A. A "company town" is one with only one (main) employer. In this town, to get a number of people to show up to work, one must offer a wage w(r). This supply curve satisfies w(0) > 0, and it is increasing, i.e. w'(x) > 0. If a workers are hired, they are paid w(r). Some of the workers who are receiving w(r) were willing to work for a lower wage. This means that the worker surplus when a are hired is the area between the function w(-) and w(r), WS(x) = xw(r)-fw(t) dt. This can be re-written as WS(x) = f[rw(x) - w(t)] dt. With people working, the employer makes profits 7(x) = pf(x)- rw(r) where f() is their production function, p is the price that they receive for their output, and rw(r) is their wage bill. The firm's problem and society's problem are maxrzo (r) and max>0 (x) + WS(x). A 1. Show, both graphically and with calculus, that WS() is an increasing function. A 2. Show that the firm hires fewer people than is socially optimal. A 3. Show that the solution to society's problem involves pf'(x) = w(r), that is, the marginal product of the last worker hired being equal to the wage they are paid. A 4. Company towns have used a large number of policies to make workers more desperate for employment, that is, they have tried to shift the function w() downwards so that they need to pay less to attract workers. Show that a downward shift in w() downwards increases the firms profits. A 5. What happens to the company's demand for labor if they man- age to shift w(-) downwards by decreasing w'(x) for all values of r?