Econ 551 Government Finance: Revenues Fall, 2019 Given by Kevin Milligan Vancouver School of Economics University of British Columbia Lecture 4b: Optimal Commodity Taxation, Part II ECON 551: Lecture 4b 1 of 34
Agenda: 1. Commodity taxation with many households 2. Commodity taxation and production efficiency 3. Application: Sugar-sweetened beverage taxation. ECON 551: Lecture 4b 2 of 34
Diamond and Mirrlees 1971 AER: This paper heralded the resurgence of interest in optimal taxation throughout the 1970s. See Myles (p. 108-111); Atkinson-Stiglitz (p. 386-390). Take a planner’s perspective: intro is about using taxes to control the economy. o This emphasizes a key point of this literature: we are simply choosing a set of prices that is optimal. This is an important way to think about taxes for reasons that will become clearer in a few minutes. Study both the production and consumption sides of the economy. In general equilibrium. Used contemporary micro tools. Duality, etc. They start with production, then move on to the consumer side. We’ll do it in backwards order, though. ECON 551: Lecture 4b 3 of 34
The setup: H households. ℎ ∈ {1, … , 𝐼} Linear commodity taxation available; no lump sum taxes, no income taxes. No direct revelation problems; everyone reports the truth. (This will be different for income taxation.) Labour is the untaxed numeraire good. Individual utility evaluated using indirect utility function (prices, wages, exogenous income). 𝑊 ℎ = 𝑊 ℎ (𝑟, 𝑥 ℎ 𝐽 ℎ ) Welfare is aggregated using a Bergson-Samuelson SWF. Ψ(𝑊) ECON 551: Lecture 4b 4 of 34
Today’s lemmata and notation : 𝜖𝑊 ℎ 𝜇 ℎ = 𝜖𝐽 ℎ Marginal utility of income. Ψ ℎ = 𝜖Ψ 𝜖𝑊 ℎ Weight put on household h in the SWF. 𝛾 ℎ = Ψ ℎ 𝜇 ℎ Social weight on household h ’s marginal income. ℎ ℎ 𝜖𝑦 𝑗 ℎ − 𝑦 𝑙 ℎ 𝜖𝑦 𝑗 ℎ is derivative of expenditure function with 𝜖𝑟 𝑙 = 𝑓 𝑗𝑙 𝜖𝐽 ℎ Slutsky equation, where 𝑓 𝑗𝑙 respect to i and k . ℎ ℎ 𝜖𝑌 𝑗 ℎ − ∑ 𝑦 𝑙 ℎ 𝜖𝑦 𝑗 𝜖𝑟 𝑙 = 𝐹 𝑗𝑙 Slutsky aggregated over households. (Capital letters for ℎ 𝜖𝐽 ℎ aggregate quantities; lower-case for household level.) 𝜖𝑊ℎ ℎ = − 𝜖𝑟𝑙 Roy’s identity 𝑦 𝑙 𝜖𝑊ℎ 𝜖𝐽ℎ ECON 551: Lecture 4b 5 of 34
Start with the optimization problem: 𝑟 𝑙 Ψ(𝑊) 𝑡. 𝑢. ∑ 𝑢 𝑗 𝑌 𝑗 ≥ 𝑆 max 𝑗 Set up a Lagrangian: ℒ = Ψ(𝑊) + 𝜈 (∑ 𝑢 𝑗 𝑌 𝑗 − 𝑆) 𝑗 First order conditions: 𝜖𝑊 ℎ 𝜖ℒ = 0 = ∑ 𝜖Ψ 𝜖𝑌 𝑗 + 𝜈 (𝑌 𝑙 + ∑ 𝑢 𝑗 ) 𝜖𝑊 ℎ 𝜖𝑢 𝑙 𝜖𝑢 𝑙 𝜖𝑟 𝑙 ℎ 𝑗 ECON 551: Lecture 4b 6 of 34
Manipulate the results using Roy’s identity and Slutsky equation : Plug in R oy’s identity to get: 𝜖𝑊 ℎ ∑ 𝜖Ψ 𝜖𝑌 𝑗 ℎ = ∑ 𝛾 ℎ 𝑦 𝑙 ℎ 𝑦 𝑙 = 𝜈 (𝑌 𝑙 + ∑ 𝑢 𝑗 ) 𝜖𝑊 ℎ 𝜖𝐽 ℎ 𝜖𝑟 𝑙 ℎ ℎ 𝑗 Now plug in Slutsky equation to derivative on the far right, and rearrange: ℎ 𝐹 𝑙𝑗 = ∑ 𝛾 ℎ ℎ 𝜖𝑦 𝑗 ℎ ∑ 𝑢 𝑗 𝜈 𝑦 𝑙 + ∑ 𝑢 𝑗 ∑ 𝑦 𝑙 𝜖𝐽 ℎ − 𝑌 𝑙 𝑗 ℎ 𝑗 ℎ Now just divide both sides by 𝑌 𝑙 and we’ll have something to interpret. = ∑ 𝛾 ℎ ℎ ℎ ℎ ∑ 𝑢 𝑗 𝐹 𝑙𝑗 𝑦 𝑙 ∑ 𝑦 𝑙 𝜖𝑦 𝑗 𝑗 + ∑ 𝑢 𝑗 𝜖𝐽 ℎ − 1 𝑌 𝑙 𝜈 𝑌 𝑙 𝑌 𝑙 ℎ 𝑗 ℎ ECON 551: Lecture 4b 7 of 34
Interpretation of the result: ℎ = ∑ 𝛾 ℎ ℎ ℎ ∑ 𝑢 𝑗 𝐹 𝑙𝑗 𝑦 𝑙 ∑ 𝑦 𝑙 𝜖𝑦 𝑗 𝑗 + ∑ 𝑢 𝑗 𝜖𝐽 ℎ − 1 𝑌 𝑙 𝜈 𝑌 𝑙 𝑌 𝑙 ℎ 𝑗 ℎ 1) The LHS is similar to what we saw for the Mirrlees index of discouragement. 2) On RHS, last part (-1) is a constant. Middle part represents extra tax revenue received when h gets one more dollar. First part is what we want to interpret. 𝛾 ℎ 3) 𝜈 is the marginal benefit to society of giving a dollar to h, divided by the cost of giving a dollar to h. Say we get 15 utils for this h for an extra dollar, but the extra dollar costs us 1.5. Then this ratio comes out as 10 dollars-worth of social benefit. ℎ 𝑦 𝑙 4) 𝑌 𝑙 is the proportion of good k consumed by this household. ECON 551: Lecture 4b 8 of 34
…continued ℎ = ∑ 𝛾 ℎ ℎ ℎ ∑ 𝑢 𝑗 𝐹 𝑙𝑗 𝑦 𝑙 ∑ 𝑦 𝑙 𝜖𝑦 𝑗 𝑗 + ∑ 𝑢 𝑗 𝜖𝐽 ℎ − 1 𝑌 𝑙 𝜈 𝑌 𝑙 𝑌 𝑙 ℎ 𝑗 ℎ ℎ 𝛾 ℎ 𝑦 𝑙 ∑ 5) is therefore like a correlation of welfare weight with proportion consumed. ℎ 𝜈 𝑌 𝑙 6) The LHS is negative, so a larger value for the first term on the RHS means that the taxes must be smaller so that proportionate demand change is smaller. 7) The RHS becomes more positive when the first term is larger. So, lower tax rates. Summarized, this means that good consumed more by those who have high social weight should be taxed less. ECON 551: Lecture 4b 9 of 34
Comparing this result to one-consumer case: If social marginal utility of income is the same for everyone ( 𝛾 ℎ = 𝛾 ∀ℎ ), then the 1) correlation will be zero and there is no gain from attempts to redistribute. Things reduce to the one-person case. 2) If all individuals demand goods in the same proportions, then there will be no variation in the proportion across households leading to a zero correlation and again it reduced to the one-household case. Think of this as a signaling problem. The state is trying to infer whom to tax by looking at who disproportionately consumes which goods. If there is no signal, it c an’t do anything about equity. ECON 551: Lecture 4b 10 of 34
Agenda: 1. Commodity taxation with many households 2. Commodity taxation and production efficiency 3. Application: Sugar-sweetened beverage taxation. ECON 551: Lecture 4b 11 of 34
Breaktime! Hungarian Mathematician Paul Erdős (1913-1996) traveled the world couch- surfing and coauthoring. Wrote 1525 published papers. Had 511 coauthors. People like to calculate their Erdős number, which counts the ‘collaboration distance’ to Erdős . Peter Diamond has Erdős number of 3 . o Peter Diamond-Eric Maskin-Peter Fishburn- Paul Erdős. Mine is 5 : Milligan---Courtney Coile---Peter Diamond---Maskin--- Fishburn--- Erdős . IIRC, some other ‘entry points’ for economists to Erdő s come from econometricians. ECON 551: Lecture 4b 12 of 34
Now we allow for a production side of the economy Recall, this is the first part of Diamond Mirrlees (1971). The question being posed here is whether we should tax intermediate inputs or not. Diamond and Mirrlees show that, in general, it is not optimal to tax inputs. Instead, only final goods should be taxed. This is referred to as the Production Efficiency Lemma . We’re going to look at a diagram rather than go through all of the math… ECON 551: Lecture 4b 13 of 34
Production efficiency and taxation U 0 E 0 Offer Curve Distortionary tax through origin E 1 U 1 Input Production Lumpsum tax budget Possibility line, not through Frontier origin. Taken from Myles (p. 128) ECON 551: Lecture 4b 14 of 34
Walking through the result Two goods; one input one output. Households supply input and consume output. Production possibility set defined for some convex technology. Ignoring taxes, we would be at a place like E 0 where the MRTS =MRS. However, tax revenue requirements shifts the set out to the left from the origin. With lumpsum taxes, we can still get to E 0 . How? Adjust endowment along the input axis until we are at the tangent point to E 0 . This is shown with dark black lumpsum tax budget line. With distortionary taxes, price line must go through the origin. Offer curve represents the locus of tangencies for ICs and price lines through the origin. Optimal choice is at E 1 . Why? o Want to get the most output possible for the least input. o At interior point below E 1 , could have increased output without more input. o At interior point to the left of E 1 , could have had same output with less input. ECON 551: Lecture 4b 15 of 34
The lemma and its implications Stating it formally: If an optimum exists, then the optimum has production on the frontier of the production possibilities set . Implications: 1) Input taxes should not be differentiated among firms; all should face the same post-tax input prices. (Why is this the case? If you are taxing inputs, you’re not going to be on the efficient production frontier. Don’t tax inputs. ) 2) We can mimic any set of input taxes using a tax on final goods, so why don’t we just tax final goods instead? If we do that, we’re not distorting production decisions. That is, do not tax intermediate goods. This is a major motivation for value-added taxation. Tax final consumption; not intermediate goods. Since capital is an input, we shouldn’t tax the return to capit al. This is a conjecture; we’ll 3) see how it holds up…. ECON 551: Lecture 4b 16 of 34
Recommend
More recommend
