Consider a soft $($cola$)$ drinks market where two firms$$Firm A and Firm B$$compete for market shares given by the multinomial logit $($MNL$)$ functions below:

$\backslash$mu

A

$=$

e

$$

$0\mathrm{.}2$

$$

p

A

e

$$

$0\mathrm{.}2$

$$

p

A

$+$

e

$$

b

B

$$

p

B

and

$\backslash$mu

B

$=$

e

$$

b

B

$$

p

A

e

$$

$0\mathrm{.}2$

$$

p

A

$+$

e

$$

b

B

$$

p

B

$,$

where

$\backslash$mu

A

and

$\backslash$mu

B

respectively denote the market shares of Firms A and B when its corresponding products are priced at

p

A

and

p

A

; moreover, price sensitivity parameter for Firm B$\text{'}$s customers, i$.$e$,$

b

B

$,$ is unknown to you$$Firm A$\text{'}$s marketing$/$pricing specialist. However, your past marketing$/$pricing data suggest that

b

B

is between $0\mathrm{.}2$ and $0\mathrm{.}8.$

a$)$ Using the data set given in the table below, determine the value of

b

B

$.$

Price of Firm A

Price of Firm B

Market Share of Firm A

Market Share of Firm B

$$1\mathrm{.}00$

$$1\mathrm{.}00$

$59\mathrm{.}87\%$

$40\mathrm{.}13\%$

$$2\mathrm{.}00$

$$2\mathrm{.}00$

$69\mathrm{.}00\%$

$31\mathrm{.}00\%$

Table: Market shares of Firms A and B under two different price combination

b$)$ As a newly hired marketing$/$pricing manager of Firm A$,$ your goal is to increase your market share up to $80\%$ when both firms price the product at $$2\mathrm{.}50.$ You decide to launch a new marketing$/$branding campaign to improve the product awareness and brand loyalty of your product, and thereby reduce the value of

b

A

to increase the market share $($

$\backslash$mu

A

$).$

What is the target value of

b

A

that takes your firm$$s market share to $80\%?$