Answers to Chapters 2 and 3, Debraj Ray Solutions, Development of Economics PDF Download

 Page 1


Professor Debraj Ray
2002 Topics in Development Economics
Sketches of Answers to Problems, Chapters 2 and 3.
The answers below are brief and try to give you the basic idea of how to approach these
problems. You will gain a lot more from studying these answers if you spend some time
independently trying to work on the problems.
Chapter 2.
(1) A traded good is one that can be bought and sold through the international market. A
nontraded good cannot. Of course, these are extreme descriptions of reality, and some goods
may be partially but not fully tradeable.
Equilibrium exchange rates are determined by the supply of and demand for a country’s
currency. The supply of a country’s currency is determined by that country’s purchases of
imports on the world market. The demand for its currency is determined by the purchase of
that country’s exports by the world. The exchange rate acts to equalize these two (thereby
creating trade balance).
Notice that a rich and productive country is likely to have a stronger currency and a
higher income. The higher income, in turns, pulls up the prices of those products within
the country that are nontraded. Thus measured in terms of exchange rate income, a rich
country looks richer than it really is, because, we are not accounting for the fact that it faces
(on average) higher donestic prices for the nontraded goods. This is why PPP measurements
typically bring down the relative income of a rich country, and pull up the relative income of
a poor country.
(2) The price of a Big Mac is, to a large extent, determined by the prices in competing
restaurants. Thus Big Macs will sell for a higher price in rich countries (where nontraded
restaurant prices are likely to be higher). Thus Big Mac prices incorporate, to some extent
— and often to a better extent than exchange rates do — the “true” cost of living within a
country. Therefore, using the relative prices of Big Macs to create “exchange rates” across
country currencies will often serve as a good approximation of relative PPP income.
(3) The setting up of infrastructure or industrial standards involves a “sunk cost” and a
“variable cost”. The sunk cost is the cost of the entire infrastructure: in the example of
tv systems, this would be all kinds of interconnected senders and receivers that broadcast
at a particular type and re?nement of resolution. The variable cost refers to the individual
purchase of tv sets. Now given that the infrastructure is not set up for it, it makes a little
sense for customers to buy high-de?nition tv sets. Moreover, given the huge sunk costs,
countries which have already invested in a particular standard may not want to tear up this
standard and start all over again, unless the new technology is much, much better.
However, a country that is starting afresh will obviously want to use the latest technology:
there is no past, no sunk cost, to be borne in mind. This is why countries which have been
early innovators are often saddled with older systems, and newcomers can leapfrog over them
1
Page 2


Professor Debraj Ray
2002 Topics in Development Economics
Sketches of Answers to Problems, Chapters 2 and 3.
The answers below are brief and try to give you the basic idea of how to approach these
problems. You will gain a lot more from studying these answers if you spend some time
independently trying to work on the problems.
Chapter 2.
(1) A traded good is one that can be bought and sold through the international market. A
nontraded good cannot. Of course, these are extreme descriptions of reality, and some goods
may be partially but not fully tradeable.
Equilibrium exchange rates are determined by the supply of and demand for a country’s
currency. The supply of a country’s currency is determined by that country’s purchases of
imports on the world market. The demand for its currency is determined by the purchase of
that country’s exports by the world. The exchange rate acts to equalize these two (thereby
creating trade balance).
Notice that a rich and productive country is likely to have a stronger currency and a
higher income. The higher income, in turns, pulls up the prices of those products within
the country that are nontraded. Thus measured in terms of exchange rate income, a rich
country looks richer than it really is, because, we are not accounting for the fact that it faces
(on average) higher donestic prices for the nontraded goods. This is why PPP measurements
typically bring down the relative income of a rich country, and pull up the relative income of
a poor country.
(2) The price of a Big Mac is, to a large extent, determined by the prices in competing
restaurants. Thus Big Macs will sell for a higher price in rich countries (where nontraded
restaurant prices are likely to be higher). Thus Big Mac prices incorporate, to some extent
— and often to a better extent than exchange rates do — the “true” cost of living within a
country. Therefore, using the relative prices of Big Macs to create “exchange rates” across
country currencies will often serve as a good approximation of relative PPP income.
(3) The setting up of infrastructure or industrial standards involves a “sunk cost” and a
“variable cost”. The sunk cost is the cost of the entire infrastructure: in the example of
tv systems, this would be all kinds of interconnected senders and receivers that broadcast
at a particular type and re?nement of resolution. The variable cost refers to the individual
purchase of tv sets. Now given that the infrastructure is not set up for it, it makes a little
sense for customers to buy high-de?nition tv sets. Moreover, given the huge sunk costs,
countries which have already invested in a particular standard may not want to tear up this
standard and start all over again, unless the new technology is much, much better.
However, a country that is starting afresh will obviously want to use the latest technology:
there is no past, no sunk cost, to be borne in mind. This is why countries which have been
early innovators are often saddled with older systems, and newcomers can leapfrog over them
1
with the newer technology. Television and telephone systems are only two examples of these.
Chapter 5 discusses these considerations in detail.
(4) Look at footnote 7 on page 17 in the book and make sure you understand the details.
Using this formula, a country growing at 10% per year will double its income in seven years,
while a country growing at 5% will take 14 years to do so. Now try a direct argument using
a calculator. If a country has income x today and is growing at 10%, it will have an income
of x(1 +
1
10
) next year. If you understand this, you can see that thinking of this number as
the “new x”, income the year after will be scaled up by the same formula. This means that
income the day after is just x(1 +
1
10
)
2
. Plodding on in this vein, we see that income after t
years is x(1 +
1
10
)
t
. Now think of t as an unknown, and we wish to know: for what value of
t is x(1 +
1
10
)
t
equal to 2x? How would you solve this using a calculator?
(5) A mobility matrix with no mobility should show 100 on its principal diagonal and zero
everywhere else. This means that countries which were at a certain relative category of world
income at the base date would be in the same relative category at the later date. In contrast,
a mobility matrix that exhibits perfect mobility should have equal percentages along its rows
(adding to 100). This means that countries in some relative category at the base date have
an equal chance of being in any relative category at some later date.
If poor countries grow faster than rich countries, then two things happen: both poor
countries and rich countries would be pushed closer to the world average, the former upward
and the latter downward. This means that the mobility matrix would tend to have its larger
numbers (on each row) closer to the categories that are clustered around world average
income.
You are encouraged to write down speci?c mobility matrices that capture this notion of
convergence, as well as examples of those that capture divergence.
(6) Using Table 2.1 to construct a Kuznets ratio, I get the following sequence of numbers,
corrected to one decimal place (written in order of ascending income): 2.5, 2.8, 2.0, 2.4, 5.4,
4.6, 1.9, 4.4, 1.7, 3.6, 7.9, 9.3, 4.5, 3.8, 7.5, 5.4, 6.0, 4.2, 5.4, 2.1, 2.2, 2.4, 1.5, 2.0, 2.2, 2.3,
2.9.
If income were distributed almost equally, then I would expect that the poorest 40%
would obtain almost 40% of the total income, while the richest 20% would earn a little bit
more than 20% of the total income. The ratio, then, should be around 0.5. This is the index
of perfect inequality.
In contrast, note that for the countries in Table 2.1, the ratio attains a low of 1.7 (for
Sri Lanka), and this is signi?cantly above the perfect equality mark. The high (for Brazil) is
a staggering 9.3, which means that the richest 20% of the population earn over nine times
that of the poorest 40%.
In our sample, there is a distinct tendency for the ratio to ?rst rise and then fall. Whether
this is a “law” of development or just an artifact of the observation that most Latin American
countries are middle-income and have high inequality remains to be seen, however.
(7)–(9): omitted.
Chapter 3
2
Page 3


Professor Debraj Ray
2002 Topics in Development Economics
Sketches of Answers to Problems, Chapters 2 and 3.
The answers below are brief and try to give you the basic idea of how to approach these
problems. You will gain a lot more from studying these answers if you spend some time
independently trying to work on the problems.
Chapter 2.
(1) A traded good is one that can be bought and sold through the international market. A
nontraded good cannot. Of course, these are extreme descriptions of reality, and some goods
may be partially but not fully tradeable.
Equilibrium exchange rates are determined by the supply of and demand for a country’s
currency. The supply of a country’s currency is determined by that country’s purchases of
imports on the world market. The demand for its currency is determined by the purchase of
that country’s exports by the world. The exchange rate acts to equalize these two (thereby
creating trade balance).
Notice that a rich and productive country is likely to have a stronger currency and a
higher income. The higher income, in turns, pulls up the prices of those products within
the country that are nontraded. Thus measured in terms of exchange rate income, a rich
country looks richer than it really is, because, we are not accounting for the fact that it faces
(on average) higher donestic prices for the nontraded goods. This is why PPP measurements
typically bring down the relative income of a rich country, and pull up the relative income of
a poor country.
(2) The price of a Big Mac is, to a large extent, determined by the prices in competing
restaurants. Thus Big Macs will sell for a higher price in rich countries (where nontraded
restaurant prices are likely to be higher). Thus Big Mac prices incorporate, to some extent
— and often to a better extent than exchange rates do — the “true” cost of living within a
country. Therefore, using the relative prices of Big Macs to create “exchange rates” across
country currencies will often serve as a good approximation of relative PPP income.
(3) The setting up of infrastructure or industrial standards involves a “sunk cost” and a
“variable cost”. The sunk cost is the cost of the entire infrastructure: in the example of
tv systems, this would be all kinds of interconnected senders and receivers that broadcast
at a particular type and re?nement of resolution. The variable cost refers to the individual
purchase of tv sets. Now given that the infrastructure is not set up for it, it makes a little
sense for customers to buy high-de?nition tv sets. Moreover, given the huge sunk costs,
countries which have already invested in a particular standard may not want to tear up this
standard and start all over again, unless the new technology is much, much better.
However, a country that is starting afresh will obviously want to use the latest technology:
there is no past, no sunk cost, to be borne in mind. This is why countries which have been
early innovators are often saddled with older systems, and newcomers can leapfrog over them
1
with the newer technology. Television and telephone systems are only two examples of these.
Chapter 5 discusses these considerations in detail.
(4) Look at footnote 7 on page 17 in the book and make sure you understand the details.
Using this formula, a country growing at 10% per year will double its income in seven years,
while a country growing at 5% will take 14 years to do so. Now try a direct argument using
a calculator. If a country has income x today and is growing at 10%, it will have an income
of x(1 +
1
10
) next year. If you understand this, you can see that thinking of this number as
the “new x”, income the year after will be scaled up by the same formula. This means that
income the day after is just x(1 +
1
10
)
2
. Plodding on in this vein, we see that income after t
years is x(1 +
1
10
)
t
. Now think of t as an unknown, and we wish to know: for what value of
t is x(1 +
1
10
)
t
equal to 2x? How would you solve this using a calculator?
(5) A mobility matrix with no mobility should show 100 on its principal diagonal and zero
everywhere else. This means that countries which were at a certain relative category of world
income at the base date would be in the same relative category at the later date. In contrast,
a mobility matrix that exhibits perfect mobility should have equal percentages along its rows
(adding to 100). This means that countries in some relative category at the base date have
an equal chance of being in any relative category at some later date.
If poor countries grow faster than rich countries, then two things happen: both poor
countries and rich countries would be pushed closer to the world average, the former upward
and the latter downward. This means that the mobility matrix would tend to have its larger
numbers (on each row) closer to the categories that are clustered around world average
income.
You are encouraged to write down speci?c mobility matrices that capture this notion of
convergence, as well as examples of those that capture divergence.
(6) Using Table 2.1 to construct a Kuznets ratio, I get the following sequence of numbers,
corrected to one decimal place (written in order of ascending income): 2.5, 2.8, 2.0, 2.4, 5.4,
4.6, 1.9, 4.4, 1.7, 3.6, 7.9, 9.3, 4.5, 3.8, 7.5, 5.4, 6.0, 4.2, 5.4, 2.1, 2.2, 2.4, 1.5, 2.0, 2.2, 2.3,
2.9.
If income were distributed almost equally, then I would expect that the poorest 40%
would obtain almost 40% of the total income, while the richest 20% would earn a little bit
more than 20% of the total income. The ratio, then, should be around 0.5. This is the index
of perfect inequality.
In contrast, note that for the countries in Table 2.1, the ratio attains a low of 1.7 (for
Sri Lanka), and this is signi?cantly above the perfect equality mark. The high (for Brazil) is
a staggering 9.3, which means that the richest 20% of the population earn over nine times
that of the poorest 40%.
In our sample, there is a distinct tendency for the ratio to ?rst rise and then fall. Whether
this is a “law” of development or just an artifact of the observation that most Latin American
countries are middle-income and have high inequality remains to be seen, however.
(7)–(9): omitted.
Chapter 3
2
(1) (a) The running costs are for labor ($2000 times 100) and for cotton fabric, which is
$600,000. Thus total costs are $800,000 per year. Total revenues are $1 million. Thus
pro?ts, not counting setup investment, are $200,000 per year.
(b) To ?gure out income generated, we must count the wage payments to workers as well,
which are $200,000. Thus income generated is wages plus pro?ts (there are no rents here),
which is $400,000 per year.
(c) The output of the ?rm is $1 million per year. The ?rm’s installed capital is $4 million.
Therefore the capital-output ratio is 4. Notice that the capital equipment can be used over
and over again (though it might depreciate over time). Therefore a capital-output ratio larger
than one is perfectly compatible with the notion of pro?tability.
(2): Omitted. Question (3) is very similar.
(3) (a) Neglecting depreciation in this exercise, The Harrod-Domar model leads us to the
equation: g = s/?, where g is the aggregate growth rate, s is the rate of savings, and ? is the
capital-output ratio. Here s=1/5 and ? =4. So g =1/20, or 5% per year.
(b) We know that the per-capita growth rate is the aggregate growth rate minus the popula-
tion growth rate. Therefore, if the required per-capita growth rate is 4% and the population
growth rate is 3%, the required aggregate growth rate is 7% per year, or 7/100. Using the
Harrod-Domar equation, we see, therefore, that the required rate of savings is g × ?, which
in this case is (7/100) × 4, or 28% of income.
(c) The trick in this problem is to calculate what is, e?ectively, the capital-output ratio in
Xanadu because of the labor problems. Basically, if ? is the amount of capital you need to
produce a single unit of output, you will now e?ectively end up using more than that. How
much more? Well, it must be ? × (4/3). If you take away a quater of this, you will get back
exactly ?. So the e?ective capital-output ratio is now 4 × (4/3)=16/3. Using this in the
Harrod-Domar equation with a rate of savings is 1/5, we see that g =3/80, which is 3.75%
per year. Subtract the population growth rate. The answer for per-capita growth is 1.75%
per year.
(d) Economic well-being comes from a mix of both current consumption and future consump-
tion. A higher savings rate bene?ts future consumption at the expense of current consump-
tion. So our objective should not be to always raise savings rates, but ?nd some intermediate
rate of savings that permits a desirable combination of current and future consumption.
(4) (a) This is simply a review of your understanding of growth rates.
(b) All you have to do is scale down $20,000 by 1.5% per year for 200 years. This leads to
the answer x, where x is given by
x=20, 000(1 - 0.015)
200
.
Solving this, we see that 200 years ago, the average income of a now-developed country was
$973. India’s per-capita income today (in PPP dollars) is about $1200.
A small variant on this problem is as follows: if growth occured at 1.5% over the last
200 years and income today stands at $20,000, what must income have been 200 years ago?
3
Page 4


Professor Debraj Ray
2002 Topics in Development Economics
Sketches of Answers to Problems, Chapters 2 and 3.
The answers below are brief and try to give you the basic idea of how to approach these
problems. You will gain a lot more from studying these answers if you spend some time
independently trying to work on the problems.
Chapter 2.
(1) A traded good is one that can be bought and sold through the international market. A
nontraded good cannot. Of course, these are extreme descriptions of reality, and some goods
may be partially but not fully tradeable.
Equilibrium exchange rates are determined by the supply of and demand for a country’s
currency. The supply of a country’s currency is determined by that country’s purchases of
imports on the world market. The demand for its currency is determined by the purchase of
that country’s exports by the world. The exchange rate acts to equalize these two (thereby
creating trade balance).
Notice that a rich and productive country is likely to have a stronger currency and a
higher income. The higher income, in turns, pulls up the prices of those products within
the country that are nontraded. Thus measured in terms of exchange rate income, a rich
country looks richer than it really is, because, we are not accounting for the fact that it faces
(on average) higher donestic prices for the nontraded goods. This is why PPP measurements
typically bring down the relative income of a rich country, and pull up the relative income of
a poor country.
(2) The price of a Big Mac is, to a large extent, determined by the prices in competing
restaurants. Thus Big Macs will sell for a higher price in rich countries (where nontraded
restaurant prices are likely to be higher). Thus Big Mac prices incorporate, to some extent
— and often to a better extent than exchange rates do — the “true” cost of living within a
country. Therefore, using the relative prices of Big Macs to create “exchange rates” across
country currencies will often serve as a good approximation of relative PPP income.
(3) The setting up of infrastructure or industrial standards involves a “sunk cost” and a
“variable cost”. The sunk cost is the cost of the entire infrastructure: in the example of
tv systems, this would be all kinds of interconnected senders and receivers that broadcast
at a particular type and re?nement of resolution. The variable cost refers to the individual
purchase of tv sets. Now given that the infrastructure is not set up for it, it makes a little
sense for customers to buy high-de?nition tv sets. Moreover, given the huge sunk costs,
countries which have already invested in a particular standard may not want to tear up this
standard and start all over again, unless the new technology is much, much better.
However, a country that is starting afresh will obviously want to use the latest technology:
there is no past, no sunk cost, to be borne in mind. This is why countries which have been
early innovators are often saddled with older systems, and newcomers can leapfrog over them
1
with the newer technology. Television and telephone systems are only two examples of these.
Chapter 5 discusses these considerations in detail.
(4) Look at footnote 7 on page 17 in the book and make sure you understand the details.
Using this formula, a country growing at 10% per year will double its income in seven years,
while a country growing at 5% will take 14 years to do so. Now try a direct argument using
a calculator. If a country has income x today and is growing at 10%, it will have an income
of x(1 +
1
10
) next year. If you understand this, you can see that thinking of this number as
the “new x”, income the year after will be scaled up by the same formula. This means that
income the day after is just x(1 +
1
10
)
2
. Plodding on in this vein, we see that income after t
years is x(1 +
1
10
)
t
. Now think of t as an unknown, and we wish to know: for what value of
t is x(1 +
1
10
)
t
equal to 2x? How would you solve this using a calculator?
(5) A mobility matrix with no mobility should show 100 on its principal diagonal and zero
everywhere else. This means that countries which were at a certain relative category of world
income at the base date would be in the same relative category at the later date. In contrast,
a mobility matrix that exhibits perfect mobility should have equal percentages along its rows
(adding to 100). This means that countries in some relative category at the base date have
an equal chance of being in any relative category at some later date.
If poor countries grow faster than rich countries, then two things happen: both poor
countries and rich countries would be pushed closer to the world average, the former upward
and the latter downward. This means that the mobility matrix would tend to have its larger
numbers (on each row) closer to the categories that are clustered around world average
income.
You are encouraged to write down speci?c mobility matrices that capture this notion of
convergence, as well as examples of those that capture divergence.
(6) Using Table 2.1 to construct a Kuznets ratio, I get the following sequence of numbers,
corrected to one decimal place (written in order of ascending income): 2.5, 2.8, 2.0, 2.4, 5.4,
4.6, 1.9, 4.4, 1.7, 3.6, 7.9, 9.3, 4.5, 3.8, 7.5, 5.4, 6.0, 4.2, 5.4, 2.1, 2.2, 2.4, 1.5, 2.0, 2.2, 2.3,
2.9.
If income were distributed almost equally, then I would expect that the poorest 40%
would obtain almost 40% of the total income, while the richest 20% would earn a little bit
more than 20% of the total income. The ratio, then, should be around 0.5. This is the index
of perfect inequality.
In contrast, note that for the countries in Table 2.1, the ratio attains a low of 1.7 (for
Sri Lanka), and this is signi?cantly above the perfect equality mark. The high (for Brazil) is
a staggering 9.3, which means that the richest 20% of the population earn over nine times
that of the poorest 40%.
In our sample, there is a distinct tendency for the ratio to ?rst rise and then fall. Whether
this is a “law” of development or just an artifact of the observation that most Latin American
countries are middle-income and have high inequality remains to be seen, however.
(7)–(9): omitted.
Chapter 3
2
(1) (a) The running costs are for labor ($2000 times 100) and for cotton fabric, which is
$600,000. Thus total costs are $800,000 per year. Total revenues are $1 million. Thus
pro?ts, not counting setup investment, are $200,000 per year.
(b) To ?gure out income generated, we must count the wage payments to workers as well,
which are $200,000. Thus income generated is wages plus pro?ts (there are no rents here),
which is $400,000 per year.
(c) The output of the ?rm is $1 million per year. The ?rm’s installed capital is $4 million.
Therefore the capital-output ratio is 4. Notice that the capital equipment can be used over
and over again (though it might depreciate over time). Therefore a capital-output ratio larger
than one is perfectly compatible with the notion of pro?tability.
(2): Omitted. Question (3) is very similar.
(3) (a) Neglecting depreciation in this exercise, The Harrod-Domar model leads us to the
equation: g = s/?, where g is the aggregate growth rate, s is the rate of savings, and ? is the
capital-output ratio. Here s=1/5 and ? =4. So g =1/20, or 5% per year.
(b) We know that the per-capita growth rate is the aggregate growth rate minus the popula-
tion growth rate. Therefore, if the required per-capita growth rate is 4% and the population
growth rate is 3%, the required aggregate growth rate is 7% per year, or 7/100. Using the
Harrod-Domar equation, we see, therefore, that the required rate of savings is g × ?, which
in this case is (7/100) × 4, or 28% of income.
(c) The trick in this problem is to calculate what is, e?ectively, the capital-output ratio in
Xanadu because of the labor problems. Basically, if ? is the amount of capital you need to
produce a single unit of output, you will now e?ectively end up using more than that. How
much more? Well, it must be ? × (4/3). If you take away a quater of this, you will get back
exactly ?. So the e?ective capital-output ratio is now 4 × (4/3)=16/3. Using this in the
Harrod-Domar equation with a rate of savings is 1/5, we see that g =3/80, which is 3.75%
per year. Subtract the population growth rate. The answer for per-capita growth is 1.75%
per year.
(d) Economic well-being comes from a mix of both current consumption and future consump-
tion. A higher savings rate bene?ts future consumption at the expense of current consump-
tion. So our objective should not be to always raise savings rates, but ?nd some intermediate
rate of savings that permits a desirable combination of current and future consumption.
(4) (a) This is simply a review of your understanding of growth rates.
(b) All you have to do is scale down $20,000 by 1.5% per year for 200 years. This leads to
the answer x, where x is given by
x=20, 000(1 - 0.015)
200
.
Solving this, we see that 200 years ago, the average income of a now-developed country was
$973. India’s per-capita income today (in PPP dollars) is about $1200.
A small variant on this problem is as follows: if growth occured at 1.5% over the last
200 years and income today stands at $20,000, what must income have been 200 years ago?
3
This is di?erent from the “backwards” argument of the previous paragraph, where we reduced
income by 1.5% a year. This leads to the answer y, where y is given by
20, 000 = y(1+0.015)
2
00.
Solving this gives us starting income equal to $1018. Not very di?erent from the previous
answer, but nevertheless di?erent. Make sure you understand why.
(5) (a) E?ective labor grows at the rate of labor force growth plus the rate of labor-augmenting
technical progress, so the answer is 5% per year.
(b) I am going to skip the graph which you should be able to do without a problem. Let’s
calculate capital-output ratios. At k = 2, total output is y = 1. So the ratio of capital to
output is 2. At k = 6, we have to ?gure out what total output is. The ?rst three units of k
produce y =1.5 units of output. The next three produce an additional 3/7 units of output.
So total output when k = 6 is (3/2)+(3/7) = 27/14. The ratio is, therefore, 6 × (14/27),
which is approximately 3. Note that this is di?erent from the “marginal” capital-output ratio
in this region of the production function: each additional unit of output is requiring 7 units
of capital, not 3. But the average ratio is less than the marginal ratio, because the former
includes capital applied in the earlier phase of the production function, where its marginal
product is higher. This captures the notion of diminsihing returns to physical capital.
Similarly, you can work out the capital-output ratio for k = 12. Omitted here, as the
way of calculating it is exactly the same.
(c) Let’s go through the derivation of the Solow model. New capital is simply old capital
plus extra investment. But savings equals investment. So capital in period t + 1 is related
to what happens in period t by the equation
K(t+1) = K(t)+ sY (t)
where s is the savings rate, and Y (t) is income in period t. Now, we divide by the e?ective
labor force at time t, let’s call it L(t) (equal to E(t)P(t) in the text). Remember that
k(t)= K(t)/L(t) and y(t)= Y (t)/L(t) for all t. Soweget
K(t+1)
L(t)
= k(t)+ sy(t)
(This is all a rehash of stu? done in the lectures.) Now we play with the left-hand side:
[K(t+1)/E(t)]=[K(t)/L(t)]×[L(t+1)/L(t)], which is just k(t+1)×1.05 (using part (a)).
Substitute this in the equation:
(1.05)k(t+1) = k(t)+ sy(t)
To ?nish the formula, we know that k(t) and y(t) are linked by whatever the capital-output
ratio is at date t, but unlike the Harrod-Domar model, this ratio is not a constant but varies
with whatever the going value of k(t) is (recall the previous problem). Let’s just call it ?(t);
so that k(t)= ?(t)y(t). Using this in the formula above, and recalling that s=1/5, we get
(1.05)k(t+1) = k(t)[1 +
1
5?(t)
]
4
Page 5


Professor Debraj Ray
2002 Topics in Development Economics
Sketches of Answers to Problems, Chapters 2 and 3.
The answers below are brief and try to give you the basic idea of how to approach these
problems. You will gain a lot more from studying these answers if you spend some time
independently trying to work on the problems.
Chapter 2.
(1) A traded good is one that can be bought and sold through the international market. A
nontraded good cannot. Of course, these are extreme descriptions of reality, and some goods
may be partially but not fully tradeable.
Equilibrium exchange rates are determined by the supply of and demand for a country’s
currency. The supply of a country’s currency is determined by that country’s purchases of
imports on the world market. The demand for its currency is determined by the purchase of
that country’s exports by the world. The exchange rate acts to equalize these two (thereby
creating trade balance).
Notice that a rich and productive country is likely to have a stronger currency and a
higher income. The higher income, in turns, pulls up the prices of those products within
the country that are nontraded. Thus measured in terms of exchange rate income, a rich
country looks richer than it really is, because, we are not accounting for the fact that it faces
(on average) higher donestic prices for the nontraded goods. This is why PPP measurements
typically bring down the relative income of a rich country, and pull up the relative income of
a poor country.
(2) The price of a Big Mac is, to a large extent, determined by the prices in competing
restaurants. Thus Big Macs will sell for a higher price in rich countries (where nontraded
restaurant prices are likely to be higher). Thus Big Mac prices incorporate, to some extent
— and often to a better extent than exchange rates do — the “true” cost of living within a
country. Therefore, using the relative prices of Big Macs to create “exchange rates” across
country currencies will often serve as a good approximation of relative PPP income.
(3) The setting up of infrastructure or industrial standards involves a “sunk cost” and a
“variable cost”. The sunk cost is the cost of the entire infrastructure: in the example of
tv systems, this would be all kinds of interconnected senders and receivers that broadcast
at a particular type and re?nement of resolution. The variable cost refers to the individual
purchase of tv sets. Now given that the infrastructure is not set up for it, it makes a little
sense for customers to buy high-de?nition tv sets. Moreover, given the huge sunk costs,
countries which have already invested in a particular standard may not want to tear up this
standard and start all over again, unless the new technology is much, much better.
However, a country that is starting afresh will obviously want to use the latest technology:
there is no past, no sunk cost, to be borne in mind. This is why countries which have been
early innovators are often saddled with older systems, and newcomers can leapfrog over them
1
with the newer technology. Television and telephone systems are only two examples of these.
Chapter 5 discusses these considerations in detail.
(4) Look at footnote 7 on page 17 in the book and make sure you understand the details.
Using this formula, a country growing at 10% per year will double its income in seven years,
while a country growing at 5% will take 14 years to do so. Now try a direct argument using
a calculator. If a country has income x today and is growing at 10%, it will have an income
of x(1 +
1
10
) next year. If you understand this, you can see that thinking of this number as
the “new x”, income the year after will be scaled up by the same formula. This means that
income the day after is just x(1 +
1
10
)
2
. Plodding on in this vein, we see that income after t
years is x(1 +
1
10
)
t
. Now think of t as an unknown, and we wish to know: for what value of
t is x(1 +
1
10
)
t
equal to 2x? How would you solve this using a calculator?
(5) A mobility matrix with no mobility should show 100 on its principal diagonal and zero
everywhere else. This means that countries which were at a certain relative category of world
income at the base date would be in the same relative category at the later date. In contrast,
a mobility matrix that exhibits perfect mobility should have equal percentages along its rows
(adding to 100). This means that countries in some relative category at the base date have
an equal chance of being in any relative category at some later date.
If poor countries grow faster than rich countries, then two things happen: both poor
countries and rich countries would be pushed closer to the world average, the former upward
and the latter downward. This means that the mobility matrix would tend to have its larger
numbers (on each row) closer to the categories that are clustered around world average
income.
You are encouraged to write down speci?c mobility matrices that capture this notion of
convergence, as well as examples of those that capture divergence.
(6) Using Table 2.1 to construct a Kuznets ratio, I get the following sequence of numbers,
corrected to one decimal place (written in order of ascending income): 2.5, 2.8, 2.0, 2.4, 5.4,
4.6, 1.9, 4.4, 1.7, 3.6, 7.9, 9.3, 4.5, 3.8, 7.5, 5.4, 6.0, 4.2, 5.4, 2.1, 2.2, 2.4, 1.5, 2.0, 2.2, 2.3,
2.9.
If income were distributed almost equally, then I would expect that the poorest 40%
would obtain almost 40% of the total income, while the richest 20% would earn a little bit
more than 20% of the total income. The ratio, then, should be around 0.5. This is the index
of perfect inequality.
In contrast, note that for the countries in Table 2.1, the ratio attains a low of 1.7 (for
Sri Lanka), and this is signi?cantly above the perfect equality mark. The high (for Brazil) is
a staggering 9.3, which means that the richest 20% of the population earn over nine times
that of the poorest 40%.
In our sample, there is a distinct tendency for the ratio to ?rst rise and then fall. Whether
this is a “law” of development or just an artifact of the observation that most Latin American
countries are middle-income and have high inequality remains to be seen, however.
(7)–(9): omitted.
Chapter 3
2
(1) (a) The running costs are for labor ($2000 times 100) and for cotton fabric, which is
$600,000. Thus total costs are $800,000 per year. Total revenues are $1 million. Thus
pro?ts, not counting setup investment, are $200,000 per year.
(b) To ?gure out income generated, we must count the wage payments to workers as well,
which are $200,000. Thus income generated is wages plus pro?ts (there are no rents here),
which is $400,000 per year.
(c) The output of the ?rm is $1 million per year. The ?rm’s installed capital is $4 million.
Therefore the capital-output ratio is 4. Notice that the capital equipment can be used over
and over again (though it might depreciate over time). Therefore a capital-output ratio larger
than one is perfectly compatible with the notion of pro?tability.
(2): Omitted. Question (3) is very similar.
(3) (a) Neglecting depreciation in this exercise, The Harrod-Domar model leads us to the
equation: g = s/?, where g is the aggregate growth rate, s is the rate of savings, and ? is the
capital-output ratio. Here s=1/5 and ? =4. So g =1/20, or 5% per year.
(b) We know that the per-capita growth rate is the aggregate growth rate minus the popula-
tion growth rate. Therefore, if the required per-capita growth rate is 4% and the population
growth rate is 3%, the required aggregate growth rate is 7% per year, or 7/100. Using the
Harrod-Domar equation, we see, therefore, that the required rate of savings is g × ?, which
in this case is (7/100) × 4, or 28% of income.
(c) The trick in this problem is to calculate what is, e?ectively, the capital-output ratio in
Xanadu because of the labor problems. Basically, if ? is the amount of capital you need to
produce a single unit of output, you will now e?ectively end up using more than that. How
much more? Well, it must be ? × (4/3). If you take away a quater of this, you will get back
exactly ?. So the e?ective capital-output ratio is now 4 × (4/3)=16/3. Using this in the
Harrod-Domar equation with a rate of savings is 1/5, we see that g =3/80, which is 3.75%
per year. Subtract the population growth rate. The answer for per-capita growth is 1.75%
per year.
(d) Economic well-being comes from a mix of both current consumption and future consump-
tion. A higher savings rate bene?ts future consumption at the expense of current consump-
tion. So our objective should not be to always raise savings rates, but ?nd some intermediate
rate of savings that permits a desirable combination of current and future consumption.
(4) (a) This is simply a review of your understanding of growth rates.
(b) All you have to do is scale down $20,000 by 1.5% per year for 200 years. This leads to
the answer x, where x is given by
x=20, 000(1 - 0.015)
200
.
Solving this, we see that 200 years ago, the average income of a now-developed country was
$973. India’s per-capita income today (in PPP dollars) is about $1200.
A small variant on this problem is as follows: if growth occured at 1.5% over the last
200 years and income today stands at $20,000, what must income have been 200 years ago?
3
This is di?erent from the “backwards” argument of the previous paragraph, where we reduced
income by 1.5% a year. This leads to the answer y, where y is given by
20, 000 = y(1+0.015)
2
00.
Solving this gives us starting income equal to $1018. Not very di?erent from the previous
answer, but nevertheless di?erent. Make sure you understand why.
(5) (a) E?ective labor grows at the rate of labor force growth plus the rate of labor-augmenting
technical progress, so the answer is 5% per year.
(b) I am going to skip the graph which you should be able to do without a problem. Let’s
calculate capital-output ratios. At k = 2, total output is y = 1. So the ratio of capital to
output is 2. At k = 6, we have to ?gure out what total output is. The ?rst three units of k
produce y =1.5 units of output. The next three produce an additional 3/7 units of output.
So total output when k = 6 is (3/2)+(3/7) = 27/14. The ratio is, therefore, 6 × (14/27),
which is approximately 3. Note that this is di?erent from the “marginal” capital-output ratio
in this region of the production function: each additional unit of output is requiring 7 units
of capital, not 3. But the average ratio is less than the marginal ratio, because the former
includes capital applied in the earlier phase of the production function, where its marginal
product is higher. This captures the notion of diminsihing returns to physical capital.
Similarly, you can work out the capital-output ratio for k = 12. Omitted here, as the
way of calculating it is exactly the same.
(c) Let’s go through the derivation of the Solow model. New capital is simply old capital
plus extra investment. But savings equals investment. So capital in period t + 1 is related
to what happens in period t by the equation
K(t+1) = K(t)+ sY (t)
where s is the savings rate, and Y (t) is income in period t. Now, we divide by the e?ective
labor force at time t, let’s call it L(t) (equal to E(t)P(t) in the text). Remember that
k(t)= K(t)/L(t) and y(t)= Y (t)/L(t) for all t. Soweget
K(t+1)
L(t)
= k(t)+ sy(t)
(This is all a rehash of stu? done in the lectures.) Now we play with the left-hand side:
[K(t+1)/E(t)]=[K(t)/L(t)]×[L(t+1)/L(t)], which is just k(t+1)×1.05 (using part (a)).
Substitute this in the equation:
(1.05)k(t+1) = k(t)+ sy(t)
To ?nish the formula, we know that k(t) and y(t) are linked by whatever the capital-output
ratio is at date t, but unlike the Harrod-Domar model, this ratio is not a constant but varies
with whatever the going value of k(t) is (recall the previous problem). Let’s just call it ?(t);
so that k(t)= ?(t)y(t). Using this in the formula above, and recalling that s=1/5, we get
(1.05)k(t+1) = k(t)[1 +
1
5?(t)
]
4
Now we have a formula that can precisely compute k(t + 1), given any value of k(t).
(d) At k(t) = 3, the value of ?(t)is3/2. So, using our trusty formula, we see that (1.05)k(t+
1) = 3[1 + (2/15)], which will give you a value of k(t + 1) that exceeds 3, what we started
with in the previous period. The idea, as discussed in class, is that capital is more productive
at the margin when its level is low, so that the economy tends to accumulate capital more
quickly than labor, raising the capital-labor ratio.
Likewise, at k(t + 1) = 10, ?gure the value of ?(t): you will see that it is 4. Using our
formula, (1.05)k(t+1) = 10[1+(1/20)], which means that k(t + 1) is also 4. By a stroke of
luck, we have found the steady-state ratio k
*
.
You should appreciate the point that this is just pure luck. What would have happened
had you started with k(t) > 10?
(e) Let’s try to set up an equation. At the steady state value of k
*
, we will simply have
k(t+1) = k(t)= k
*
. Use this in the formula: you see that k
*
drops out (it appears on both
sides of the equation), so that 1.05=[1+(1/5?
*
)], where ?
*
is the capital-output ratio at
the steady state. Solve this equation to see again that ?
*
= 4. Unlike the previous problem,
this answer is not luck but comes from the insight of using k(t)= k(t + 1) in our formula.
(6) This is a more mathematical exercise that will help you understand how the steady state
in the Solow model is described. We have the equation
Y (t)= AK(t)
a
L(t)
1-a
describing how total output is produced with capital and labor. In the ?rst step, we transform
this into a per-capita magnitude by dividing through by the labor force L (there is no technical
progress here so that labor is just the same as e?ective labor). If we de?ne y =
Y
L
and k =
K
L
,
we see that
y(t)= Ak
a
.
Therefore the equation describing the Solow model is
(1 + n)k(t+1) = sAk(t)
a
+(1 - d)k(t).
In the steady state
ˆ
k, k(t)= k(t+1) =
ˆ
k. Consequently,
(1 + n)
ˆ
k = sA
ˆ
k
a
+(1 - d)
ˆ
k.
Now we solve this equation to ?gure out what the value of
ˆ
k must be. Try it: you will see
that
(n + d)
ˆ
k
1-a
= sA,
or that
ˆ
k = {
sA
n + d
}
1/(1-a)
.
Now using this equation, you should be able to easily tell the direction in which
ˆ
k moves, in
response to all the changes asked about in the question.
(7) Omitted. Fully discussed in text and in class.
5