8 Oligopoly - Luiscabral.net

p = a − b Q, where Q = ∑ n
i=1 q i.
(a) Show that firm i’s best-response mapping is given by qi ∗(Q −i) = a−c i
1
2 Q −i, where Q −i ≡ ∑ j≠i q j.
2 b −
(b) Show(
that, in equilibrium, total output is given by
̂Q = n a − ∑ )
n
i=1 c i / ( b (n + 1) ) . (Hint: add up all n first-order
conditions for profit maximization.)
(
(c) Show that equilibrium price is given by ̂p = a + ∑ )
n
i=1 c i /(n + 1)
(d) Show(
that, in equilibrium, firm i’s output level is given by
ˆq i = a − n c i + ∑ )
j≠i c j / ( b (n + 1) )
(e) Show that, in equilibrium, firm i’s profit is given by
(
ˆπ i = a − n c i + ∑ 2/
j≠i j) c (n + 1) 2 / b − F i
(f) Show that, in equilibrium, consumer surplus is given by
(
CS = 1
2 b
n a − ∑ ) 2
n
i=1 c i /(n + 1)
2
8.12. Elasticity rule (reprise). Show that the elasticity rule derived in Chapter 3, that
is (p − MC )/MC = −1/ɛ, holds under Cournot competition with linear demand and costs,
where MC is firm i’s marginal cost and ɛ its demand elasticity (not the market elasticity).
8.13. Efficiency loss under Cournot. Consider a market for a homogeneous product
with demand given by Q = 37.5 − P/4. There are two firms, each with constant marginal
cost equal to 30.
(a) Determine output and price under a Cournot equilibrium.
(b) Compute the efficiency loss as a percentage of the efficiency loss under
monopoly.
8.14. Equilibrium price under Cournot. Show analytically that equilibrium price under
Cournot is greater than price under perfect competition but lower than monopoly price.
8.15. Cournot with increasing marginal cost. Consider a duopoly for a homogenous
product with demand Q = 10−P/2. Each firm’s cost function is given by C = 10+q(q +1).
(a) Determine the values of the Cournot equilibrium.
(b) Re-compute the equilibrium values assuming that one of the firms —
say, firm 2 — has a cost function given by C = 10 + q(q + 1).
8.16. Cement. Two firms compete (a la Cournot) in the cement market. Demand for
cement is given by Q = 450 − 2 P . Firm 1’s marginal cost is constant at 50, firm 2’s at 40.