Professor Debraj Ray 19 West 4th Street, Room 608 Office Hours: Mondays 2.30–5.00pm email: debraj.ray@nyu.edu, homepage: http://www.econ.nyu.edu/user/debraj/ Webpage for course: click on “Teaching”, and then on the course name. Econ-UA 323 Development Economics Problem Set 7 (1) Review the concepts of birth rates, death rates, and age distributions, and the way in which these notions interact with one another. 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. (2) Here is a specific example shows how birth and death rates affect age distributions. 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 dy , dm , with do = 1. Finally, suppose that only M ’s give birth: let b be the fraction of newborns relative to the entire M -population. Let ny (t), nm (t) and no (t) be the populations of the three ages. Make sure you understand the following relationships (you don’t need to write anything for this): ny (t + 1) = bnm (t), nm (t + 1) = (1 − dy )ny (t), and no (t + 1) = (1 − dm )nm (t). (a) Now suppose that the population is growing steadily at the rate of g, with pop shares of each age group given by σy , σm and σo , and suppose that these shares are unchanging over time. Using the above relationships, show that (i) σy (1 + g) = bσm , (ii) σm (1 + g) = (1 − dy )σy , and (iii) σo (1 + g) = (1 − dm )σm . (b) Use part (a) to show that σy 1 − dy 1 − dm 1+ + = 1. 1+g b p (c) Use equations (i) and (ii) of part (a) to show that 1 + g = b(1 − dy ), and then combine with part (b) to conclude that " # p 1 − dy 1 − dm √ σy 1 + + = 1. b b Now show that as the birth rate b goes up, or as the death rates dy and dm go up, the share of the young, σy , must go up as well. 1 2 (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. To answer this question I want you to read the example of the “Prisoner’s Dilemma” in the game theory appendix of the main text, and then try and create an application of that to the question above. (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. (b) Calculate the expected number of surviving children for this family, under various values of q. (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? (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. (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. (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. (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. (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. (d) Birth rates may be high even when death rates may be falling. 3 (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. (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 50-50. (a) Will the country as a whole have more girl births than boy births (or vice versa) under this targeting rule? (b) Will larger families have more daughters or sons? (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? (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 500 100 Two 750 100 Three 840 100 Four 890 100 Five 930 100 Six 950 100 Seven 960 100 Eight 960 100 (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? (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.