--topic2Imperfect Competition 微观经济学高级讲义

steeper than the negative slopping 450 line and R2(q1) is ‡atter than that line.)
Thus
equilibrium
is
unique
if
¯¯¯
@2¼i @ qi @ qj
¯¯¯
<
¯¯¯
@2¼i @qii
¯¯¯
:
This
condition
I will organize the lectures for this topic in such a way that introduces some basic models of oligopoly theory and discuss some of the current research in this area. My coverage will necessarily be incomplete—a more comprehensive treatment can easily take up an entire course.
3
Suppose …rst that there are two …rms, with constant marginal costs 0 · c1 · c2: Assuming equilibrium uniquely exists, we can write the equilibrium outputs and pro…ts as qi(c1; c2) and ¼i(c1; c2); i = 1; 2:
j6=i
j6=i
In most applications, we are interested in NE where all …rms produce positive
outputs. The second-order su¢cient conditions at an interior equilibrium are:
also
ensures
that
the
equilibrium is stable.
Comparative statics.— When …rms have constant marginal costs, it is often useful to see how equilibrium changes as costs change.
Problem 2.1. For n …rms with constant marginal cost 0 · c1 · ::: · cn;
P
let c¹ = i ci: Show that (a) In any interior Cournot equilibrium, the market price depends on c¹ but not on
su¢cient condition for this is that jRi0(qj)j < 1 at the relevant regions. (This implies
that the two reaction functions, both downward slopping, are such that R1(q2) is
2.1 STATIC MODELS OF OLIGOPOLY
The traditional static models of oligopoly include the Cournot model and the Bertrand model for homogeneous goods and the Bertrand model for di¤erentiated goods. These basic models are then used to study strategic behavior of …rms, often in a dynamic context.
We can show: 1. qi(c1; c2) decrease in ci and increases in cj: To show this, notice that Ri(qj) shifts to the left when ci increases, while Rj (qi) is not a¤ected by the change in ci: Since the reaction curves are downward slopping, qi(c1; c2) decrease in ci and increases in cj: 2. ¼i(c1; c2) decrease in ci and increase in cj: This is because
@¼i(Ri(qj); qj) = 0 : @qj
@2¼i @qii
Ri0
(qj
)
+
@2¼i @qi@qj
=
0
Thus
jRi0 (qj )j
=
¯¯¯¯¯¯
@ 2 ¼i @qi@qj
@ 2 ¼i @qii
¯¯¯¯¯¯
For a unique equilibrium, the reaction function should only intersect once. A
2.1-1 Quantity Competition (Cournot Model)
Suppose that there are n > 1 …rms producing a homogeneous product in a market with inverse demand P (Q) ; which is usually assumed to be twice di¤erentiable: Firm
P (0) > c0i(0) Ri(qjm) > 0;
where Ri(qj) is …rm i0s optimal reaction function and qjm is …rm j0s output if it were a monopoly.
2
If these are two …rms with constant marginal costs 0 · c1 · c2; I think (need to check in class) that we have q2¤ > 0 if
1
i0s cost is a function of its output qi; ci(qi): Firm i0s pro…t is
0
1
X
¼i = qiP @qi + qjA ¡ ci (qi) :
j6=i
In a model of quantity (Cournot) competition, …rms simultaneously choose (non-
Oligopoly theory is itself a very broad topic. It is an area that has had enormous progress in the last twenty years, mostly parallel to the advances in game theory. There are several ways one can classify oligopoly theory. There are static and dynamic models; complete and incomplete information models; single market models and multimarket models, etc. While oligopoly theory has traditionally been studied by industrial organization economists, it has increasingly been applied to other areas of economics, such as in international economics.
¼1(c1; c2) = q1(c1; c2) [P (q1(c1; c2) + q2(c1; c2)) ¡ c1]
@¼1(c1; c2) @c1
=
@¼1(c1; c2) @q2
@q2 @c1
¡
q1(c1; c2)
<
0;
@¼1(c1; c2) @c2
=
q1(c1; c2)P 0 (q1(c1; c2) + q2(c1; c2))
0
1
0
1
2P 0 @qi¤ + X qj¤A + qi¤P 00 @qi¤ + X qj¤A ¡ c0i0 (qi¤) < 0:
j6=i
j6=i
Existence of Equilibrium.— One typical way to ensure (a pure-strategy) equilibrium existence is to assume that each …rm’s pro…t function is concave with respect to its own output. A su¢cient condition for this is that c0i0 (¢) ¸ 0 and P 00 () · 0: Since P 0 () < 0; we know that the su¢cient conditions can also be satis…ed if c0i0 (¢) ¸ 0 and demand is “not too” convex. Note that constant marginal cost and linear demand always satisfy these conditions. To ensure equilibrium to be interior, each …rm’s c0(0) need be su¢ciently small. If n = 2; we can impose the conditions that
@q2 @c2
>
0:
3. [q1(c1; c2) + q2(c1; c2)] decreases in ci:
These are useful results in analyzing …rms’ strategies that may a¤ect ci:
You are asked to prove another interesting result, as a homework:
ECON 7050: Topic 2. Imperfect Competition
Imperfect competition is a very broad topic. It covers oligopoly theory and the theory of monopolistic competition. I will focus on oligopoly theory for this topic.
negative) outputs, and we look for a Nash equilibrium, (q1¤; :::; qn¤) ; which necessarily
solves the following …rst-order conditions:
0
1
源自文库
0
1
P @qi¤ + X qj¤A+qi¤P 0 @qi¤ + X qj¤A¡c0i (qi¤) · 0; with “ = " if qi¤ > 0; for i = 1; :::; n:
P (0) > c2
and R2(q1m) > 0:
Uniqueness of Equilibrium.— Assume that each …rm’s pro…t function is strictly concave with respect to its own output, so that (a pure-strategy) equilibrium exist, it may not be unique. In the case of two …rms, we can di¤erentiate the …rst-order condition