Let us assume that all economies were at the steady-state in 1820 and the output per capita at time t was described by the following aggregate production function:

yt = A_t (X/N_t)^1/3

where A_t stands for the aggregate productivity level at time t, X denotes the fixed stock of land used in agriculture and N_t represents the aggregate population size at time t.

a. Derive a relationship between the growth rate of the output per capita from time t to t+1, and the growth rate of the aggregate population from time t to t+1 under the assumptions that productivity grows at a constant rate .