Economics 141B Problem Set #3 Spring 2024

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Economics 141B

Problem Set #3

Spring 2024

1.  Consider two homogeneous communities, one of which contains rich consumers (type 1), who are high demanders of the public good z, while the other contains poor consumers (type 2), who are low demanders. The demand curves for the types are given by D1  = 6 – z and D2  = 4 – z. The public good is financed by a property tax in each community, but since the communities  are homogeneous, the property tax is equivalent to a head tax.  The level of z is decided by majority vote. Suppose that c/n, the cost per unit of z on a per capita basis, equals 3 in each community.

a) Find the public good level in each community, and illustrate the levels in a diagram. Show the surplus levels for both types of consumers.

Now consider what happens if one poor consumer moves into the rich community. Because the poor consumer buys a small house, his cost per unit of z is lower than 3. Using some algebra, it can be shown that the cost per unit of z for the poor consumer equals

3H2/H1,

where H2  gives the size of a poor consumer’s house and H1  gives the size of a rich consumer’s house. Since H2 < H1, this cost is smaller than 3. By contrast, the cost per unit of z for a rich consumer remains at 3 (this requires that just one poor person moves into the community).

b)  Suppose that H2/H1 = 1/3, so that a poor consumer’s house is only one-third as large as a rich consumer’s house. Using the above formula, compute the cost per unit of z for the poor household. Remembering that majority rule prevails,  compute the change in surplus for the poor consumer if he moves into the rich community. Illustrate your calculation in a diagram. Based on your answer, will the poor consumer relocate?

c)  Repeat part (b) under the assumption that H2/H1 = 2/3.

d)  Repeat part (b) under the assumption that H2/H1 = 5/6.

e)  Provide an intuitive explanation of your results.

2.  Suppose the marginal damage and marginal benefit curves in a polluted neighborhood are MD = P/3 and MB = 4 – P.  Also, suppose that  transactions costs are low, so that the consumers and the firm can bargain. We saw that in this case, the socially-optimal level of pollution is achieved.  Start by computing the socially-optimal P. Then, for each of the following cases, compute the amount of money transferred through the bargaining process, and indicate who pays whom (i.e., whether consumers pay the firm, or vice versa). Also, compute the gains to each party relative to the status quo (i.e., the starting point of the bargaining process).

a)   Consumers have the right to clean air; firm is dominant in the bargaining process.

b)  Consumers have the right to clean air; consumers are dominant in the bargaining process.

c)  Firm has the right to pollute; firm is dominant in the bargaining process.

d)  Firm has the right to pollute; consumers are dominant in the bargaining process.

3.   Suppose there  are two polluting factories, surrounded by identical residential neighborhoods. The marginal damage curves are identical for the two neighborhoods, and they are given by MD1  = P and MD2  = P, where P is the level of pollution.   The marginal benefit curves for the factories, however, are different.  The marginal benefit curve for the first factory is MB1  = 8 – P, while the curve for the second factory, which uses a cleaner production process, is given by MB2  = 4 – P (both curves become zero once they hit they reach the horizontal axis).

a)     Illustrate  the  curves  for  the  two  neighborhoods  in  two  diagrams,  and  identify  the pollution levels chosen by the firms in the absence of government intervention.  Find the level of social surplus achieved in this case, which equals the sum of surpluses in the two neighborhoods.

b)     Find  the   socially-optimal  pollution  levels  in  the  two  neighborhoods. Explain intuitively why they are different.  Compute social surplus in each neighborhood, and sum the values across the two neighborhoods to get total surplus.  This is the surplus level that would result from imposition of separate pollution standards in the two neighborhoods.

c)     Suppose the government institutes a common pollution standard, which applies to both neighborhoods.  This standard restricts pollution from any factory to a maximum value of 3 units.  Under this standard, how much does each factory pollute?  Recall that each factory  will  want  to  pollute  as  much  as  possible  without  exceeding  the  pollution standard.  Compute the resulting level of social surplus in each neighborhood, and add the values.

d)     Considering the social surplus from parts (a), (b) and (c), comment on the wisdom of using a common pollution  standard. How does the standard compare to separate pollution standards, and to the case where the government does not intervene at all?

4.  Suppose that pollution in a neighborhood comes from two factories, with marginal benefit curves given by MB1 = 12 – P1  and MB2  = 8 – P2.  The level of pollution in the neighborhood is given by P = P1  + P2.  The government wants to limit pollution by instituting a pollution-rights market. The government’s desired level of P is 10, so it prints 10 pollution rights and offers them for sale to the firms.

a)     Find the amount of pollution that each firm wants to generate as a function of the price s of pollution rights.  Then find the equilibrium value of s (which yields total pollution of 10), as well as the equilibrium pollution levels for the two factories.

b)    Repeat part (a) for the case where the government’s desired level of pollution equals 14.

c)     Comment on the usefulness of a pollution rights market in achieving efficient levels of pollution abatement.

5.  Suppose that the N rich residents of a jurisdiction care about the M poor people who live there. Their utility  function is given by x + ln(Y + T), where x  is the consumption level  for  a rich household and Y + T is the income of each poor household in the jurisdiction.   Y gives the poor household’s own income (which is low) and T is the welfare payment per household.  Taking the derivative of this utility function, the marginal benefit to a rich resident of an extra dollar of income for the poor is equal to  1/(Y + T).   This  formula gives the “demand curve”  for T, a downward sloping function of T.   Since the per capita cost to the rich of making a welfare payment of T to each poor household is MT/N, the per capita cost of increasing T by a dollar is M/N.

a)  As in the case of other public goods, the welfare payment chosen through the voting process lies at the intersection of the median voter’s demand curve and per capita cost line.  In this case, the rich are all identical, having the same demand curve, so everyone agrees on the best T (in other words, there are no non-median voters).  Using the above information, write down the equation that determines the chosen T.

b)  Rearrange this equation to express T as a function of N, M, and Y.

c)   Suppose  Y  increases.    Does  T  rise  or  fall?    Give  an  intuitive  explanation  for  your answer.

d)  Suppose N  increases.   Does  T rise  or  fall?    Give  an  intuitive  explanation  for  your answer.

e)   Suppose  M  increases.   Does  T rise  or  fall?   Give  an  intuitive  explanation  for your answer.