Problem Set 11 Due Lecture 13 in class on paper

Transcription

1 Microeconomics for Public Policy I Fall Market Demand for Public Goods Problem Set 11 Due Lecture 13 in class on paper Suppose that Masie and Jeff both use the services of national defense. However, Masie is more concerned with the nuclear threat than Jeff. Therefore, Masie s marginal benefit from investments in a nuclear shield is MB m = 100 Q, where Q is the quality of the nuclear shield. Jeff is also worried about nuclear attack, but somewhat less than Masie and values the shield at MB j = 50 Q. The marginal cost of each unit of quality of deterrence is $80. (a) Suppose there were a private market for nuclear deterrence. Would be expect that this market would provide the efficient amount of nuclear deterrence? Why or why not? Note that nuclear deterrence is fundamentally a public good. If Masie purchases it, Jeff can use it this means that it is non-excludable. It is also the case that Masie s consumption of nuclear defense does not impede Jeff s consumption of it it is non-rival. We expect public goods to be under-provided by the private market because of the free rider problem. In this example, it seems likely that Jeff would be satisfied with any amount of nuclear deterrence that Masie purchases because he wants it less. This would lead him to never purchase nuclear deterrence himself. However, Masie will not buy as much nuclear deterrence as both she and Jeff demand. She will purchase what she, privately, demands. (b) Supposing that it could be provided, what is the efficient quantity of nuclear deterrence? Draw a picture that includes Masie s demand curve, Jeff s demand curve, and the market demand curve. To find the efficient quantity of nuclear deterrence, we need to know the total market demand for the public good. We do this by adding up Masie s demand and Jeff s demand. However, remember that for public goods we add willingness to pay, not quantities (as we do for private goods). Therefore, the total market demand is the marginal benefit received by both Masie and Jeff, MB m : MB m = MB m + MB j = (100 Q) + (50 Q) = 150 Q 1

2 Setting this market demand equal to marginal cost, we can solve for the optimal level of provision of nuclear deterrence: See the picture at the end.. GLS Chapter 17, Question Q = = Q Q = 70/ = 35 (a) Graph demand, marginal cost, and external marginal cost. See graph at end. (b) To figure out how many songs the market would provide without considering external costs, set P = MC: P = MC 0.75Q = 1000 Q 1.75Q = 1000 Q 571 We can find the price using either equation. From the marginal cost equation, P = (0.75)(Q) = (0.75)(571) 49. (i) Consumer surplus is the area below the demand curve and above the price line. We can find the area of the triangle as CS = 1 bh = 1 (571)( ) 163, 01. (ii) Producer surplus is the area above the supply curve and below the market price. Here, this is P S = 1 bh = 1 (571)(49 0) 1, 480. (iii) Total surplus is the sum of producer and consumer surplus: T S = P S + CS = 85, 500. (iv) Total damage is the area under the external marginal cost curve as far as the market quantity. We need to know the height of the EMC curve at Q = 571 to do this. Recall that our equation for EM C relates quantity to price. We can therefore write the damage equation as damage = 1 bh = 1 (571)(0.5(571)) 40, 755.

3 (v) Find the total social welfare: P S + CS damage = 85, 500 = 40, 755 = 44, 745. (c) Find the social marginal cost by summing private and external marginal costs: SMC = MC + EMC = 0.75Q + 0.5Q = Q (d) To find the optimal quantity when we take external marginal costs into account, set the social marginal cost equal to the social marginal benefit: Q = 1000 Q Q = 1000 Q = 500 (e) To find the price at this quantity, plug into either the supply or the demand curve: P = 1000 Q = 500. (f) The new consumer surplus is CS = 1 (500)(500) = 15, 000 The new producer surplus is the area between private marginal cost and the price. Notice that this is a rectangle plus a triangle. P S = (500)(500 (0.75)(500)) + 1 (500)(0.75(500)) = 156, 50 Total damage is the area under the external marginal cost curve: damage = 1 (500)((0.5)(500)) = 31, 50 (i) Total surplus decreases: P S new + CS new < P S original + CS original. (ii) Damage declines from around $40k to slightly about $30k. Notice that damage does not go to zero. (iii) The net value for society from the marching band is P S new + CS new damage = 81, 5031, 50 = 50, 000. The net value to society has actually increased, even though less is being produced. 3. Public Goods 3

4 Give an example of a good that has some public good aspects. Please give an example that we have not covered in class and is not in the textbook. Explain what elements of this good are public (in the sense of being a public good) and which are private. A good answer to this question must properly apply the definition of public goods (non-rival and non-excludable) to the example. 4

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