Book solution economic growth david n weil chapters 1 8 PDF

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Economic Growth and Development EC 375

Prof. Murphy

Problem Set 1 Answers Chapter 1 #2, 3, 4, 5, 6, 7 (on pages 24-25) and Appendix problems A.1 and A.2 (on pages 28-29). 2.

Let g be the rate of growth. The rule of 72 says that 72/g ! 9. So g ! 8%.

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 three percent 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 1,000 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 withincountry income inequality in both Country A and Country B. Because there is no betweencountry inequality, world inequality can be entirely attributed to within-country inequality.

5.

We can solve for the average annual growth rate, g, by substituting the appropriate values into the equation: (Y1900) × (1 + g)100 = Y2000. Letting Y1900 = $1,433, Y2000 = $23,971, and rearranging to solve for g , we get: g = ($23,971/$1,433)(1/100) – 1, g ! 0.0286. Converting g into a percent, we conclude that the growth rate of income per capita in Japan over this period was approximately 2.86 percent per year. To find the income per capita of Japan 100 years from now, in 2100, we solve (Y2000) × (1 + g)100 = Y2100. Letting Y2000 = $23,971 and g = 0.0286, ($23,971) × (1 + 0.0286)100 = Y2100,

2

Y2100 = $402,103.76. That is, if Japan grew at the average growth rate of 2.86 percent per year, we would find the income per capita of Japan in 2100 to be about $402,103.76. 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 2005 and the year U.S. income per capita equaled that of 2005 Sri Lanka income per capita. Equating income per capita of Sri Lanka in year 2005 to income per capita of the United States in year 2005 – t, we now write an equation for the United States as (YU.S., 2005 – t) × (1 + g)t = YU.S., 2005. Since YU.S., 2005 – t = YSri Lanka, 2005 = $4,650, YU.S., 2005 = $36,806, and g = 0.019, we then substitute in these values and solve for t. ($4,650) × (1 + 0.019)t = $36,806. (1 + 0.019)t = ($36,806/$4,650). One can solve for t by simply trying out different values on a calculator. Alternatively, taking the natural log of both sides, and noting that ln(x y ) = y ln(x), we get t ln(1 + 0.019) = ln($36,806/$4,650) t = 109.92. That is, 109.92 years ago, the income per capita of the United States equaled that of Sri Lanka’s income in the year 2005. This year was roughly 2005 – t, i.e., the year 1895.

7.

In order to calculate the year in which income per capita in China will overtake the income per capita in the United States, we first need to find t , the number of years it will take for the income per capita in both countries to be equal. That is, (YU.S., 2005 ) × (1 + .022)t = (YChina, 2005) × (1 + .073)t. Since YU.S., 2005 = $36,806, YChina, 2005 = $5,955, we then substitute in these values and solve for t. (1 + 0.073/1 + .022)t = ($36,806/$5,955). We can solve for t by trying out different values on a calculator. Alternatively, taking the Natural Log of both sides, and noting that ln(x y ) = y ln(x), we get t ln(1.05) = ln($36,806/$5,955) t = 37.33. That is, in 37.33 years, assuming they grow at the current growth rates, the income per capita of China will surpass that of the United States. This year will roughly be 2005 + t , i.e., the year 2042.

3

Appendix Questions A.1.

The number of people living on less than a dollar a day will be larger if we calculate it using market exchange rates instead of purchasing power exchange rates because market exchange rates only take into account the relative value of traded goods, which are relatively more expensive in poorer countries. Individuals in these countries will have low purchasing power for traded goods. By using the market exchange rate, we are assuming that traded goods and non-traded goods are the same price, and therefore individuals in poor countries will have low purchasing power for non-traded goods as well, which will make them appear poorer than they actually are.

A.2.

a. The level of GDP per capita in each country, measured in its own currency is (Computers per capita × Price) + (Ice Cream per capita × Price) = GDP per capita. Therefore, Richland’s GDP per capita is 40 and Poorland’s GDP per capita is 4. b. The market exchange rate is determined by the law of one price. As computers 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 the world consumption basket: 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.

4

Chapter 2 # 1, 2, 3, 4, 7, 8 (on page 46). 1.

Proximate causes are causes that directly affect the variable of interest. Low levels of physical and human capital, technology, and efficiency are all examples of a proximate cause of low GDP per capita.

2.

Fundamental causes are causes that indirectly affect the variable of interest by systematically affecting one or many other causes that in turn affect the variable of interest. Possible fundamental causes may be government, culture, ethnic composition, rule of law, geography, climate, resources, and so forth. These causes affect GDP per capita by affecting the proximate causes of low GDP per capita.

3.

To show different levels of factors of production, the figures must not intersect at the same level of output. To show different levels of productivity, the figures must have different slopes. In the figure below, Country 1 and Country 2 have the same level of output per worker. However, Country 1 has a higher level of factors of production than does Country 2, and Country 1 has a lower level of productivity than does Country 2.

4.

In the long run, the two countries would be expected to have the same levels (and thus growth rates) of income, because they have the same fundamentals. In the short run Country B would be expected to have faster growth because the two countries are moving toward having similar income levels, but Country B is starting out with a lower level.

7.

a.

Although the majority of right-wing voters may live longer, the inference that being a political conservative is good for you is incorrect because correlation does not imply causation. A majority of right-wing voters may live longer, not because they are conservative but rather, because they lead healthier lifestyles that right-wing policies promote. Thus, we have an omitted third variable affecting both the choice of party affiliation and the length of life. b. Although people in hospitals are generally less healthy than those outside hospitals, the inference that one should avoid hospitals is incorrect because of reverse causation. That is, a majority of people go to the hospital because they are unhealthy in contrast to the reverse inference, whereby going to the hospital makes one unhealthy.

5

8.

a.

Positive Correlation. It is reasonable to assume that higher (lower) GDP per capita increases (decreases) available expenditure for printing books. Moreover, it is also reasonable to assume that a greater (smaller) number of books printed per capita increases (decreases) the level of education within a country, translating into higher (lower) levels of GDP per capita. b. Negative Correlation. The higher GDP is per capita, the more likely it is that basic nutrition needs of the population will be met, and the smaller the number of people suffering from malnutrition, the more likely it is that there will be a healthier labor force to produce higher levels of GDP per capita. Hence, higher GDP per capita should be correlated with lower fractions of people suffering from malnutrition and vice versa. c. No Correlation or Positive Correlation. There are two things to consider. First, does eyesight progressively deteriorate with age? Second, does the level of GDP positively affect both one’s ability to diagnose and correct vision problems and one’s life expectancy through access to better nutrition, health care, and so on? If one does not assume the above to be true, then there should be no correlation between life expectancy and the population that wears eyeglasses. On the other hand, if one does assume the above to be true, then one should see high life expectancy figures when one sees a high fraction of people wearing eyeglasses, for the simple reason that there is a large elderly population with poor vision able to afford glasses. d. No Correlation. There is no obvious relationship between the number of letters in a country’s name and the number of automobiles per capita.

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 offset 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 1 γy = γk 2 . If capital depreciates at rate δ, then the steady state capital stock (k ss ) is given by the following equality: 1 γk ss 2 = δk 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

1

∆k = γk 2 − δk = 0.5 ∗ (400) 2 − 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 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: γAk α = δk. Which yields the following steady-state capital stock: 1 µ ¶ 1−α γA ss . k = δ Inserting this value in the output function, we get the following SS: ³γ ´ α 1 1−α y ss = Ak ssα = A 1−α . δ If two countries differ solely by their investment rate: ¢ α α 1 ¡ µ µ ¶1−α ¶ 1/3 A 1−α γδi 1−α 0.05 1−1/3 γi yiss = = = 0.5. ¡γ ¢ α = 1 yjss γj 0.2 A 1−α δj 1−α

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 == yjss

µ

γi γj

α ¶ 1−α

=

µ

0.05 0.2

¶ 1−2/23/3

= 0.0625 =

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: µ ¶ α µ ¶ 1/3 yTss γT 1−α 0.303 1−1/3 = = = 1.75. yBss γB 0.099 In reality, the income per capita ratios is: 14260 = 2.06, 6912 which is somewhat close to the Solow model prediction. 5.b) In this case, the Solow model predicts: α ¶ 1/3 µ µ ¶ 1−α yNss 0.075 1−1/3 γN = 0.717. = = 0.146 yTss γT 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: µ ¶ α µ ¶ 1/3 yJss γJ 1−α 0.313 1−1/3 = = = 1.23. ss yN γN 0.207 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 (k ss ) in SS must respect: γk ss1/2 = δk ss . Hence ¡ ¢2 1 k ss = δγ = 25 and y SS = 25 2 = 5. 7.b) In period 2, you should get: k = 16.2, y = 4.02, γy = 1.005, δk = 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, γy = 1.041, δk = 0.87, ∆k = 0.174. 7.c) The growth rate between years 1 and 2 is: X2 − X1 4.02 − 4 = 0.005 = 0.5%. = g= 4 X1 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. 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 affected 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 k 0ss < 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.

SOLUTIONS CHAP 4 1. As seen in chapter one, the formula is (

Xt+n g= Xt

)1

n

(

6400000000 −1= 2

)

1 100000

− 1 = 0.000218888 = 0.0218%.

2. The graphs accompanying this answer are in the file solutions_graph_chap4b.pdf. 2.a) The initial equilibrium is at point A, where population does not grow. With the new seed variety, each worker can produce more; curve Z shifts to the right at Z ′ . In the short run, output jumps to point B, resulting in a higher per capita output. This higher output per capita causes the population to grow, as seen on the lower graphic. In the long run, the new steady-state equilibrium is at point C, where consumption per capita is the same as before the introduction of the new seed variety. ′ At LSS , the total population size is however larger than initially. 2.b) The initial equilibrium is at point A, where population does not grow. Population size suddenly drops by half: the economy jumps to point B suddenly. This increases per capita output because there is more land available per worker. As a result, population starts to grow, as indicated by the lower graph. In the long run, the economy returns to its initial point A. 2.c) The initial equilibrium is at point A, where population does not grow. With the destruction of half of the land, output per capita reduces by half for a given population size (assuming constant returns to land). This displaces curve Z to the ′ left to Z ′ . But population size also decreases by half simultaneously, from LSS a` LSS . The economy thus jumps from one steady-state to another one without any transition period. Income per capita is the same at the new steady-state but the population size has reduced by half. 3. The initial equilibrium is at point A, where population does not grow. The population growth curve suddenly shifts from V to V ′ : at each given income level, people want more kids. Population growth suddenly jumps to point B and becomes positive. As population size increases, income per capita decreases. In the long run, ′ income per capita reaches level y SS , which is lower than initially and population size ′ is larger at LSS . 5. The ratio of incomes per capita in the steady-state is given by: 1

2

y˜i = y˜j

A A

1 1−α

1 1−α

(

(

γi δ+ni γj δ+nj

α ) 1−α

)

α 1−α

=

(

γi δ+ni γj δ+nj

α ) 1−α

=

(

0.20 0.05+0 0.05 0.05+0.04

)

1/3 1−1/3

= 2.683.

6. The graphs accompanying this answer are in the file solut...