Question 4. Simple model with overlapping generations and income growth

Consider a simple model with overlapping generations. 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 Ny,t and No,t for the young and old generation, respectively. Hence, the size of the total population is: Nt = Ny,t + No,t. The young generation (and population) grows at the fixed growth rate n, so that: Ny,t = (1 + n)Ny,t1. 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 y2 = (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 cy and co denote consumption in each period for the young and the old generation, respectively. Any individual savings st between periods earn the interest rate rt. Individuals discount the future at rate , while preferences are given by the lifetime utility function:

U(cy, co) = ln(cy) + ln(co).

a. Given the description of this economy, solve for the optimal allocations of a typical individual: cy, co, st. 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 accu- mulation of aggregate capital in the economy? Provide some economic intuition for this result