The economy consists of a single individual who represents a large number of homogenous individuals. This representative individual is assumed to live for two periods. Her preferences over consumption in periods 1 and 2 are given by U = ln(C1) + ln(C2). The individual receives an income of $100 in period 1 and $50 in period 2. The individual can save as much as she wants at the given interest rate of r=10% per period. There is no bequest motive, so the individual spends all her money before the end of period 2.

a. Write the individual's per period budget constraints (i.e. constraints in periods 1 and 2). Derive and plot the individual's lifetime budget constraint.

b. Solve for the optimal consumption in both periods.

c. How much saving does the individual do in the first period?

Now, suppose that the government decides to set up a social security system. This system will take $10 amount from individual's income in the first period, put it in the bank, and transfer it to the individual with interest in the second period.

d. What are the individual's budget constraints in periods 1 and 2 and the new lifetime budget constraint?

e. Solve for the optimal consumption in both periods.

f. How does the system affect the optimal amount of private savings?

d. Suppose instead that the government uses the $10 contribution from each individual to start paying out benefits to current retirees (who did not pay in to a social security when they were working). It still promises to pay current workers their $10 (plus interest) back when they retire using contributions from future workers. Similarly, it will pay back future workers interest on their contributions using the contributions of the next generation of workers. An influential politician says: "This is a free lunch: we help out current retirees, and current and future workers will still make the same contributions and receive the same benefits, so it doesn't harm them, either." Do you buy this argument? If not, what is wrong with it?