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 5b: Optimal Income Taxation, Part II ECON 551: Lecture 5b 1 of 41

Agenda: 1. Indirect vs direct taxation 2. Atkinson-Stiglitz Theorem 3. Joint income taxation ECON 551: Lecture 5b 2 of 41

Indirect and Direct Taxation John Stuart Mill (1806-1873) ‘direct taxes’ those that were levied on the peo ple who actually  must pay them.  ‘Indirect taxes’ on the other hand, could be passed onto other people through higher prices.  This definition is far from satisfactory, because for many taxes some part of the burden can be shared. This is true for both income and commodity taxes. A better definition is given by Atkinson and Stiglitz (p. 427): “Direct taxes may be adjusted to the individual characteristics of the taxpayer, whereas indirect taxes are levied on transactions irrespective of the circumstances of the buyer and seller.” Even this falls apart: ECON 551: Lecture 5b 3 of 41

 Point of sale reductions in sales tax for ‘children’s clothes’ in BC’s PST. ECON 551: Lecture 5b 4 of 41

Indirect and Direct Taxation Which is better? The debate about which to use – or both – has persisted. Here is an excerpt from the British House of Commons debate, spoken by William Gladstone, four-time Prime Minister in the 19 th Century: “I never can think of direct and indirect taxation except as I should think of two attractive sisters . .. differing only as sisters may differ. I cannot conceive any reason why there should be unfriendly rivalry between the admirers of these two damsels. I have always thought it not only allowable, but even an act of duty, to pay my address to t hem both.” ECON 551: Lecture 5b 5 of 41

Indirect and Direct Taxation Arguments one way or the other include:  Two instruments; two targets. Use direct taxation for equity and indirect for efficiency.  Direct taxation would be swell, but it requires a high level of administrative sophistication. So, indirect taxation is desirable because it is easier to administer.  Indirect taxes are undesirable because they distort relative prices whereas direct taxes don’t. This is only true if labour is inelastically supplied, of course! ECON 551: Lecture 5b 6 of 41

Agenda: 1. Indirect vs direct taxation 2. Atkinson-Stiglitz Theorem 3. Joint income taxation ECON 551: Lecture 5b 7 of 41

The Atkinson-Stiglitz Theorem We will approach the question systematically, building on the work of Atkinson and Stiglitz (1976).  These authors developed an eponymously named theorem to describe the circumstances in which direct and indirect taxation should be chosen.  Good background reading on this topic includes the A&S paper itself, as well as the Boadway and Pestieau article on the reading list. o B&P provide a nice summary of advances in the area since A&S published their original work. ECON 551: Lecture 5b 8 of 41

“The result is arguably the most relevant result for policy purposes to emerge from the optimal income tax literature initiated by Mirrlees (1971)” ECON 551: Lecture 5b 9 of 41

Notation: Two types: 𝑗 ∈ {1,2} Indexes ability. So, type 2 doesn’t have to work as hard…  𝑜 𝑗 size of population of type i.  𝑑 𝑗 disposable income.  x and z goods.  𝑚 𝑗 labour income  𝑥 𝑗 wage: 𝑥 1 < 𝑥 2  𝑧 𝑗 earned income, 𝑧 = 𝑥𝑚 .  𝑣(𝑕(𝑦, 𝑨), 𝑚) direct utility — note separability in labour/leisure.  ℎ(𝑟, 𝑑 𝑗 ) indirect utility of household i over goods  t tax on good z ; no tax on good x .  q tax inclusive price of z . ECON 551: Lecture 5b 10 of 41

if 𝑢 = 0 , then we should use income taxation since that implies tax on z and x should be the same ⟷ uniform commodity tax. ECON 551: Lecture 5b 11 of 41

Planner’s objective: Maximize social utility (using utilitarian SWF) subject to revenue constraint and self- selection constraint. (ℎ(𝑟, 𝑑 𝑗 ), 𝑧 𝑗 𝑑 1 ,𝑑 2 ,𝑟 ∑ 𝑜 𝑗 𝑤 𝑗 max 𝑥 𝑗 ) 𝑗=1,2 subject to revenue constraint: ∑ 𝑜 𝑗 [(𝑧 𝑗 − 𝑑 𝑗 ) + 𝑢𝑨 𝑗 (𝑟, 𝑑 𝑗 )] ≥ 𝑆 𝑗=1,2 (multiplier λ) And selection constraint: 𝑤 2 (ℎ(𝑟, 𝑑 2 ), 𝑧 2 𝑥 2 ) ≥ 𝑤 2 (ℎ(𝑟, 𝑑 1 ), 𝑧 1 𝑥 1 ) (multiplier γ) For the selection constraint, the utility to type 2 must be higher when he puts in more effort than when he puts in less and mimics the type 1. ECON 551: Lecture 5b 12 of 41

ECON 551: Lecture 5b 13 of 41

Set up the Lagrangian and solve (ℎ(𝑟, 𝑑 𝑗 ), 𝑧 𝑗 𝑥 𝑗 ) + 𝜇 [ ∑ 𝑜 𝑗 [(𝑧 𝑗 − 𝑑 𝑗 ) + 𝑢𝑨 𝑗 (𝑟, 𝑑 𝑗 )] ℒ = ∑ 𝑜 𝑗 𝑤 𝑗 ] 𝑗=1,2 𝑗=1,2 + 𝛿 [𝑤 2 (ℎ(𝑟, 𝑑 2 ), 𝑧 2 𝑥 2 ) − 𝑤 2 (ℎ(𝑟, 𝑑 1 ), 𝑧 1 𝑥 1 )] FOCs wrt 𝑑 1 , 𝑑 2 , 𝑟 : 1 − 𝜇𝑜 1 (1 − 𝑢 𝜖𝑨 1 2 = 0 ̂ 𝑑 1 ℎ 𝑑 2 ℎ 𝑜 1 𝑤 ℎ 𝜖𝑑 1 ) − 𝛿𝑤 ̂ ℎ (We use the ‘hats’ for a type 2 mimicking a t ype 1. Which is true with 𝛿 > 0 . 2 − 𝜇𝑜 2 (1 − 𝑢 𝜖𝑨 2 2 = 0 2 ℎ 𝑑 2 ℎ 𝑑 𝑜 2 𝑤 ℎ 𝜖𝑑 2 ) + 𝛿𝑤 ℎ + 𝜇 ∑ 𝑜 𝑗 (𝑨 𝑗 + 𝑢 𝜖𝑨 𝑗 𝑗 ℎ 𝑟 2 − 𝑤 ̂ 𝑟 2 ℎ 𝑟 2 ℎ 𝑗 2 ) = 0 ∑ 𝑜 𝑗 𝑤 ℎ 𝜖𝑟 ) + 𝛿(𝑤 ℎ ̂ ℎ 𝑗=1,2 𝑗=1,2 ECON 551: Lecture 5b 14 of 41

ECON 551: Lecture 5b 15 of 41

Combine the FOCs and solve Start by multiplying the first two FOCs by 𝑨 1 and 𝑨 2 . Also remember how to use 𝜖𝑤𝑗 𝑗 ℎ 𝑟 𝑗 = −𝑨 𝑗 ℎ 𝑑 𝑗 . 𝜖𝑟 Roy’s identity here: = 𝑗 = −𝑨 𝑗 . So, ℎ 𝑟 𝜖𝑤𝑗 ℎ 𝑑 𝜖𝑑𝑗 Using all this, we can plug the first two FOCs into the first term of the third: −𝜇𝑜 1 𝑨 1 (1 − 𝑢 𝜖𝑨 1 2 𝑨 1 − 𝜇𝑜 2 𝑨 2 (1 − 𝑢 𝜖𝑨 2 ̂ 𝑑 2 ℎ 2 ℎ 𝑑 2 𝑨 2 + 𝜖𝑑 1 ) − 𝛿𝑤 ̂ ℎ 𝜖𝑑 2 ) + 𝛿𝑤 ℎ 𝜇 ∑ 𝑜 𝑗 (𝑨 𝑗 + 𝑢 𝜖𝑨 𝑗 2 − 𝑤 ̂ 𝑟 2 ℎ 𝑟 2 ℎ 2 ) = 0 𝜖𝑟 ) + 𝛿(𝑤 ℎ ̂ ℎ 𝑗=1,2 Let’s move lamdas to the left and gammas to the right…. ECON 551: Lecture 5b 16 of 41

ECON 551: Lecture 5b 17 of 41

−𝜇𝑜 1 𝑨 1 (1 − 𝑢 𝜖𝑨 1 𝜖𝑑 1 ) − 𝜇𝑜 2 𝑨 2 (1 − 𝑢 𝜖𝑨 2 𝜖𝑑 2 ) + 𝜇 ∑ 𝑜 𝑗 (𝑨 𝑗 + 𝑢 𝜖𝑨 𝑗 2 − 𝑤 ̂ 𝑑 ̂ 𝑟 2 ℎ 2 𝑨 1 − 𝛿𝑤 ℎ 2 ℎ 𝑑 2 𝑨 2 − 𝛿(𝑤 ℎ 2 ℎ 𝑟 2 ℎ 2 ) 𝜖𝑟 ) = 𝛿𝑤 ̂ ℎ ̂ ℎ 𝑗=1,2 Collect terms ; use Roy’s identity again on RHS… 𝜇𝑢 ∑ 𝑜 𝑗 [(𝜖𝑨 𝑗 𝜖𝑨 𝑗 ̂ 𝑑 2 ℎ 2 (𝑨 1 − 𝑨̂ 2 ) − (𝑤 ℎ 2 ℎ 𝑑 2 𝑨 2 + 𝑤 ℎ 2 ℎ 𝑟 2 )] 𝜖𝑟 ) + 𝑨 𝑗 𝜖𝑑 𝑗 ] = 𝛿[𝑤 ̂ ℎ 𝑗=1,2 One more application of Roy’s identity on RHS pushes last term to zero… 𝜇𝑢 ∑ 𝑜 𝑗 [(𝜖𝑨 𝑗 𝜖𝑨 𝑗 ̂ 𝑑 2 ℎ 2 (𝑨 1 − 𝑨̂ 2 )] 𝜖𝑟 ) + 𝑨 𝑗 𝜖𝑑 𝑗 ] = 𝛿[𝑤 ̂ ℎ 𝑗=1,2 Note that on the LHS, the 2 nd term inside the square bracket is compensation for a price 𝜖𝑨 𝑗 change. (If there were only an income effect from a tax change, 𝑨 𝑗 𝜖𝑑 𝑗 would put you right back where you started. Use the Slutsky equation … .) ECON 551: Lecture 5b 18 of 41

ECON 551: Lecture 5b 19 of 41

Atkinson and Stiglitz result: Let’s call 𝑨̃ 𝑗 the compensated demand for good i ….and we’re done. 𝑢 ∑ 𝑜 𝑗 [(𝜖𝑨̃ 𝑗 = 𝛿 ̂ 𝑑 2 ℎ 2 (𝑨 1 − 𝑨̂ 2 )] 𝜖𝑟 )] 𝜇 [𝑤 ̂ ℎ 𝑗=1,2  Since type 2 is mimicking type 1 at 𝑨̂ 2 , incomes are the same. Type 2 gets more leisure, but they report same income.  If incomes are the same, and if leisure is separable from consumption, then both types will choose the exact same consumption; 𝑨 1 = 𝑨̂ 2 . The RHS collapses to zero, so therefore t must be zero. We don’t impose a commodity tax; just income tax.  Recall that the RHS is negative (own price elasticities…) so that means if 𝑨 1 < 𝑨̂ 2 then 𝑢 > 0 . ECON 551: Lecture 5b 20 of 41

 If leisure and z are complements, then mimicker will likely choose more z than type 1. Why? Because he has more leisure when they both have the same income since he is higher ability. o So 𝑨 1 < 𝑨̂ 2 and 𝑢 > 0 . ECON 551: Lecture 5b 21 of 41

Implications  If leisure and goods are separable, then we should tax income only. Commodity taxation is useless. o Mimickers are earning and consuming the same as low types, so planner just can’t see any way to distinguish them.  Saez (2002) and others have re-examined the case. For example, heteroegenous preferences will generate 𝑢 > 0 if high types prefer the good.  Big implications for capital income taxation. If z is future consumption, we will only want to tax z different than x if z is complementary to leisure. Is retirement consumption complementary to leisure? In Saez’s model, do high types have higher preference for saving or future consumption?  But why do we see both direct and indirect coexisting in the real world? Administration, ignorance, factors outside the model, not separable, income taxes aren’t optimal so A - S doesn’t apply. All possible explanations. (This is from Boadway and Pestieau.) ECON 551: Lecture 5b 22 of 41