eco204_summer_2009_practice_problem_25_solution

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 25 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Question 1 In the US, TV networks' profits are a function of advertising which in turn is a function of ratings. TV shows ratings are proportional to the number of viewers. In this problem, you will analyze two TV networks' decision to schedule a "hit" show: each network's viewership depends on whether they go "head to head" (i.e. the same time slot). The TV networks ABC and NBC each have a "hit" show. ABC's hit show is Grey's Anatomy and NBC's hit show is Heroes. The profits of the two networks depend on viewership. For simplicity, suppose the hit shows can be scheduled at 8 pm or 9 pm: ABC "Grey's Anatomy" NBC "Heroes" 8 pm 9 pm 8 pm 36, 33 30, 36 9 pm 39, 28 32, 30 If the networks must announce the schedules simultaneously, what is the time slot of each hit show? Do not assume mixed strategies. Answer: This is a one shot simultaneous move game. We can solve this by (pure strategy) Nash equilibrium. See Table below for the best responses: 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain NBC "Heroes" 8 pm 9 pm ABC "Grey's Anatomy" 8 pm 36, 33 30, 36 9 pm 39, 28 32, 30 The only pair of actions which are mutual best responses are for NBC to schedule the hit at 8 pm and for ABC to schedule the hit at 8 pm. You could have also solved the game by dominant strategies method. Observe that for both NBC and ABC, 8 pm dominates 9 pm. Thus, say ABC eliminates 9 pm, then by common knowledge, NBC too knows that the ABC will not play 9 pm. Thus: ABC "Grey's Anatomy" NBC "Heroes" 8 pm 9 pm 8 pm 36, 33 30, 36 9 pm Turning to NBC's perspective, note how 8 pm dominates 9 pm. Eliminating 9 pm for NBC yields: ABC "Grey's Anatomy" NBC "Heroes" 8 pm 9 pm 8 pm 36, 33 9 pm Which is the same answer using Nash Equilibrium. Question 2 In 1992, Saudi Arabia and Iran (both members of OPEC) produced an average of 5m and 2m barrels of oil a day. Production costs were about $10 per barrel and the price of oil averaged 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain about $20 per barrel. Each country had the capacity to produce an additional 1m barrels a day. At that time, it was estimated that each 1m barrels increase in supply would depress the price of oil by $2 (in practice, this is estimated using price elasticities). (a) Fill in the missing profit entries in the table below: Iran 2m 3m Saudi Arabia 5m 6m Answer: Each country's profits are given by the equation: = R C = P Q AC Q = (P AC) Q The question tells us that AC = $10/barrel and independent of volume. That is, the total cost function is linear and has no fixed cost. Start with the baseline case where Saudi Arabia produces 5m barrels and Iran produces 2m barrels. The price is $20. Now: Saudi Arabia = (P AC) Q Saudi Arabia = (20 10) 5 Saudi Arabia = $50m Similarly: Iran = (P AC) Q Iran = (20 10) 2 Iran = $20m The table becomes: 3 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 5m 6m Note: Figures in cells are millions of dollars Saudi Arabia Iran 2m 50, 20 3m Next consider the case where Saudi Arabia produces 5m barrels and Iran produces 3m barrels. Relative to the baseline case, with an extra 1m barrels on the market, the price falls by $2 per barrel. Now: Saudi Arabia = (P AC) Q Saudi Arabia = (18 10) 5 Saudi Arabia = $40m Similarly: Iran = (P AC) Q Iran = (18 10) 3 Iran = $24m The table becomes: Iran 2m 3m Saudi Arabia 5m 50, 20 40, 24 6m Note: Figures in cells are millions of dollars Note how Iran gains at Saudi Arabia's expense by producing more oil. This is the classic cartel problem you've seen in ECO 100 where each party can gain by "cheating" and producing more than its quota. Next consider the case where Saudi Arabia produces 6m barrels and Iran produces 2m barrels. Relative to the baseline case, with an extra 1m barrels on the market, the price falls by $2 /barrel. Now: Saudi Arabia = (P AC) Q Saudi Arabia = (18 10) 6 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Saudi Arabia = $48m Similarly: Iran = (P AC) Q Iran = (18 10) 2 Iran = $16m The table becomes: Saudi Arabia 5m 6m Note: Figures in cells are millions of dollars Iran 2m 50, 20 48, 16 3m 40, 24 Finally, consider the case where Saudi Arabia produces 6m barrels and Iran produces 3m barrels. Relative to the baseline case, with an extra 2m barrels on the market, the price falls by $4 /barrel. Now: Saudi Arabia = (P AC) Q Saudi Arabia = (16 10) 6 Saudi Arabia = $36m Similarly: Iran = (P AC) Q Iran = (16 10) 3 Iran = $18m The table becomes: 5 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 5m 6m Note: Figures in cells are millions of dollars (b) What actions will each country take? Answer: The payoff matrix is: Saudi Arabia 5m 6m Note: Figures in cells are millions of dollars Saudi Arabia Iran 2m 50, 20 48, 16 3m 40, 24 36, 18 Iran 2m 50, 20 48, 16 3m 40, 24 36, 18 Let's solve this game by Nash equilibrium. The following table shows the best responses: Iran 2m 3m Saudi Arabia 5m 50, 20 40, 24 6m 48, 16 36, 18 Note: Figures in cells are millions of dollars The Nash equilibrium has Saudi Arabia sticking to its quota of 5m barrels a day and Iran "cheating" by producing 3m barrels a day. (c) Is this a prisoner's dilemma game? Answer: No. Recall that a prisoner's dilemma is when the Nash equilibrium is sub optimal. This is not the case here there is no other outcome that is better for both parties. 6 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Question 3 (20072008 Problem) Two hospitals compete on the basis of service to patients. The following table gives the profits (in millions of $): Basic Hospital A's Services All Purpose Specialty Basic 5, 7 4, 5 6, 10 Hospital B's Services All Purpose 5, 4 8, 7 3, 12 Specialty 12, 6 7, 4 3, 3 (a) Does either hospital have a dominant strategy? If so, solve the game by dominant/dominated strategies. Show all steps below. Answer: While A does not have a dominant strategy, B does: Basic dominates Specialty. The game becomes: Basic Hospital A's Services All Purpose Specialty Basic 5, 7 4, 5 6, 10 Hospital B's Services All Purpose 5, 4 8, 7 3, 12 Specialty But now, neither hospital has a dominant strategy and therefore, the game cannot be solved by dominant/dominated strategies. (b) Solve for the pure strategy Nash equilibrium. Show your reasoning in the matrix below: Answer: 7 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Basic Basic 5, 7 4, 5 6, 10 Hospital B's Services All Purpose 5, 4 8, 7 3, 12 Specialty 12, 6 7, 4 3, 3 Hospital A's Services All Purpose Specialty (c) If these hospitals merge and coordinate their hospital services, what actions should they take? Explain briefly. If necessary, show your reasoning below: Basic Hospital A's Services All Purpose Specialty Basic 5, 7 = 12 4, 5 = 9 6, 10 = 16 Hospital B's Services All Purpose 5, 4 = 9 8, 7 = 15 3, 12 Specialty 12, 6 = 18 7, 4 = 11 3, 3 = 6 Now the hospitals will want to maximize the joint profits. Simply add up the payoffs and choose the cell with highest total. See above. A does Basic services, B does Specialty services. 8 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto.

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