a. Write down the equation of the production function in per capita units and explain how an increase in the aggregate population size at time t affects the output per capita at that period given ! . (10 points) b. Using your answer to question a, derive the exact relationship showing how the growth rate of the output per capita from time t to t+1 is affected by the growth rate of the aggregate population from time t to t+1 given the productivity growth rate a. (10 points) c. Identify an equation showing how the output per capita at time t affects the growth rate of the aggregate population from time t to t+1. (5 points) d. Using your answers to questions b and c, derive a relationship showing how the output per capita at time t affects its rate of change from time t to t+1 given the productivity growth rate a. Explain how a higher output per capita affects its growth rate. (10 points) e. Using your answer to question d, derive the steady-state level of output per capita. (5 points) f. Derive the population growth rate at the steady-state solution. (5 points) g. How does a zero productivity growth rate: a=0 affect your answer to questions e and f. (5 points)

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ECON*3810: ADVANCED MACROECONOMICS Fall Semester 2022 Homework Assignment 1 (Due Day: October 5th,2022 ) I. A Malthusian Model with Productivity Growth (50 points) Let us consider a Malthusian economy in which the aggregate level of output at time t can be described by the following aggregate Cobb-Douglas production function: Yt=AtXNt1 where (0,1),At denotes the productivity level at time t,X>0 stands for a fixed aggregate stock of land used in agriculture and Nt represents the aggregate population. Let us assume that productivity grows at a constant rate a(0,1) : AtAt+1At=a Let xt stand for the land per capita at time t defined as: xtNtX At time t, let us consider that the birth rate: bt and the death rate: dt can be written as: bt=(yt)dt=1(yt) where ,>0,(0,1) are parameters and yt denotes the output per capita at time t defined as: ytNtYt The rate of population growth is given by: NtNt+1Nt=btdt ECON*3810: ADVANCED MACROECONOMICS Fall Semester 2022 Homework Assignment 1 (Due Day: October 5th,2022 ) I. A Malthusian Model with Productivity Growth (50 points) Let us consider a Malthusian economy in which the aggregate level of output at time t can be described by the following aggregate Cobb-Douglas production function: Yt=AtXNt1 where (0,1),At denotes the productivity level at time t,X>0 stands for a fixed aggregate stock of land used in agriculture and Nt represents the aggregate population. Let us assume that productivity grows at a constant rate a(0,1) : AtAt+1At=a Let xt stand for the land per capita at time t defined as: xtNtX At time t, let us consider that the birth rate: bt and the death rate: dt can be written as: bt=(yt)dt=1(yt) where ,>0,(0,1) are parameters and yt denotes the output per capita at time t defined as: ytNtYt The rate of population growth is given by: NtNt+1Nt=btdt