Question

There are two firms competing in the market for Airplanes -Boeing and Airbus. 4. 120-p. Boeing has lower Marginal Costs of The market demand is given by Q $0 for both production than Airbus. Thus MC"-$20, MCA-$40. Assume that TFC firms. (Think of price being in thousands.) Boeing: Derive Boeing's residual demand curve, assuming that Airbus produces qA units. a. b. What is Boeing's Marginal Revenue? Derive Boeing's Reaction Function c. Airbus: Derive Airbus' residual demand curve, assuming that Boeing produces q8 units. d. What is Airbus' Marginal Revenue? e. f. Derive Airbus' Reaction Function Find the duopoly quantities sold by the two firms. g. h. Who produces the greater output and why? What is the total quantity of airplanes sold in the market? What is the i. equilibrium price? Calculate the profit for each firm and the industry profit. j. What will happen to this market if the market price is $35? k.

Answer 1

The total quantity would be , and the inverse demand curve would be , and the market residual demand would be .

(a) Boeing's residual demand would be .

(b) Boeing's MR would be as $MR_B = \frac{\partial (TR_B)}{\partial q^B} = \frac{\partial }{\partial q^B}(pq^B)$ or $MR_B = \frac{\partial }{\partial q^B}(120q^B - q^A q^B - (q^B)^2)$ or .

(c) Boeing's RF would be where its MR is equal to the MC, ie or or or .

(d) Airbus's residual demand would be .

(e) Airbus's MR would be as $MR_A = \frac{\partial (TR_A)}{\partial q^A} = \frac{\partial }{\partial q^A}(pq^A)$ or $MR_A = \frac{\partial }{\partial q^A}(120q^A - (q^A)^2 - q^A q^B)$ or .

(f) Airbus's RF would be where its MR is equal to the MC, ie or or or .

(g) The quantities produced by each firm would be where their reaction functions intersect. Solving for the RF's, putting the RF of Airbus in Boeing, we have or or or , and since , we have or . These are the required quantities produced by the firms.

(h) Boeing produces greater quantities than Airbus since Boeing have lower marginal cost than Airbus.

(i) The total quantity sold in the market would be or . At this quantity, the market price would be or or dollars.

(j) The marginal cost of Boeing is or $\frac{\mathrm {d} }{\mathrm {d} q^B}(TC_B) = 20$ or $TC_B = \int 20 \mathrm{d} q^B$ or , and since fixed cost is zero, we have . The profit of Boeing would be $\pi_B = TR_B - TC_B$ or $\pi_B = pq^B - 20q^B$ or $\pi_B = 60*40 - 20*40 = 2400 - 800 = 1600$ dollars.

The marginal cost of Boeing is or $\frac{\mathrm {d} }{\mathrm {d} q^A}(TC_A) = 20$ or $TC_A = \int 40 \mathrm{d} q^A$ or , and since fixed cost is zero, we have . The profit of Boeing would be $\pi_A = TR_A - TC_A$ or $\pi_A = pq^A - 40q^A$ or $\pi_A = 60*20 - 40*20 = 1200 - 800 = 400$ dollars.

The market profit would be $\pi = \pi_A + \pi_B$ or $\pi = 400 + 1600 = 2000$ dollars.

(k) Supposing that the market price is fixed at $35. In that case, Boeing's MR would be , and the optimal output would be where or or . Similarly, Airbus's MR would be , and the optimal output would be where or or . In case the price is not fixed, it would be reestablished to the equilibrium price found before as market price. But, if the price is fixed at $35, then the quantity produced would be and .

Answer 2

The total quantity would be , and the inverse demand curve would be , and the market residual demand would be .

(a) Boeing's residual demand would be .

(b) Boeing's MR would be as $MR_B = \frac{\partial (TR_B)}{\partial q^B} = \frac{\partial }{\partial q^B}(pq^B)$ or $MR_B = \frac{\partial }{\partial q^B}(120q^B - q^A q^B - (q^B)^2)$ or .

(c) Boeing's RF would be where its MR is equal to the MC, ie or or or .

(d) Airbus's residual demand would be .

(e) Airbus's MR would be as $MR_A = \frac{\partial (TR_A)}{\partial q^A} = \frac{\partial }{\partial q^A}(pq^A)$ or $MR_A = \frac{\partial }{\partial q^A}(120q^A - (q^A)^2 - q^A q^B)$ or .

(f) Airbus's RF would be where its MR is equal to the MC, ie or or or .

(g) The quantities produced by each firm would be where their reaction functions intersect. Solving for the RF's, putting the RF of Airbus in Boeing, we have or or or , and since , we have or . These are the required quantities produced by the firms.

(h) Boeing produces greater quantities than Airbus since Boeing have lower marginal cost than Airbus.

(i) The total quantity sold in the market would be or . At this quantity, the market price would be or or dollars.

(j) The marginal cost of Boeing is or $\frac{\mathrm {d} }{\mathrm {d} q^B}(TC_B) = 20$ or $TC_B = \int 20 \mathrm{d} q^B$ or , and since fixed cost is zero, we have . The profit of Boeing would be $\pi_B = TR_B - TC_B$ or $\pi_B = pq^B - 20q^B$ or $\pi_B = 60*40 - 20*40 = 2400 - 800 = 1600$ dollars.

The marginal cost of Boeing is or $\frac{\mathrm {d} }{\mathrm {d} q^A}(TC_A) = 20$ or $TC_A = \int 40 \mathrm{d} q^A$ or , and since fixed cost is zero, we have . The profit of Boeing would be $\pi_A = TR_A - TC_A$ or $\pi_A = pq^A - 40q^A$ or $\pi_A = 60*20 - 40*20 = 1200 - 800 = 400$ dollars.

The market profit would be $\pi = \pi_A + \pi_B$ or $\pi = 400 + 1600 = 2000$ dollars.

(k) Supposing that the market price is fixed at $35. In that case, Boeing's MR would be , and the optimal output would be where or or . Similarly, Airbus's MR would be , and the optimal output would be where or or . In case the price is not fixed, it would be reestablished to the equilibrium price found before as market price. But, if the price is fixed at $35, then the quantity produced would be and .