21 Model Building

\[ U_A = V - P_A - t |x-a| \\ U_B = V - P_B - t|b-x| \]

where

  • \(\bar{x}\) = marginal consumer = indifferent consumer

  • \(-t|x-a|\) = 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 \(U_A = U_B\)

\[ V - P_A - t(\bar{x}-a) = V - P_B - t (b-\bar{x}) \\ P_B - P_A = 2t \bar{x}+ at - tb \\ \bar{x} = \frac{b-a}{2} + \frac{P_B - P_A}{2t} \]

Market Share for firm A = \(\bar{x}\)

Market Share for firm B = \(1 - \bar{x}\)

\[ \pi_A = P_A (\frac{b-a}{2} + \frac{P_B - P_A}{2t}) \\ \pi_B = P_B (1 - \frac{b-a}{2} - \frac{P_B - P_A}{2t}) \]

\[ \frac{d\pi_A}{d P_A} = \frac{p \pi_B}{d P_B} = 0 \]

\[ \frac{P_B}{2t} - \frac{2 P_A}{2t} + \frac{b-a}{2} = 0 \]

and

\[ \frac{d^2\pi_A}{dP^2_A} = \frac{1}{t} < 0 \]

\[ 1 - \frac{b-a}{2} - \frac{P_B}{t} + \frac{P_A}{2t} = 0 \\ 1/2 + \frac{b-a}{4} - \frac{3 P_A}{4t} = 0 \\ \frac{t(2 + b- a)}{3}= P_A \\ \frac{1}{2} - \frac{b-a}{4} + \frac{2 + b - a}{12} = \frac{P_B}{2t} \\ \frac{4/3} - \frac{b-a}{3} = \frac{P_B}{t} \\ \frac{t(4 - b+a)}{3} = P_B \]