Two firms supply an industry with a demand curve of Q(P)=200-5P and each firm has an constant marginal (and average) cost of $1. Please fill in the following chart and attach a sheet of paper showing all formulas and work.

Graduate Students (and Extra Credit for Undergraduates): Derive the Stackelberg result (output) for the example used in the text (attached is the mathematical derivation from another book):

Market Demand: Q=1000 - 1000P

MC=$0.28

Caution: I'm not sure that the answer listed in Table 7.1 is correct.

Collusion

The collusion solution is the found by finding the monopoly solution and dividing output among the firms (in this case 2). The monopoly solution is found by MR=MC.

P=40-1/5Q

TR = PQ = (40-1/5Q)Q

MR = 40-2/5Q = MC = 1

39=2/5Q

Q=97.5

q=48.75

P=40-1/5Q=40-1/5(97.5)=20.5

Profit of the firm = (P-AC)q = 19.5*48.75 = 950.625

Consumer Surplus = ½ hb = ½ (40-20.5) 97.5 = 950.625 (40 comes from finding P when Q=0)

Cournot

Profit1 = TR=TC = Pq1-1q1 = (40-1/5Q)q1-q1 = (40-1/5(q1+q2))q1-q1

dprofit1/dq1 = 40-2/5q1-1/5q2-1 = 0

39=2/5q1+1/5q2

q1=95-1/2q2 (reaction curve)

by symmetry q2=95-1/2q1

solving both reaction curves simultaneously

-39=3/5q1

q1=65=q2

Q=130

Find P by solving P=40-1/5(Q)=40-1/5(130)=14

profit for the firm = (P-AC) q = 13*65=845


Consumer Surplus = ½ (40-14) 130 = 1690

Bertrand Model

Bertrand is similar to the competitive outcome where P=MC, P=MC=1, Q=200-5P = 195

q = Q/2 = 97.5

Profit for the firm = (P-AC) q = (0) *97.5 = 0

Consumer Surplus = ½ (40-1) (195) = 3802.5

Stackelberg n-firms

P=1-.001Q=1-.001(q1+(n-1)qf)

profit f = P(Q)qf-.28qf

dprofit f/dqf = [d(P(Q))/dqf]qf + P(Q)dqf/dqf-.28=.001qf+(1-.001(q1+(n-1)qf))-.28=

.72-.001q1-.001qf(n-1+1)=0

qf=(.72-.001q1)/.001n

Firm L (1) recognizes all n followers will behave acording to this reaction function:

(n-1)dqf/dq1=(n-1)(-1/n)=-(n-1)/n

Firm L's profit maximization is

profit 1 = p(Q)q1-.28q1

dprofit 1/dq1 = p(Q)+ql*dP(Q)/dq1-.28

dP(Q)/dq1=-.001 * dQ/dq1=-.001 *(dq1/dq1+dqf/dq1)=-.001* (1-(n-1)/n)

dprofit 1/dq1=(1-.001(q1+(n-1)qf))-.001(1-(n-1)/n)q1-.28=0

.72-.001q1-.001(n-1)(.72-.001q1)/.001n-.001(1-(n-1)/n)q1=0

.72(1-(n-1)/n)-.001q1(1-(n-1)/n+1/n)=0

.72/n-.001q1(2/n)=0

q1=360

qf=(.72-.001(360))/.001n=360/n