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

ECON 551: Lecture 4a 1 of 36

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 2 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 3 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 4 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.

• Exogenous; assume burned or spent on separable public good.
• How could we enrich this assumption?

 Goods xk, numbered from 1 to K.  Prices pk and taxes tk.  For now, let’s assume ad valorem multiplicative taxes.

ECON 551: Lecture 4a 5 of 36

Budget constraints:

Individual’s budget constraint: ∑ 𝑞𝑙(1 + 𝑢𝑙)𝑦𝑙 ≤ 𝐽

𝑙

Government’s revenue constraint: ∑ 𝑢𝑙

𝑙

𝑞𝑙𝑦𝑙 ≥ 𝑆

ECON 551: Lecture 4a 6 of 36

Result: uniform commodity tax equivalent to flat income tax

The government can get its revenue on the income side 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 constraint: ∑ 𝑞𝑙(1 + 𝑢𝑙)𝑦𝑙 ≤ 𝐽

𝑙

Impose uniform commodity tax 𝑢𝑙 = 𝑢 ∀𝑙, and rearrange: ∑ 𝑞𝑙

𝑙

𝑦𝑙 ≤ 𝐽 (1 + 𝑢) = (1 − 𝑢𝐽)𝐽 (With income tax 𝑢𝐽 set to 𝑢𝐽 =

𝑢 (1+𝑢).)

ECON 551: Lecture 4a 7 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 negative, you’re a seller. We’ll use the same notation; commodities xk, numbered from 1 to K.  But let’s use good k=0 to stand for labour.

ECON 551: Lecture 4a 8 of 36

Result: Uniform commodity tax raises no revenue

Write down the individual budget constraint: ∑ 𝑞𝑙𝑦𝑙

𝐿 𝑙=0

≤ 0 Take the usual government budget constraint: ∑ 𝑢𝑙

𝑙

𝑞𝑙𝑦𝑙 ≥ 𝑆 But now, 𝑢𝑙 = 𝑢, ∀𝑙. So…. 𝑢 ∑ 𝑞𝑙𝑦𝑙

𝑙

≥ 𝑆 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 9 of 36

Digression on the taxation of leisure

In order to get out of the Sandmo box, we need to leave

• ne commodity untaxed.

 This means we enter the world of differential commodity taxation. Conventionally, the assumption 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 10 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 11 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 12 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 13 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 xk, numbered from 0 to K.  Prices pk and taxes tk.

• Taxes are linear, multiplicative, but potentially differentiated.
• Tax on good 0 is zero: 𝑢0 = 0
• Let’s call the tax-inclusive price 𝑟𝑙 = 𝑞𝑙 + 𝑢𝑙.

 Quasilinear preferences, in order to isolate income effect: 𝑣(𝑦, 𝐽) = ∑ 𝑤𝑙(𝑦𝑙) + 𝐽

𝑙

 For now, assume no cross-price effects:

𝜖𝑦𝑘 𝜖𝑢𝑙 = 0, ∀𝑙 ≠ 𝑘.

ECON 551: Lecture 4a 14 of 36

Partial equilibrium analysis:

Dark shaded area is tax revenue. Stripey area is the excess burden. Our math will aim to minimize this stripey area.

xk Price Demand tk x0k pk qk xtk

ECON 551: Lecture 4a 15 of 36

Set up for partial equilibrium problem

Define the excess burden: 𝛾𝑙(𝑢𝑙) = ∫ 𝑟𝑙(𝑦𝑙)𝑒

𝑦𝑙 𝑦𝑙

𝑢

𝑦𝑙 − 𝑞𝑙(𝑦𝑙

0 − 𝑦𝑙 𝑢)

Our goal: Minimize this, subject to the revenue constraint by choosing tax rates. min

𝑢𝑙 ∑ 𝛾𝑙(𝑢𝑙) 𝑙

𝑡. 𝑢. ∑ 𝑢𝑙𝑦𝑙

𝑢 𝑙

≥ 𝑆

ECON 551: Lecture 4a 16 of 36

Some necessary lemmata

• 1. Derivative of excess burden wrt tax rate:

𝜖𝛾𝑙 𝜖𝑢𝑙 = −𝑟𝑙 𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 + 𝑞𝑙 𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 = −𝑢𝑙 𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 (First term results from derivative wrt variable on bound of integral; Leibnitz’s rule.)

• 2. State the elasticity in a convenient way:

𝜁𝑙 = − 𝑟𝑙 𝑦𝑙 𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙

ECON 551: Lecture 4a 17 of 36

Write out the Lagrangian:

Slap a minus sign out front to make it a maximization for greater ease. ℒ = − ∑ 𝛾𝑙(𝑢𝑙) + 𝜈 (∑ 𝑢𝑙𝑦𝑙

𝑙

− 𝑆)

𝑙

First order conditions: 𝜖ℒ 𝜖𝑢𝑙 = 0 = 𝑢𝑙 𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 + 𝜈 (𝑦𝑙

𝑢 + 𝑢𝑙

𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 ) Collect terms and solve for 𝑦𝑙

𝑢:

𝑦𝑙

𝑢 = −𝑢𝑙

𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 (1 + 𝜈) 𝜈

ECON 551: Lecture 4a 18 of 36

Clean up the result:

𝑦𝑙

𝑢 = −𝑢𝑙

𝜖𝑦𝑙

𝑢

𝜖𝑢𝑙 (1 + 𝜈) 𝜈 Substitute in 𝜄 =

𝜈 1+𝜈 and use the elasticity definition.

𝑦𝑙

𝑢 = 𝑢𝑙

𝑦𝑙

𝑢

𝑟𝑙 𝜁𝑙 1 𝜄 Rearrange and that’s it: 𝑢𝑙 𝑟𝑙 = 𝜄 𝜁𝑙

ECON 551: Lecture 4a 19 of 36

Interpretation of the partial equilibrium result:

𝑢𝑙 𝑟𝑙 = 𝜄 𝜁𝑙 1.

𝑢𝑙 𝑟𝑙 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 20 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 21 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, p0=w.  Net demand environment, so no more endowment income. I=0. The budget constraint is now: ∑ 𝑟𝑙𝑦𝑙 + 𝑥𝑦0 ≤ 0

𝑙

Indirect utility function 𝑊(𝑟, 𝑥) = max 𝑣(𝑦) 𝑡. 𝑢. ∑ 𝑟𝑙𝑦𝑙 + 𝑥𝑦0 = 0

𝑙

ECON 551: Lecture 4a 22 of 36

Some more lemmata:

a) We can rewrite the indirect utility function in the dual as an expenditure function. 𝐹(𝑟, 𝑥, 𝑣) = min ∑ 𝑟𝑙𝑦𝑙 + 𝑥𝑦0 = 0

𝑙

𝑡. 𝑢. 𝑣 ≥ 𝑣 b) Remember Shephard’s lemma: derivative of expenditure function gives Hicksian demand. 𝐹𝑙 = 𝜖𝐹 𝜖𝑟𝑙 = 𝑦𝑙(𝑟, 𝑥, 𝑣) c) Rewrite cross-price elasticity using Slutsky symmetry. 𝜁𝑙𝑗 = − 𝑟𝑗 𝑦𝑙 𝜖𝑦𝑙 𝜖𝑢𝑗 = − 𝑟𝑗 𝑦𝑙 𝜖𝑦𝑗 𝜖𝑢𝑙

ECON 551: Lecture 4a 23 of 36

Write down the problem:

The minimization problem. min

𝑟

𝐹(𝑟, 𝑥, 𝑣) 𝑡. 𝑢. ∑ 𝑢𝑙𝑦𝑙

𝑙

≥ 𝑆 The Lagrangian. ℒ = −𝐹(𝑟, 𝑥, 𝑣) + 𝜈 (∑ 𝑢𝑙𝑦𝑙

𝑙

− 𝑆) First order conditions: 𝜖ℒ 𝜖𝑢𝑙 = 0 = −𝐹𝑙 + 𝜈 (𝑦𝑙 + ∑ 𝑢𝑗 𝜖𝑦𝑗 𝜖𝑢𝑙

𝑗

)

ECON 551: Lecture 4a 24 of 36

Rearrange the results:

Define 𝜄 =

1−𝜈 𝜈 and solve for theta.

Solve for theta 𝜄 = ∑ 𝑢𝑗 𝜖𝑦𝑗 𝜖𝑢𝑙

𝑗

𝑦𝑙 Use the Slutsky switcheroo to give this some more meaning. 𝜄 = ∑ 𝑢𝑗 𝜖𝑦𝑙 𝜖𝑢𝑗

𝑗

𝑦𝑙 And one final manipulation to get tax rates and elasticities: 𝜄 = ∑ 𝑢𝑗 𝑟𝑗 𝜁𝑙𝑗

𝑗

ECON 551: Lecture 4a 25 of 36

Interpret the results:

• 1. This says: How far does xk move proportionally when taxes are increased? That’s the RHS.

On the LHS we have a constant. Put these together and you get the idea: This equation must hold for all goods, so the quantity of all goods must move proportionally the same amount when taxes are changed.

• 2. Mirrlees called this the “index of discouragement”. Consumption of all goods should be

discouraged by the introduction of taxes in the same way.

• 3. The right way to think about minimizing distortion is to equalize the distortion to quantities,

not prices. That is, this argues against uniform taxation.

• 4. Again, we should tax low elasticity goods, but now that means ones with low elasticities in

aggregate across all goods.

• 5. This is a one-person economy. No concern for equity here…

ECON 551: Lecture 4a 26 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 27 of 36

Corlett-Hague (1953): A three-good example

If we must leave one good untaxed, is there a way to improve welfare by getting ‘close’ to taxing it? Would that be desirable?  These questions were addressed in a contribution by Corlett and Hague (1953). Let’s start by defining an ad valorem tax rate: 𝜐𝑙 ≡ 𝑢𝑙 𝑟𝑙 Also, let’s write down the cross-price elasticity: 𝜁𝑙𝑗 = − 𝑟𝑗 𝑦𝑙 𝜖𝑦𝑙 𝜖𝑟𝑗

ECON 551: Lecture 4a 28 of 36

Still more necessary lematta:

Remember Euler’s formula (see Myles p. 124). If a set of Hicksian demand functions x(q) are homogeneous of degree λ, then: ∑ 𝜖𝑦𝑙 𝜖𝑟𝑗 𝑟𝑗 = 𝜇𝑦𝑙

𝑂 𝑗=1

(𝑟1, … , 𝑟𝑂) If these are homogenous of degree 0, then it must be that ∑ 𝜖𝑦𝑙 𝜖𝑟𝑗 𝑟𝑗 = 0 ⋅ 𝑦𝑙

𝑂 𝑗=1

(𝑟1, … , 𝑟𝑂) = 0 Divide each side by 𝑦𝑙. ∑ 𝜖𝑦𝑙 𝜖𝑟𝑗 𝑟𝑗 𝑦𝑙 = 0 = ∑ 𝜁𝑙𝑗

𝑂 𝑗=1 𝑂 𝑗=1

ECON 551: Lecture 4a 29 of 36

Corlett-Hague (1953): How to tax three goods?

From our Euler lemma: 0 = 𝜁𝑗0 + 𝜁𝑗1 + 𝜁𝑗2 Re-write the Ramsey-Mirrlees index of discouragement result for this case: 𝜄 = ∑ 𝑢𝑗 𝑟𝑗 𝜁𝑙𝑗

𝑗

= ∑ 𝜐𝑗𝜁𝑙𝑗

𝑗

= {𝜐1𝜁11 + 𝜐2𝜁12 𝜐1𝜁21 + 𝜐2𝜁22 Set these two equal to each other and solve for the tax rates. 𝜐1 𝜐2 = 𝜁12 − 𝜁22 𝜁21 − 𝜁11 Remembering that cross-price elasticities sum to zero, we can rewrite this as: 𝜐1 𝜐2 = −(𝜁11 + 𝜁22)−𝜁10 −(𝜁11 + 𝜁22)−𝜁20 …which implies that if 𝜁10 > 𝜁20 (i.e. 𝜁20 is more negative than 𝜁10) then 𝜐1 < 𝜐2.

ECON 551: Lecture 4a 30 of 36

Corlett-Hague (1953): Implications

If 𝜁10 > 𝜁20 then 𝜐1 < 𝜐2.

• 1. Tax rate should be higher on the good more complementary with leisure. Think of this as an

‘indirect’ way of taxing leisure. (Or more generally, the untaxed good.)

• 2. If leisure is untaxed good, we could try to tax complements of leisure. E.g. vacations, fancy

cars.

• 3. If goods 1 and 2 are consumption now and in the future, and the future is retirement, then we

might want to tax more heavily consumption in the future. This suggests capital income taxation should be positive.

ECON 551: Lecture 4a 31 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 32 of 36

Back to the case for uniform taxation….

As a second application of the Ramsey-Mirrlees result, let’s re-examine the case for uniform taxation. Recall that Ramsey’s surprising conclusion was that uniform taxation was in general inferior to differentiated commodity taxes. However, does there exist a special case in which uniform commodity taxation is better? This is important to consider for two reasons: a) Many countries do have uniform taxation in the form of VATs. b) The uniform tax may mimic direct taxation, providing a nice link between direct and indirect taxes. To see more on this question, refer to Smart (2002); Baker et al. (2014).

ECON 551: Lecture 4a 33 of 36

When is uniform taxation optimal?

Proposition: If 𝜁𝑗0 = 𝜁

𝑘0 ∀𝑗, 𝑘 ∈ {1, … , 𝐿} then uniform taxation is optimal.

If all cross-price elasticities with leisure are equal, then uniform taxation is optimal. Lemma: 0 = ∑ 𝜁𝑙𝑗

𝐿 𝑗=0

⟹ 𝜁𝑙0 = − ∑ 𝜁𝑙𝑗

𝐿 𝑗=1

This is the Euler result we showed earlier extended to the many goods case. It will be used in the proof below.

ECON 551: Lecture 4a 34 of 36

When is uniform taxation optimal?

Proof: Start with the Ramsey-Mirrlees result, then show that equal cross-prices make it hold for uniform taxation: Impose uniform taxation 𝜐𝑗 = 𝛽 , ∀𝑗. ∑ 𝜐𝑗𝜁𝑙𝑗

𝑗

= 𝜄 ⟹ ∑ 𝛽𝜁𝑙𝑗

𝑗

= 𝜄 Using the lemma, substitute in for the sum: ∑ 𝛽𝜁𝑙𝑗

𝑗

= −𝛽𝜁𝑙0 = 𝜄 Since 𝜄 is constant, this only holds true if 𝜁𝑙0 is the same for all k.

ECON 551: Lecture 4a 35 of 36

When is uniform taxation optimal?

What this says: uniform commodity taxation is only optimal if the cross-price elasticity of all goods with the price of leisure(labour) is the same. Otherwise, you want to differentiate. More generally, the tradeoff here is how big the ‘one big distortion’ is compared to all the little

• nes.

Let’s review the path we’ve followed on uniform taxation: 1. Can’t tax all goods uniformly or else no revenue raised. (The Sandmo box.) 2. When one good not taxed, Ramsey showed that uniform taxation is not generally desirable for the taxed goods. 3. Ramsey-Mirrlees index of discouragement: taxation should affect goods’ demand equi-

• proportionately. Proportional effect on quantities, not prices.
• 4. Only if cross-price elasticities for all goods with leisure is equal does uniform taxation become
• ptimal.

ECON 551: Lecture 4a 36 of 36

Next time:

We will continue on commodity taxation:  consider the multi-household case.  Consider the production side.  Examine some empirical evidence. In advance, you could read the paper on soft drink taxation: P * Fletcher, Jason M., David E. Frisvold, and Nathan Tefft (2010), “The effects of soft drink taxes on child and adolescent consumption and weight outcomes,” Journal of Public Economics,

• Vol. 94, No. 11-12, pp. 967-974. Available online.