2. (Malthus Model with productivity shocks) Thomas Malthus, in 1798, argued that while population grew exponentially, the food supply increased only arithmetically, so that humanity was doomed to exceed its own means of subsistence. In order for the population to be placed back below its carrying capacity, Malthus belived that natural measures would be taken to lower the population such as disease, plague, and famine and that people should abstain from marriage to keep the fertility rate down. We formalize this idea. Consider an economy devoted to agriculture only. Output (Y) is produced with productivity (A+), labor (N) and land (D), and we suppose that the amount of land is fixed and equal to 1, D = 1. Then the relationship between inputs and output is described as follows: The population growth (N+++) is given by Y = ANa N++1 Nt = We where w is wage at time t, ws is subsistence wage. Notice that labor is assumed to be equal to population (N). a) Firms determine their demand for labor through a lens of profit maximiation. Write down the condition for optimal labor demand by firms. b) Derive the law of motion for population (Hint: Plug labor demand in (a) into the population growth equation to get rid of we, and express N++1 (left-hand side) in terms of N and other variables (right-hand side). c) Suppose that A = 1 for all t. Find out the steady state of the population using the law of motion for population from b). d) Suppose now that the discovery of a new ferilizer improves A+ from 1 to 2. Following this change, will the economy expand? Why? Explain what will happen to the population N and wage w over time