Econ 551 Government Finance: Revenues Fall 2019 Given by Kevin - PowerPoint PPT Presentation

Econ 551 Government Finance: Revenues Fall 2019 Given by Kevin Milligan Vancouver School of Economics University of British Columbia Lecture 2b: Positive Political Economy ECON 551: Lecture 2b 1 of 34

Agenda: 1. Models of Direct Democracy: Median Voter Theorem 2. Application: Voting over taxes 3. Models of Representative Democracy ECON 551: Lecture 2b 2 of 34

Models of Direct Democracy: Direct democracy means citizens have a choice over the outcomes, not representatives.  Think of referenda. The notation and proof we go through come from Persson and Tabellini (2000). The examples of voting systems come from Levin and Nalebuff (1995). ECON 551: Lecture 2b 3 of 34

Black 1948: Originated (or re-discovered) the idea to apply economic reasoning to politics  You can see how novel it is (or at least he thinks it is) in his tone. The theory, indeed, would appear to present the basis for the development of a pure science of politics . This would employ the same theory of relative valuation as economic science. It would employ a different definition of equilibrium. Equilibrium would now be defined in terms of voting, in place of the type of definition employed in economic science. We could move from the one science to the other with the alteration of a single definition. This, in the view of the writer, would be the main function of the theory. Offered first proof of the median voter theorem — the workhorse of many modern models. Note that this pre-dated Arrow — which may account for the optimistic tone of his paper. ECON 551: Lecture 2b 4 of 34

The environment:  Direct Democracy. Citizens vote for outcomes, rather than for candidates who will implement outcomes.  Sincere Voting. Citizens vote for their preferred outcome. Note that this is an assumption, not a result.  Open agenda. Decisions are made by placing policies against each other in sequential pairwise comparisons. It is ‘open’ because any policy can be put forward against any other; no agenda-setting. ECON 551: Lecture 2b 5 of 34

Notation:  Set of social alternatives: X  Set of voters: Θ = {1,2, … , 𝑂}  For decisions 𝑦, 𝑧 ∈ 𝑌 𝑦𝑄 𝜄 𝑧 means that x is strictly preferred to y by guy θ. 𝑦𝑆 𝜄 𝑧 means that x is weakly preferred to y by guy θ. 𝜈(𝑦, 𝑧) is the fraction of voters for whom 𝑦𝑆 𝜄 𝑧 ∗ is guy θ’ s bliss policy; 𝑦 ∗ 𝑆 𝜄 𝑧, ∀𝑧 ∈ 𝑌 𝑦 𝜄 ECON 551: Lecture 2b 6 of 34

Definition: A Condorcet winner A policy 𝑦 ̃ ∈ 𝑌 is a Condorcet Winner if it beats all other policies 𝑧 ∈ 𝑌 in pairwise votes. 𝜈(𝑦 ̃, 𝑧) ≥ 𝜈(𝑧, 𝑦 ̃), ∀𝑧 ∈ 𝑌 ECON 551: Lecture 2b 7 of 34

Definition: Single-Peakedness Preferences are single-peaked if the distance from the bliss point dictates the preference ordering. ∗ (or 𝑦 ∗ ) then 𝑦̅𝑆 𝜄 𝑦 If 𝑦 ̂ ≤ 𝑦̅ ≤ 𝑦 𝜄 ̂ ≥ 𝑦̅ ≥ 𝑦 𝜄 ̂ This just says the closer you are to the bliss-point, the better the option is (weakly). ECON 551: Lecture 2b 8 of 34

Illustration: Single-Peakedness Valuation of x Jones Singh Zhu Quantity of x ECON 551: Lecture 2b 9 of 34

Definition: Median voter ∗ . The median voter is the θ with 𝑦 𝜄 ∗ ranked Order all voters 𝜄 ∈ Θ by their 𝑦 𝜄  (N+1)/2 if N is odd  (N/2) and (N/2) +1 if N is even. ECON 551: Lecture 2b 10 of 34

Proposition: The Median Voter Theorem If all voters have single peaked preferences and the set of policies X is one dimensional, then:  ∃ a Condorcet winner, 𝑦 ̃ . ∗  The Condorcet winner is the bliss point of the median voter: 𝑦 ̃ = 𝑦 𝑛 ECON 551: Lecture 2b 11 of 34

Proof: The Median Voter Theorem The following is the proof of the case for which N is odd. (The even case can be proved, but is harder to do.) ∗ , 𝑦 ∈ 𝑌 Take a policy 𝑦 < 𝑦 𝑛 ∗ > 𝑦 𝑛 ∗ 𝑆 𝜄 𝑦 . This is (N-1)/2 ∗ will have 𝑦 𝑛 Single peakedness implies that all θ with 𝑦 𝜄 voters. ∗ 𝑆 𝜄 𝑦 . This is one more voter. Median voter has 𝑦 𝑛 So, we have (N-1)/2 + 1 = N/2 +1/2 > N/2 ∗ , 𝑦) > 0.5 and 𝑦 𝑛 ∗ is a Condorcet winning policy. ∴ 𝜈(𝑦 𝑛 ECON 551: Lecture 2b 12 of 34

Comments on the Median Voter Theorem:  The social ordering works. How does this square with Arrow IT? MVT requires restriction on preferences to be single peaked. This violates the (U) axiom.  We have social transitivity. This is great. Without transitivity, the order in which pairwise comparisons are made determines the outcome; agenda-setting matters. This conveys a lot of power on the agenda setter.  We restricted attention to the case of a single policy dimension. What if policies are multi-dimensional? Stronger restrictions on preferences are required. ECON 551: Lecture 2b 13 of 34

Example 1: Θ = {𝑇𝑏𝑚𝑚𝑧, 𝐶𝑝𝑐𝑐𝑧, 𝐿𝑓𝑜𝑜𝑧} 𝑌 = {𝐵, 𝐶, 𝐷} with 𝐵 < 𝐶 < 𝐷 Rank preferences look like this: A B C Sally 1 2 3 Bobby 2 1 3 Kenny 3 2 1 ECON 551: Lecture 2b 14 of 34

Example 1: Kenny Bobby Sally A B C  Are all three single peaked?  Who is the median voter?  What is the median voter’s pref erred policy.  Can other policies beat that one? Try it.  Transitive?  Agenda-independent? ECON 551: Lecture 2b 15 of 34

Example 2: Θ = {𝑇𝑏𝑚𝑚𝑧, 𝐶𝑝𝑐𝑐𝑧, 𝐿𝑓𝑜𝑜𝑧} 𝑌 = {𝐵, 𝐶, 𝐷} with 𝐵 < 𝐶 < 𝐷 Rank preferences look like this: A B C Sally 1 3 2 Bobby 2 1 3 Kenny 3 2 1 ECON 551: Lecture 2b 16 of 34

Example 2: Sally A B C  Are all three single peaked?  Who is the median voter?  What is the median voter’s preferred policy.  Can other policies beat that one? Try it.  Transitive?  Agenda-independent? ECON 551: Lecture 2b 17 of 34

Example 3: Ends against the middle Epple and Romano (1996) Imagine the following:  Paying for a local public good (e.g. schooling) out of local tax.  Choice is X = {𝑚𝑝𝑥, 𝑛𝑓𝑒𝑗𝑣𝑛, ℎ𝑗𝑕ℎ} , where low means you get low taxes and low quality; high is opposite.  Population is Θ = {𝑞𝑝𝑝𝑠, 𝑛𝑗𝑒𝑒𝑚𝑓, 𝑠𝑗𝑑ℎ} .  Say each group has the ‘obvious’ bliss point. (poor likes low, middle likes medium, rich likes high.)  Rich has outside option — private school — which they will take if X<high .  But Rich has to pay taxes, no matter what X is chosen. Q: Who is median voter? Is median voter decisive? Q: What is Rich’s preference ranking? Q: Are there any signs of trouble here? Q: What problems could arise? ECON 551: Lecture 2b 18 of 34

Agenda: 1. Models of Direct Democracy: Median Voter Theorem 2. Application: Voting over taxes 3. Models of Representative Democracy ECON 551: Lecture 2b 19 of 34

Application: Voting on Taxes This application draws on what’s known as the Meltzer -Richards (1981) model. Consider:  A society with N individuals indexed by i , ∈ {1, … , 𝑂} .  Each has an identical utility function which takes only consumption as an argument: 𝑉 𝑗 = 𝑉(𝑑 𝑗 ) .  Each individual has exogenous income y i .  Income is taxed at a flat rate of t . Taxed income is returned to individuals through a uniform lump sum transfer d . The budget is balanced, so that 𝑂𝑒 = ∑ 𝑢𝑧 𝑗 𝑗 ∑ 𝑧 𝑗  Define 𝑧 = 𝑗 𝑂 , so that 𝑒 = 𝑢 × 𝑧 .  Define utility level when t=0 to be U 0 . ECON 551: Lecture 2b 20 of 34

Application: Voting on Taxes Decision faced by electorate is whether to institute the tax/transfer plan.  Each voter evaluates whether 𝑉 𝑗 ≤≥ 𝑉 0 𝑗 . Write down the budget constraint: 𝑑 𝑗 = 𝑧 𝑗 (1 − 𝑢) + 𝑒 From the government budget constraint we can substitute in for d : 𝑑 𝑗 = 𝑧 𝑗 (1 − 𝑢) + 𝑢𝑧 and simplify to 𝑑 𝑗 = 𝑧 𝑗 + 𝑢(𝑧 − 𝑧 𝑗 ) 𝑗 = 𝑧 𝑗 , if there is no transfer. Note: 𝑑 0 ECON 551: Lecture 2b 21 of 34

Application: Voting on Taxes If the distribution of y i is single-peaked, then the median voter will be decisive. Call the median voter i=m . Then redistribution will pass the referendum iff 𝑧 > 𝑧 𝑛 (by the median voter theorem). The predictions from this model therefore are: 1) if the mean income is higher than the median, then government will be larger. 2) If the mean income is higher than the median, there will be more redistribution. ECON 551: Lecture 2b 22 of 34