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EC185-PS2-3

# EC185-PS2-3 - The Theory of Economic Growth Problem set 2-3...

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The Theory of Economic Growth Problem set 2-3 Theories of the Demographic Transition (DT) Solutions ¨ Omer ¨ Ozak * October 11, 2008 A The rise in income in the process of industrialization triggered DT Suppose that parents generate utility from consumption, c; and children, n: Their utility function is u = n γ c 1 γ Parents are endowed with one unit of time. They optimally allocate this unit- time endowment between work and raising children. If they work full time their wage income is w . The time cost of raising a child is a fraction τ of parental time and consequently the opportunity cost of raising n children is wτn . The cost of consumption is normalized to 1. Parental’s budget constraint is therefore wτn + c w. Problem 1. Find the optimal level of consumption, c ; and the optimal number of children, n . (You may use the fact that when preferences are homothetic the share of income allocated to consumption is given by the exponent that is associated with consumption in the utility function, 1 - γ , and the share of income allocated to children is the exponent associated with children, γ . Solution: We know that the budget constraint has to hold with equality (why?), so that wτn + c = w. * Department of Economics, Brown University. Email: [email protected] 1

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¨ Omer ¨ Ozak On the other hand, the marginal rate of substitution between children and consumption has to equal the ratio of their prices, so that γn γ 1 c 1 γ (1 - γ ) n γ c γ = 1 which implies that γ 1 - γ c = wτn. Replacing this into the budget constraint, we get c = (1 - γ ) w and, thus, n = γ τ . Problem 2. Could a rise in wage income generate a decline in fertility in this economy? Why? Solution: We have shown that in this economy the optimal number of children per worker is independent of income. The reason for this outcome is that the utility function is homothetic, so that income and substitution effects cancel each other. Problem 3. What changes in the utility function will permit a rise in income to generate a decline in fertility? Solution: We need the substitution effect to dominate at higher levels of income, e.g. allowing for a declining exponent in the utility function. Problem 4. What are the testable predictions of the Beckerian theory? Solution: It should be the case that among countries (similar in social-political-environmental fac- tors), richer countries will experience the transition earlier. Also, within an economy, richer individuals will have lower number of surviving children than poorer ones. Problem 5. 2
¨ Omer ¨ Ozak Evidence suggests that countries that are similar in their sociopolitical en- vironment, but differ in income, experienced the DT in the same time period. How does it reflect on the Beckerian theory?

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EC185-PS2-3 - The Theory of Economic Growth Problem set 2-3...

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