Solutions debraj ray-1 PDF

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Development economics...

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Sketches of Answers to Problems

Note The answers below are brief and are only designed 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.

(1) A traded good is one that can be bought and sold through the international market, while a nontraded good cannot. Of course, these are extreme descriptions of reality, and some goods may be partially but not fully tradable. Equilibrium exchange rates are determined by the supply of and demand for a country’s currency. More specifically, the supply of a country’s currency is determined by that country’s purchases of imports in the world market, and the demand for its currency is determined by the purchases 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 level. The higher income, in turns, pulls up the prices of those nontraded products within that country. Thus, measured in terms of exchange rate income, a rich country looks richer than it really is since we are not accounting for the fact that it faces (on average) higher domestic 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. Therefore, Big Macs will sell for a higher price in richer countries, where nontraded restaurants are likely to have higher prices. Intuitively, 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. Thus, using the relative prices of Big Macs to create “exchange rates” across currencies can often serve as a good approximation of relative PPP income.

(3) The setting up of infrastructure or industrial standards involves a “fixed cost” and a “variable cost”, while the benefit of a system is often paid off over a long period of time. In the example of television transmission systems, the fixed cost would be all kinds of interconnected senders and receivers that broadcast at a particular type and refinement of resolution, while the variable cost would be individual TV sets purchased by consumers. 1

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Development Economics

The difference between countries starting afresh and those that already have a preexisting infrastructure system is that for the latter, upgrading the system would not only be fiercely challenged by those who had been benefiting from the old system, but would also incur extra (opportunity) costs from foregoing the benefits that are to come from the existing system (see Problem 10 in Chapter 5), not to mention the inevitable frictions during the transition, such as disruption of services and forcing people to adapt to a new system. Therefore, countries which have already invested in a particular standard may not want to tear up this standard and start all over again, unless the benefit of adopting the new technology is great enough to justify the huge cost. With no such preexisting systems and hence no such concerns, many developing countries would jump directly to 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 with the newer technology. Television and telephone systems are only two examples of these. Chapter 5 discusses these considerations in detail.

(4)(a) 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 1 ) next year. If you understand this, you can see that thinking of have an income of x(1 + 10 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 + 101 )2 . Plodding on in this vein, we see that income after t years is x(1 + 101 )t . Now think of t as an unknown, and we wish to know: for what value of t is x(1 + 101 )t equal to 2x? How would you solve this using a calculator? (4)(b) ??? “shaves an additional percentage point off its population growth rate for the next 20 years”;not sure what this means...something like growthratet = 0.99t , or growthratet = n0 − t, or simply growthratenext20years = n0 − 0.01? Also, when asked “how much richer” are we comparing to the counterfactual outcome (natural population growth), or the initial condition? If it were the last case, and we are comparing to the counterfactual outcome, then the 20 −1.0520 ≃ 20.9% richer. country would be 1.061.05 20 . (4)(c) Compounding monthly, we get (1 + 0.3)12 = 23.3; hence the equivalent annual inflation rate would be approximately (23.3− 1) ×100 = 2230%. (Compare with the withoutcompounding case and feel the power of exponential growth!)

(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.

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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 specific 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 equality. 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 significantly 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 first 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) Open-ended question. Regarding the last point, it is often a matter of tradeoff: Presenting the indicators separately gives people a full picture of what is happening but can sometimes seem redundant (since in many of the cases they display the same trend anyway) and overwhelming; while a single aggregated number abstracts much information away from reality, it often comes in handy for decision making purposes such as determining a cutoff rule. (8) Most famines are caused by local rather than global factors, whether they be political, climatic, or simply bad luck. Hence, on a global scale, where the per capita availability of food grain is usually quite stable (if not growing), it is hard to make sense of so many people suffering from famine when there is adequate food grain. (9)(a) Open-ended question. It is possible that instead of looking only at the number of children, parents also care about the quality of their children. A (probably inappropriate) analogy would be, when you get richer, instead of purchasing two low-end cell phones, you might want to simply purchase a trendy high-end model. (9)(b) Higher population growth rates are usually the results of higher rates of newborn children, who are and will remain under 15 for 15 years. Accumulating overtime, they can make up a rather large proportion of the overall population.

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Development Economics

(9)(c) Open-ended question. Poorer countries may be more likely to be rural because it often requires a certain level of trade and productivity (either industrial or agricultural) to induce urbanization. On the other hand, rural countries face many challenges to grow rich; for instance, rural areas are usually less densely populated, rendering it harder for the government to provide education and health services to the local residents. (9)(d) As we have seen, prices of primary goods typically fluctuates more than those of manufactured goods. One possible explanation is that primary goods are more susceptible to factors largely beyond human control (such as rainfall). Manufactured goods are less likely to experience large price fluctuation since price fluctuations in a few of its inputs can only affect its price to a certain degree, if not cancelling out each other completely.

(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 profits, not counting setup investment, are $200,000 per year. (1)(b) To figure out income generated, we must count the wage payments to workers as well, which are $200,000. Thus income generated is wages plus profits (there are no rents here), which is $400,000 per year. (1)(c) The output of the firm is $1 million per year. The firm’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 profitability. (2)(a) δ = 0, hence s/θ = g. When θ = 4, s = 4g, and hence the savings rate should be 0.32 and 0.4 when g = 0.08 and 0.1, respectively. Similarly, when s = 0.2, θ = s/g = 0.2/g , and therefore θ needs to be 2.5 and 2 when g is 0.08 and 0.1, respectively. A fall in the capital-output raito means that for every unit of capital the firm can produce more output. (2)(b) Using the approximation provided by equation (3.7), we see that the growth rate falls by the same amount as the increase in depreciation rate. When depreciation rate is 3%, we need s/θ to be 3%+5%=8% to achieve a 5% growth rate. This translates into a savings rate of 4×8%=32%. s

−δ−n

(2)(c) Rearranging equation (3.6) we get g ∗ = θ 1+n . Plugging the numbers we get . g ∗ = 0.0196 = 1.96%. It is obvious from the expression above that g ∗ = 0 when n = θs −δ=4% (notice that g ∗ could be negative if population growth is even higher). It makes sense because g ∗ is a per capita measure, and hence gets smaller as the population grows faster. (2)(d) Please appreciate the beauty of the equations.

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(3)(a) Neglecting depreciation in this exercise, The Harrod-Domar model leads us to the equation: g = s/θ, where g is the aggregate growth rate, s is the rate of savings, and θ is the capital-output ratio. Here s = 1/5 and θ = 4. So g = 1/20, or 5% per year.

(3)(b) Using the approximation provided by equation (3.7), we know that the per capita growth rate is the aggregate growth rate minus the population 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%. Then the required rate of savings is g × θ, which in this case is 0.07 × 4=28% of total income. (3)(c) Due to the labor strikes, the effective capital-output ratio is now θ ∗ = θ × 4/3 = 4 × (4/3) = 16/3. Using this in the Harrod-Domar equation with the rate of savings equal to 0.2, we see thatg = s/θ∗ = 3/80 = 3.75%. Subtract the population growth rate to get the per-capita growth rate: 1.75%.

(3)(d) Economic well-being is determined by both current consumption and future consumption. A higher savings rate benefits future consumption at the expense of current consumption. Hence, the objective should not always be to raise savings rates, but to find an intermediate rate of savings that balances current and future consumption.

(4)(a) For example, in the first year, the growth rate of the professor’s salary is 1000/100000=1%, while the rate for the teacher is 1000/50000=2%. . (4)(b) It would be $20000 × (1 − 1.5%)200 = $973. Notice that there is a subtle difference from asking “If growth occurred at 1.5% over the last 200 years and income today stands at $20,000, what must income have been 200 years ago?”. This variation would yield . $20000/(1 + 1.5%)200 = $1018.

(5)(a) Effective labor grows at the rate of labor force growth plus the rate of laboraugmenting technical progress, so the answer is 5% per year.

(5)(b) At k = 2, total output is y = 1. So the ratio of capital to output is 2. At k = 6, we have to figure out what total output is. The first 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 different from the “marginal” capital-output ratio in this region of the production function: each additional unit of output is requiring 7 units of capital rather than 3. 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 diminishing returns to capital. Similarly, the capital-output ratio for k = 12 is approximately 4.4.

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Development Economics

(5)(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 both sides by the eective labor force at time t, L(t). Remember that k(t) = K(t)/L(t) and y(t) = Y (t)/L(t) for all t. So we get K(t + 1)/L(t) = k(t) + sy(t). Notice that the left-hand side is K(t + 1)/L(t) = [K(t + 1)/L(t + 1)] × [L(t + 1)/L(t)], which is just k(t + 1) × 1.05(L(t + 1)/L(t) = 1.05 from part (a)). Substitute this back in the equation we get (1.05)k(t + 1) = k (t) + sy (t). Let θ(t) be the capital-output ratio at date then k(t) = θ(t)y (t). Using this in the 1 formula above, and plugging in s = 1/5, we get (1.05)k(t + 1) = k(t)[1 + 5θ(t) ]. Now we have a formula that can precisely compute k(t + 1), given any value of k(t)

(5)(d) Using the above equation, and θ(t) derived from the relationship between k(t) and y(t) given in part(b), we have two series of k’s: From k(t)=3, we get:3, 3.143, 3.2828,... From k(t)=10,we get:10, 10, 10, 10,... By a stroke of luck, we have found the steady-state k ∗ =10. You should appreciate the point that this is just pure luck. What would have happened had you started with k(t) > 10?

(5)(e) We have already found k ∗ =10 by luck in part(d). To solve for it rigorously, notice that in the long run, k(t) = k(t + 1). Plugging this in to the equation that defines their relationship, we get θ ∗ = 4, which allows us to solve for the unique k ∗ from the production function in part(b). The solution is indeed k ∗ =10.

(6) Dividing the production function by L(t) on both sides, we get y(t) = Ak(t)α. Plugging this into equation (3.9) we get (1 + n)k(t + 1) = (1 − δ)k (t) + sAk(t)α. In the steady state, 1 sA 1−α ) . Now k(t + 1) = k (t) = k ∗ ; solving for k ∗ from the equation above we get k ∗ = (n+δ using this equation, you should be able to easily tell the direction in which k ∗ moves, in response to all the changes asked about in the question.

(7)(a) As discussed in the text, an increase in the savings rate will increase the output and per capita capital stock in every subsequent period. Using Figure 3.4, we see that the steady state capital stock per capita must go up as well, so the net effect of an increase in the savings rate is to push capital and output to a higher long-run level, while leaving the long-run income growth just as before - equal to the population growth rate. The effect of depreciation rate is similar. In contrast, higher population growth lowers the steady-state level of per capita income (while it increases the total income growth rate). See text for details.

(7)(b) Recall equation (3.5) and (3.6). It is straight forward to see that in the HarrodDomar model, savings and depreciation rate have growth effects on both income and per capita income.

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(8)(a) True. Here write down the Harrod-Domar equation. And then go on to mention that in the Solow model, long-run growth rate is determined simply by the exogenous rate of technical progress. The savings rate only determines long-run capital stocks per capita and the level of per capita output, not its rate of growth. (8)(b) False. An increase in the capital-output ratio will lower the rate of growth. (8)(c) False. Studying countries that are currently rich introduces a bias towards convergence, as you are simply selecting ex post countries that were successful and so similar. (8)(d) True. Quah’s study of mobility of countries shows that both very poor and very rich countries are unlikely to change world rankings all that much. In contrast, countries that were middle-income in 1960 have shown remarkable mobility. A large fraction of them have become dramatically richer, while another large fraction have become dramatically poorer. (8)(e) True. In the Solow model, population growth has only a level effect on long-run per capita income. Here you may draw a quick diagram that describes the steady state in the Solow model and show what happens as population growth increases. Notice that in the long-run, the rate of growth in the Solow model is just the rate of technical progress. (8)(f ) False. Output per head increases as capital per head increases; it does so at a diminishing rate, but it increases nevertheless. (9) Draw this diagram. Now the idea is to show that if you start from very low levels of income, then per capita growth tends to first decelerate as income increases. This is because the growth of total income is constant but the population growth rate is increasing. There is no trap because at the point where the population growth reaches its peak, the per capita growth rate reaches its lowest point, but this point is still positive (because the entire population growth rate lies below the total income growth line). After this stage, as the population growth rate falls, the per capita growth rate of the economy begins to accelerate once again. (10) A country with a low ratio of capital to labor might grow faster for two reasons: (1) Its marginal product of capital may be higher because there are lots of labor to work with the capital; (2) Low capital is also likely to indicate easier upgrades or adoption of new technologies because the old ones are hopelessly out of date or nonexistent to start with (phone, computer, television networks for example). This is more difficult for richer countries which have (perhaps not f...