2024年3月16日发(作者：比特彗星官网下载)
Dominant Firm and Competitive Fringe
The behavior of a dominant firm with a competitive fringe can be analyzed using calculus. This
appendix illustrates such an analysis with the no-entry model with n fringe firms, using more
general demand and cost functions than were (implicitly) assumed in Figure 4.6 in your
textbook and concentrating on a long-run analysis in which average variable costs and
average costs are equal.
If the cost function of a fringe firm is C
f
(q
f
), then its average cost is AC = C
f
(q
f
)/q
f
and its
marginal cost is MC = C'
f
(q
f
), where the prime indicates differentiation. The fringe firm's
objective is to maximize its profits,
f
, through its choice of its output level, q
f
:
(Equation 1)
max
f
= pq
f
- C
f
(q
f
),
q
f
where pq
f
is its total revenues. This firm believes it is a price-taker that can sell as much as it
wants at the going price and that it cannot affect the price through its own actions.
The first-order condition for profit maximization for a fringe firm is:
(Equation 2)
p = C'
f
(q
f
).
That is, the firm sets its output where price (the firm's marginal revenue) equals its marginal
cost. The second-order condition is C"
f
(q
f
) > 0; that is, the marginal cost curve must be
upward sloping at the equilibrium quantity for profits to be maximized. [The theory of the
competitive firm requires that, in addition to meeting the first- and second-order conditions, a
firm must make sure that its profits are positive (or else it should go out of business). Profits
are positive if the market price is above the minimum average cost ( in Figure 4.6a).]
The combined output of the fringe (Q
f
= nq
f
) and the dominant firm (Q
d
) determines the
market price: p(Q) = p(Q
f
+ Q
d
). Thus, all else the same, as the dominant firm increases its
quantity, the price falls. As the price falls, each fringe firm chooses to produce less (because
its marginal cost is increasing in q
f
and it sets price equal to marginal cost from Equation 2).
We can show formally that the fringe supply falls as Q
d
rises. First, rewrite Equation 2 to
reflect how price varies with Q
d
:
(Equation 2')
p(nq
f
+ Q
d
) = C'
f
(q
f
).
Then, totally differentiate Equation 2' to show that p'ndq
f
+ p'dQ
d
= C"
f
dq
f
or (rearranging)
(Equation 3)
dq
f
- p'
------ = ----------- < 0,
dQ
d
np' - C"
f
where the inequality follows because np' < 0 and -C"
f
< 0 (by the second-order condition).
That is, the quantity supplied by a fringe firm falls as Q
d
rises. As Q
d
rises, all else the same,
price must fall, and as price falls, the quantity supplied by a fringe firm falls. We can
write Q
f
(Q
d
) to show that Q
f
is a function of Q
d
. From Equation 3, we know that
dQ
f
/dQ
d
= ndq
f
/dQ
d
= - np'/(np' - C"
f
) < 0, which says Q
f
falls as Q
d
rises.
The dominant firm takes the relationship (2') into account when trying to maximize its profits
through its choice of output level:
(Equation 4)
max
p(Q
d
+ Q
f
(Q
d
))Q
d
- C
d
(Q
d
),
Q
d
where C
d
(Q
d
) is the dominant firm's cost function. The first-order condition for a profit
maximization is
(Equation 5)
dQ
f
p(Q
d
+ Q
f
) + p'(Q
d
+ Q
f
)Q
d
[1 + ------ ] = C'
d
(Q
d
).
dQ
d
According to Equation 5, profits are maximized if the dominant firm sets its output so that its
marginal revenue conditional on the response of the competitive fringe, the left-hand side of
the equation, equals its marginal cost, the right-hand side of the equation. From Equation 3,
dQ
f
/dQ
d
= - np'/(np' - C"
f
), so the term in brackets in Equation 5 can be rewritten as- C"
f
/(np'
- C"
f
). This ratio is positive but less than 1.
If Q
f
0 and dQ
f
/dQ
d
0, the dominant firm is a monopoly. Then Equation 5 is the
monopoly's profit maximization condition: Marginal revenue (corresponding to the market
demand curve) equals marginal cost. The monopoly's p is a function of only the monopoly's
output, and Q
d
p'(Q
d
) is multiplied by 1; whereas in the dominant firm model, price is a
function of the dominant firm's and the competitive fringe's output, and Q
d
p'(Q
d
+ Q
f
) is
multiplied by a term that is less than 1.
We can also express the effect of the fringe's supply on the dominant firm using elasticities.
The fringe's supply affects the elasticity of demand that the dominant firm faces and hence
helps determine the dominant firm's price. Using slightly different notation, the dominant
firm's residual demand, Q
d
= D
d
(p), can be written as the market demand, D(p), minus the
supply, S(p), of the fringe:
(Equation 6)
D
d
(p) = D(p) - S(p).
The dominant firm's marginal revenue corresponding to this residual demand curve is obtained
by differentiating Equation 6 with respect to p:
(Equation 7)
dD
d
= dD - dS .
----- ------ ------
dp dp dp
Equation 7 can be expressed in terms of elasticities by multiplying both sides of the equation
by p/Q, multiplying the left-side by Q
d
/Q
d
, and multiplying the last term on the right side
byQ
f
/Q
f
:
(Equation 7')
Q
d
Q
f
( ---- )
d
= - ( ---- )
f
,
Q Q
where
d
= [(D
d
/p)(p/Q
d
)] is the residual demand elasticity, is the elasticity of the
market demand curve,
f
is the fringe's supply elasticity, Q
d
/Q is the dominant firm's share of
output, and Q
f
/Q is the fringe's share. This expression may be rewritten as
(Equation 7'')
Q Q
f
d
= ----- - ( ----- )
f
,
Q
d
Q
d
where Q/Q
d
is the ratio of total industry output to that of the dominant firm and Q
f
/Q
d
is the
ratio of the fringe's output to that of the dominant firm. Thus, all else the same, the absolute
value of the elasticity of the residual demand facing the dominant firm is higher (and hence
the lower the price it charges), the higher is the supply elasticity of the fringe, the higher is
the fringe's relative share of the market Q
f
/Q
d
, and the higher is the absolute value of the
industry elasticity of demand. If the fringe does not exist (n = 0), the dominant firm's residual
demand elasticity equals the industry demand elasticity, and it charges the monopoly price.
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