Chemistry Reference
In-Depth Information
solves the optimization problem
max.U.q/
pq/:
Assuming an interior optimum, the first order condition implies that
0
D
U
0
.q/
p
D
a
bq
p;
so that the individual demand at the price p is therefore
a
b
1
b
p:
q.p/
D
Consider now n heterogenous consumers with quadratic utility and preference
parameters a
i
and b
i
. From the previous description we know that for any fixed price
consumer i will buy the amount q
i
D
.a
i
p/=b
i
, so the total demand becomes
n
n
n
X
X
X
a
i
b
i
1
b
i
p;
D
D
q
i
D
i D1
i D1
i D1
and hence the relationship between total demand and market price is linear. Notice
that if price increases, demand decreases and that there is a maximum price, usually
referred to as the
reservation price
, above which demand reduces to zero. If we
denote by Q
D
P
kD1
x
k
the quantity supplied by the N firms in the industry and
we assume that at the price p the market clears, that is D
D
Q, then it also follows
that the relation between industry output and price is linear. Hence, by inverting this
relationship we finally obtain
p
D
f.Q/
D
A
BQ;
where
X
n
X
n
X
n
a
i
b
i
1
b
i
; B
D
1
1
b
i
:
A
D
i D1
i D1
i D1
Obviously, this representation is only valid for Q
A=B, that is as long as the
industry output is below the
market saturation
point. Otherwise, we have p
D
0.
In the case of a general inverse demand function the profit of firm k.1
k
N/
is the difference between its revenue and its cost and so is given by
'
k
.x
1
;:::;x
N
/
D
x
k
f
N
!
X
x
l
C
k
.x
1
;:::;x
N
/;
(1.1)
l
D
1
where C
k
is the cost function of firm k.
1
Our formulation takes into account the fact
that the cost of each firm depends not only on its own output but also on the outputs
1
In the game theory context the profit functions are usually called the payoff functions, and the
firms are called the players. We will occasionally make use of these terms throughout this topic.
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