Econ 551 Government Finance: Revenues Fall 2019 Given by Kevin Milligan Vancouver School of Economics University of British Columbia Lecture 4a: Optimal Commodity Taxation ECON 551: Lecture 4a 1 of 36
Agenda: 1. Uniform commodity taxation 2. The Ramsey Rule: Partial Equilibrium 3. The Ramsey Rule: General Equilibrium 4. Application: Corlett-Hague (1953) 5. Application: Revisiting the case for uniform taxation. ECON 551: Lecture 4a 2 of 36
Uniform commodity taxation: If you trace through the history of economic thought, a common theme is the optimality (or not) of taxing all commodities at the same rate — uniform commodity taxation. The debate continues today with questions over what to do with value added taxes like Canada’s GST/HST. Q: What advantages come to mind for uniform vs. differentiated commodity taxes? We’re going to start with a basic case and then sequentially add more richness. Let’s see how the answer changes…. ECON 551: Lecture 4a 3 of 36
Notation and framework: We’re going to start simple. Consider an economy with: One person Exogenous income I . Revenue to be raised 𝑆 . o Exogenous; assume burned or spent on separable public good. o How could we enrich this assumption? Goods x k , numbered from 1 to K . Prices p k and taxes t k . For now, let’s assume ad valorem multiplicative taxes. ECON 551: Lecture 4a 4 of 36
Budget constraints: Individual’s budget constraint: ∑ 𝑞 𝑙 (1 + 𝑢 𝑙 )𝑦 𝑙 ≤ 𝐽 𝑙 Government’s revenue constraint: ∑ 𝑢 𝑙 𝑞 𝑙 𝑦 𝑙 ≥ 𝑆 𝑙 ECON 551: Lecture 4a 5 of 36
Result: uniform commodity tax equivalent to flat income tax The government can get its revenue on the income sid e or expenditure side of the individual’s budget. With exogenous income and uniform commodity taxes, the income side and expenditure side taxes are equivalent; uniform commodity taxes are a lumpsum tax on endowment! Rearrange the individual’s budget cons traint: ∑ 𝑞 𝑙 (1 + 𝑢 𝑙 )𝑦 𝑙 ≤ 𝐽 𝑙 Impose uniform commodity tax 𝑢 𝑙 = 𝑢 ∀𝑙 , and rearrange: 𝐽 (1 + 𝑢) = (1 − 𝑢 𝐽 )𝐽 ∑ 𝑞 𝑙 𝑦 𝑙 ≤ 𝑙 (With income tax 𝑢 𝐽 set to 𝑢 𝐽 = 𝑢 (1+𝑢) .) ECON 551: Lecture 4a 6 of 36
Result: Uniform commodity taxes raise no revenue What if we allow labour to be supplied elastically, and we allow for choices in production? Turns out that a uniform commodity tax will raise ZERO revenue. See Sandmo 1974, or Myles Ch 4, section 7, pp. 122-124. Agnar Sandmo Consider a system of net demands for ‘commodities’, where one of those commodities is labour or time. If your net demand is positive, you’re a buyer. If your net demand is negativ e, you’re a seller. We’ll use the same notation; commodities x k , numbered from 1 to K . But let’s use good k =0 to stand for labour. ECON 551: Lecture 4a 7 of 36
Result: Uniform commodity tax raises no revenue Write down the individual budget constraint: 𝐿 ∑ 𝑞 𝑙 𝑦 𝑙 ≤ 0 𝑙=0 Take the usual government budget constraint: ∑ 𝑢 𝑙 𝑞 𝑙 𝑦 𝑙 ≥ 𝑆 𝑙 But now, 𝑢 𝑙 = 𝑢 , ∀𝑙 . So…. 𝑢 ∑ 𝑞 𝑙 𝑦 𝑙 ≥ 𝑆 𝑙 Examine individual budget constraint. The government budget can only hold if 𝑆 = 0 . Let’s call this the ‘Sandmo zero- revenue box’. Or just the Sandmo box. ECON 551: Lecture 4a 8 of 36
Digression on the taxation of leisure In order to get out of the Sandmo box, we need to leave one commodity untaxed. This means we enter the world of differential commodity taxation. Conventionally, the a ssumption is that we don’t tax leisure. (In above example, we subsidized individuals selling labour at rate t . In terms of relative prices, this is the same as taxing leisure.) You’ll read things like, “Since we can’t tax leisure, we cannot have uniform taxation…” Sandmo’s analysis does not rely on the premise that we cannot tax leisure. This is not some assumed technical constraint about government’s ability to observe leisure. Instead, it is a mathematical constraint. We could choose any commodity — say pencils — as the untaxed commodity and get out of the Sandmo box. ECON 551: Lecture 4a 9 of 36
Agenda: 1. Uniform commodity taxation 2. The Ramsey Rule: Partial Equilibrium 3. The Ramsey Rule: General Equilibrium 4. Application: Corlett-Hague (1953) 5. Application: Revisiting the case for uniform taxation. ECON 551: Lecture 4a 10 of 36
Optimal differential commodity taxation With leisure untaxed, we are faced with choice: a) Taxing all taxed goods at uniform rates b) Using differentiated taxes for the taxed goods. Choice is between one big distortion and a bunch of mini-distortions. Q: What does the ‘General Theorem of the Second Best’ have to say about this? Conventional economic wisdom: uniform commodity taxation ideal. Even Musgrave (1959) argued in favour of uniform ad valorem rate. ECON 551: Lecture 4a 11 of 36
The mysteries and intrigue of optimal differentiated taxation Frank Ramsey (1903-1930) was student of Keynes and Pigou at Cambridge. Challenged to Pigou to consider whether conventional wisdom on uniform commodity taxes held up to greater scrutiny. Wrote article in 1927 EJ, but didn’t attract much attention at the time. Paul Samuelson wrote a memo for the US Treasury in 1951. Tattered old copies were passed around for decades like a Talmudic scroll. Built on Ramsey; had access to better assumptions on income In 1986, Jim Poterba published the original in J. Public Economics so that everyone could have it. ECON 551: Lecture 4a 12 of 36
Notation and framework: Start same as before, but some new twists. Consider an economy with: One person; fixed producer prices. Exogenous income I . Revenue to be raised 𝑆 . Goods x k , numbered from 0 to K . Prices p k and taxes t k . o Taxes are linear, multiplicative, but potentially differentiated. o Tax on good 0 is zero: 𝑢 0 = 0 o Let’s call the tax -inclusive price 𝑟 𝑙 = 𝑞 𝑙 + 𝑢 𝑙 . Quasilinear preferences, in order to isolate income effect: 𝑣(𝑦, 𝐽) = ∑ 𝑤 𝑙 (𝑦 𝑙 ) + 𝐽 𝑙 𝜖𝑦 𝑘 For now, assume no cross-price effects: 𝜖𝑢 𝑙 = 0, ∀𝑙 ≠ 𝑘 . ECON 551: Lecture 4a 13 of 36
Partial equilibrium analysis: Price q k t k p k Demand x tk x 0k x k Dark shaded area is tax revenue. Stripey area is the excess burden. Our math will aim to minimize this stripey area. ECON 551: Lecture 4a 14 of 36
Set up for partial equilibrium problem Define the excess burden: 0 𝑦 𝑙 0 − 𝑦 𝑙 𝑢 ) 𝛾 𝑙 (𝑢 𝑙 ) = ∫ 𝑟 𝑙 (𝑦 𝑙 )𝑒 𝑦 𝑙 − 𝑞 𝑙 (𝑦 𝑙 𝑢 𝑦 𝑙 Our goal: Minimize this, subject to the revenue constraint by choosing tax rates. 𝑢 min 𝑢 𝑙 ∑ 𝛾 𝑙 (𝑢 𝑙 ) 𝑡. 𝑢. ∑ 𝑢 𝑙 𝑦 𝑙 ≥ 𝑆 𝑙 𝑙 ECON 551: Lecture 4a 15 of 36
Some necessary lemmata 1. Derivative of excess burden wrt tax rate: 𝑢 𝑢 𝑢 𝜖𝛾 𝑙 𝜖𝑦 𝑙 𝜖𝑦 𝑙 𝜖𝑦 𝑙 = −𝑟 𝑙 + 𝑞 𝑙 = −𝑢 𝑙 𝜖𝑢 𝑙 𝜖𝑢 𝑙 𝜖𝑢 𝑙 𝜖𝑢 𝑙 (First term results from derivative wrt variable on bound of integral; Leibnitz’s rule .) 2. State the elasticity in a convenient way: 𝑢 𝜁 𝑙 = − 𝑟 𝑙 𝜖𝑦 𝑙 𝑦 𝑙 𝜖𝑢 𝑙 ECON 551: Lecture 4a 16 of 36
Write out the Lagrangian: Slap a minus sign out front to make it a maximization for greater ease. ℒ = − ∑ 𝛾 𝑙 (𝑢 𝑙 ) + 𝜈 (∑ 𝑢 𝑙 𝑦 𝑙 − 𝑆) 𝑙 𝑙 First order conditions: 𝑢 𝑢 𝜖ℒ 𝜖𝑦 𝑙 𝜖𝑦 𝑙 𝑢 + 𝑢 𝑙 = 0 = 𝑢 𝑙 + 𝜈 (𝑦 𝑙 ) 𝜖𝑢 𝑙 𝜖𝑢 𝑙 𝜖𝑢 𝑙 𝑢 : Collect terms and solve for 𝑦 𝑙 𝑢 (1 + 𝜈) 𝜖𝑦 𝑙 𝑢 = −𝑢 𝑙 𝑦 𝑙 𝜖𝑢 𝑙 𝜈 ECON 551: Lecture 4a 17 of 36
Clean up the result: 𝑢 (1 + 𝜈) 𝜖𝑦 𝑙 𝑢 = −𝑢 𝑙 𝑦 𝑙 𝜖𝑢 𝑙 𝜈 𝜈 Substitute in 𝜄 = 1+𝜈 and use the elasticity definition. 𝑢 𝑦 𝑙 1 𝑢 = 𝑢 𝑙 𝑦 𝑙 𝜁 𝑙 𝜄 𝑟 𝑙 Rearrange and that’s it: 𝑢 𝑙 = 𝜄 𝑟 𝑙 𝜁 𝑙 ECON 551: Lecture 4a 18 of 36
Interpretation of the partial equilibrium result: 𝑢 𝑙 = 𝜄 𝑟 𝑙 𝜁 𝑙 𝑢 𝑙 1. 𝑟 𝑙 is the tax rate. 2. Optimal tax rate is inversely proportional to the own- price demand elasticity. The ‘inverse elasticity rule’ is born! 3. High tax on inelastic goods; low tax on goods with elastic demands. 4. Can we just disregard the cross-price elasticities? ECON 551: Lecture 4a 19 of 36
Agenda: 1. Uniform commodity taxation 2. The Ramsey Rule: Partial Equilibrium 3. The Ramsey Rule: General Equilibrium 4. Application: Corlett-Hague (1953) 5. Application: Revisiting the case for uniform taxation. ECON 551: Lecture 4a 20 of 36
Setup for general equilibrium differential commodity taxes. Now let’s consider what happens when we allow for cross -price elasticities to change. Same setup as before, with a couple of changes. Let’s now call the price of good 0 the wage, p 0 =w . Net demand environment, so no more endowment income. I =0. The budget constraint is now: ∑ 𝑟 𝑙 𝑦 𝑙 + 𝑥𝑦 0 ≤ 0 𝑙 Indirect utility function 𝑊(𝑟, 𝑥) = max 𝑣(𝑦) 𝑡. 𝑢. ∑ 𝑟 𝑙 𝑦 𝑙 + 𝑥𝑦 0 = 0 𝑙 ECON 551: Lecture 4a 21 of 36
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