Chapter 11: Problems 2,4,6
2. Ajax Cleaning Products is a medium-sized firm operating in an industry dominated by one large
firm Tile King. Ajax produces a multi-headed tunnel wall scrubber that is similar to a model
produced by Tile King. Ajax decides to charge the same price as Tile King to avoid the possibility
of a price war. The price charged by Tile King is $20,000.
Ajax has the following short-run cost curve:
,000 - 5,000Q + 100Q2.
A. Computer the marginal cost curve for Ajax.
B. Given Ajax’s pricing strategy, what is the marginal revenue function for Ajax?
Ajax Cleaning Products versus Tile King.
a. MC = d(TC)/dQ = 5000 + 200 Q
b. P = MR = $20,000
c. MC = MR is profit maximizing output, where 5000 + 200 Q = 20,000. Therefore, 200 Q =
25,000, so Q* = 125.
d. * = 20,000(125)  800,000 + 5000(125)  100(125)2
* = $762,500
4. Unique Creations holds a monopoly position in the production and sale of manometers. The
cost function facing Unique is estimated to be
TC = $100,000 + 20Q
A. What is the marginal cost for Unique?
C. What is the marginal revenue at the price computed in Part (b)?
Unique Creations holds a monopoly in magnometers.
a. MC = dTC/dQ = 20
b. MR = P(1 + 1/Ed)
20 = P [1+ (1/1.5)]
P* = $60
c. MR = 60(1 + 1/1.5) = $20
d.
MR = P (1 + 1/3) = 20 = MC, so P* = $30. This is a simple (or non-discriminating)
monopoly problem.
6. Wyandotte Chemical Company sells various chemicals to the automobile industry. Wyandotte
currently sells 30,000 gallons of polyol per year at an average price of $15 per gallon. Fixed costs
of manufacturing polyol are $90,000 per year and total variable costs equal $180,000. The
operations research department has estimated that a 15 percent increase in output would not
affect fixed costs but would reduce average variable costs by 60 cents per gallon. The marketing
department has estimated the arc elasticity of demand for polyol to be –2.0.
A. How much would Wyandotte have to reduce the price of polyol to achieve a 15 percent increase
in the quantity sold?
B. Evaluate the impact of such price cut on (i) total revenue, (ii) total costs, and (iii) total profits.
Wyandotte Chemical Company sales 30,000 gallons of polyol at $15/gallon.
a. ED = %ΔQD / %ΔP, –2.0 = +15% / %ΔP. To get 15% more sales, it can %ΔP = –7.5%. Using the
arc price formula, we can get the new price and new quantity:
–.075 = (P2 – 15.00)/ [(P2 + 15)/2]
P2 = $13.92
ΔP = $15 –$13.92 = $1.08
Also: .15 = (Q2 – 30,000)/ [(Q2 + 30,000)/2]
Q2 = 34,865 gallons
b. i). On TR: Before: TR, = 15(30,000) = $450,000
After: TR2 = 13.92(34,865) = $485,321, so ΔTR = +$35,321
ii). On TC: Before: FC1 =$90,000
After: FC2 = $90,000
VC/unit = $6.00 – .60 = $5.40
VC2 = $5.40 × 34,865 = $188,271
TC2 = 90,000 + 188,271 = $278,271, so ΔTC = +$8,271
iii). On Total Profits: Before: π, = $450,000 – $270,000 = $180,000
After: π2 = $485,321 – $278,271 = $207,050, or
Δπ = + 27,050
Check: ΔTR - ΔTC = Δπ: $35,321 - $8,271 = + 27,050.
Chapter 12: Problems 1,2,5
1. Assume that two companies (C and D) are duopolists that produce identical products. Demand
for the products is given by the following linear demand function:
– Qc- Qd
Where Qc and Qd are the quantities sold by the respective firms and P is the selling price. Total
cost functions for the two companies are
,000 + 100Qc
,000 + 125Qd.
Assume that the firms act independently as in the Cournot model (i.e., each firm assumes that the
other firm Â’s output will not change).
A. Determine the long-run equilibrium output and selling price for each firm.
B. Determine the total profits for each firm at the equilibrium output found in Part (a). This is
answer for part (a).
Cournot duopoly problem for firms C and D with different TC curves.
a. C = PQC  TCC = (600  QC  QD)QC  (25,000 + 100QC)
= 25,000 + 500QC  QC2  QCQD
D = (600  QC  QD) QD  (20,000 + 125QD) = 20,000 + 475QD  QD2  QCQD
C/QC = 500  2QC  QD
and D/QD = 475  2QD  QC
Conditions for an optimum require that both partials be set equal to zero and the resulting
equations be solved simultaneously for optimal values of QC and QD:
QD* = 150 units and QC* = 175 units. P = 600 – 150 – 175 = $275, so P* = $275.
b. C* = 25,000 + 500(175)  (175)2  175(150) = $5,625.
D* = 20,000 + 475(150)  (150)2  175(150) = $2,500.
2. Assume that two companies (A and B) are duopolists who produce identical products.
Demand for the products is given by the following linear demand function:
– Qa - Qb
Where Qa and Qb are the quantities sold by the respective firms and P is the selling price. Total
cost functions for the two companies are:
,500 + 55Qa + Q^2a
,200 + 20Qb + 2Q^2b .
Assume that the firms act independently as in the Cournot model (i.e., each firm assumes that the
other firm Â’s output will not change).
B. Determine Firm A, Firm B, and total industry profits at equilibrium solution found in Part (a).
Cournot duopoly problem for firms A and B, also with different cost curves.
a. A = PQA  TCA = (200  QA QB)∙QA (1500 + 55 QA + QA2)
A = 1500 + 145QA  2QA2  QAQB
A/QA = 145  4QA  QB = 0
B = PQB  TCB = (200  QA QB)∙QB (1200 + 20 QB + 2QB2)
B = 1200 + 180QB  3QB2  QAQB
B/QB = 180  6QB  QA = 0
Solving these two equations simultaneously results in: QB* = 25 and QA* = 30, accordingly, P =
200 -30 -25 = $145, or P* = $145 for both firms.
b. A* = 1500 + 145(30)  2(30)2  30(25) = $300.
B* = 1200 +180(25)  3(25)2 30(25)= $675.
Also * = 300 + 675 = $975.
5. Alchem (L) is the price leader in the polyglue market. All 10 other manufacturers (follower [F]
firms) sell polyglue at the same price as Alchem. Alchem allows the other firms to sell as much as
they wish at the established price and supplies the remainder of the demand itself. Total demand
for polyglue is given by the following function ( +QF):
,000 – 4 QT
Alchem marginal cost function for the manufacturing and selling polyglue is
,000 + 5QL
The aggregate marginal cost function for the other manufacturers of polyglue is
?,000 + 4QF
B. What is the total market demand for polyglue at the price established by Alchem in Part (a)?
How much of total demand do the follower firms supply?
Alchem is a dominant firm price leader in polyglue.
a.
Alchem's (L) profit-maximizing output occurs where: MRL = MCL
MRL is found as follows: MRL = d(TRL)/dQL
TRL is given by the following expression: TRL = P  QL
Also, QL is given by: QL = QT  QF
Using the total demand function, one can solve for QT:
P = 20,000  4QT or QT = 5000  .25P
In order to find QF, one notes that Alchem lets the follower firms (F) sell as much polyglue as
they wish at the given market price (P). Therefore, the follower firms are faced with a horizontal
demand function and hence: MRF = P
In order to maximize profits, the follower firms will operate where: MRF = MCF , or where: P =
2000 + 4QF
Solving for QF yields: QF = .25 P  500
Substituting QF and QT into the expression above for QL gives:
QL = (5000  .25P)  (.25P  500) = 5500  .50P
Solving for P gives: P = 11,000  2QL
Substituting P into the TRL expression above gives: TRL = (11,000  2QL)QL = 11,000QL 
2Q 2
L
MRL is therefore equal to: MRL = 11,000  4QL
Setting MRL = MCL gives: 11,000  4QL = 5000 + 5QL so QL* = 666.7.
Substituting this value into the expression for P gives: P* = 11,000  2(666.7) = $9,666.70.
b.
From the expression above for QT, one obtains: QT* = 5000  .25(9666.7) = 2,583.3.
The followers supply the rest: QF* = .25(9666.7)  500 = 1,916.7.