Discuss the endowment overlapping generations model

Question

Assignment:

Problem: Overlapping Generations Model

1. Consider the following endowment Overlapping Generations model where people live two periods. Suppose people are endowed with 1-ε units of the good when young and ε units when old. Assume ε < 1/2 Each generation has the same number of people.

Preferences are ln(c_y ) βln(c_old ) where β<1

a. Solve out the equilibrium where the government implements a pay as you go social security system with lump sum taxes so that agents consume 1/2 when young and 1/2 when old.

b. Can the government implement a fully funded system that gives the same consumptions for young and old as in the pay as you go in part a? Explain. Note the government plans of rolling over its debt.

c. Parts (a) and (b0 should convince you that not every pay-as-you-go/defined benefits system has a fully-funded/defined contributions equivalent. Let's see if we can find a pay-as-you-go/DB pension for which there is an equivalent fully-funded/DC system. In particular, what should the government's pay as you go system be in order for there to be an equivalent fully funded system where the government rolls over its debt every period that does not change over time?

2. Consider the Overlapping Generations model with production studied in the chapter. Namely, Each generation has the same number of people. Preferences are ln(c_y )-a_y [1-h_y ] + ln(c_old ) + a_old [1-h_old]. Each agent is endowed with one unit of time in both the first period and second period of his life. Production is given by : c^s= Ah^d.

a. Normalize A=1 and calibrate the value of αy so that in the equilibrium with the pay as you go system young people work .25. (This is the second row in the Summary and Conclusion Table).

b. Suppose α_old =2 α_y . Verify that people are better off in the pay as you go system compared to autarky?

c. By how much do you have to scale up c_y and c_old in the case of no government policy so that people have the same utility as in the pay as you go equilibrium.

3. Use the same Overlapping Generations model with production studied in the chapter with α_old > α_y. Consider a slightly different pay as you go system. Specifically, suppose the government imposes a tax on young consumption to fund the transfers to the old alive in the period. Suppose the tax on consumption and transfers to the old are such that (1) old people do not work and (2) people enjoy the same amount of consumption when young and when old. Solve for the equilibrium.