Simple model with overlapping generations and income growth Consider a simple model with overlapping generations, similar to question 1. The following description remains exactly the same:

Assume that the life cycle has two time periods: youth-denoted by subscript y and old age-denoted by subscript o. Therefore, in each period the economy is populated by a young and an old generation. In each time period t, population sizes are N_y,t and N_o,t for the young and old generation, respectively. Hence, the size of the total population is:

N_t = N_y,t+ N_o,t.

The young generation (and population) grows at the fixed growth rate n, so that:

N_y,t = (1 + n) N_y,t1.

The part that is different from question 1 is that both young and old generations earn an exogenous income in each period: young earn y1 and old y2. In addition, income grows at rate g, so y_2 = (1 + g) y1 (and there is no pension system or production.).

Finally a typical individual makes inter-temporal decisions spanning the two periods of life. Let C_y and co denote consumption in each period for the young and the old generation, respectively. Any individual savings S_t between periods earn the interest rate rt. Individuals discount the future at rate , while preferences are given by the lifetime utility function:

U(c_y, co) = ln(c_y) + ln(c_o).

a. Given the description of this economy, solve for the optimal allocations of a typical individual: C_y, C_o, S_t. b. Compute the share of each generation in the total population and aggregate savings. c. How would an increase in the income growth rate g impact aggregate savings, that is the accumulation of aggregate capital in the economy? Provide some economic intuition for this result.