Please show all work for all parts

Make note in 4(a), to compare the profit from charging 29.99, with the profit from charging 30 (when the other firm charges 30), to demonstrate that charging 29.99 is better. In 4(b), find the best price to charge as a function of all rival's prices P that are at least 20.02, rather than just 20.02. That is, your answer will have the variable P in it. Similarly, in 4(c), your answer will be a function of P. For 4(d), your prices are a Nash equilibrium, but firm 2 is charging 20, which is exactly its cost per unit. There is another Nash equilibrium in which firm 2 charges more than 20, find that aswell 4. Two firms compete selling identical goods. Each firm can charge any price it wants, as long as the price can be denominated in pennies (for instance, a firm could charge 19.99, but not 19.999); they choose their prices simultaneously (they are playing a "Bertrand game"). Because the firms' goods are identical, all consumers who buy will buy from the firm with the lower price if they charge different prices. If they charge the same price, consumers who want to buy will divide themselves equally between the firms, i.e., each firm will sell the same quantity of output. When the lowest price any firm charges is P, consumers demand Q=50-P total units of output. Firm 1 has constant marginal and average cost of production of 10 per unit of output, firm 2 has constant marginal and average cost of production of 20 per unit of output. Under these assumptions, firm I would charge price 30 if it had a monopoly in the market; firm 2 would charge price 35 if it had a monopoly in the market. (a) Suppose that one (either) firm chose a price of 30. What price for the other firm would maximize the other firm's profit? If your answer is not 30, show that the other firm makes more profit the price that is your answer than the other firm makes by charging 30. The calculation will depend on which firm you pick as the "other firm"; you can choose either one. (b) In general, if one firm charges a price P that is at least 20.02, what price for the other firm would maximize its profit, as a function of P? Give an ex lanation of why this is true; you don't need calculations in your explanation, though you can provide them if you want. (c) Suppose that firm 2 charges a price P2 between 10.02 and 20.01. Find the price for firm 1, Pl, that maximizes firm 1 's profit, as a function of P2, (d) Find a Nash equilibrium of this game in which both firms charge more than their cost per unit (i.e., firm I charges more than 10, and firm 2 charges more than 20). That is, find a price for firm 1 and a price for firm 2, such that firm I 's price is maximizing 1 's profit given 2's price, and firm 2's price is maximizing 2's profit given I 's choice. Provide confirmation that 2's price is a best response, i.e., that 2 could not get more profit charging a different price.