Carl and Simon are two rival pumpkin growers who sell their pumpkins at the Farmers’ Market in Lake Witchisit, Minnesota. They are the only sellers of pumpkins at the market, where the demand function for pumpkins is q = 3, 200 − 1, 600p. The total number of pumpkins sold at the market is q = qC + qS, where qC is the number that Carl sells and qS is the number that Simon sells. The cost of producing pumpkins for either farmer is $.50 per pumpkin no matter how many pumpkins he produces.
(a) The inverse demand function for pumpkins at the Farmers’ Market is p = a − b(qC + qS), where a = _____ and b = _____. The marginal cost of producing a pumpkin for either farmer is _______.
(b) Every spring, each of the farmers decides how many pumpkins to grow. They both know the local demand function and they each know how many pumpkins were sold by the other farmer last year. In fact, each farmer assumes that the other farmer will sell the same number this year as he sold last year. So, for example, if Simon sold 400 pumpkins last year, Carl believes that Simon will sell 400 pumpkins again this year. If Simon sold 400 pumpkins last year, what does Carl think the price of pumpkins will be if Carl sells 1,200 pumpkins this year? _______. If Simon sold qt−1 S pumpkins in year t−1, then in the spring of year t, Carl thinks that if he, Carl, sells qtC pumpkins this year, the price of pumpkins this year will be 2 − _______.
(c) If Simon sold 400 pumpkins last year, Carl believes that if he sells qtC pumpkins this year then the inverse demand function that he faces is p = 2 − 400/1, 600 – qtC /1, 600 = 1.75 – qtC /1, 600. Therefore if Simon sold 400 pumpkins last year, Carl’s marginal revenue this year will be 1.75 – qtC /800. More generally, if Simon sold qSt−1 pumpkins last year, then Carl believes that if he, himself, sells qtC pumpkins this year, his marginal revenue this year will be _______.
(d) Carl believes that Simon will never change the amount of pumpkins that he produces from the amount qSt−1 that he sold last year. Therefore Carl plants enough pumpkins this year so that he can sell the amount that maximizes his profits this year. To maximize this profit, he chooses the output this year that sets his marginal revenue this year equal to his marginal cost. This means that to find Carl’s output this year when Simon’s output last year was qt−1 S , Carl solves the following equation. _______.
(e) Carl’s Cournot reaction function, RtC (qSt−1), is a function that tells us what Carl’s profit-maximizing output this year would be as a function of Simon’s output last year. Use the equation you wrote in the last answer to find Carl’s reaction function, _______.
(f) Suppose that Simon makes his decisions in the same way that Carl does. Notice that the problem is completely symmetric in the roles played by Carl and Simon. Therefore without even calculating it, we can guess that Simon’s reaction function is _______. (Of course, if you don’t like to guess, you could work this out by following similar steps to the ones you used to find Carl’s reaction function.)
(g) Suppose that in year 1, Carl produced 200 pumpkins and Simon produced _______ pumpkins. In year 2, how many would Carl produce? _______. How many would Simon produce? _______. In year 3, how many would Carl produce? _______. How many would Simon produce? _______. Use a calculator or pen and paper to work out several more terms in this series. To what level of output does Carl’s output appear to be converging? _______ How about Simon’s? _______.
(h) Write down two simultaneous equations that could be solved to find outputs qS and qC such that, if Carl is producing qC and Simon is producing qS, then they will both want to produce the same amount in the next period. qs = _______.
(i) Solve the two equations you wrote down in the last part for an equilibrium output for each farmer. Each farmer, in Cournot equilibrium, produces _______ units of output. The total amount of pumpkins brought to the Farmers’ Market in Lake Witchisit is _______. The price of pumpkins in that market is _______. How much profit does each farmer make? _______.