In the land of TwoOcc, there are just two occupations. You can become a worker at no cost, earning a wage of w, or you can become an entrepreneur. If the latter, you set up a factory at cost S, and then hire outside labor ` at the worker wage w, producing y = A l in total value. Everyone lives for just one round of production, and everyone seeks to maximize their income net of costs.
(a) If the wage is w, show that every profit-maximizing entrepreneur will hire & units of labor, given by 2 (1) () 2u and that each entrepreneurs's net income (her profit) will be given by A2 (2) Profit = S 4w (b) If everyone can freely choose to become a worker or a entrepreneur (and can pay the setup cost up-front), then they must be indifferent between these two occupations. Using this fact, show that the equilibrium wage must be given by A2 (3) wa + Sw= and using this formula (either graphically or algebraically), show that the equilibrium wage rises with A and falls with S. Intuitively explain these results in words. c) Using equation (1), show that the number of entrepreneurs - call it e- must be given by the formula (4) 1 + (A/[24])2' where n is the total working population. Using the equations you have so far, prove that the number of entrepreneurs must increase as A increases, and so must also the profits of each entrepreneur. (If you have trouble, try at least giving an intuitive explanation.) (d) Indicate very briefly which of equations (1-4)) might fail to hold if the setup cost must be borrowed, even if that borrowing is done at a zero rate of interest? n e (a) If the wage is w, show that every profit-maximizing entrepreneur will hire & units of labor, given by 2 (1) () 2u and that each entrepreneurs's net income (her profit) will be given by A2 (2) Profit = S 4w (b) If everyone can freely choose to become a worker or a entrepreneur (and can pay the setup cost up-front), then they must be indifferent between these two occupations. Using this fact, show that the equilibrium wage must be given by A2 (3) wa + Sw= and using this formula (either graphically or algebraically), show that the equilibrium wage rises with A and falls with S. Intuitively explain these results in words. c) Using equation (1), show that the number of entrepreneurs - call it e- must be given by the formula (4) 1 + (A/[24])2' where n is the total working population. Using the equations you have so far, prove that the number of entrepreneurs must increase as A increases, and so must also the profits of each entrepreneur. (If you have trouble, try at least giving an intuitive explanation.) (d) Indicate very briefly which of equations (1-4)) might fail to hold if the setup cost must be borrowed, even if that borrowing is done at a zero rate of interest? n e
