23 Model Building
\[ U_A = V  P_A  t xa \\ U_B = V  P_B  tbx \]
where
\(\bar{x}\) = marginal consumer = indifferent consumer

\(txa\) = Disutility of consuming a product away from ideal
 transportation cost parameters, weights on the disuility incur when they purchase a product different from the ideal point.
V = reservation price = max max that the customer is willing to take
x = location of marginal customer  customer who is indifference from buying from firm A and firm B.
To find the point that a consumer is indifference from buying from firm A and buying from firm B, we equate \(U_A = U_B\)
\[ V  P_A  t(\bar{x}a) = V  P_B  t (b\bar{x}) \\ P_B  P_A = 2t \bar{x}+ at  tb \\ \bar{x} = \frac{ba}{2} + \frac{P_B  P_A}{2t} \]
Market Share for firm A = \(\bar{x}\)
Market Share for firm B = \(1  \bar{x}\)
\[ \pi_A = P_A (\frac{ba}{2} + \frac{P_B  P_A}{2t}) \\ \pi_B = P_B (1  \frac{ba}{2}  \frac{P_B  P_A}{2t}) \]
\[ \frac{d\pi_A}{d P_A} = \frac{p \pi_B}{d P_B} = 0 \]
\[ \frac{P_B}{2t}  \frac{2 P_A}{2t} + \frac{ba}{2} = 0 \]
and
\[ \frac{d^2\pi_A}{dP^2_A} = \frac{1}{t} < 0 \]
\[ 1  \frac{ba}{2}  \frac{P_B}{t} + \frac{P_A}{2t} = 0 \\ 1/2 + \frac{ba}{4}  \frac{3 P_A}{4t} = 0 \\ \frac{t(2 + b a)}{3}= P_A \\ \frac{1}{2}  \frac{ba}{4} + \frac{2 + b  a}{12} = \frac{P_B}{2t} \\ \frac{4/3}  \frac{ba}{3} = \frac{P_B}{t} \\ \frac{t(4  b+a)}{3} = P_B \]