1. Public Choice Theory-Direct Democracy
ECON 453
Public Choice Theory-Direct Democracy
E. Nketiah-Amponsah
Department of Economics
Room W.18
12/7/2018

2. Learning Outcomes Appraise the raison d'être for voting
Explain the unanimity rule (Lindahl equilibrium) of direct democracy
Explain the majority voting rule of direct democracy
Explain the concept of cycling and voting paradox
Explain single and double-peaked preferences and how double-peaked preferences lead to a voting paradox
Distinguish between logrolling and cycling
Do a critique of Arrow’s Impossibility Theorem
12/7/2018
Dr. E. Nketiah-Amponsah

3. Introduction
The theory of public choice studies how decisions to allocate resources and redistribute income are made through a nation’s political system.
It enables us to examine how economic choices are made in the public sector by government
Public choice theory is the study of the processes through which government policies are determined and implemented
The field of public choice examines how a political process is used to determine the quantity of goods and services supplied by governments.
The supply of a public good through political institutions requires agreements on the quantity of the public good and the means of finance.

4. Public Choice Theory-What is it ?
Mueller’s (2003) definition of public choice: “Public choice can be defined as the economic study of nonmarket decision- making, or simply the application of economics to political science”.
This definition suggests that public choice is an inherently
interdisciplinary field
The subject matter of public choice is the
theory of the state
voting rules
voter behavior
party politics
bureaucracy.
The methodology of public choice is that of economics; i.e., individuals are rational utility maximizers.

5. Modus Operandi of Collective
Decision Making
Public choices are made formally through elections in
which each individual is usually allowed one vote.
Voting is a technique that allows individuals to reveal their preferences for public goods or collective action. It Constitutes an alternative for determining the level of public expenditure.
A rational person’s most preferred political outcome is the quantity of the government supplied good corresponding to the point at which the person’s tax share is exactly equal to the marginal benefits of the good.
This level of output of the good provides the maximum
possible satisfaction to the person.

6. Tax Share vs Marginal BenefitZ ti
Tax per unit
of output
MBi Q*
Output per year

7. The Rational Voter Model
A person’s decision to vote depends on the cost and benefits of doing so, as well as on the probability that voting will help achieve the anticipated benefits
Why do people vote?
According to Downs (1957) it could not be considered rational to vote yet many people do (voting paradox).
The decision to vote is assumed to reflect the utility from doing so. However people vote even when the expected utility from voting is apparently negative – the paradox of voting.
The rational voter model is as follows:
12/7/2018

8. The Rational Voter Theorem
Given that there is a policy that a voter would like to see implemented, X
Each voter therefore calculates the value of that policy being implemented which yields a utility, U(X)
The voter also calculates the probability, P that their vote is decisive and will make a difference to the outcome of the election, where 0≤P≤1.
The value of P reflects the size of the electorate and how closely contested the election will be.
Assuming all voters are rational then P will be approximately:
12/7/2018

9. Value of P N – the size of the electorate
q – voters’ expectation of the proportion of votes each party/policy is likely to receive
P will be largest when q=1/2 (when election is close) and N is small.
12/7/2018

10. Expected Utility of Voting
Cost of voting will include:
Time spent considering various alternatives and their likely impact on the voter
Opportunity cost of the time spent in voting
Other associated costs
Thus the expected utility of voting combines the gains from voting and the cost of voting
The Expected Utility: E(U) = P[U(X)] - C
12/7/2018

11. Rational Voter Model
No single individual is likely to influence the outcome and thus P will be very small.
Given that the cost of voting is positive and relatively big, E(U) < 0.
Thus, it is irrational to vote.
This analysis is particularly applicable to countries with majority voting. It can however be rational to cast a vote for a minority party under some system of proportional representation since there is a greater chance that your party/policy will have a representative elected than under majority voting system
12/7/2018

12. Explanations for the Paradox
Minimax strategy: Since decision to vote is made under uncertainty individuals may vote to minimize regret instead of voting to maximize their expected utility.
Imagine two states S1 where the vote has no impact and S2 where the vote has impact. The voter examines the value of regret associated with the two states. In S1, the outcome is independent of the voter’s strategy, therefore a vote results in regret C (the cost of voting) and there is no regret associated with voting S1 S2
Vote C
Abstain
U-C
However, if in the second state, S2 the vote is decisive, then there is no regret associated with voting but if the voter abstains, there will be a regret equal to the utility one could have attained less the cost of voting, U-C
12/7/2018

13. Explanations for the Paradox
Ethical Voter theorem:
Voting behaviour is a consequence of altruism
The voter is not interested in the impact of the vote on his/her utility but also the utility of others – altruism
This theorem explains why a wealthy person might vote for a party which will increase the tax rate to increase transfers to the poor.
Individuals derive utility from voting:
Voting seen as a right or a civic duty
Vote to preserve democracy
Utility from expressing your political preference
Voting as a social activity
Voting based on Ideologies
12/7/2018

14. Explanations for the Paradox
Considering above reasons it seems that voting yields a positive utility which increases the likelihood of E(U) >0, making voting a rational activity
12/7/2018

15. Democracy as a Conduit for Resource Allocation
Democracy could be viewed twofold
Direct democracy
Representative democracy
Direct democracy refers to where social choices are determined directly by citizens using a particular voting procedure.
12/7/2018
Dr. E. Nketiah-Amponsah

16. Direct Democracy
Under direct democracy various voting procedures are used to decide on public expenditure.
These include:
Unanimity Rule
Majority voting Rule
Simple majority
Qualified majority
logrolling
12/7/2018
Dr. E. Nketiah-Amponsah

17. Unanimity Rule
This is a voting procedure designed to elicit unanimous agreement for a particular public expenditure
It was designed by an economist called Lindahl in the early 20th century.
It is therefore referred to as the Lindahl procedure which involves three sets of assumptions
Determining the total amount of public expenditure and taxes
Determining the allocation of total public expenditure among goods and services providing for the satisfaction of various social wants
Determining the allocation of total taxes among various individuals
12/7/2018
Dr. E. Nketiah-Amponsah

18. Unanimity Rule
Lindhal contends that this approach complies with both the fundamental principles of public finance: benefits principle and ability-to-pay.
Lets see how the Lindahl procedure works
Lets assume we have only two individuals (Adam and Eve) and the government wants to seek a unanimous decision on the number of street lights it should construct.
We also assume Adam and Eve share the cost of provision(in the form of tax)
Let Sa denote Adam’s share of the cost of provision
12/7/2018
Dr. E. Nketiah-Amponsah

19. Unanimity Rule
For any particular value of Sa, Adam demands some quantity of street lights. As his tax shares increase and street lights become more expensive to him, he demands a smaller quantity.
Figure 7.1 plots the relationship between Adam's share of cost and quantity demanded.
Tax share is measure on the vertical axis while the horizontal axis measures the quantity demanded.
12/7/2018
Dr. E. Nketiah-Amponsah

20. Unanimity Rule
Figure 7.1
12/7/2018
Dr. E. Nketiah-Amponsah

21. Unanimity Rule
Adam’s share is measured by the vertical distance from point 0.
The Da curve in the figure shows how the quantity of street lights demanded by Adam decrease as his tax share increases.
In the same way if we denote the tax share of Eve as Se, then when Se goes up the quantity demanded by eve decreases.
Note that Sa +Se =1
12/7/2018
Dr. E. Nketiah-Amponsah

22. Unanimity Rule
In the figure, Eve’s tax share increases as we move down along the vertical axis from 0’.
Thus the vertical distance 00’ is 1
Eve’s demand schedule is denoted De. It slopes upward because upward movements along the vertical axis represent a lower price to her.
The tax shares are referred to as Lindahl prices
12/7/2018
Dr. E. Nketiah-Amponsah

23. Unanimity Rule
An equilibrium is a set of Lindahl prices that at those prices each person votes for the same quantity of the public good.
This occurs when Adam’s tax share is 0S* and Eve’s tax share is 0’S*.
At these prices both parties agree that r* should be provided.
The question one may ask: how feasible is the unanimity rule? Discuss
12/7/2018
Dr. E. Nketiah-Amponsah

24. Unanimity Rule Limitations
The analysis assumed that people vote sincerely. However there could be strategic behavior by the parties and as a result equilibrium will never be achieved.
It may take a lot of time to find the mutually agreeable tax shares. In most cases, many people are involved and therefore getting everyone’s consent may involve very high decision making costs.
12/7/2018
Dr. E. Nketiah-Amponsah

25. Majority Voting Rule
As shown unanimity is difficult to attain. Therefore we need a voting system not requiring unanimity.
One such system is the majority voting rule.
In the case of the majority voting rule more than half of the voters must favor a measure for it to be approved.
12/7/2018
Dr. E. Nketiah-Amponsah

26. Majority Voting Rule
Lets look at some aspects of the majority voting rule.
Lets consider a community with three voters, Cosmo, Elaine and George who have to choose among three levels (A, B and C) of public good provision.
Level A is small budget, Level B is moderate budget, and Level C is large budget.
12/7/2018
Dr. E. Nketiah-Amponsah

27. Majority Voting Rule
The voter preferences of these individuals are depicted in Table 7.1.
Table 7.1.
12/7/2018
Dr. E. Nketiah-Amponsah

28. 12/7/2018
Dr. E. Nketiah-Amponsah

29. Majority Voting Rule
Suppose an election were held on whether to adopt A or B. B will win by 2 to 1 because Elaine and George will vote for B but Cosmo will vote A.
Similarly if an election were held between B and C, still B wins by a vote of 2 to 1.
Level B wins any election against its opposition and thus is the option selected by majority rule
12/7/2018
Dr. E. Nketiah-Amponsah

30. MAJORITY VOTING RULE
However majority decision rules do not always yield such clear-cut results.
Suppose the preferences are rather depicted by table 7.2.
Again lets imagine a series of pair elections to determined the most preferred level.
Pair-wise elections are those held between any two alternatives when three or more alternatives are possible
An election between A and B, A would win, between B and C, B would win, between A and C, C would win.
12/7/2018
Dr. E. Nketiah-Amponsah

31. 12/7/2018
Dr. E. Nketiah-Amponsah

32. Majority Voting Rule This result is disconcerting.
Conventional notions of consistency suggests that A should be preferred to C but it is rather the opposite.
Although individual voter’s preferences are consistent, the community’s preferences are not.
This phenomenon is referred to as the Voting Paradox
12/7/2018
Dr. E. Nketiah-Amponsah

33. Majority Voting Rule
In situations like this, the ultimate outcome depends crucially on the order in which the votes are taken.
If the first election is between A and B and the winner runs against C, then C is the ultimate choice.
If the first election is between B and C and the winner B runs against A, then A is chosen.
12/7/2018
Dr. E. Nketiah-Amponsah

34. Majority Voting Rule
This means that the ability to control the order of voting (agenda) plays a very significant role.
The problem here is that paired voting can go on forever without reaching a decision.
In pair-wise/cycling elections, no political equilibrium exists. No one alternative can defeat all others wherever it appears on the ballot box
Between A and B, A wins but if C challenges A then C wins. If B then challenges C, B wins.
12/7/2018
Dr. E. Nketiah-Amponsah

35. Majority Voting Rule
The process can continue indefinitely a phenomenon called Cycling.
The question is: what is the cause of these problems?
It is as result of the structure of individual preferences for various levels of public good provision.
12/7/2018
Dr. E. Nketiah-Amponsah

36. Majority Voting Rule
Figure 7.2 shows the relationship between the various levels.
12/7/2018
Dr. E. Nketiah-Amponsah

37. 12/7/2018
Dr. E. Nketiah-Amponsah

38. Majority Voting Rule
A peak in an individual’s preferences is defined as the point at which all the neighboring points are lower.
A voter is said to have a Single-peaked Preferences if as he/she moves away from his /her most preferred outcome in any and all direction his/her utility consistently falls.
12/7/2018
Dr. E. Nketiah-Amponsah

39. Majority Voting Rule
A voter has Double-Peaked Preferences if as he/she moves away from the most preferred outcome utility goes down but then goes up again.
From the figure, Cosmo has a single peak at point A, George has a single peak at point B and Elaine has two peaks one at A and one at C.
The voting paradox is as a result of Elaine's Double –Peaked Preferences
12/7/2018
Dr. E. Nketiah-Amponsah

40. Majority Voting Rule
If Elaine had any set of single peaked preferences majority voting would lead to a consistent decision.
The conclusion is that if all voters have single peaked preferences, then majority voting will always lead to consistent decision i.e. no voting paradox will occur.
12/7/2018
Dr. E. Nketiah-Amponsah

41. Rules by which Winning Votes are determined
Simple Majority: Each individual has one vote and the simple majority wins. Where more than two alternatives are considered, they must be voted upon by successive elimination among surviving pairs.
Qualified Majority: Usually two-thirds of the vote is needed for a win.
Minority Rule: Less than 50% of the votes are needed for a decision to be taken.
A community that agrees on a minority rule runs the risk of making decisions that are disliked by majority voters.
In such a situation majority of voters will bear external cost imposed by the minority.
In some instances however, such decisions may be thought off as more efficient since less resources especially time are devoted to arriving at a decision.
12/7/2018

42. Voting Decision Rule Spectrum
Unanimity
Qualified Majority
2/3N
N/2 +1
Simple Majority
N/2
Minority
12/7/2018

43. Costs of Social Choice Process
Every decision making rule has expected external cost with the exception of the unanimous rule. E.g., If a voter votes against an issue but the decision carries, then he/she will be dissatisfied with the outcome.
According to Buchanan and Tullock (1962) the rational voter will prefer a decision rule that minimizes his total expected cost of the social choice process.
The cost of the social choice process consists of:
Expected external cost
Expected decision making cost
12/7/2018

44. Costs of Social Choice Process
For an individual voter, the external cost of the political process associated with any one particular activity may be defined as the present value of the cost he expects to bear as a result of the action of others through the political process.
External costs are assumed to be inversely related to the number of voters required for an option to be chosen.
Decision making costs on the other hand refers to those cost incurred by the individual during the decision making process.
The decision making cost is assumed to be directly related to the number of individuals needed for a public choice to be reached. This is because more time and effort are needed in order to reach a decision if the number of voters is large.
12/7/2018

45. Costs Cont’d – External Costs
Present Value of
Expected External Costs C
No. of Individuals needed for a
collective decision to be taken
12/7/2018

46. Costs – Decision Costs D N Present Value of Expected
Decision Making Costs D
No. of Individuals needed for a
collective decision to be taken N
12/7/2018

47. External and Decision Making Costs
Present Value of
Expected Costs
C+D D C N K*
12/7/2018
No. of Individuals needed for a
collective decision to be taken

48. External and Decision Making Costs
The U-shaped diagram, C + D, represent the total expected cost of the political decision making process on one voter.
K* minimizes the voter’s PV of total expected cost.
This implies that the voting rule requires that a minimum of K*/N of the population must agree before a collective action can be taken in order to minimize the total cost of collective decision-making.
12/7/2018

49. Summary
Lindahl prices can be used to elicit unanimous agreement for public expenditure
The decision to vote is largely explained by the ethical voter theorem and minimax strategy.
Voting involves both external and decision costs
Voting paradox occurs as a result of some voters having multiple peaked preferences.
If all voters preferences are single peaked, majority voting decision reflects the median voter’s preferences.
12/7/2018
Dr. E. Nketiah-Amponsah