ECON-115 Take Home Final Problem Set
NAME Kids in Prison Program v. 1
DUE May 3rd, 2016
Assumptions Market Demand: P = 14 – ½
Two Groups:
Incumbent/Entrant:
Qtotal
P = 16 – Q1[HD]
P = 14 – QI - qe
Duopoly: P = 14 – ½(q1 + q2)
P = 12 – Q1[LD]
Entrant’s Total
Marginal Cost: MC = 4
[HD = high demand; LD =
Cost = 9 + 4qe
low demand]
Capacity Constraint: 5/firm
Steps
Perfect
1. Set P = MC
Competition
Monopoly
1. Calculate MR
2. Set MR = MC
Group
Discount
1. Determine MR for each
group’s demand function.
2. Set MR1 = MR2 = MC
Cournot
Duopoly
1. Take derivative of
demand function in terms
of q1 (dq1/dP) = MR1.
2. Set MR1 = MC. That’s
Firm 1’s reaction
function.
3. Repeat for Firm 2.
4. Where the reaction
functions cross is the
Nash/Cournot
Equilibrium. Therefore,
plug q2’s reaction
function into q 1.
5. Solve for both quantities.
6. Add q1 + q2 = Qtotal
7. Use Qtotal to find price
1. Set P = C
Bertrand
Duopoly
(standard)
Bertrand
Duopoly
(capacity
constraints)
1. Determine P if Q = both
firms’ capacity.
Solution
P = MC = 14 – 1/2Q = 4
10 = 1/2 Q
Q = 20
P=4
TR = PQ = 14Q – 1/2Q2
MR = 14 – Q = 4
Q = 10
P = 14 – ½(10) = 9
TR [HD] = PQ1 = 16Q1-Q12
MR[HD] = 16-2Q1 = 4
Q1 = 6
TR [LD] = PQ2 = 12Q2-Q22
MR[LD] = 12-2Q2 = 4
Q2 = 4
P[HD] = 16 – 6 = 10
P[LD] = 12 – 4 = 8
P = 14 – 1/2q1 – 1/2q2
TR1 = 14q1 – 1/2q12 - 1/2q1q2
MR1 = 14 - q1 - 1/2q2 = 4
q1 = 10 – 1/2q2
TR2 = 14q2 – 1/2q1q2 - 1/2q22
MR2 = 14 - 1/2q1 - q2 = 4
q2 = 10 – 1/2q1
q1 = 10 – ½ (10 – 1/2q1)
q1 = 10 – 5 + ¼ q1
¾ q1 = 5
q1* = 20/3 ≈ 6
q2* = 10 – ½ (20/3) = 20/3 ≈ 6
Q = 6 + 6 = 12
P = 14 – ½ (12) = 8
P = 14 – 1/2q1 – 1/2q2 = 4
10 – 1/2q1 – 1/2q2 = 0
q1 = 10; q2 = 10; Q = 20
P=4
P = 14 – 1/2q1 – 1/2q2
P = 14 – ½(5) – ½(5) = 9
Q = 10
P=9
Stackelberg
Duopoly
Limit
Pricing
1. Determine Second
Mover’s reaction function
(See Cournot above).
2. Plug Firm 2’s reaction
function into the overall
Demand Function.
3. Use that Demand
Function to determine the
First Mover’s (Firm 1’s)
Marginal Revenue
Function.
4. Calculate Firm 1’s profit
maximizing quantity.
5. Use Firm 1’s profit
maximizing quantity to
calculate Firm 2’s
quantity.
1. Find Entrant’s Residual
Demand function by
aggregating Incumbent’s
quantity with intercept.
2. Use the residual demand
function to calculate
Entrant’s profit
maximizing quantity, q*.
3. Write Entrant’s profit
equation (TR – TC). Set 0
in order to prevent entry.
4. Solve for Q*,
Incumbent’s Limit
Output.
5. Calculate Limit Price
using Q*.
FORM OF COMPETITION
Perfect Competition
Monopoly
Group Price Discrimination
Cournot Duopoly
Bertrand (standard)
Bertrand (capacity constraints)
Stackelberg (quantity)
Limit Pricing
P = 14 – 1/2q1 – 1/2q2
TR2 = (14 – 1/2 q1) q2 – 1/2q22
MR2 = 14 – 1/2q1 – q2 = 4
q2 = 10 – ½ q1
P = 14 – 1/2q1 – ½ (10 – ½ q1)
P = 14 – 1/2q1 – 5 + 1/4q1 = 9 – 1/4q1
TR1 = 9q1 – 1/4q12
MR1 = 9 – 1/2q1 = 4
q1* = 10
q2* = 10 – ½(10) = 5
P = 14 – ½(10) – ½(5) = 6.5
P = 14 – QI – qe
Pqe = (14 – QI) q2 – qe2
MR = 14 – QI – 2q2 = 4
2q2 = 10 - QI
qe = 5 – QI/2
Profit = (TR-TC) = Pqe – (9 + 4qe) = 0
9 = q (P-4)
P – 4 = 9/qe
14 – Q – q – 4 = 9/qe
14 – Q I – (5 – QI/2) – 4 = 9/(5-QI/2)
(5-Q/2) = 9/(5-Q/2)
(5-Q/2)2 = 9
5-Q/2 = 3
qe = 3
Q=4
P = 14 – 4 – 3 = 7
7 * 3 – 9 – 4(3) = 0
Market Price
Market Quantity
4
20
9
10
P[LD] = 8 P[HD] = 10
Q = 10
8
12
4
20
9
10
6.5
15
7
4