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If you have access to a computer with some software that will solve systems of linear equations (e.g., Mathematica, Maple, or Gauss), you should try this question.With the differentiated demand system given by equation (23.1), we can write down the system of equations that defines the equilibrium premerger (technically, the set of first-order conditions) as Ap = b where the matrix A is consists of the following coefficients:
A=

2b

d

e

d

2f

g

e

g

2h

p is the [3X1] vector of prices and the coefficient vector b consists of

b=

bc1

a1

f c2

a2

hc3

a3

The set of equations that defines the equilibrium after the merger (of products 1 and 2) is the same, but with the matrix A replaced by A_ where
A_

2b

2d

e

2d

2f

g

e

G

2h

Begin initially with the parameter values given in the text. Suppose now that products 1 and 2 are closer substitutes than either of them is to product 3. We can represent this by increasing the substitution parameter d between products 1 and 2 from a value of 2 to 3. Now recalculate the pre- and post merger equilibrium prices. What happens and why? Now suppose that the opposite is truethe merging products are not close substitutes. Set the parameter d equal to 2 again and the parameters e and g equal to 3. Recalculate the post merger equilibrium and once again explain the effects on prices.
The system of equations defining the equilibrium premerger is Ap = b. Begin initially with the parameter values given in the text. Products 1 and 2 are closer substitutes than either of them is to product 3. This can be represented by increasing the substitution parameter d between products 1 and 2 from a value of 2 to 3. Calculate again the pre-merger and…

ger equilibrium prices. Assuming that the Products 1 and 2 are not close substitutes. Set the parameter d equal to 2 again and the parameters e and g equal to 3. Calculate the post-merger equilibrium and once again explain the effects on prices.

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