Homework answers / question archive / University of Pittsburgh - ECON 1100 Chapter 13 1)a)      P = MC implies 70 – 2Q = 10, or Q = 30 and P b)         A monopolist produces until MR = MC yielding 70 – 4Q = 10 so Qm = 15 and Pm = 40

University of Pittsburgh - ECON 1100 Chapter 13 1)a)      P = MC implies 70 – 2Q = 10, or Q = 30 and P b)         A monopolist produces until MR = MC yielding 70 – 4Q = 10 so Qm = 15 and Pm = 40

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University of Pittsburgh - ECON 1100

Chapter 13

1)a)      P = MC implies 70 – 2Q = 10, or Q = 30 and P

b)         A monopolist produces until MR = MC yielding 70 – 4Q = 10 so Q^m = 15 and P^m = 40.  Thus π^m

c)         For Amy, MR_A = MC implies 70 – 4q_A – 2q_B = 10.  We could either calculate Beau’s profit-maximization condition (and solve two equations in two unknowns), or, inferring that the equilibrium will be symmetric since each seller has identical costs, we can exploit the fact that q_A = q_B in equilibrium. (Note: You can only do this after calculating marginal revenue for one Cournot firm, not before.)  Thus 70 – 6q_A­ = 10 or q_A = 10.  Similarly, q_B = 10.  Total market output under Cournot duopoly is Q^d = q_A + q_B = 20, and the market price is P^d

2          a)         With two firms, demand is given by .  If , then  or .  Setting  implies

 

                                               

 

b)         For Firm 1, .  Setting  implies

 

 

 

 

c)         Because of symmetry, in equilibrium both firms will choose the same level of output.  Thus, we can set  and solve

 

 

d)         If this market were perfectly competitive, then equilibrium would occur at the point where

 

e)         If the firms colluded to set the monopoly price, then

 

                                               

 

 

f)          If the firms acted as Bertrand oligopolists, the equilibrium would coincide with the perfectly competitive equilibrium of

g)         Suppose Firm 1 has  and Firm 2 has .  For Firm 1, .  Setting  implies

 

 

                                               

 

3          For Zack, MR_Z = MC_Z implies 100 – 2q_Z – q_A = 1, so Zack’s reaction function is q_Z­ = ½*(99 – q_A).  Similarly, MR_A = MC_A implies 100 – 2q_A – q_Z = 10 so Andon’s reaction function is q_A = ½*(90 – q_Z).  

 

4          a)         The inverse market demand curve is P = 100 – (Q/40) = 100 – (Q₁ + Q₂)/40.  Firm 1’s reaction function is found by equating MR₁ = MC₁:

             

b)         The two reaction functions give us two equations in two unknowns. Using algebra we can solve these to get: Q₁ = 1,333.33 and Q₂ = 533.33.

 

5          a)         Market demand is given by .  Firm 1’s reaction function can be found by equating its marginal revenue to its marginal cost.

 

 

b)         As discussed in Chapter 11, a multiplant monopolist will equate the marginal cost of production across all its plants.  At any level of marginal cost MC_T, each plant would operate so that MC_T = Q_i + 10, or Q_i = MC_T – 10.  Thus, total output Q = Q₁ + Q₂ = 2MC_T – 20.  So the multiplant marginal cost curve is given by MC_T = 0.5Q + 10.  Equating MR with MC_T yields

Thus, both plants will produce Q_i = 8 units.  Industry price is found by substituting these quantities into the market demand function.  This implies P = 34.

 

c)         If the firms acted as price takers, the market would see the perfectly competitive solution.  Setting  for both firms implies

                                               

 

 

 

 

6          As you read the answer to this, think of the reaction functions being graphed in coordinate system with Besanko’s Q on the horizontal axis and Schmedders’ Q on the vertical axis.

 

 

 

 

 

 

7          The table below summarizes the answer to this problem. The solution details follow.

 

	Firm 1 output	Firm 2 output	Market Price	Firm 1  Profit	Firm 2 Profit
Cournot	40	40	120	3,200	3,200
Stackelberg with Firm 1 as leader	60	30	100	3,600	1,800

 

a)         Firm 1’s marginal revenue is MR = 280 – 2Y – 4X.  Equating MR to MC gives us:

b)         To find the Stackelberg equilibrium in which Firm 1 is the leader, we start by writing the expression for Firm 1’s total revenue:

                       

 

 

8          a)         Begin by inverting the market demand curve: Q = 600 – 3P Þ P = 200 – (1/3)Q. The marketing-clearing price if firm 1 produces Q₁ and firm 2 produces Q₂ is:        

 

b)         To find the Stackelberg equilibrium, we begin by substituting firm 2’s reaction function into the expression for the market-clearing price to get firm 1’s residual demand curve. This gives us:

 

c)         Let’s now compute the profit of each firm under Stackelberg leadership and compare to the profit under Cournot. The leader’s profit is:

 

9          The table below summarizes the solution. The details follow.

 

	Firm 1 output	Firm 2 output	Market Price	Firm 1  Profit	Firm 2 Profit
Cournot	6	3	9	36	9
Stackelberg with Firm 1 as leader	9	1.5	7.5	40.5	2.25

 

a)         For Firm 1, equating MR to MC yields 18 – Y – 2X = 3, or X = 7.5 – 0.5Y.

b)         If Firm 1 is the Stackelberg leader, we plug Firm 2’s reaction function into the expression for Firm 1’s total revenue: TR = [18 – (6 – 0.5X) – X]X = (12 – 0.5X)X.

10        a)         The leader would produce more and the follower would produce less.

 

b)         The leader would produce less and the follower would produce more.

11        a)         We will first solve for Alpha’s reaction function. We begin by solving Alpha’s demand function for P_A in terms of Q_A and P_B: P_A = 15 – (1/10)Q_A + (9/10)P_B. The corresponding marginal revenue equation is:

 

b)         We can find Bravo’s reaction function by following steps identical to those followed to derive Alpha’s reaction function. Following these steps gives us:

 

c)         The following diagrams show how each change affects the reaction functions.

 

 

d)         The above diagrams can used to verify how each of the changes affect the Bertrand equilibrium:
 

12        Jerry’s demand curve can be written as p_J = (1/3)(100 + p_T) – (1/3)q_J.  Hence, MR_J = MC_J implies (1/3)(100 + p_T) – (2/3)*q_J = 0 yielding Jerry’s reaction function: q_J = 50 + 0.5p_T.  At this quantity, Jerry will charge a price of p_J = (100 + p_T)/6.  Similarly, we can find that Teddy’s best response function is p_T = (100 + p_J)/6.  Solving the two equations yields p_T = p_J = 20 in equilibrium.  We can also see this graphically:

 

13        a)         If American sets a price of $200, we can plug this price into United’s demand curve to get United’s perceived demand curve.

 

b)         If American sets a price of $400, then United’s perceived demand curve is

                                               

 

 

 

c)         American’s demand can be rewritten as

                                               

 

d)         The Bertrand equilibrium will occur where these price reaction functions intersect.  Substituting the expression for  into the expression for  implies

                                                 

 

14 a)    Monopoly price = $6 per pair of shoes, Q

b)         Let’s begin by deriving Firm 1’s reaction function.