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1 FIN 501: Financial Economics Professor Deryugina Problem Set 5 Solutions TA Sarah Kim Question 1 a. Suppose that coal is produced in a competitive world market. The world price of coal is a constant $40 per ton (similar to our trade example). Assume that US firms are price takers in this world market. The marginal private cost of coal production in the US, aggregated across all firms, is 𝑀𝐶 =0.8 ? , where ? is tons of coals per day. Fixed costs are zero. How many tons of coal will US firms produce per day? To maximize profits, firms will choose to produce the quantity such that MC=MR. In a competitive market, price is constant, so total revenue TR=P*Q and marginal revenue MR=P. 𝑀𝐶 = ? → 0.8? = 40 ? ∗ = 40 0.8 = 50 b. Maintain the same general set up as before (competitive world market, price-taking firms). The world price of coal is a constant $40 per ton. The marginal private cost of coal production in the US is 𝑀𝐶 =0.8 ? . Now let’s take into account the fact that coal produces sulfur dioxide when burned. The US government has estimated that the external social cost of coal to the US is ??𝐶 =1.4 ? 2 . This does not include the marginal private cost of production. What is the socially optimal level of coal production by US firms, assuming all the coal that is produced in the US gets burned in the US? ????? ???𝑔𝑖??? ???𝑖?? ???? (𝑀?𝐶) = ???𝑔𝑖??? ??𝑖𝑣??? ???? (𝑀?𝐶) + ???𝑔𝑖??? ?𝑥?????? ???𝑖?? ???? (𝑀𝐸?𝐶) MPC=0.8Q and 𝑀𝐸?𝐶 = 𝜕??𝐶 𝜕? = 2.8? , so MSC=3.6Q. Thus, the socially optimal level of coal production satisfies the condition MSC=P 3.6? = 40 → ? ?𝑜𝑐 ∗ = 11.11 c. Maintain the same general set up as before (competitive world market, price-taking firms). The world price of coal is a constant $40 per ton. The marginal private cost of coal production in the US is 𝑀𝐶 =0.8 ? . The external social cost of coal to the US is ??𝐶 =1.4 ? 2 . This does not include the marginal private cost of production. The US government wants to impose a tax on each ton of coal that the US produces. What size tax will induce US firms to produce the socially optimal level of output? As found in (b), ? ?𝑜𝑐 ∗ = 11.11 , so MPC will be 0.8*11.11=8.888 at the socially optimal Q. This MPC is much lower than the world price, $40. If taxes are collected, firms will internalize the tax as additional costs, and they will still be incentivized to produce Q=11.11. The optimal tax amount satisfies the condition, 8.888+tax=40. Therefore, tax=40-8.888 ≈ $31.11

2 d. Maintain the same general set up as before (competitive world market, price-taking firms). The world price of coal is a constant $40 per ton. The marginal private cost of coal production in the US is 𝑀𝐶 =0.8 ? . The external social cost of coal to the US is ??𝐶 =1.4 ? 2 . This does not include the marginal private cost of production. If the government decides to solve the externality problem with tradable coal-mining permits, how many daily permits should it issue (it’s ok to have fractional permits)? The government would want the quantity of production to be at the socially optimal level. As determined in (b), this level is ? ?𝑜𝑐 ∗ = 11.11. This corresponds to the exact number of daily permits the government would issue.

3 Question 2 a. Apple is a monopolist selling iPhones. The demand for iPhones in the United States is ? ? = 300−5 ? . Apple’s cost function is 𝐶 ( ? )=3.5 ? 2 . How much should Apple charge per iPhone to maximize profits? Inverse demand function is ? = 60 − 1 5 ? , so 𝑀? = 𝜕{?(?)∗?} 𝜕? = 60 − 2 5 ? . Additionally, MC=7Q. Setting MC=MR, 7? = 60 − 2 5 ? 37 5 ? = 60 ? 𝑀 ∗ = 8.1081 Plugging ? 𝑀 ∗ into the inverse demand function yields P=58.38 b. Apple is a monopolist selling iPhones. The demand for iPhones in the United States is ? ? = 5 00− 10 ? . Apple’s cost function is 𝐶 ( ? )=3 ? 2 . What is the deadweight loss from the monopoly? Under perfect competition, assuming that C(Q) represents total cost in the US market, MC will be the equivalent to a market inverse supply. Hence, the optimal market level of production will be at the point of intersection between the inverse demand and supply curves: 50 − 1 10 ? = 6? Therefore, 61 10 ? = 50 or ? 𝑐 ∗ = 8.1967 . If Apple is a monopoly, then it produces where MC=MR, where MR = 𝜕{?(?)∗?} 𝜕? = 50 − 2 10 ? = 50 − 1 5 ? . 𝑀𝐶 = 𝑀? → 50 − 1 5 ? = 6? Hence, ? 𝑀 ∗ = 8.065 and ? 𝑀 ∗ = 50 − 8.065 10 = 49.1935 . Also at this ? 𝑀 ∗ , 𝑀𝐶 = 6(8.065) = 48.39 Therefore, the area of the triangle representing DWL is 1 2 ∗ ? ∗ ℎ = 1 2 (49.1935 − 48.39)(8.1967 − 8.065) = 0.05

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4 c. Apple expands to also selling iPhones in China. The demand for iPhones in the United States is ? 𝑼𝑺 = ??? − ?? . The demand for iPhones in China is ? 𝑪??𝒏𝒂 = ??? − ?? . Assume consumers in one country cannot resell the phones in the other country. Apple’s cost function is 𝑪(?) = ?(? 𝑼𝑺 + ? 𝑪??𝒏𝒂 ) ? . How much should Apple charge for an iPhone in China? The inverse demand functions for iPhones in the US and China are ? ?? = 200 − ? ?? and ? 𝐶 = 200 − 1 2 ? 𝐶 , respectively. Therefore, profit 𝜋 = (200 − ? ?? )? ?? + (200 − 1 2 ? 𝐶 ) ? 𝐶 − 3(? ?? + ? 𝐶 ) 2 . Setting 𝜕𝜋 𝜕? 𝑈𝑆 equal to zero gives 200 − 2? ?? − 6(? ?? + ? 𝐶 ) = 0 100 − ? ?? = 3? ?? + 3? 𝐶 ? ?? = 100 − 3? 𝐶 4 Setting 𝜕𝜋 𝜕? 𝐶 =0 gives 200 − ? 𝐶 = 6(? ?? + ? 𝐶 ) , and hence, ? 𝐶 = 200−6? 𝑈𝑆 7 . Plugging Q US into Q C , ? 𝐶 = 200 − 6 ∗ 100 − 3? 𝐶 4 7 = 800 − 600 + 18? 𝐶 28 28? 𝐶 = 200 + 18? 𝐶 ? 𝐶 ∗ = 200 10 = 20 Since ? 𝐶 = 200 − 1 2 ? 𝐶 , price charged for iPhone in China will be ? 𝐶 = 200 − 1 2 (20) = $190 . d. Apple expands to also selling iPhones in China. The demand for iPhones in the United States is ? 𝑼𝑺 = ??? − ?. ?? . The demand for iPhones in China is ? 𝑪??𝒏𝒂 = ??? − ?. ?? . Apple’s cost function is 𝑪(?) = ?. 𝟕?(? 𝑼𝑺 + ? 𝑪??𝒏𝒂 ) ? . Suddenly, a law is passed that requires Apple to set a single price for the iPhone in both markets. What price will Apple charge? ? ?𝑜?𝑎𝑙 = ? ?? + ? 𝐶 = 600 − 2? ? = 300 − 1 2 ? ?𝑜? 𝑀? = 300 − ? ?𝑜? 𝐶(?) = 0.75(? ?? + ? 𝐶 ) 2 = 0.75? ?𝑜? 2 𝑀𝐶 = 1.5? ?𝑜?

5 Setting MC=MR, 1.5? ?𝑜? = 300 − ? ?𝑜? → ? ?𝑜? ∗ = 300 2.5 = 120 ∴ ? ∗ = 300 − 1 2 (120) = 300 − 60 = $240 Question 3 a. In yellow, player 1 ’ s payoffs are highlighted to represent the best response of player 1 given player 2 ’ s actions. In light blue, player 2 ’ s payoffs are highlighted to represent his best response given player 1 ’ s actions. The overlap of the highlights occurs only at one strategy, (M, L). b. Consider the simultaneous game from (a). Is there any dominant or dominated strategy? From looking at the highest payoffs highlighted above, M is strongly dominant because player 1 ’ s payoffs are always highest and strictly greater than other payoffs when choosing M. Also, regardless of whether player 2 chooses L, C, or R, player 1 always gets the strictly least payoff from choosing B. Hence, B is strongly dominated . Player 2 has no dominant or dominated strategy.

6 c. Backwards induction: • For the 3 rd turn o If previous strategies are (T, T): player 1 chooses T (5>-1) o If previous strategies are (B, B): player 1 chooses B (10>3) • For the 2 nd turn o If previous strategy is T: player 2 chooses B (7>5; comparison with 2 as the payoff for (T,T,B) is out of the question because player 1 will never choose B given (T,T).) o If previous strategy is B: player 2 chooses T (2>0; similar to the above, payoff of 4 from (B,B,T) is not considered.) • For the 1 st turn o Player 1 chooses T (4>2; P1 is faced with the payoffs from two strategies (T,B) for choosing T in the first turn, or (B,T) for choosing B ) Therefore, because player 1 will choose T in the first turn, player 2 will respond by choosing B and the game will end before getting to the third turn.

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7 d. If player 1 chooses L, player 2 ’ s rational choice is to choose the strategy that yields a higher payoff. Therefore, player 2 would only choose X if ? ≥ ? , and her promise would not be credible if d > b .