Children, Parents, Baby Booms and Baby Busts: 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 2-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.
A: 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.
(a) Suppose first that there is one parent and one child. Illustrate an Edgeworth Box with current consumption c1 on the horizontal and future consumption c2 on the vertical axes. Indicate where the endowment allocation lies.
(b)Given that everyone has homothetic tastes (and assuming that consumption now and in the future are not perfect substitutes), where does the region of mutually beneficial trades lie?
(c)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.
(d) Suppose that there are now two identical children and one parent. Keep the Edgeworth Box the same dimensions as in (a). However, because there are now two children, every action on the a child’s part must be balanced by twice the opposite action from the one parent that is being modeled in the Edgeworth Box. Does p go up or down? (An equilibrium is now characterized by the parent moving twice as far on the equilibrium budget as each child.)
(e) What happens to child consumption now and parent consumption in the future?
(f) Instead, suppose that there are two parents and one child. Again show what happens to the equilibrium price p.
(g)What happens to child consumption now and parent consumption in the future?
(h) Would anything have changed in the original one-child/one-parent equilibrium had we assumed two children and two parents instead?
(i) While it might be silly to apply a competitive model to a single family, we might interpret the model as representing generations that compete for current and future resources. Based on your analysis above, will parents enjoy a better retirement if their children were part of a baby boom or a baby bust? Why?
(j) Will children be more spoiled if they are part of a baby boom or a baby bust? Why?
(k) 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?
B: Suppose the set-up is as described in A and A(a), with y = 100, and let tastes be described by the utility function u(c1,c2) = c1c2.
(a) Is it true that, given these tastes, the entire inside of the Edgeworth Box is equal to the area of mutually beneficial allocations relative to the endowment allocation?
(b) Let p be defined as in A(c). Derive the parent and child demands for c1 and c2 as a function of p.
(c) Derive the equilibrium price p∗ in the case where there is one parent and one child.
(d)What is the equilibrium allocation of consumption across time between parent and child?
(e) Suppose there are 2 children and one parent. Repeat (c) and (d).
(f ) Suppose there are 2 parents and one child. Repeat (c) and (d).
(g) Suppose there are 2 children and 2 parents. Repeat (c) and (d).