econ105_10s_ps4sol

econ105_10s_ps4sol - Problem Set 4 Solutions ECON105...

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Problem Set 4 Solutions ECON105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego 1. Claudia operates a fruit stand and sells bananas and coconuts. Marginal cost is zero as Claudia just picks up the fruit as it falls from the trees onto her land. There are two consumers, Bavanee and Cecilia. Cecilia values a banana at $4 and a coconut at $9. For Bavanee, her valuations are $6 and $3 respectively. Show that Cecilia could make more money by bundling one banana and one coconut into a fruit basket and selling the basket rather than selling the fruits separately. Separate: p banana = 8, p coconut = 9, TR separate = 17. Bundled: p basket =9, TR basket = 18. 2. Two firms compete Cournot. The market demand curve is Q = 16/p and each firm faces a cost function C(qi) = cqi. a. Find the equilibrium firm quantities, industry quantity, price and firm profits. Let us do maximize the profits of either firm (i) by choosing this firm’s quantity (that is what makes the game ―Cournot‖) as a function of the opponent’s quantity (j): 0 2 16 0 2 0 ) ( ) ( 0 ) ( : * ) ( * ) ( max 2 2 2 2 2 2 cQ j Q iQ j c cQ i Qj cQ j Q j c cQ i Qi c c FOC cQ i C P Qi This last equation is still just the first-order condition which implicitly defines the optimal quantity for firm i as a function of the quantity of firm j and the parameter c. If we wanted to, we could express this relationship explicitly by solving this quadratic equation for Qi(Qj) which would give us the best-response function of firm i: Ri(Qj;c)=Qi(Qj). But what we are looking for ultimately is Qi and Qj only as functions of the parameter c and not each other. We could find those optimal quantities by solving a system of two equations, where the two equations are just the two FOCs for the two firms. One FOC is just the one we found above for firm i, and another one is just like it but with switched
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subscripts everywhere which would come out if we re-solved the above problem for firm j instead: 0 2 16 : 0 2 : 2 2 2 2 cQi Qi Qj c cQj Qi FOCj cQj c cQi Qj FOCi However, since the two firms are symmetrical (have the same MC=c), we know that there must be a symmetric equilibrium (i.e., equal quantities), so we can replace one of those two equations with the much simpler equation: Qi=Qj. cQ j Q iQ j c cQ i 0 2 2 2 0 2 16 2 2 cQi Qi Qi c 0 2 2 2 2 c 0 4 2 cQ i 0 ) 4 ( 4 When both firms are producing at Q1=Q2=4/c, neither has any incentive to change its quantity so these quantities must be a Nash Equilibrium. (Note that it is not an equilibrium for both firms to produce zero, because either one would have the incentive to behave like a monopolist if they thought their opponent dropped out.) Total quantity is Q=Q1+Q2=8/c. Prices are P=16/Q=2c.
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econ105_10s_ps4sol - Problem Set 4 Solutions ECON105...

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