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 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