Parents, Children and the Degree of Substitutability across Time: Consider again exactly the same scenario as in exercise 16.6.
A: This time, however, suppose that parent and child tastes treat consumption nowand consumption in the future as perfect complements.
(a) Illustrate in an Edgeworth Box an equilibrium with a single parent and a single child.
(b) Is the equilibrium you pictured in (a) the only equilibrium? If not, can you identify the set of all equilibrium allocations?
(c) Now suppose that there were two children and one parent. Keep the Edgeworth Box with the same dimensions but model this by recognizing that, on any equilibrium budget line, it must now be the case that the parent moves twice as far from the endowment E as the child (since there are two children and thus any equilibrium action by a child must be half the equilibrium action by the parent). Are any of the equilibrium allocations for parent and child that you identified in (b) still equilibrium allocations?
(d) Suppose instead that there are two parents and one child. How does your answer change?
(e) Repeat (a) through (d) for the case where consumption now and consumption in the future are perfect substitutes for both parent and child.
(f ) Repeat for the case where consumption now and consumption in the future are perfect complements for parents and perfect substitutes for children.
(g) True or False: The more consumption is complementary for the parent relative to the child, and the more children there are per parent, the more gains from trade will accrue to the parent.
B: Suppose that parent and child tastes can be represented by the CES utility ³ function u(c1,c2) =

(a) Letting p denote the price of consumption now with price of future consumption normalized to 1, derive parent and child demands for current and future consumption as a function of ρ and p.
(b)What is the equilibrium price€”and what does this imply for equilibrium allocations of consumption between parent and child across time. Does any of your answer depend on the elasticity of substitution?
(c) Next, suppose there are 2 children and only 1 parent. How does your answer change?
(d) Next, suppose there are 2 parents and only 1 child. How does your answer change?
(e) Explain how your answers relate to the graphs you drew for the extreme cases of both parent and child preferences treating consumption as perfect complements over time.
(f ) Explain how your answers relate to your graphs for the case where consumption was perfectly substitutable across time for both parents and children.

Transcribed Image Text:

11p 0.5c-ρ +0.5c2ρ