Professor David Popp Solutions to Problem Set #6 Fall 2018

Transcription

1 p. 1 of 6 PAI 723 Professor David Popp Solutions to Problem Set #6 Fall a) The fixed costs are the costs that do not change as the number of attendees changes. This includes the exhibit hall space and speaker fees. The total fixed costs are 11,500 Galleons (= 5, , ,000). If 500 wizards attend, the average fixed costs are 11,500/500 = 23 Galleons. b) Variable costs are the costs that do change as the number of attendees changes. These are all the costs labeled per wizard. Total variable costs per wizard are 45 Galleons. This is the average variable cost. Note that since these are per wizard costs, you do not need to divide by the number of attendees. The average variable cost is the same no matter how many attend. Because it is the same, it is also the marginal cost of an attendee. c) To break even, the price must equal the average total cost. This is equal to the average fixed cost plus the average variable cost. Thus, to break even, charge a price of 68 Galleons. d) There will now be 625 wizards attending the conference. The new average fixed cost is 11,500/625 = Galleons. e) For wizards not attending the reception, the remaining variable costs are the brochures, lunches and snacks. These come to 25 Galleons per wizard. f) In both cases, each group should be charged their average cost of attendance. This is the new average fixed costs plus the appropriate average variable cost. For wizards attending the reception, they should be charged Galleons (= ). For wizards not attending the reception, they should be charged Galleons (= ). This is an example of how price discrimination can help all consumers. By increasing attendance and lowering the average fixed cost, attendees who do attend the reception also benefit from a lower price.

2 p. 2 of 6 2. a) The marginal product of a single leaflet is the additional votes generated by one leaflet. As noted above, this value is Note that this is simply 1/200 we get 1 new vote for each 200 potential voters reached. b) The effectiveness of the local office depends on the number of voters, V. Your candidate earns one additional vote for each 16 voters reached. For example, if there are 160 voters in a community, you will generate 10 additional votes (= 160/16). Thus, the marginal product is the number of voters in a community divided by 16: MMPP DD = VV 16 = VV Note that marginal product is simply the additional output from another worker. It does not include information about costs. Thus, the 250,000 cost of an office doesn t go in this expression. That comes next, in part (c) of the question. c) Costs are minimized when the marginal product per dollar spent on each option is equal. Thus, it will be more cost effective to open an office when the marginal product per dollar spent on door-to-door canvasing is larger. Letting MP L represent the marginal product of leaflets and MP D represent the marginal product of door-to-door canvasing, cost minimization requires: MMPP LL PP LL = MMPP DD PP DD From part (a), we know that the marginal product of leaflets is The price of each leaflet is $1. We can use our expression from part (b) for the marginal product of door-to-door canvasing. The price of each office is $250,000. This gives us: = VV 250,000 It will be more cost-effective to open an office for door-to-door canvasing when the expression on the right is larger. Thus, we solve for V and open an office in each community where the number of voters is higher: VV = (250,000)(0.005) = 20,000 Thus, we should open a local office in any community with more than 20,000 potential voters to reach.

3 p. 3 of 6 3. In a perfectly competitive market, all firms are price takers. Thus, the price equals marginal revenue. To maximize profits, each firm equates marginal revenue to marginal cost. Thus, we have P = MR = MC. Now, consider the demand curve of consumers. The demand curve tells us the value that people place on the good that is, the marginal benefit of the good. Since consumers pay the price, P, for the good, it must be true that the value of the last good purchased is P. Thus, MB = P. Putting the behavior of firms and consumers together, note that the price in perfect competition is a mechanism that equates the marginal benefits that consumers get from the product to the marginal cost of producing the good. That is, in equilibrium, MB = MC. There is no way to change output and make welfare higher. If we increased output, the marginal cost would be greater than the marginal benefit, and welfare would fall. Similarly, if we decreased output, the marginal benefit would be greater than marginal cost. Total welfare would fall because of lost opportunities. b) Monopolies are a market failure because the price of the good sold is greater than marginal cost. This occurs because P > MR for a monopolist. Thus, P > MC for a monopolist. As a result, we get MB = P > MR = MC, so that MB > MC. There is too little of the good produced by the monopolist. 4. a) At a price of $700, the quantity sold is 15. Total profits are $9,375. At this price, marginal revenue is greater than marginal cost. If the firm sold another good, total costs would increase by the marginal cost, which is $150. Revenues would increase by the marginal revenue, which is $400. Since the added revenue is greater than the added cost, profits will increase. Thus, the firm cannot be maximizing profits now. b) If the firm sells another unit, its marginal revenue will fall. The firm will need to lower the price that it sells each good at. Thus, it will receive less than before for goods that it was already selling. The firm gets the price for the additional good sold, but it loses some money on what it was already selling. As a result, MR is always less than the price. Marginal costs are likely to increase as it produces more, because of diminishing returns to the inputs. Indeed, when price is lowered to $680, the quantity increased to 16. Profits increased to $9,600. Marginal revenue is still greater than marginal cost, but they are closer. c) At a price of $500, the quantity sold is 25. Now, marginal cost is greater than marginal revenue. If we produce one less unit, the cost savings will be greater than the lost revenue. The firm has now set its price too low, and is selling too much. The firm should raise its price and sell less. d) Profits are maximized when marginal revenue and marginal cost are equal. At this point, the additional revenue brought in by selling another good would be exactly offset by the additional cost. For this firm, marginal costs and marginal revenue are equal at a price of $ units are sold at this price, and total profits are $10,000.

4 p. 4 of 6 5. a) Profits are maximized where MR=MC. If the city acts as a monopolist, the marginal revenue curve will bisect the demand curve. Thus, MR = 400 6Q. MR = 400 6Q = 40 = MC 360 = 6Q Q = 360/6 = 60 To get the price, we need to look at the demand curve, to see how much citizens are willing to pay for 60 MWh of electricity. We get: P = 400 3(60) P = $220 The graph for this market is shown below. Note that the equilibrium quantity is found where MC=MR, and the price is found from the demand curve. The price charged is how much consumers are willing to pay for 60 MWh of electricity. P MC 40 D 60 Q MR b) To calculate the profit, note that we need to consider the fixed cost. Thus, profit is not just the producer surplus from the graph. Rather, we must calculate profit as total revenue minus total cost. Total costs include the per unit costs (= 60 MWh at $40/per MWh) plus the fixed costs: profit = TR TC profit = PxQ TC profit = (220)(60) 40(60) 5000 profit = $5,800

5 p. 5 of 6 c) Consumer surplus is the area above the price and below demand. It is equal to areas A and B. Although you didn t need to calculate the area, for those who are curious, it is a triangle with base of 60, and a height of 180 (= ). Thus, consumer surplus = 0.5(60)(180) = $5,400. The deadweight loss is area E on the graph. Again, for those who are curious, this area is a triangle with base of 60 (= ) and a height of 180 (= ). Thus, the deadweight loss = 0.5(60)(180) = $5,800. Note that you need the quantity sold in perfect competition, which we will find in part (d), to calculate this area. P 400 A B 220 C D E 40 MC D Q MR d) To completely eliminate this deadweight loss, the city should set the price equal to marginal cost. We find the new quantity by equating marginal cost and demand: 400 3Q= = 3Q Q = 120 P = 40 Unfortunately, at this price, the city will lose money, because of the fixed costs: profit = TR TC profit = PxQ TC profit = (40)(120) 54(120) 5000 profit = -$5,000

6 p. 6 of 6 e) The problem with marginal cost pricing is that it does not cover the fixed costs of providing electricity in Mount Washington. One solution would be average cost pricing, where the price equals the average cost of each MWh of electricity provided. This way, the price covers both the marginal cost of $40 per MWh and each person s share of the fixed costs, allowing the city to break even. While there is some deadweight loss associated with average cost pricing, the deadweight loss will be less than with monopoly pricing. Another possibility is to consider price discrimination. For example, low income users could be charged the marginal cost of electricity generation, and higher income users could be charged a higher price. This second price simply needs to be high enough so that it covers the fixed costs as well as the marginal costs of the municipal power plant. Another price discrimination idea suggested by several students is to vary the price by time of day. Users could be charged more to use electricity during peak demand periods, such as hot summer days. Not only does this pricing scheme bring in more money when demand is high, it also discourages households from using electricity when the power plant is under stress from heavy demand. Finally, tiered pricing is another option. Each MWh could be priced at marginal cost, but users could also be charged an annual connection fee. The connection fee could be set based on the number of households to cover the fixed costs of the municipal power plant.