arl and Simon are two rival pumpkin growers who sell their 27.1 (0) C pumpkins at…

arl and Simon are two rival pumpkin growers who sell their 27.1 (0) C 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 qs, where qc is the number that Carl sells 323 NAME mber that Simon sells. The cost of producing pu mpkins 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 2and b 1/1,600. The c+qs), where a – $.50 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 ar. 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 were sold by the other farmer last ye pumpkins will be if Carl sells 1,200 pumpkins this year? 1 If pumpkins will be if Carl sells 1.200 pumpkins this vear? Simon sold pumpkins in year t -1, then in the spring of year t, Carl thinks that if he, Carl, sells pumpkins this year, the price of pumpkins If 2-41 + /1, 600 his year will be (c) If Simon sold 400 pumpkins last year, Carl believes that if he sells ae pumpkins this year then the inverse demand function that he faces is p 2-400/1.600-__/ 1.600 1.75-GC / 1.600. Therefore if Simon sold 400 pumpkins last year, Carl’s marginal revenue this year will be 1.75 /800. More generally, if Simon sold ds pumpkins last year, then Carl believes that if he, himself, sells pumpkins this year, his marginal reveme this year will be 2/1,600 /800 1.600 (d) Carl believes that Simon will never change the amount of pumpkins that he produces from the amount js 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 q^1, Carl solves the following equation. 800-.5 Is (e) Carl’s Cournot reaction function. (qS-1), _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._,(qs-1)_ 1.200-G1/2. (Hint This is a linear expression of the form a – bds1. constants a and b.) You have to find the (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 (q i) = 1.200- 2. (Of course, if you don’t like to guess, you could work this out by following similar steps to the ones vou used to find Carl’s reaction function (g) Suppose that in year 1, Carl produced 200 pumpkins and Simon pro- duced 1,000 pumpkins. In year 2, how many would Carl produce? 700. How many would Simon produce? 1,100. In year 3, how many would Carl produce? 650. How many would Simon produce? 850. 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? 800 How about Simon’s? 800 (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 produc- ing qs, then they will both want to produce the same amount in the next 1,200-90/2 period. (Hint: _se the reaction functions.) _ and gc= 1.200-qs/2 i) Solve the two equations you wrote down in the last part for an equi- librium output for each farmer. Each farmer, in Cournot equilibrium, 800 units of output. The total amount of pumpkins brought produces to the Farmers Market in Lake Witchisit is 1,600. The price of Solve the two equations you wrote down in the last part for an equi- librium output for each farmer. Each farmer, in Cournot equilibrium, produces 800 units of output. The total amount of pumpkins brought to the Farmers Market in Lake Witchisit is 1,600. The price of pumpkins in that market is $1. How much profit does each farmer make? $400