Econ-UA 323 Development Economics. Outline of Answers to Problem Set 7
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1 Professor Debraj Ray 19 West 4th Street, Room 608 Office Hours: Mondays pm homepage: Webpage for course: click on Teaching, and then on the course name. Econ-UA 323 Development Economics Outline of Answers to Problem Set 7 (1) Construct an example of countries A and B, where A has higher death rates than B in every age category, yet has an overall lower death rate. Consider the following hypothetical example: For country A, 44% of the population are under 15 and display a death rate of 10 (per thousand, same for death rates below), 53% of the population are under 64 but older than 15 and display a death rate of 5, and finally 3% of the population are above 64 and display a death rate of 50. For country B, 19% of the population are under 15 and display a death rate of 7, 67% of the population are under 64 but older than 15 and display a death rate of 4, and finally 14% of the population are above 64 and display a death rate of 40. In this hypothetical example, country A has higher death rates in every category compared to country B. Let us calculate the overall death rates, which are given by a weighted average of the age-specific death rates, using population shares as weights. For A it is = 8.55, while for B it is = 9.61, higher than that of A. (2) Suppose there are just three ages; Y (young), M (middle-aged) and O (old). Suppose that death at each age is given by the fractions d y, d m, with d o = 1. Finally, suppose that only M s give birth: let b be the fraction of newborns relative to the entire M-population. Let n y (t), n m (t) and n o (t) be the populations of the three ages. Make sure you understand the following relationships (you don t need to write anything for this): n y (t + 1) = bn m (t), n m (t + 1) = (1 d y )n y (t), and n o (t + 1) = (1 d m )n m (t). (a) Now suppose that the population is growing steadily at the rate of g, with pop shares of each age group given by σ y, σ m and σ o, and suppose that these shares are unchanging over time. Using the above relationships, show that (i) σ y (1 + g) = bσ m, 1
2 2 Take the equation n y (t + 1) = bn m (t), and divide both sides by the population n(t) to get n y (t + 1) n t+1 n t+1 n t = b n m(t) n(t), where on the left hand side we have used the old Solow model-like trick to multiply and divide by n t+1. This is easily written as the desired answer just by using the definitions of σ and of population growth. (ii) σ m (1 + g) = (1 d y )σ y, and (iii) σ o (1 + g) = (1 d m )σ m. Follow the same kind of argument as in part (i). (b) Use part (a) to show that [ σ y d y 1 + g + 1 d ] m = 1. b We know that σ y + σ m + σ o = 1 by definition (these are the three population shares after all). Now simply substitute the answers you got in part (a), and you are done. (c) Use equations (i) and (ii) of part (a) to show that 1 + g = b(1 d y ), and then combine with part (b) to conclude that ] 1 dy σ y [ d m = 1. b b From part (ii), we see that σ m = (1 d y ) σ y 1 + g. Substitute this into part (i) to eliminate σ m, so that σ y (1 + g) = b(1 d y ) σ y 1 + g. Remove the common term σ y, transpose 1 + g and then take the square root to see that 1 + g = b(1 d y ). Now just substitute this result into part (b). Now show that as the birth rate b goes up, or as the death rates d y and d m go up, the share of the young, σ y, must go up as well. This should now be obvious from the equation in part (c). (3) [A Prisoner s Dilemma for population: Explain why each country might want to take a pro-natalist stand for military or political reasons, but the combination of all countries taking the same pro-natalist stance may make all countries worse off relative to a neutral stance on population. Here is one very simple example. You can do this in several ways, though. Suppose that there are two countries, 1 and 2. Suppose that each country can pursue a high population
3 policy (H) or a low population policy (L). Suppose that in isolation, each country prefers low population to high population; say the payoff from a low population is 125, while the payoff from a high population is 100. But now in a world with both countries there, suppose that there is an additional payoff of 50 to be gained from military supremacy, which depends on one country having a larger population than the other. If that happens, the other country loses 50. If both countries pursue H or both L, then there is no military supremacy and this additional payoff can be set to 0. Now use this example to show that both countries must pursue the high population policy, and both are worse off for it, even though both countries are making rational decisions. (4) We studied a model where a family wants one surviving child to provide old-age security. Let us say that the probability of any one child living to look after its parents in old age is 1/2 (i.e., 50 50). However, the family wants this security level to be higher, say a probability of q > 1/2. (a) Describe the family s fertility choices for different values of q, and examine the results for different values of q. The probability of x children all failing to survive is (.5) x, and hence the probability of at least one child surviving is 1 (.5) x. Setting this to q we get x = log(1 q)/ log(1/2). Clearly as q increases (from 0 to 1), the corresponding x goes from 0 to infinity. (b) Calculate the expected number of surviving children for this family, under various values of q. The expected number of surviving children would be the survival rate (.5) times x:.5 x =.5 log(1 q)/ log(1/2). (5) In the land of Oz, there are three inputs to production: capital, physical labor, and mental labor. Men in Oz have more physical labor power than women, but both men and women have the same amount of mental labor power. (a) Who earns more in Oz, men or women? What do these differences depend upon? Men would earn more since they have more physical power than women (and the same mental power). The magnitude of the difference would depend on the difference between the marginal productivity of physical labor and that of mental labor. (b) Now imagine that the technology is such that more capital raises the marginal product of mental labor faster than it raises physical labor. As the economy of Oz grows over time, its stock of physical capital is steadily increasing. How would you expect the relative wage of men to women to change over time? Explain. Due to the type of technology in this society, increases in capital result in a relative increase of return to mental labor (compared to the return to physical labor). Since women have a comparative advantage in mental labor, one would expect the relative wages of women to rise. 3
4 4 (c) Women have one unit of labor time that they can allocate between raising children and being part of the workforce. How would this allocation be affected by the changes over time that you found in your answer to (b)? Discuss the implications for fertility levels in the population. Since the changes would result in an increase in women s wages, we should expect to see women allocate more time in the workforce and less time in child-bearing. Hence, the fertility rate in the population would likely decrease. This is discussed in more detail in class and in the text. (6) True or False? (a) A developing country is likely to have an overall death rate that is lower than that of a developed country. True. Although developing countries usually have a higher age-specific death rate in each age group, it is not fully captured by the overall death rate since developing countries usually also have a younger population, which biases downward the overall death rate. (b) The populations of Europe and North America grew at a combined rate between 1750 and 1900 that significantly exceeded the population growth rates of developing countries at that time. True; see text and slides; you should know enough of this table to give some idea of the magnitudes in an exam question. (c) If country A has a population growth rate that is lower than country B, then the average woman in country A has less children than her counterpart in country B. False. It depends on the age distribution of the population. It is possible for the fertility rate in one country to be lower than that in another country, and yet for the population growth rate in the former country to be higher. You can try to construct an example as in Problem 1. (d) Birth rates may be high even when death rates may be falling. True. Discussed in the demographic transition (phase 2, in particular). (e) If total mortality among children remained constant, but the incidence of that mortality shifted from late childhood to early childhood, fertility rates should decline. True. This would shift parents fertility decisions from hoarding to targeting, and the latter usually leads to lower fertility rates. (7) Suppose that families have a gender bias; that is, they have children until a son is born. Suppose that at each birth, the probability of a child being a boy is (a) Will the country as a whole have more girl births than boy births (or vice versa) under this targeting rule? The overall ratio of girl to boy births will be unchanged.
5 5 (b) Will larger families have more daughters or sons? If families choose to have children until they have one son, and then they stop, the possible birth patterns are: (boy); (girl, boy); (girl, girl, boy); (girl, girl, girl, boy), and so on. Now it is obvious that larger families (families with more children) have more daughters (among the offsprings). Note: Since girls are also more likely to be in a family that has more children to take care of, even if she is treated equally within the family, she is still more likely to die (on average) compared to other boys in the society (because larger families on average have less resource to spend on each child, not just the female children). (c) If you have information on the sex and birth order of each child born to each family in the village. How would you use the data to test your hypothesis that there is gender bias? I would test the hypothesis that among all the last child within a household, the male-female ratio is 1:1. If the data reject this hypothesis (in favor of a ratio larger than 1), it would suggest the existence of potential gender bias in favor of boys. (8) This is a question on joint families, externalities, and fertility choice. Suppose that Ram and Rani are the heads of a nuclear family, making their fertility decisions. For simplicity, assume away gender bias and issues of child survival. The following table details the costs and benefits (in dollars, say) of different numbers of children. (a) Based on the information in the table, how many children would Ram and Rani have in order to maximize their net benefit? Number of children Total benefit ($) Additional cost One Two Three Four Five Six Seven Eight Note that the marginal benefit of each additional child is decreasing. That of the second child is $250, but that of the third is $90, so the couple will have two children. (b) Now consider two identical nuclear families: Ram and Rani (as above), and Mohan and Mona. Ram and Mohan are brothers and the two couples form a joint family. Both couples have exactly the same costs and benefits of having children as in the table. Now suppose that 50% of the upbringing costs of each child (e.g., child care) can be passed on to the other family. Each couple makes independent decisions, taking only its own welfare into account. Now how many children will each couple have?
6 6 If 50% of the costs are passed on, the marginal cost becomes $50, so that now they will have four kids each (by the same reasoning as in part (a)). (c) Explain the reason for this seemingly paradoxical result, using the concept of externalities, and try and understand why larger families (either integrated across generations or between siblings in the same generation), will tend to have a larger number of children per couple. Since now the cost is shared with the relatives, while the benefit is retained by the couple, when doing the cost-benefit analysis, each couple would choose to have more children. In this case, each additional child creates negative externality to the joint family, and causes the size of the family to increase to a sub-optimal level.
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