Instructor: Shane Murphy
Econ 100 Duopoly practice worksheet
1. Let the inverse demand function and the cost function be given by P = 50 − 2Q and C = 10 + 2q
respectively, where Q is total industry output and q is the firm’s output. If this is a monopoly, find
the profit function and find the maximum profit Q, P , and π.
Answer: Profit is π = T R − T C = P ∗ q − (10 + 2q) = (50 − 2q)q − (10 + 2q) = −2q 2 + 48q − 10.
dπ
Profit is maximized where dπ
dq = 0. dq = −4q + 48 = 0 ⇒ q = 12. Plug this q into P = 50 − 2Q =
50 − 2 ∗ 12 = 26. Plug q into π = −2q 2 + 48q − 10 = −2(122 ) + 48 ∗ 12 − 10 = 278.
2. Now suppose this is a duopoly - two firms. The inverse demand function is the same, but Q = q1 + q2 .
The cost functions are giving by C1 = 10 + 2q1 and C2 = 10 + 2q2 , so the goods are homogenous.
Find each firms profit function, best response function, q1 , q2 , Q, P , π1 , and π2 .
Answer: π1 = q1 [50 − 2(q1 + q2 )] − 10 − 2q1 and π2 = q2 [50 − 2(q1 + q2 )] − 10 − 2q2 . To find the
best responce function for firm 1, find where firm 1s profit is maximized for a given level of quantity
25−2q2 −1
∂π
1
= 0. So, ∂π
produced by firm 2, so find ∂q
. Likewise for
∂q1 = 50 − 4q1 − 2q2 − 2 = 0 ⇒ q1 =
2
1
2
firm 2, ∂π
∂q2 = 50 − 4q2 − 2q1 − 2 = 0 ⇒ q2 =
π1 = π2 = 118.
25−2q1 −1
.
2
These cross at q1 = q2 = 8 so Q = 16, P = 18,
3. Now assume the same inverse demand function, but two different cost functions (not homogenous
processes): C1 = 10 + 2q1 and C2 = 12 + 8q2 . Find each firms profit function, best response function,
q1 , q2 , Q, P , π1 , and π2 .
Answer: Now the profit functions are: π1 = q1 [50 − 2(q1 + q2 )] − 10 − 2q1 and π2 = q2 [50 − 2(q1 +
q2 )] − 12 − 8q2 . The best response functions are q1 = 25−q22 −1 and q2 = 25−q21 −4 . These intersect
where q1 = 9 and q2 = 6. So Q = 16, P = 20, π1 = 152, π2 = 60. The low-cost firm produces more
and makes higher profits than the high-cost firm.
4. Now consider the general form. Let the inverse demand function be P = a − bQ and the cost function
be T Ci = c + dqi . Find each firms profit function, best response function, q1 , q2 , Q, P , π1 , and π2 .
2 −c
Answer: πi = [a − b(q1 + q2 ) − c]qi . The best response function for firm 1 is q1 = a−bq
and for firm
2b
2 is q2 =
a−bq1 −c
.
2b
So, q1 = q2 =
a−c
3b .
Q=
2(a−c)
3 ,
P =
a+2c
3 ,
and π1 = π2 =
(a−c)2
16b .
5. Let two firms sell homogeneous goods. Let the inverse demand function for each firm depend on the
price the firm offers. If firm one offers a lower price, the inverse demand functions are q1 = 10 − 2p1
and q2 = 0. The functions are switched if firm 2 offers the lower price. If the two firms offer the same
price, the inverse demand functions are q1 = (10 − 2p1 )/2 and q2 = (10 − 2p2 )/2. The cost to each
firm is c1 = 2q1 and c2 = 2q2 . Find each firms profit function, sketch best response functions, and
find q1 , q2 , Q, p1 , p2 , π1 , and π2 .
Answer: the profit function for firm 1 is π1 = (p1 − 2)(10 − 2p1 ) if p1 < p2 , π1 = (p1 − 2)(10 − 2p1 )/2
if p1 = p2 and π1 = 0 if p1 > p2 . For firm 2, profit is π2 = (p2 − 2)(10 − 2p2 ) if p2 < p1 ,
π2 = (p2 − 2)(10 − 2p2 )/2 if p2 = p1 and π2 = 0 if p2 > p1 . Solving, p1 = p2 = M C = 2., q1 = q2 = 3,
Q = 6, π1 = π2 = 0.
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6. Let two firms sell homogeneous goods. Let the inverse demand function for each firm depend on the
price the firm offers. If firm one offers a lower price, the inverse demand functions are q1 = 18 − 2p1
and q2 = 0. The functions are switched if firm 2 offers the lower price. If the two firms offer the same
price, the inverse demand functions are q1 = (18 − 2p1 )/2 and q2 = (18 − 2p2 )/2. The cost to each
firm is c1 = 6q1 and c2 = 6q2 . Find each firms marginal cost, p1 , p2 , q1 , q2 , Q, π1 , and π2 .
Answer: M C = 6, p1 = p2 = 6, q1 = q2 = 6, Q = 18, π1 = π2 = 0.
7. Let two firms sell differentiated goods. The inverse demand function for firm 1 is q1 = 400 − 4p1 + 2p2
and for firm 2 is q2 = 240 − 3p2 + 1.5p1 . Assume zero marginal cost. For each firm, find the total
revenue function and the marginal revenue function. Set M R = M C and find the best responce
function for each firm. Find the equilibrium prices, quantities, and profit.
Answer: For firm 1, T R1 = p1 q1 = p1 (400 − 4p1 + 2p2 ) = 400p1 − 4p21 + 2p1 p2 . To find the best
R1
2
response, set 0 = ∂T
∂p1 = 400−8p1 +2p2 ⇒ p1 = 50+0.25pb . For firm 2, T R2 = 240p2 −3p2 +1.5p1 p2 ,
R2
and 0 = ∂T
∂p2 = 240−6p2 +1.5p1 ⇒ p2 = 40+0.25p1 . The two best response functions intersect where
p1 = 64, p2 = 56, so q1 = 256, q2 = 232, Q = 488. and π1 = 64 ∗ 256 = 16384, π2 = 56 ∗ 232 = 12992.
8. In Stackelberg’s model, one firm gets to go first. In this case, each firm knows the other firms best
response function (or can find it from the other firms profit function). Suppose firm 1 goes first, the
inverse demand function for a homogeneous good is p = 500 − q, q = q1 + q2 , and both firms have a
marginal cost of 150 and no fixed costs. Firm 2s best response function is q2 = 175−0.5q1 . Substitute
this into the inverse demand function to get a function of price in terms of firm 1s quantity only. You
should get p = 325 − 0.5q1 . Find firm 1s profit function and find the optimal level q1 by finding where
the slope of the profit function equals zero. Use firm 2s best response function to find q2 . Solve for
q and p and for profit for each firm. Which firm produces more? Which firm gets higher profits?
Answer: π1 = T R1 − T C1 = (p − M C1 )q1 = (325 − 0.5q1 − 150)q1 = 175q1 − 0.5q12 . Find where
dπ1
dq1 = 175 − q1 = 0, so optimal q1 = 175. Plug this into 2s best response, q2 = 175 − 0.5q1 =
175 − 0.5(175) = 87.5. So Q = 262.5 and p = 500 − Q = 500 − 262.5 = 237.5. For profits,
π1 = 237.5(175)150(175) = 15312.5, while π2 = 237.5(87.5)150(87.5) = 7656.25. So the first firm
produces a higher quantity and gains more profits.
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