Game Theory and Macro Practice Questions with Answers

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FIN 501 Additional Practice Problems & Solutions Professor Tatyana Deryugina Game Theory Discrete actions (a) Find the best responses of each player, identify any dominant or dominated strategies, and solve for the pure-strategy Nash Equilibrium, if any. (b) Re-draw the games above as sequential games with player 1 moving first and find the outcome of each game. a. Best responses of player 1 are indicated in red circles; best responses of player 2 indicated in dark blue circles. Game 1 There is no dominant strategy for either player because there is no strategy that is always a best response. “Bottom” is strictly dominated because playing “top” always gives higher payoffs than playing “bottom”. “Left” is weakly dominated because playing “right” always gives at least as high of a payoff as playing “left” and sometimes higher. (Note that because these choices are unique to each player, it is not necessary to specify which player we are referring to). The Nash equilibrium is (Top, Right) because both players are best responding to each other; thus, neither has an incentive to deviate if the other one is not deviating. b. The game tree on the left below shows the above game as a sequential move game, with player 1 moving first. Note that I have omitted the labels for some of player 2’s actions in the second stage, for space reasons. After player 1 moves, the top arrow corresponds to player 2 choosing “left”, the middle arrow to player 2 choosing “middle”, and the bottom arrow to player 2 choosing “right”. The game tree on the right below shows the backward induction solution to the game, with arrows that are not chosen in grey. If P1 chooses “top”, P2 will choose “right” because that gives P2 the highest payoff. If P1 chooses “middle”, P2 will choose “right” for the same reason. Finally, if P1 chooses “bottom”, P2 will choose “middle”. Knowing this, P1 will choose “top” because this gives her the highest payoff. Thus, the outcome of this game will be “top”, ”right”. Left Middle Right Top 10 , 1 5, -2 3 , 3 Middle 5, 5 9 , 1 1, 9 Bottom 7, 3 2, 4 0, 3 Player 2 Player 1

(10,1) Left P2 (5,-2) Middle Top Right (3,3) P2 (5,5) Player 1 Middle (9,1) (1,9) Bottom P2 (7,3) (2,4) (0,3) (10,1) Left P2 (5,-2) Middle Top Right (3,3) P2 (5,5) Player 1 Middle (9,1) (1,9) Bottom P2 (7,3) (2,4) (0,3)

Game 2 a. Player 1 has a strictly dominant strategy – to play “Top” . “Top” always does better than any other strategy, regardless of what player 2 chooses. Thus, both “middle” and “bottom” are dominated by “top”. Player 2 has no dominant or dominated strategies – each strategy is a best response for some action of player 1. The Nash equilibrium is (Top, Left) (same reasoning as game 1). b. The game tree on the left below shows the above game as a sequential move game, with player 1 moving first. Note that I have omitted the labels for some of player 2’s actions in the second stage, for space reasons. After player 1 moves, the top arrow corresponds to player 2 choosing “left”, the middle arrow to player 2 choosing “middle”, and the bottom arrow to player 2 choosing “right”. The game tree on the right below shows the backward induction solution to the game, with arrows that are not chosen in grey. If P1 chooses “top”, P2 will choose “left” because that gives P2 the highest payoff. If P1 chooses “middle”, P2 will choose “middle” for the same reason. Finally, if P1 chooses “bottom”, P2 will choose “bottom”. Knowing this, P1 will choose “middle” because this gives her the highest payoff. Thus, the outcome of this game will be “middle”, ”middle”. (1,6) Left P2 (5,-2) Middle Top Right (3,3) P2 (-2,2) Player 1 Middle (3,8) (1,0) Bottom P2 (0,1) (2,2) (0,3) (1,6) Left P2 (5,-2) Middle Top Right (3,3) P2 (-2,2) Player 1 Middle (3,8) (1,0) Bottom P2 (0,1) (2,2) (0,3) Left Middle Right Top 1 , 6 5 , -2 3 , 3 Middle -2, 2 3, 8 1, 0 Bottom 0, 1 2, 2 0, 3 Player 2 Player 1 Left: 1 5 3 最大 strictly dominate : 同一排全圈

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(c) Find the outcomes in the following sequential move games, where player 1 moves first, player 2 moves second, and then player 1 moves again. Game 1 The game tree on the right below shows the backward induction solution to the game, with arrows that are not chosen in grey. When the outcome is “top”, “up”, P1 will choose B because that gives P1 the highest payoff. Following “top”, ”down”, P1 will choose “B” for the same reason. After “bottom”, ”up”, P1 will choose “B”, and after “bottom”, “down”, P1 will choose “T”. Anticipating this, Player 2 will choose “down” following “top” and choose “up” following “bottom”. Given these future choices, player 1 will choose “bottom” at the beginning of the game. Thus, the outcome will be “bottom”, “up”, “B”. (1,3) T B (4,1) Up Player 1 T Top Down (3,5) B Player 1 Player 2 (4,4) Bottom Up T (-2,1) B (8,4) Down Player 1 T (10,3) B (3,7) (1,3) T B (4,1) Up Player 1 T Top Down (3,5) B Player 1 Player 2 (4,4) Bottom Up T (-2,1) B (8,4) Down Player 1 T (10,3) B (3,7)

(5,1) T B (4,2) Up Player 1 T Top Down (8,2) B Player 1 Player 2 (2,4) Bottom Up T (3,7) B (9,4) Down Player 1 T (4,3) B (5,7) Game 2 The game tree on the right below shows the backward induction solution to the game, with arrows that are not chosen in grey. When the outcome is “top”, “up”, P1 will choose T because that gives P1 the highest payoff. Following “top”, ”down”, P1 will choose “T” for the same reason. After “bottom”, ”up”, P1 will choose “B”, and after “bottom”, “down”, P1 will also choose “B”. Anticipating this, Player 2 will choose “down” following “top” or following “bottom”. Given these future choices, player 1 will choose “top” at the beginning of the game. Thus, the outcome will be “top”, “down”, “top” . (5,1) T B (4,2) Up Player 1 T Top Down (8,2) B Player 1 Player 2 (2,4) Bottom Up T (3,7) B (9,4) Down Player 1 T (4,3) B (5,7)

Continuous actions Note: I am showing rounded answers, but using exact answers in subsequent steps. If you are using a rounded answer in subsequent steps (which is fine), some of your answers may be slightly different. (a) Two firms are competing. Each firm has MC = 10. Neither firm has fixed costs. Demand is 𝑄𝑄 𝐷𝐷 = 70 − 𝑃𝑃 . Find the outcomes (prices, quantities, and profits) if they compete (a) via Bertrand, (b) via Cournot, (c) via Stackelberg, with firm 1 moving first. Bertrand Competition: 𝑝𝑝 = 𝑀𝑀𝑀𝑀 = 10 . Each firm produces 𝑄𝑄 = 30 (such that total quantity is 𝑄𝑄 𝐷𝐷 = 70 − 10 = 60 ). Profits are zero. Note: be sure you can explain why this is a Nash Equilibrium and why there is no other Nash equilibrium. Cournot Competition: First, we find the inverse demand curve as a function of the quantities of both firms: 𝑃𝑃 = 70 − 𝑄𝑄 1 − 𝑄𝑄 2 . Profit of firm 1 is then (70 − 𝑄𝑄 1 − 𝑄𝑄 2 ) 𝑄𝑄 1 − 10 𝑄𝑄 1 = 60 𝑄𝑄 1 − 𝑄𝑄 1 2 − 𝑄𝑄 2 𝑄𝑄 1 . To maximize, we take the derivative and set it equal to zero: 60 − 2 𝑄𝑄 1 − 𝑄𝑄 2 = 0 𝑄𝑄 1 = 30 − 𝑄𝑄 2 2 This is firm 1’s best response function or reaction function. Because firm 2 has the same marginal cost, it will have a symmetric reaction function: 𝑄𝑄 2 = 30 − 𝑄𝑄 1 2 . Solving this for 𝑄𝑄 1 , we get 𝑄𝑄 1 = 60 − 2 𝑄𝑄 2 . Set this equal to the best response function of firm 1 and solve for 𝑄𝑄 2 : 30 − 𝑄𝑄 2 2 = 60 − 2 𝑄𝑄 2 3 𝑄𝑄 2 2 = 30 𝑄𝑄 2 = 20 𝑄𝑄 1 = 30 − 𝑄𝑄 2 2 = 20 Each firm will produce a quantity of 20 . The price will be 70 − 40 = 30 . Profits will be (30 − 10) ∗ 20 = 400 for each firm. Stackelberg Competition: solve by backward induction. Because firm 2 moves second, it takes 𝑄𝑄 1 as given. The best response of firm 2 will then be the same as in Cournot competition, 𝑄𝑄 2 = 30 − 𝑄𝑄 1 2 . Knowing this, firm 1 chooses 𝑄𝑄 1 to maximize (70 − 𝑄𝑄 1 − 𝑄𝑄 2 ) 𝑄𝑄 1 − 10 𝑄𝑄 1 = � 70 − 𝑄𝑄 1 − 30 + 𝑄𝑄 1 2 � 𝑄𝑄 1 − 10 𝑄𝑄 1 = 30 𝑄𝑄 1 − 𝑄𝑄 1 2 2 . Take the derivative, set = 0, and solve: 30 − 𝑄𝑄 1 = 0 𝑄𝑄 1 = 30 ; 𝑄𝑄 2 = 30 − 𝑄𝑄 1 2 = 15 The price will be 70 − 45 = 25 . Firm 1’s profits will be (25 − 10)30 = 450 . Firm 2’s profits will be (25 − 10) ∗ 15 = 225 .

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(b) Two firms are competing. Firm 1 has MC = 10 and firm 2 has MC = 20. Neither firm has fixed costs. Demand is 𝑄𝑄 𝐷𝐷 = 100 − 1 2 𝑃𝑃 . Find the outcomes (prices, quantities, and profits) if they compete (a) via Bertrand, (b) via Cournot, (c) via Stackelberg, with firm 1 moving first. Bertrand Competition: we know that firm 1 can shut firm 2 out of the market by charging a price slightly under firm 2’s marginal cost. Whether it will be optimal to undercut firm 2 by a little bit or by a lot depends on whether the optimal monopoly price is below firm 2’s marginal cost (if it is, firm 1 will charge the monopoly price). So we calculate and solve the monopoly solution: (200 − 2 𝑄𝑄 𝐷𝐷 ) 𝑄𝑄 𝐷𝐷 − 10 𝑄𝑄 𝐷𝐷 = 190 𝑄𝑄 𝐷𝐷 − 2 𝑄𝑄 𝐷𝐷 2 190 − 4 𝑄𝑄 𝐷𝐷 = 0 𝑄𝑄 𝐷𝐷 = 190 4 𝑃𝑃 = 200 − 190 2 = 105 But 𝑃𝑃 = 105 is above firm 2’s marginal cost, so this solution is unattainable. Thus, firm 1 charges a price of 19 (assuming price has to be an integer). Firm 2 produces nothing (and “charges” some price 20 or above). Firm 1 produces to satisfy the demand of the entire market: 𝑄𝑄 𝐷𝐷 = 100 − 19 2 = 90.5 . Firm 2 makes zero profits. Firm 1 makes profits of (19 − 10) ∗ 90.5 = 814.5 . Cournot competition: Find the inverse demand function: 𝑃𝑃 = 200 − 2 𝑄𝑄 𝐷𝐷 . Firm 1’s profit function is (200 − 2 𝑄𝑄 1 − 2 𝑄𝑄 2 ) 𝑄𝑄 1 − 10 𝑄𝑄 1 = 190 𝑄𝑄 1 − 2 𝑄𝑄 1 2 − 2 𝑄𝑄 2 𝑄𝑄 1 . Take the derivative, set equal to zero to find firm 1’s reaction function. 190 − 4 𝑄𝑄 1 − 2 𝑄𝑄 2 = 0 𝑄𝑄 1 = 47.5 − 1 2 𝑄𝑄 2 Firm 2’s profit function is: (200 − 2 𝑄𝑄 1 − 2 𝑄𝑄 2 ) 𝑄𝑄 2 − 20 𝑄𝑄 1 = 180 𝑄𝑄 2 − 2 𝑄𝑄 2 2 − 2 𝑄𝑄 2 𝑄𝑄 1 . Take the derivative, set equal to zero to find firm 2’s reaction function: 180 − 4 𝑄𝑄 2 − 2 𝑄𝑄 1 = 0 𝑄𝑄 2 = 45 − 1 2 𝑄𝑄 1 Solve this for 𝑄𝑄 1 and set equal to firm 1’s reaction function: 𝑄𝑄 1 = 90 − 2 𝑄𝑄 2 = 47.5 − 1 2 𝑄𝑄 2 . 42.5 = 3 2 𝑄𝑄 2 𝑄𝑄 2 = 28.33 𝑄𝑄 1 = 33.83 Price is 200 − 2(28.33 + 33.83) = 76.67 . Profits of firm 1 are ( 76.67 − 10) ∗ 33.83 = 2222.2 . Profits of firm 2 are ( 76.67 − 20) ∗ 28.33 = 1605.6 .

(c) Two firms are competing. Firm 1 has MC = 15 and firm 2 has MC = 5. Neither firm has fixed costs. Demand is 𝑄𝑄 𝐷𝐷 = 200 − 3 𝑃𝑃 . Find the outcomes (prices, quantities, and profits) if they compete (a) via Bertrand, (b) via Cournot, (c) via Stackelberg, with firm 1 moving first. Bertrand Competition: we know that firm 2 can shut firm 1 out of the market by charging a price slightly under firm 1’s marginal cost. Whether it will be optimal to undercut firm 1 by a little bit or by a lot depends on whether the optimal monopoly price is below firm 1’s marginal cost (if it is, firm 2 will charge the monopoly price). So we calculate and solve the monopoly solution: � 200 3 − 𝑄𝑄 𝐷𝐷 3 � 𝑄𝑄 𝐷𝐷 − 5 𝑄𝑄 𝐷𝐷 = 61.67 𝑄𝑄 𝐷𝐷 − 𝑄𝑄 𝐷𝐷 2 3 61.67 − 2 3 𝑄𝑄 𝐷𝐷 = 0 𝑄𝑄 𝐷𝐷 = 92.5 𝑃𝑃 = 200 3 − 92.5 3 = 35.83 But 𝑃𝑃 = 35.83 is above firm 1’s marginal cost, so this solution is unattainable. Thus, firm 2 charges a price of 14 (assuming price has to be an integer). Firm 1 produces nothing (and “charges” some price =15 or above). Firm 2 produces to satisfy the demand of the entire market: 𝑄𝑄 𝐷𝐷 = 200 − 3 ∗ 14 = 158 . Firm 1 makes zero profits. Firm 2 makes profits of (14 − 5) ∗ 158 = 1422 . Cournot competition: Find the inverse demand function: 𝑃𝑃 = 200 3 − 𝑄𝑄 𝐷𝐷 3 . Firm 1’s profit function is � 200 3 − 𝑄𝑄 1 3 − 𝑄𝑄 2 3 � 𝑄𝑄 1 − 15 𝑄𝑄 1 = 155 3 𝑄𝑄 1 − 𝑄𝑄 1 2 3 − 1 3 𝑄𝑄 2 𝑄𝑄 1 . Take the derivative, set equal to zero to find firm 1’s reaction function. 155 3 − 2 3 𝑄𝑄 1 − 1 3 𝑄𝑄 2 = 0 𝑄𝑄 1 = 155 2 − 1 2 𝑄𝑄 2 Firm 2’s profit function is: � 200 3 − 𝑄𝑄 1 3 − 𝑄𝑄 2 3 � 𝑄𝑄 2 − 5 𝑄𝑄 1 = 61.67 𝑄𝑄 2 − 𝑄𝑄 2 2 3 − 1 3 𝑄𝑄 2 𝑄𝑄 1 . Take the derivative, set equal to zero to find firm 2’s reaction function: 185 3 − 2 3 𝑄𝑄 2 − 1 3 𝑄𝑄 2 𝑄𝑄 1 = 0 𝑄𝑄 2 = 185 2 − 1 2 𝑄𝑄 1 Solve this for 𝑄𝑄 1 and set equal to firm 1’s reaction function: 𝑄𝑄 1 = 185 − 2 𝑄𝑄 2 = 155 2 − 1 2 𝑄𝑄 2 . 107.5 = 3 2 𝑄𝑄 2 𝑄𝑄 2 = 71.67

𝑄𝑄 1 = 41.67 Price is 200 3 − 1 3 (71.67 + 41.67) = 28.89 . Profits of firm 1 are ( 28.89 − 15) ∗ 41.67 = 578.7 . Profits of firm 2 are ( 28.89 − 5) ∗ 71.67 = 1712.0 . Stackelberg competition: Firm 2’s reaction function is the same. Firm 1 maximizes: � 200 3 − 𝑄𝑄 1 3 − 185 6 + 𝑄𝑄 1 6 � 𝑄𝑄 1 − 15 𝑄𝑄 1 = 125 6 𝑄𝑄 1 − 1 6 𝑄𝑄 1 2 125 6 − 1 3 𝑄𝑄 1 = 0 𝑄𝑄 1 = 125 2 = 62.5 𝑄𝑄 2 = 185 2 − 62.5 2 = 61.25 Price is 200 3 − 1 3 (62.5 + 61.25) ≈ 25.4 . Profits of firm 1 are (25.4 − 15) ∗ 62.5 = 651.0 . Profits of firm 2 are (25.4 − 5) ∗ 61.25 = 1250.5 .

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Macroeconomics ( Note: I will not provide detailed answers to these questions, as most of them are in Mankiw’s macroeconomics textbook, in our lecture notes or can easily be deduced from our lecture notes. If you have further questions, please email me, come to office hours, or see the TA). (a) What happens to short-run and long-run per-capita output, per-capita consumption, and per-capita capital in a simple Solow growth model after each of the following scenarios? Assume that the economy starts out in a long-run equilibrium. Illustrate your answers with diagrams. 1. An increase/decrease in the savings rate (2 scenarios). Short-run consumption per capita decreases/increases. Long-run consumption increases/decreases. Long-run capital per capita increases/decreases. Output per capita and capital per capita unchanged in the short run. 2. An increase/decrease in the population growth rate (2 scenarios). Short-run consumption per capita is unchanged. Long-run consumption per capita decreases/increases. Long-run capital per capita decreases/increases. Output per capita and capital per capita unchanged in the short run. 3. Technological progress. Short-run and long-run consumption per capita increase. Long-run capital per capita increases. Output per capita increases in the short run; capital per capita unchanged in the short run. 4. A natural disaster that destroys half the capital stock. Capital per capita, output per capita, and consumption per capita fall in the short run. In the long run, nothing changes (economy returns to initial equilibrium). 5. A natural disaster that kills some of the population. Capital per capita, output per capita, and consumption per capita increase in the short run (intuition: same amount of capital, but fewer people to divide it amongst). In the long run, nothing changes (economy returns to initial equilibrium). 6. An increase in the economy’s price level (inflation). Nothing happens in the short run or the long run; prices play no role in the Solow growth model. (b) Determine what happens in the foreign currency exchange market and the market for loanable funds in the following scenarios for a closed economy (loanable funds market only), a small open economy , and a large open economy (including what happens to world interest rates, savings, investment, and equilibrium real exchange rate, if applicable). Illustrate your answers with diagrams. 1. Country’s own government increases/reduces national savings (2 scenarios). o All economies: private savings increases/falls. o Closed/large open economy: interest rate falls/rises; investment increases/decreases. Small open economy: no change in interest rate or investment. o Large/small open economy: net exports increase/decrease. Real exchange rate decrease/increases. 2. A large foreign government increases/reduces national savings (2 scenarios). o Note: not applicable to closed economy. o World interest rates decreases/increases. o All open economies: investment increases/decreases. Savings are unchanged (by assumption). Net exports fall/rise. Real exchange rate increases/decreases.

3. The investment demand function shifts in/out (2 scenarios). o Closed/large open economy: lower/higher real interest rate, lower/higher investment. Small open economy: no change in real interest rate, lower/higher investment. Closed economy: savings are lower/higher (no change in other economies by assumption). o Large/small open economy: net exports rise/fall. Equilibrium exchange rate falls/increases. 4. Country’s own government eliminates/imposes tax on earned interest (2 scenarios). o By assumption, no change in savings in open economies. Savings in closed economy increase/decrease, real interest rate falls/increases, investment increases/falls. 5. Country’s own government increases/decreases investment subsidies (2 scenarios). o Note: this works in the same way as if the investment function shifted out/in. o Closed/large open economy: higher/lower real interest rate, higher/lower investment. Small open economy: no change in real interest rate, higher/lower investment. Closed economy: savings are higher/lower (no change in other economies by assumption). o Large/small open economy: net exports fall/rise. Equilibrium exchange rate increases/falls. 6. Country’s government restricts imports/exports (2 scenarios). o No change in savings, investment, or the real interest rate. All open economies: net export demand shifts up/down, real exchange rate increases/falls. Net exports are unchanged in either scenario. (c) Use the aggregate demand/aggregate supply model to illustrate and describe what happens in the short- and long-run to output, unemployment, and price levels, with and without optimal government intervention following a negative/positive supply/demand shock (4 scenarios). Describe what the optimal government intervention looks like (e.g., should the government raise spending/cut taxes or decrease spending/increase taxes)? • Negative supply shock: in the short run output falls below potential output, prices are higher (inflation), unemployment rate is above its natural rate. • Positive supply shock: in the short run output rises above potential output, prices are lower (deflation), unemployment rate falls above its natural rate. • Negative demand shock: in the short run output falls below potential output, prices are lower (deflation), unemployment rate is above its natural rate. • Positive demand shock: in the short run output rises above potential output, prices are higher/inflation, unemployment rate falls above its natural rate. • Optimal government intervention depends on whether the output gap (potential minus actual output) is positive (negative supply shock or negative demand shock) or negative (the other two scenarios). With a positive output gap, the government should increase spending and/or decrease taxes; with a negative output gap, the government should decrease spending and/or increase taxes. • Regardless of the scenario, in the long run, output will return to potential output and the unemployment rate will return to its natural rate. Prices may be lower/higher/or the same as before.

Game Theory and Macro Practice Questions with Answers

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