Economists often think of parents and children trading with one another across time. When children are young, parents take care of children; but when parents get old, children often come to take care of their parents. We will think of this in a two-period model in which children earn no income in period 1 and parents earn no income in period 2. For purposes of this problem, we will assume that parents have no way to save in period 1 for the future and children have no way to borrow from the future when they are in period 1. Thus, parents and children have to rely on one another.

Suppose that, during the periods when they earn income (i.e., period 1 for parents and period 2 for children), parents and children earn the same amount y. Suppose further that everyone has homothetic tastes with MRS = -1 when c1 = c2.

1. Let p be the price of current consumption in terms of future consumption (and let the price of future consumption be normalized to 1). Illustrate a competitive equilibrium.

2. Consider two types of government spending: (1) spending on social security benefits for retirees, and (2) investments in a clean environment for future generations. When would this model predict will the environment do better: During baby booms or during baby busts? Why?