Why do we typically consider economies where the endowment when young is greater than the
endowment when old? Consider a model economy as described in Module 1, but assume that workers are endowed with y units of the consumption good when old and 0 units when young. Assume the
population is constant over time, i.e. Nt = Nt−1 = N for all t ≥ 0.
(a) Explain the planner’s problem in words.
(b) Derive the planner’s feasible set for a given period t and do not impose the assumption that the
allocation is stationary.
(c) What is a stationary allocation? Explain.
(d) Assume the planner wants to implement a stationary allocation. Draw a figure that depicts the
solution to the planner’s problem using the feasible set and sample indifference curves that satisfy
the properties discussed in Lecture 1.
Now consider a decentralized version of this economy where individuals make their own consumption
and trading decisions. Do you think there is a way to sustain trade between the young and the old in this
case? Explain.
Some important info:
We focus on the case in which conversion is the same for each generation: c1,t=c1c1,t=c1 and c2,t+1=c2c2,t+1=c2.
This is known as a stationary equilibrium.
The relative mix of consumption when young and consumption when old depends on people’s behaviour and the conversion rate.
Assume every member of generation tt is given the same lifetime allocation (c1,t,c2,t+1)(c1,t,c2,t+1). Then,
Total consumption by young in period tt is Ntc1,tNtc1,t
Total consumption by old in period tt is Nt−1c2,tNt−1c2,t
Then, total consumption by young and old at time tt is Ntc1,t+Nt−1c2,t≤NtyNtc1,t+Nt−1c2,t≤Nty
For simplicity, assume a constant population and drop the time subscript: Nc1,t+Nc2,t≤NyNc1,t+Nc2,t≤Ny.
Dividing by NN, we obtain the per capita constraint the central planner is facing: c1,t+c2,t≤yc1,t+c2,t≤y.
We are looking for a stationary allocation, i.e., an allocation that gives the members of the same generation the same lifetime consumption pattern.
Then, the per capita constraint becomes: c1+c2≤yc1+c2≤y. This is a very simple linear equation in c1c1 and c2c2.
Graphically, the set of feasible per capita allocations is bounded by the triangle in the diagram below (Figure 1.8). We refer to the triangular region as the feasible set. The feasible set line shows the per capita constraint binding.