Why did output per capita growth not start before the late 19th century? One story, memorably told by Thomas Malthus, is that population growth responds to increases in output, driving consump- tion down to a subsistence level. To formalize the Malthusian story, let's consider an economy where The production function is Y = A, L, where L, is labor supply and a < 1 is a parameter capturing returns to scale. There is no capital, saving or investment and as a result, all output is consumed each period, Ct = Yt. The population evolves according to the equation 4 (2-c) L+1 = 4+74 | where y is a parameter capturing how consumption translates into a higher birth rate, and c is a parameter capturing subsistence consumption. Assume that hours worked per worker is fixed at 1, so movements in labor supply come entirely from movements in the population. 1. Assume y > 0. Is this a reasonable assumption to make? Justify your answer. 2. Calculate the growth rate of the population, 9 = -1, as a function of A, L, a... For a given level of A,, does 9Lt increase, decrease or not change as L, increases? Provide intuition. 3. Suppose At = A at all dates - i.e. there is no technological progress. Calculate the value of Le such that gr=0, and call this Lo. 4. Continue to suppose A, = A at all dates - i.e. there is no technological progress. Show that if L, < Lo then gut > 0 and if L > Lo then gut