Suppose that there are three individuals with three wealth levels in the economy. Denote the wealth levels by A, B, and C, and suppose that A < B < C.

(a) Suppose that the person with wealth level A earns an annual income of w_A, and saves a fraction s_A of it. If the rate of interest on asset holdings is r, write down a formula for this person’s wealth next year.

(b) Show that if income earnings of each individual is proportional to wealth (that is, w_B/w_A = B/A and w_C/w_B = C/B), and if the savings rate is the same across individuals, inequality of wealth next year must be the same as inequality this year, as measured by the Lorenz curve.

(c) Retain the same assumptions on income and wealth as in part (b) but now suppose that the savings rates satisfy s_A < s_B < s_C. Now how does the Lorenz curve for wealth next year compare with its counterpart for the current year?

(d) Carry out the same exercise as in part (c), but now assume that all wages are equal and so are savings rates.

(e) Try and understand these results intuitively as forces that change inequality over time. What features (other than the ones studied in this question) might also affect inequality in the context of this example?