North Bend currently has one McDonald's fast-food franchise. Demand for hamburgers in North Bend is given by Q = 300 - 10P. Any McDonald's franchise has costs of C = 70 + 2Q for producing Q hamburgers. If a second McDonald's franchise were to move into North Bend (and both behaved as duopolists), the profit of the original McDonald's would fall from

Question

Quizplus · Accepted Answer

To find the original profit of the original McDonald's franchise, we need to first find the original price and quantity they were selling at. We can use the demand equation to solve for this:

Q = 300 - 10P
10P = 300 - Q
P = 30 - 0.1Q

Now we can substitute this into the cost equation to find the original profit:

π = TR - TC
π = (P*Q) - (70 + 2Q)
π = (30Q - 0.1Q^2) - 70 - 2Q
π = -0.1Q^2 + 28Q - 70

To maximize profit, we take the derivative and set it equal to zero:

dπ/dQ = -0.2Q + 28 = 0
Q = 140

So the original McDonald's was selling 140 hamburgers at a price of:

P = 30 - (0.1 * 140) = $16

And their original profit was:

π = -0.1(140)^2 + 28(140) - 70 = $1,890

Now, if a second McDonald's franchise enters the market, we have a duopoly situation where both firms will consider each other's actions when making pricing decisions. This means that the new profit-maximizing quantity will be less than 140 (the original quantity). To find this new quantity, we need to solve for the Nash equilibrium (where both firms' profits are maximized given the other firm's actions). This requires solving a system of equations:

Firm 1: Q1 = 300 - 10Ptotal
Firm 2: Q2 = 300 - 10Ptotal

where Ptotal is the total price (the price charged by both firms combined). Since we are assuming both firms behave as duopolists, they will split the total quantity equally, so Q1 = Q2 = 0.5Qtotal. Substituting this into the demand equations gives:

0.5Qtotal = 300 - 10Ptotal
Ptotal = 30 - 0.05Qtotal

Now we can find each firm's best response function (their profit-maximizing quantity given the total price charged by both firms). We start with Firm 1:

π1 = (P1 * Q1) - (70 + 2Q1)
π1 = (Ptotal/2 * 0.5Qtotal) - (70 + 2(0.5Qtotal))
π1 = (15 - 0.025Qtotal)(0.5Qtotal) - 70

Taking the derivative and setting it equal to zero gives:

dπ1/dQtotal = -0.025Qtotal + 15 - 0.5 = 0
Qtotal1 = 580

So Firm 1's best response function is Q1 = 0.5(580) = 290. By symmetry, Firm 2's best response function is also Q2 = 290. Substituting this back into the demand equation gives the total price charged:

Qtotal = Q1 + Q2 = 2(290) = 580
Ptotal = 30 - 0.05(580) = $1

Now we can find the new profit of the original McDonald's franchise using the new price and quantity:

Q1 = 140 - (1/2)(140 - 290) = 115
π1 = (P1 * Q1) - (70 + 2Q1)
π1 = ($15 * 115) - (70 + 2(115))
π1 = $801.11

So the original McDonald's profit falls from $1,890 to $801.11 if a second franchise enters the market. The answer is therefore D.

North Bend Currently Has One McDonald's Fast-Food Franchise