21
POLITICAL CHAPTER ECONOMICS
The problems in this final chapter are of two general types. First are four problems in traditional welfare economics (Problems 21.121.3 and 21.5) that illustrate various issues that arise in comparing utility among individuals. These are rather similar to the problems in Chapter 12. The other six problems in the chapter concern public choice theory. Comments on Problems 21.1 A problem utilizing two very simple utility functions to show how none of several differing welfare criteria seems necessarily superior to all the others. This clearly illustrates the basic dilemma of traditional welfare economics. This problem examines the Scitovsky bribe criterion for judging welfare improvements. Although the criterion as a general principle is not widely accepted, the notion of bribes in public policy discussions is still quite prevalent (for example, in connection with trade adjustment policies). Shows how to integrate production into the utility possibility frontier construction. In the example given here, the frontiers are concentric ellipses so the Pareto criterion suggests choosing the one that is furthest from the origin. The choice is, however, ambiguous if the frontiers intersect. Illustrates the irrelevant alternative assumption in the Arrow theorem. A further examination of welfare criteria that focuses on Rawls uncertainty issues. Shows that the results derived from a Rawls initial position depend crucially on the strategies individuals adopt in risky situations. Further examination of the Arrow theorem and of how contradictions can arise in fairly simple situations. A simple problem focusing on an individuals choice for the parameters of an unemployment insurance policy. The problem would need to be generalized to provide testable implications about voting (see the Persson and Tabellini reference). A problem in rent seeking. The main point is to differentiate between the allocational harm of monopoly itself and the transfer nature of rent-seeking expenditures. A discussion question concerning voter participation.
21.2
21.3
21.4 21.5
21.6 21.7
21.8 21.9
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21.10 An alternative specification for probabilistic voting that also yields desirable normative consequences.
Solutions
21.1
200 pounds
U1=
f1
U2=
1 2
f2
a. 100 pounds each b. f1= f1 = 40 c. U 1 + U 2 = 1 2 f2 f 1=
U1 = 10, U2 = 5 1 f 42
f2 = 160 f1+ 1 200 f 1 2
1 0.5 1 0.5 f 1 = (200 f 1) 2 4 f1 = 160, f2 = 40 d. U 2 5 , best choice is U2 = 5 f2 = 100, f1 = 100. e.
0.5 W = U 1 U 0.5 = f 1 2
11 f2= f 1 (200 f 1 ) 2 2
W 11 1 1 3/ 4 3/ 4 = f 1 [ (200 f 1 ) ] + f 1 (200 f 1 ) = 0 f1 4 2 24
3/ 4 3/ 4 f 1 (200 f 1 ) = f 1 (200 f 1 )
f1 = 200 f1 21.2
f1 = 100, f2 = 100.
If compensation is not actually made, the bribe criterion amounts to assuming that total dollars and total utility are commensurable across individuals. As an example, consider: Income in State A Individual 1 Individual 2 100,000 5,000 Income in State B 110,000 0
State B is superior to State A in that Individual 1 could bribe Individual 2. But, in the absence of compensation actually being made, it is hard to argue that State B is better.
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Chapter 21/Political Economics y 116
21.3
Pareto efficiency requires MRS1 = MRS2 y1 x1 = y2 x2 = y y1 . x x1 x1 = x x 2 = (1 ) x .
Hence, all efficient allocations have y1 = y
y 2 = (1 ) y
2 2 U 1 = xy
2 2 U 2 = (1 ) xy
2 (U 1 + U 2 ) 2 = U 1 + 2U 1 U 2 + U 2 = 2 xy + 2( )(1 ) xy + (1 ) 2 xy = xy 2
a. If y = 10, x = 160 Utility Frontier (U1 is + U2)2 = 1600. b. y = 30 x = 120 y = 45
(U1 + U2)2 = 3600 c. Maximize XY subject to X + 2Y = 180 yields x = 90 (U1 + U2)2 = 4050. d. In this problem, the utility possibility frontiers do not intersect, so there is no ambiguity in using the Pareto criterion. If they did intersect, however, one would want to use an outer envelope of the frontiers. 21.4 7 individuals with states A, B, C. Votes are A 3 A 3 B 2 B 4 C 2
If C is not available, let both C votes go to B.
This example is quite reasonable: it implies that Arrows axiom is rather restrictive. 21.5 a. D b. E, E(U) = .5(30) + .5(84) = 57 c. E(U) = .6(L) + .4(H) EUA = 50, EUB = 52, EUC = 48.6, EUD = 51.5, EUE = 50.
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So choose B. d. max E(U) | U 1 U 2 | values: A: 50 0 = 50 So choose A. e. It shows that a variety of different choices might be made depending on the criteria being used. 21.6 Suppose preferences are as follows: Individual 1 C Preference A B 2 A B C 3 B C A C: 49.5 9 = 40.5 E: 57 54 = 3 B: 55 30 = 26 D: 51.75 2.5 = 49.26
a. Under majority rule, APB (where P means is socially preferred to), BPC, but CPA. Hence, the transivity axiom is violated. b. Suppose Individual 3 is very averse to A and reaches an agreement with Individual 1 to vote for C over B if Individual 1 will vote for B over A. Now, majority rule results in CPA, CPB, and BPA. The final preference violates the nondictatorship assumption since B is preferred to A only by Individual 3. c. With point voting, each option would get six votes, so AIBIC. But that result can be easily overturned by introducing an irrelevant alternative (D). 21.7 a. So long as this utility function exhibits diminishing marginal utility of income, this person will opt for parameters that yield y1 = y2. Here that requires w(1 t) = b. Inserting this into the governmental budget constraint produces uw(1 t) = tw(1 u) which requires u = t. b. A change in u will change the tax rate by an identical amount. c. The solutions in parts a and b are independent of the risk aversion parameter, . 21.8 a. Since p = q/100 + 2, MR = q/50 + 2 MR = MC when q = 75, p = 1.25, = 56.25. The firm would be willing to pay up to this amount to obtain the concession (assuming that competitive results would otherwise obtain). b. The bribes are a transfer, not a welfare cost.
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Chapter 21/Political Economics y 118
c. The welfare loss is the deadweight loss from monopolization of this market, which here amounts to 28.125. 21.9 An essay on this topic would stress that free riding may be a major problem in elections where voters perceive that the marginal gain from voting may be quite small. If such voters are systematically different from other voters, candidates will recognize this fact and tailor their platforms to those who vote rather than to the entire electorate. The effect would be ameliorated by the extent to which platforms can affect voter participation itself.
21.10 Candidate 1s problem is to chose 1, to maximize
i = f i (U i ( 1i ) / U i ( 2i ) subject to
i =1 i =1
n
n
i =1
n
1i
=0.
' ' The first order conditions for a maximum are f i U i / U i ( * i ) = for all i = 1 . . .n . 2 ' ' Assuming f i is the same for all individuals, this yields U i / U i ( * i ) = k for i = 1 . . .n . 2
In words, the candidate should equate the ratio of the marginal utilities of any two voters (U i' / U 'j ) to the ratio of their total utilities (U i / U j ) . Since each candidate follows this strategy, they will adopt the strategies that would maximize the Nash Function, SWF.
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