Question

Consider an Overlapping generations economy with constant population N = 100 of young
individuals are born each period. Individuals are endowed with y1 units of the consumption
good when young and y2 when old, the initial old also starts with y2. The utility function
of one typical agent is: u(c1, c2) = log(c1) + 1/2 log (c2)

Consider the case in which y1 = 5 and y2 = 2.

(a) State and solve the Planner’s Problem.
(b) Suppose the initial old are endowed with a total of M = 100 units of fiat money. Write down equations that represent the constraints on first and second-period consumption for a typical individual. Combine these constraints into a lifetime budget constraint.
(c) Find the Stationary Competitive Equilibrium.

Answer 1

a)

$$5y_1+2y_2\le 100$$

b)

$$\frac {\delta U}{\delta c_1}=\frac {1}{c_1}$$ $$\frac {\delta U}{\delta c_2}=\frac {1}{2c_2}$$

c)

$$y_2=1.25 y_1$$

$$5y_1+2.5y_1=100$$

$$y_1=13$$

$$y_2=16$$