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Problems 13 1. π΄πΆ = ππΆ = 5 π = 53 β π β π = 53 β π = π΄π a. π, π, π monopoly =? ππ = π β π = (53 β π)π = 53π β π 2 πππ ππ = = 53 β 2π ππ Quantity supplied in monopoly market to maximize profit requires: ππ = ππΆ 53 β 2π = 5 β πΈ = ππ Substituting the quantity (Q) into price (P), we have: π = 53 β π = 53 β 24 = 29 Then the monopolistβs profit becomes: π = ππ β ππΆ = π β π β π΄πΆ β π = (π β π΄πΆ)π = (29 β 5)24 = 576 b. π, π competitive market =? In a perfect competitive market, we have the following relation: π = ππΆ, thus π· = π and 53 β π = 5 β πΈ = ππ c. P 53 Note: ο§ Area 1: Consumer Surplus in Monopoly Market ο§ Area 2: Monopolistβs Profit ο§ Area 3: Deadweight Loss 1 Eq. in Monopoly Market 29 D = AR 2 3 Eq. in Perfectly Competitive Market 5 AC = MC 24 26.5 48 53 Q Consumer Surplus in a Monopoly Market: πΆππππ = π΄πππ 1 = 1 × 24 × 24 = 288 2 Monopolistβs Profit: ππππ = π΄πππ 2 = 24 × 24 = 576 Deadweight Loss: π·ππΏ = π΄πππ 3 = 1 × 24 × 24 = 288 2 Consumer Surplus in a Perfectly Competitive Market: πΆπππΆπ = π΄πππ 1 + π΄πππ 2 + π΄πππ 3 = 288 + 576 + 288 = 1152 From the calculation above, we can see that the sum of consumer surplus in a monopoly market and the monopolistβs profit is less than the amount of consumer surplus in a perfectly competitive market. 2. π = 70 β π β π = 70 β π = π΄π ππ = π β π = (70 β π)π = 70π β π 2 πππ ππ = = 70 β 2π ππ a. π΄πΆ = ππΆ = 6 Quantity supplied in monopoly market to maximize profit requires: ππ = ππΆ 70 β 2π = 6 β πΈ = ππ Substituting the quantity (Q) into price (P), we have: π = 70 β π = 70 β 32 = 38 Then the monopolistβs profit becomes: π = ππ β ππΆ = π β π β π΄πΆ β π = (π β π΄πΆ)π = (38 β 6)32 = 1024 b. πΆ(π) = 0.25π 2 β 5π + 300 ππΆ(π) ππΆ = = 0.5π β 5 ππ Quantity supplied in monopoly market to maximize profit requires: ππ = ππΆ 70 β 2π = 0.5π β 5 75 = 2.5π β πΈ = ππ Substituting the quantity (Q) into price (P), we have: π = 70 β π = 70 β 30 = 40 Then the monopolistβs profit becomes: π = ππ β πΆ(π) = ππ β πΆ(π) = 40π β (0.25π 2 β 5π + 300) = 1200 β 225 + 150 β 300 = 825 πΆ(π) = 0.0133π 3 β 5π + 250 ππΆ(π) ππΆ = = 0.0399π 2 β 5 β 0.04π 2 β 5 ππ Quantity supplied in monopoly market to maximize profit requires: ππ = ππΆ 70 β 2π = 0.04π2 β 5 0.04π 2 + 2π β 75 = 0 π 2 + 50π β 1875 = 0 (π + 75)(π β 25) = 0 π1 = β75 πππ π2 = 25 (the negative value of Q is not plausible) Substituting the positive value of Q into price (P), we have: π = 70 β 25 = 70 β 25 = 45 Then the monopolistβs profit becomes: π = ππ β πΆ(π) = ππ β πΆ(π) = 45π β (0.0133π 3 β 5π + 250) = 1125 β 207.8125 + 125 β 250 = 792.1875 c. d. P MCc 45 40 38 MCb c b a 6 10 25 30 32 MR MCa D = AR Q 9. a. The undesirable effect of a monopoly is a higher price and a lower quantity of goods if it is compared to a price competitive market. To overcome this situation, the government tries to reduce the price and increase the quantity supplied in the monopoly market. A lump-sum subsidy will not make the governmentβs goal achieved due to the following reasons: The total cost of a firm is given by ππΆ = πΉπΆ + π΄πΆ(π), and its Marginal Cost is given by ππΆ = b. πππΆ ππ = π΄πΆ β² (π) If the government gives a lump-sum subsidy to the firm with an amount of S, then its total cost becomes ππΆβ² = πΉπΆ + π΄πΆ(π) β π Therefore, its Marginal Cost function is given by πππΆ ππΆβ² = = π΄πΆ β² (π) ππ We can see that MCβ = MC. Thus, when the government gives a lump-sum subsidy into a firm, it would not change the Marginal Cost function of the firm. As a result, the equilibrium price and quantity in the monopoly market would not change because to find the optimal quantity supplied in the monopoly market, we need to set MR=MC. If the government give a per-unit-of-output subsidy to the firm, then the total cost of the firm becomes ππΆβ² = πΉπΆ + π΄πΆ(π) β π(π) Therefore, its Marginal Cost function is given by πππΆ ππΆ β² = = π΄πΆ β² (π) β π β² (π) ππ We can see that the Marginal Cost of the firm after subsidy (MCβ) is smaller than its initial Marginal Cost (MC). The graphical proof is shown in the following picture: P MC MCβ b c a MR c. D=AR Q In a perfectly competitive market, the equilibrium occurs at point a, in the intersection of supply curve (MC) and demand curve (AR). In a monopoly market, equilibrium occurs at point b, where MC=MR. Consequently, the price is higher and the quantity supplied in the market decreases. A per-unit-of-output subsidy by the government will reduce the total cost of the firm and shift the marginal cost curve down. As a result, a new equilibrium with lower price and higher quantity than the condition in monopoly market is created. In a perfectly competitive market, the equilibrium occurs at P=MC. In a monopoly market, maximum profit is achieved when MR=MC. To achieve both condition, the government should allocate a per-unit subsidy t such that MR=MC-t. Therefore, we have the following relation: 1 ππ = ππΆ β π‘ = π (1 + ) ππ,π Since P=MC, then 1 π β π‘ = π (1 + ) ππ,π Divide both side by P, we have: π‘ 1 1β =1+ π ππ,π Thus, we have: π π =β π· ππΈ,π· A monopoly creates inefficiency in the market; as a result, the required condition P=MC is not achieved, because the price in a monopoly market is higher than MC. Consequently, a subsidy with an amount that fulfilled the condition above is required to overcome the gap between P and MC. Problems 20 P 1. a. b. 3. π = $20, ππΆ = 0.4 π In a perfectly competitive market, the condition for optimal production is: π = ππΆ β 20 = 0.4 π β π = ππ Thus, the optimal production of output for the firm is 50 widgets. π = $20, πππΆ = 0.5 π 20 The socially optimal condition requires: Tax = $4 16 π = πππΆ β 20 = 0.5 π β π = ππ Thus, the socially optimal production of output for the firm is 40 widgets. At π = 40, the marginal cost of the firm is ππΆ = 0.4 π = 0.4 β 40 = $16 Therefore, the government should impose a tax with an amount of: π‘ = π β ππΆ = $20 β $16 = $4 c. P = $10/barrel MC = AC = 1000 π = 500π β π2 π π = = 500 β π π SMC =0.5 q MC =0.4 q 40 50 q a. ππ = π β π = 10(500 β π) = 5000 β 10π Equilibrium condition requires: ππ = ππΆ 5000 β 10π = 1000 10π = 4000 β π = 400 The total output of the well is: π = 500π β π2 = 500(400) β 4002 = 40,000 Marginal Productivity of each well is: ππ π (500 β π) = β1 = ππ ππ Q 62,500 Diminishing Marginal Productivity of Q shows the existence of externalities 0 b. c. 7. ππ ππ a. b. c. 250 500 n We can see that the marginal productivity of each well diminish if there is additional wells, therefore, we can conclude that there is an externality in the production of the oil, because additional wells reduces the output of each well. From the graph above, we can also observe that the total output also experiences diminishing marginal productivity. Total Revenue from all wells is: ππ = ππ = 10(500π β π2 ) = 5000π β 10π2 πππ ππ = = 5000 β 20π ππ An optimal output requires ππ = ππΆ, hence 5000 β 20π = 1000 β π = πππ Then, the total output is: π = 500π β π2 = 500(200) β (200)2 = 60,000 And, the output per well is: π = 500 β π = 500 β 200 = 300 The optimal number of well is 200 wells, therefore, the government should limit the number of well to 200 wells. The marginal cost of each well is 1000. The total revenue of each well when n = 200 is: ππ = 5000 β 10π = 5000 β 10(200) = 3000 Therefore, the license fee for the firm is: π = π¨πΉ β π΄πͺ = ππππ β ππππ = ππππ = 100 β π β π = 100 β ππ = 200 β π β π = 200 β ππ ππΆ = $120 Total Willingness to Pay of the society is: πππ = ππ + ππ = 300 β 2π The optimal condition occurs when TWP=MC 300 β 2π = 120 π β= 90 In a private market we have P = MC Demand of mosquito control for individual a: 120 = 100 β ππ β ππ = β20 (thus, demand of individual a for mosquito control is 0) Demand of mosquito control for individual b: 120 = 200 β ππ β ππ = 80 Total demand of the society is 80, which is less than the optimal demand when mosquito control is treated as public goods. The cost for producing mosquito control: ππΆ = π β π = $120 β 90 = $10,800 Willingness to pay of individual a is: π = 100 β π β= 100 β 90 = 10. Hence, the proportion of individual a is $10 β 90 = $900 Willingness to pay of individual b is: π = 200 β π β= 200 β 90 = 110. Hence, the proportion of individual b is $110 β 90 = $9900