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In the paper "Collusive Price Leadership" by Julio J. Rotemberg and Garth Saloner in The Journal of Industrial Economics, Vol. 39, No. 1 (Sep., 1990), pp. 93-111, the following statement is made: We consider two firms. Firm 1 produces good 1 while firm 2 produces good 2. Each good is produced with a constant marginal cost c. The demands for these goods are given by: $Q_1 = x - bP_1 + d(P_2 - P_1)$ and $Q_2 = x - bP_2 + d(P_1 - P_2)$.

My question is: how can we derive these demand functions? I tried to use the method of maximizing Utility $U = U(x, y)$ given that $P_1 \cdot Q_1 + P_2 \cdot Q_2 = I$, but I cannot find a way to prove the above demand functions.

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  • $\begingroup$ Linear demand functions are usually generated by quadratic utility functions (this might be relevant). $\endgroup$
    – tdm
    Jan 31 at 14:23

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