Country B, the income disparity is 1,000. Therefore, we see within-country income inequality in both Country A and Country B. Because there is no between-country inequality, world inequality can be entirely attributed to within-country inequality. Equivalently, one could calculate the mean log deviation to attain values for within- and between-country inequality. Using the formula on page 19, the value for betweencountry inequality is 0 whereas, the value for within-country inequality for Country A and Country 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 the appropriate values into the equation: (Y 1900) * (1+g)100 = Y 2000 . Letting Y 1900 = $1,433, Y 2000 = $26,375, and rearranging to solve for g , we get : g = ($ 26,375 / $ 1,433) g
!
(1 / 100)
– 1,
0.0296.
Converting g into a percent, we conclude that the growth rate of income per capita in Japan 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 (Y 2000) * (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 the income 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 was equal to income per capita in Sri Lanka, we need to find t , the number of years that passed between the year 2000 and the year US income per capita equaled that of 2000 Sri Lanka income per capita. Equating income per capita of Sri Lanka in year 2000 to income per capita of the United States in year 2000–t , we now write an equation for the United States as (Y US , 2000-t ) * (1+g)t = Y US , 2000 . Since Y US , 2000-t = Y Sri Lanka, 2000 = $3,527 , Y US , 2000 = $35,587 , and g = 0.019, we then substitute 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 Sri Lanka’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 the only traded good, the price of computers should be the same. Consequently, the exchange 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 first convert GDP denominations into the same currency. In the analysis that follows, I choose to convert GDP denominations into Poorland dollars, but converting to Richland dollars is equally correct, similar, and will yield the same result. From part (a), we convert Richland’s GDP per capita, denominated in Richland dollars, into Poorland Dollars 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’s GDP per capita, yielding 20 Poorland Dollars. Thus, the ratio of Richland GDP per capita 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 in Richland 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 rate must be, 10 Richland Dollars : 4 Poorland Dollars (e) To find the ratio of GDP per capita between Richland and Poorland, we must first convert GDP denominations into the same currency. In the analysis that follows, I choose to convert GDP denominations into Poorland dollars, but converting to Richland dollars is equally correct, similar, and will yield the same result. From part (a), we convert Richland’s GDP per capita, denominated in Richland dollars, into Poorland Dollars 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’s GDP per capita, yielding 16 Poorland Dollars. Thus the ratio of Richland GDP per capita to 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 in capital is exactly off set by depreciation in capital. (Note: there is no population growth here). If we let the investment rate be given by γ , then the investment level is equal to γ y = γ k . If capital depreciates at rate δ , then the steady state capital stock (k ss ) is given by the following equality: γ k ss = δ k ss . With γ = 0.5 and δ = 0.05, we have k ss = 102 = 100. At 400, the present capital stock thus exceeds the steady-state stock. This means that the stock will go down over time. Indeed, we can verify this with the following: 1 2
1 2
1
∆k = γ k 2
δ k = 0.5 ∗ (400)
−
1 2
0.05 ∗ 400 = −10 < 0.
−
At k t = 400, depreciation exceeds investment.
3. An example in biology is that of the deer population on an island. The quantity of deers that can be supported by the island is limited by the food available on it. If there are very few deers, the food is abundant and their population will grow fast, i.e. births numbers exceed deaths 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 size goes down. Between these two extremes, there must be a long-run equilibrium number of deers that can be supported indefinitely into the future as the numbers of deaths and births are 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 functional form, i.e. y = Ak α , we have, in the steady-state: γ Ak α = δ k.
Which yields the following steady-state capital stock: kss =
µ A∂ γ δ
1 1−α
.
Inserting this value in the output function, we get the following SS: y
ss
= Ak
ssα
= A
1 1−α
γ δ
≥¥
α
1−α
.
If two countries diff er solely by their investment rate: A yiss = y jss A
1 1−α 1 1−α
γ i δ γ j δ
°¢ °¢
α
1−α α
1−α
=
µ ∂ γ i γ j
α
1−α
=
µ 0.05 ∂ 0.2
1/3 1−1/3
= 0.5.
In the long run, i.e. at the steady-state, income per capita in country j will be twice that of country i because the latter’s savings rate is four times lower. 1
2
But if α = 2/3, we have yiss == y jss
µ ∂ γ i γ j
α
1−α
=
µ 0.05 ∂
2/3 1−2/3
= 0.0625 =
0.2
1 . 16
In the long run, income per capita in country j will now be sixteen times that of country i because 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 model predicts that: α
1/3
ss yT 0.303 − γ T − = = 0.099 yBss γ B In reality, the income per capita ratios is: 14260 = 2.06, 6912 which is somewhat close to the Solow model prediction.
µ ∂
1
µ
α
∂
1
1/3
= 1.75.
5.b) In this case, the Solow model predicts: ss yN ss yT
=
µ ∂ γ N γ T
α
1−α
=
µ 0.075 ∂
1/3 1−1/3
0.146
= 0.717.
While in reality, the income per capita ratios is: 3648 = 0.209, 17491 which is quite far from the Solow model’s predictions.
5.c) In this case, the Solow model predicts: yJ ss = ss yN
µ ∂ γ J γ N
α
1−α
=
µ 0.313 ∂
1/3 1−1/3
0.207
= 1.23.
While in reality, the income per capita ratios is: 48389 = 1.116, 43360 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 stock is 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. The fact that both countries now have the same income per capita suggests that the investment rate in country X is higher than in country Y .
3
7.a) The per capita level of capital (kss ) in SS must respect: γ kss1/2 = δ kss . Hence 2 kss = γ δ = 25 and ySS = 25 = 5. 7.b) In period 2, you should get: k = 16.2, y = 4.02, γ y = 1.005, δ 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, γ y = 1.041, δ k = 0.87, ∆k = 0.174. 7.c) The growth rate between years 1 and 2 is: 4.02 − 4 X 2 − X 1 g= = = 0.005 = 0.5%. X 1 4 7.d) The growth rate between years 7 and 8 is: 4.16 − 4.14 g = = 0.0048 = 0.48%. 4.14 6.e) The growth rate goes down the closer is the economy to its steady-state value. 1
°¢
2
8. (See accompanying graphic.) If y = c , then investment is i = 0. If y > c then i = γ (y − c ) = γ (f (k) − c ). The output and the depreciation curves are not aff ected by this. But the investment curves shifts down as per the accompanying figure. There is a strictly positive income level below which investment becomes nil. This income level is referred to as the subsistence income level. 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 k0ss et k1ss , with k0ss < k1ss . However, only k1ss denotes a stable steady-state. Indeed, any deviation around that value will bring the economy back to it. In the case of k0ss , a deviation to the right-hand side will send the economy over to k1ss 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 k0ss , 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 the long run, capital always disappears. ∗
∗
∗
∗
