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Altruistic parents choose fertility and consumption by maximizing a dynastic utility function. The maximization implies an arbitrage condition for consumption across generations, and equality between the benefit from an extra child and the child-rearing cost. These conditions imply that fertility in open economies depends positively on the world's long-term real interest rate, the degree of altruism, and the growth of child-survival probabilities; and negatively on the rate of technical progress and the growth rate of social security. The growth of per capita consumption across generations depends on changes in the child-rearing cost, but not on interest rates or time preference.
Becker et al. (Mon,) studied this question.