UNIVERSITY OF GUELPH X Gordon S. LANG School of Business and Economics Department of Economics and Finance a. Write-down the equation governing the relationship between the growth rate of the aggregate ECON*3810.01: ADVANCED MACROECONOMICS population from time & to t+1 and the income per capita at time & (5 points) b. Plot the relationship found in a. in a two-dimensional diagram. (5 points) Winter Semester 2021 C. Show that the output per capita at time & can be written as a function of the productivity level Homework Assignment 1 at time t and the land per capita at time & (5 points) (Due: Feb 2nd 2021 11:59PM) Let us assume in the next two questions that the productivity level depends on the size of the elite class: I. A Malthusian Economy with Heterogeneous Agents. (60 points) Zt = AEP Let us consider a Malthusian economy populated at time t with an aggregate population of size Nt. where pe(0,1) is a parameter with p 0 and a =(0,1) are parameters and y, denotes the income/output per capita at time t defined as: f. According to your answers to d. and e. is it possible for an economy to be "asphyxiated" by its elite class? If so, then what would be the "optimal" fraction of the aggregate population to be part of the elite class? (10 points) Let us assume in the next two questions that productivity grows at every period at a constant where Y, denotes the aggregate level of income/output at time t which can be described by the exogenous rate: following aggregate Cobb-Douglas production function: Zit1 - 21 = DE Yt = ZXBL-6 Zt where &, pe(0,1) are parameters. where BE(0,1) is a production parameter, Z, > 0 stands for the aggregate productivity level at time t X denotes a fixed aggregate stock of land used in agriculture and Lt > 0 stands for the aggregate g. Using the production function per capita found in c., derive a relationship between the growth population of workers which is assumed to be a fraction 1 - & of the aggregate population: rate of the output per capita from time & to f+1 and the growth rate of the aggregate population from time fto f+1. (5 points) Lt = (1 - E)N. h. Derive the steady-state population growth rate and the steady-state output per capita level where &=(0,1) denotes the fraction of the aggregate population that is part of an elite class of size compatible with no income per capita growth? (10 points) Er at time t: i. Does the economy achieve a higher standard of living in h. than in e.? Explain (5 points) Et = EN. Let x't stand for the land per capita at time & defined as: 2