Demand for hot dogs on Main Street in Smalltown is:Q = 100 -20P

Assume that there are fixed costs of $1200 and that MC = $1.00. The purpose of this problem is to examine duopoly equilibria in this market (different behavioral assumptions lead to different equilibria) and to compare these equilibria to both the competitive and monopoly equilibria as well as to each other.

1. Graph the Demand and MR.
2. Indicate the competitive price and quantity with P* and Q*.
3. Indicate the monopoly price and quantity with Pm and Qm.
4. The town grants vending rights to Doug's Dogs and Frank's Franks, at no charge. Other vendors are excluded. Assume for now that both Doug and Frank are Cournot Duopolists---they take each other's production as fixed and adjust their own to maximize their profits. Let q_Drepresent Doug's production and q_Frepresent Frank's: Q = q_D+ q_F.
5. Write Doug's profits as a function of q_Dandq_{F .}[Hint:as a first step, invert the demand function and replace Q with q_{D +}q_F.] _D= (1)
6. Suppose that Doug's knows q_F. If he assumes that this will not change once Frank knows q_D, What value of q_Dshould Doug Choose to maximizehis profits as a function of q_F? This is Doug's "reaction function". Draw it one the diagram. q_D= (2)
7. Now, find Frank's reaction function and add it to the diagram. q_F= (3)
8. Suppose that Doug had been producing at the monopoly solution before Frank arrived on the scene. Indicate his output level on the reaction function graph. What output would Frank initially choose to sell? Frank's Output to sell = (4)
9. Explain why this would not be an equilibrium, and describe what would happen.
10. Draw the path that output would follow toward the new equilibrium on the reaction function diagram. What would equilibrium P, q_D,q_Fand Q be? P = (6), q_D= (7)_,q_F= (8)_,Q= (9), Doug's profits are _D= (10), Frank's profits are _F= (11), Their combined profits are = (12). Plot P and Q on the graph in a and label them Pc and Qc (C for Cournot) (13)
11. Stackelberg Model, Leader Follower.
12. Explain why the behavior of both Frank and Doug in b is nave.
13. Assume that Frank continues to behave naively, but that Doug figures out that Frank is reacting to his decisions. Specifically, Dough Knows that Frank's choice of q_Fis a function of his ownchoice of q_D,as you found in b (Frank's reaction function). Dough takes this into account by substituting Frank's reaction function into his own profit equation, to get: _D= (2)
14. Find Doug's new profit-maximizing choice of q_D.q_D=(3), Frank will choose q_F= (4), Total Quantity produced is q = (5), The price is P = (6), Doug's profits are _D= (7), Frank's profits are _F= (8), Their combined profits are = (9). Plot the new Q and P on the graph in a, labelling them P_LFand Q_LF(LF for Leader-Follower) Also, plot the point (q_D,q_F) on the reaction function diagram in b and label it LF.
15. Leader-Leader model: aka Bertrand Model. In c it was assumed that Frank continued to be nave. Suppose instead that he figures out Doug's behavior at the same time that Doug figures out his own. That is, they both simultaneously decide to lead, mistakenly thinking the other will follow.
16. This choice of output level will then be q_F= (1) [Hint: does his problem differ in any way from Doug's in c?] q_F= (1), Total output will be Q = (2), And the price will be P = (3), Doug's profits are _D= (4), Frank's profits are _F= (5), Their combined profits are = (6). Plot the new Q and P on the graph in a and label them Q_LLand P_LL. Also, plot the point (q_D,q_F)on the reaction function diagram in b and label it LL. Finally, what do you speculate will happen once they both realize that the other is not following? (7)

Notes:

All I need to be solved in this problem is finding the Producer Surplus and Total Surplus. Please Show all the steps and do not copy from other solutions as they are either incorrect or not very clear. Thank you.