A town has a bakery that sells bread and a cheese shop. It costs $1 to make a lo

Question

A town has a bakery that sells bread and a cheese shop. It costs $1 to make a loaf of bread and $2 to make a pound of cheese. If the bakery’s price for a loaf of bread is p1 and the cheese shop’s price for a pound of cheese is p2, then the number of loaves of bread sold q1 and the number of pounds of cheese sold q2 are given by the following demand functions:



q1 = 14 – p1 - .5p2



q2 = 19 - .5p1 – p2



(a) Are bread and cheese complements or substitutes in this problem? Defend your answer.



(b) Suppose the firms choose prices simultaneously. Derive each store’s best reply function. Graph the best reply functions and find the Nash equilibrium.



(c) Now suppose the two stores collude and set prices so as to maximize their joint profits. Find the prices that maximize joint profits.



(d) Are the prices in part (b) higher, lower, or equal to the prices in part (c)? What is the intuition behind this result?

Explanation / Answer

A. If the price of cheese increases, then people buy less bread. If the price of bread increases, then people buy less cheese. We know that because the signs on p1 and p2 are negative in both equations. This is characteristic of complements. B. To find your best replies, set up each firm's profit functions (they will be symmetric) take the derivative with respect to q1 and q2. These are your FOC. Substitute them into each other to find the best replies. You can graph them with q1 on both the x and y axis or q2 on both the x and y axis. The intersection will be the Nash equilibrium. C. You can solve this in one of two ways. 1. You can add the two profit functions together and take the derivative with respect to q1 and q2, substitute and find the optimal values. Then, substitute those values into your profit functions. 2. You can pretend that there is only one firm, treat it as a monopoly. Take your derivatives and find your first order conditions. Substitute them into each other to find the optimal levels of output. Plug these values into the profit function. Each firm will receive half of the monopoly profit. D. The prices are higher in part C, when the firms collude. This is always the case. Collusion yields higher prices because they behave as monopolies. Duolopolies (like the one in part B) produce greater quantities. So, they have to charge lower prices. Demand curves are downward sloping.

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