There are three firms in the chert industry. Their demand curves are linear, and

are given by the following equations.

Qi =\alpha i -\beta ii Pi +\delta ij Pj +\delta ik Pk , where (i, j, k)=(1,2,3).

You are given the following information about these firms.

Firm 1 Firm 2 Firm 3

Prices 52.1646.1356.86

Outputs 193.02249.02292.98

Gross Margins 0.6170.6750.736

Furthermore, you are given the following diversion ratios:

D12=0.33, D21=0.25, D13=0.50, D31=0.285, D23=0.50, D32=0.285.

Note 1: There will be small rounding errors in your calculations.

Note 2: From profit maximization, use the formula:

margin =1/(price elasticity of demand).

Note 3: Remember that the price elasticity of demand for a firm is given by

the own-price coefficient (\beta ii) multiplied by that firm's price divided

by that firm's quantity.

a. Suppose that Firm 1 and Firm 2 want to merge. Suppose further that you are an analyst at Charles River Associates, a well-known consulting firm, and that you are working to analyze the likely effects of a prospective merger. To do so, among many other analyses, you must calculate the UPPI indices for P1 and P2. Do so, and assume that the merged firms can reduce marginal costs by 5% as a result of merger-related efficiencies.

b. How large would merger-related efficiencies have to be in order to guarantee no post-merger price increases?