2. (a) This is a standard Cournot problem with a = 10, b=1, and c= 2. Using slide 21 from Topic 2, the total quantity and price are Q=8- and P=22 n+1 (b) Given a quantity Q, the cartel makes a revenue of P(Q)Q but incurs a cost of 20 as well as a fine of max{0, Q(P(Q) - P*)/2}. Therefore, its profit is T(Q) = (P(Q) — 2)Q - max = (10-Q-2)Q- max x{0,0 (¹0 = (8-Q)Q- max = (8-Q)Q - max x {0, Q(P(Q) - P²}} 2 10-Q 2 x {0₁0 (²³ Q (8-Q)Q-max 0, Q -9)}. 2²+5 n+1 x{₁,Q {0₁0 (5-9/-5+7)} 2 n 1 n+1 2 == 5n+5-5-n n+1 4n n+1 8n n+1 Note: Alternatively, we can consider the two cases separately. If Q> 8n/(n+1) (equivalently, P(Q) ≤ P*), then (Q) = (8-Q)Q. If Q≤ 8n/(n+1) (equivalently, P(Q) > P*), then 4n *(Q) = (8 - Q)Q- Q (41 - 2) 4n Q² = 8Q - Q²- -Q+ 2 4n 41) Q n+ n+1 +18 Q² 8n +84n + 2 n+1 Q² n+2 +4 Q 2 n+1 = (1+2-2) o. 10+ 2n 2(n+1) "-2)} (c) If the cartel produces Q> 8n/(n+1), the profit is (8-Q)Q, which is maximised by setting 8-2Q=0 so Q = 4. As n ≥ 2, })} Therefore, the best the cartel can do if it wants to avoid a fine is to set Q= 8n/(n+1), which means that it will not optimally produce more than 8n/(n+1). We can therefore focus on the case where Q≤ 8n/(n+1). In that case, the cartel maximises¹ (12+2-9) 9. Taking the first-order condition and solving for Q yields Q = 4(n+2)/(n+1). As 2. There are one supplier S that produces a good at no cost and two retailers R₁ and R₂ that compete in quantities and face the inverse demand P = 4-qR₁qR₂, where, for each i = 1,2, qr, is the quantity that R; buys from S and sells to consumers. The timing of the game is as follows. First, S chooses its price s. Then, the two retailers observes and simultaneously choose their quantities qR, and qR₂- (a) Calculate the equilibrium values for s, qR₁, and qR₂ Suppose now that S and R₁ merge to become the integrated firm I. Firm I produces at no cost, sells to R₂ at price s, and sells directly to consumers at price P=4-91-9R₂ Firm R₂ buys from firm I at prices and sells to consumers at price P=4-91-9R₂- The timing of the game is as follows. Firm I first sets the price s at which it sells to R₂. Then, having observed s, I and R₂ simultaneously set their quantities qr and qR₂. Write down the profit of both firms as functions of s, qr, and qR₂. (b) (c) (d) Calculate the equilibrium values for s, qr, and qR₂ Without doing any additional calculations, determine whether the merger is good or bad for consumers. [max: 50 words] And here is an example. When solving for the question given above, You can refer to the format: 2. There are n ≥ 2 profit-maximising firms producing a homogeneous good, competing in quantity, and facing the inverse demand function P(Q) = 10-Q, where Q = -19i is the total quantity produced in the market. Each firm i faces the same linear cost function: C(q) = 2qi. (a) Find the total quantity produced and the equilibrium price. Suppose that the n firms form a cartel that chooses a total quantity Q to produce. To avoid lengthy court proceedings, the competition authority has decided to allow cartels to be formed but issue them with a tax equal to 50% of the revenue that firms make above the competitive price. That is, letting P* be the competitive price (found in (a)) and letting Q be the quantity that the cartel chooses, the cartel has to pay max{0, Q(P(Q) - P*)/2}. (We assume that the competition commission knows P* and is able to collect the tax.) (b) Write down the formula for the profit of the cartel as a function of Q. (c) If the cartel maximises its profit, what will be the total quantity produced and the equilibrium price? (d) For what values of n are consumers strictly worse off under the cartel than under competition? (e) Comparing the competitive price (found in (a)), the cartel price (found in (c)), and the price that the cartel would set if there were no competition authority, how effective is the competition authority at keeping the price low?