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Unformatted text preview: , the prices p1 and p2
are in equilibrium, a competitive equilibrium or Nash equilibrium. Note that each firm’ optimum price depends on
s
the partial derivative of its demand with respect to its price. If firms 1 and 2 merge and set prices on the products so as to maximize the total profit
p1 , p2
p1 q1 p1 , p2 p2 q2 p1 , p2
C1 q1 p1 , p2
C2 q2 p1 , p2
1 p1 , p2
2 p1 , p2
then the optimum prices would satisfy
0 q1 p1 , p2 0 p2 p1 mc1 q2 p1 , p2 p1 p1 mc1 q1 p1 , p2
p1
q1 p1 , p2
p2 p2 mc2 p2 mc2 q2 p1 , p2
p1
q2 p1 , p2
p2 (1’ and
)
(2’
). These conditions differ from the first order conditions for the separate firms (1) and (2), the difference depending on
the cross partial derivatives q2
p1 and q1
p2 . The sensitivity of demand to price is usually expressed in terms of an elasticity. The own price elasticity of demand
for product 1 is
q1 p1
Ε1 1
pq
1 1 and similarly for 2
q2 p2
Ε2 2
pq
2 2 Own price elasticities of demand tells only part of the story of demand here. A price increase of one product causes
consumers to switch to the other product. So it may also be useful to know
q1
p2 and q2
p1 or the cross price elasticities of demand
Ε1 2 q1 p2
p2 q1 q2 p1 and Ε2 1 p1 q2 The matrix of partial derivatives of the demand functions with respect to prices at some vector of prices and corresponding vector of quantities gives a linear approximation to the demand functions at that point. Specifically, if
q1 q1 p1 p2 p1 , p2 q2 q2 p1 M q1 ,q2 p2 is the matrix of first derivatives, then
q1
p1
M
q2
p2
The matrix of elasticities
Ε1 1 Ε1 2
E
Ε2 1 Ε2 2
serves the same purpose if one considers fractional changes in prices and quantities,
q1 q1
p1 p1
E
q2 q2
p2 p2 Computation
Consider then a simple example where demand and cost functions are linear. Take m256hw03soln.nb cost1 q1_ : mc1
cost2 q2_ : mc2 q1
q2 q1 p1_, p2_ : a11
q2 p1_, p2_ : a21 3 fc1;
fc2; p1
p1 a12
a22 p2
p2 b1;
b2; Then the profits of the firms are
profit1 p1_, p2_ : p1
profit2 p1_, p2_ : p2 q1 p1, p2
q2 p1, p2 cost1 q1 p1, p2
cost2 q2 p1, p2 ;
; So for example
Simplify profit1 p1, p2
fc1 mc1 p1 b1 a11 p1 a12 p2 The partial derivative is computed by the Mathematica function D.
D profit1 p1, p2 , p1
b1 a11 mc1 2 a11 p1 a12 p2 and we note that the fixed cost fc...
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This note was uploaded on 07/12/2013 for the course MATH 256 taught by Professor Schantz during the Spring '11 term at Vanderbilt.
 Spring '11
 Schantz
 Math

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