D2L ECON3810W24HW1 - ECON* M Gmail -0 D2L ECON3810W24HW1 - ECON* courselink.uoguelph.ca/d2l/le/content/864431/fullscreen/3735782/View ECON3810W24HW1 1 of 2 90% I. Malthusian Models (60 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: Y = A+XN+-x where ae(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): At+1 - At At = a Let x+ stand for the land per capita at time t defined as: X xt = Nt as: Let us consider that the birth rate is fixed at b=(0,1) and the death rate at time t can be written d = 1 - Syt where > 0 is a parameter and yt denotes the output per capita at time t defined as: Yt Yt = Nt a. Show that the rate of population growth from time to t+1 can be written as a function of the income per capita and identify the level of income per capita compatible with no population growth. (8 points) 1 Relaunch to update : Print Download D2L ECON3810W24HW1 - ECON* D2L ECON3810W24HW1 - ECON* M Gmail -0 courselink.uoguelph.ca/d2l/le/content/864431/fullscreen/3735782/View ECON3810W24HW1 2 of 2 + 130% b. Write down the equation of the production function in per capita units. Is this relationship consistent with Malthus' assumption 1 given At? (12 points) c. Using your answer to question b., derive the exact growth relationship showing how the aggregate population growth rate from time to t+1 and the productivity growth rate of a affect the output per capita growth rate from time to t+1. (8 points) d. Using your answers to questions a and c, derive a relationship showing how the output per capita at time t affects its rate of change from time to t+1 given the productivity growth rate a. (4 points) e. Using your answer to question d, derive the steady-state level of output per capita: yss compatible with no growth in the living standard. (8 points) f. Derive the population growth rate at the steady-state solution found in question e. (8 points) t g. In the absence of productivity growth rate: a=0 where A++1 = A = Z, derive the new output per capital steady-state solution as well as the associated population growth rate and population size. (12 points) Relaunch to update : Print Download