Lecture 1: Tax avoidance and excess burden

Transcription

1 Lecture 1: Tax avoidance and excess burden Michael Smart Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 1 / 14

2 Introduction Understanding avoidance responses is a key element to analysis of tax (and other regulatory) policies: revenue forecasting understanding the equity efficiency tradeoff program evaluation and optimal policy design Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 2 / 14

3 Introduction Understanding avoidance responses is a key element to analysis of tax (and other regulatory) policies: revenue forecasting understanding the equity efficiency tradeoff program evaluation and optimal policy design In these lectures: Measuring excess burden from avoidance sufficient statistics for policy analysis real v. accounting responses alternative approaches to estimation Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 2 / 14

4 Today s lecture: Tax avoidance and theory of excess burden Outline: 1 consumer surplus and excess burden 2 equivalent variation and excess burden 3 analytical results 4 approximation formulas Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 3 / 14

5 Consumer surplus Consider a single consumer with demand for a single good x(q, I). We seek a monetary measure of the change in consumer welfare resulting from a price increase from q 0 to q 1. Define CS = q 1 q 0 x(q, I)dq as the change in Marshalian consumer surplus from the reform. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 4 / 14

6 Consumer surplus Consider a single consumer with demand for a single good x(q, I). We seek a monetary measure of the change in consumer welfare resulting from a price increase from q 0 to q 1. Define CS = q 1 q 0 x(q, I)dq as the change in Marshalian consumer surplus from the reform. Intuition: Recall that x 1 (X, I) represents marginal willingness to pay for X. So p x(q, I)dq represents total willingness to pay for right to purchase at p. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 4 / 14

7 Consumer surplus and excess burden p Revenue 1 + t Excess burden 1 X(p) X 1 X 0 X

8 Consumer surplus and excess burden p Revenue 1 + t Excess burden 1 X(p) X 1 X 0 X Notice that CS includes the additional revenue generated by the price increase, which is a transfer. A better measure is therefore the excess burden of the price change: EB m = CS (q 1 q 0 )x(q 1, I)

9 Application: A subway fare increase In January 2010, the Toronto subway increased the price of a trip from $2.25 to $2.50. Budget documents show that the anticipated impact of the reform was to increase revenue by $50 million, reduce ridership by 11.5 million, and reduce operating costs by $9 million. What is EB?

10 Application: A subway fare increase In January 2010, the Toronto subway increased the price of a trip from $2.25 to $2.50. Budget documents show that the anticipated impact of the reform was to increase revenue by $50 million, reduce ridership by 11.5 million, and reduce operating costs by $9 million. What is EB? fare p+t p p c c D(p) X Trips

11 Solving for EB EB X(2.50 c) where c is marginal cost, estimated to be $0.78 per trip. So EB = Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 7 / 14

12 Solving for EB EB X(2.50 c) where c is marginal cost, estimated to be $0.78 per trip. So EB = To gain $50 million in revenue, the TTC created $20 million in excess burden, or 40 cents per dollar of marginal revenue. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 7 / 14

13 Marshallian EB: Pro and Con Advantages of the Marshallian approach: 1 Requires minimal preference information x(q, I) 2 Aggregates across consumers with only market demand information X(q) = h x h(q, I h ) Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 8 / 14

14 Marshallian EB: Pro and Con Advantages of the Marshallian approach: 1 Requires minimal preference information x(q, I) 2 Aggregates across consumers with only market demand information X(q) = h x h(q, I h ) Disadvantages of the Marshallian approach: 1 Includes income effects of price changes which even lump-sum taxes would have 2 What if many prices are changing? Line integrals like EB m (π) = π(i) p 1 i p 0 i x i (p, y)dp i R are generally path-dependent 3 Distributional insensitivity Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 8 / 14

15 Equivalent variation EV measures consumer s willingness to pay (as a lump-sum tax) for the right to purchase x at q 0 instead of q 1. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 9 / 14

16 Equivalent variation EV measures consumer s willingness to pay (as a lump-sum tax) for the right to purchase x at q 0 instead of q 1. Let U 1 be utility at prices (q 1, 1), and let (ˆx, ŷ) be the cheapest way to attain U 1 if prices are (q 0, 1). Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 9 / 14

17 Equivalent variation EV measures consumer s willingness to pay (as a lump-sum tax) for the right to purchase x at q 0 instead of q 1. Let U 1 be utility at prices (q 1, 1), and let (ˆx, ŷ) be the cheapest way to attain U 1 if prices are (q 0, 1). Then EV = (p 0 x 0 + y 0 ) (p 0ˆx + ŷ) = I (p 0ˆx + ŷ) Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 9 / 14

18 Equivalent variation EV measures consumer s willingness to pay (as a lump-sum tax) for the right to purchase x at q 0 instead of q 1. Let U 1 be utility at prices (q 1, 1), and let (ˆx, ŷ) be the cheapest way to attain U 1 if prices are (q 0, 1). Then EV = (p 0 x 0 + y 0 ) (p 0ˆx + ŷ) = I (p 0ˆx + ŷ) Y E v R U X 1 X^ X 0 X Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 9 / 14

19 EV and the expenditure function We can generalize this using the consumer expenditure function e(p, u) = min{p x : U(x) u} EV is the change in lump-sum income required to attain the post-change utility at pre-change prices. So EV = e(p 0, u 0 ) e(p 0, u 1 ) EV is a money-metric index of the utility change. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 10 / 14

20 EV and the expenditure function We can generalize this using the consumer expenditure function e(p, u) = min{p x : U(x) u} EV is the change in lump-sum income required to attain the post-change utility at pre-change prices. So EV = e(p 0, u 0 ) e(p 0, u 1 ) EV is a money-metric index of the utility change. But if I is (fixed) lump-sum income then I = e(p 0, u 0 ) = e(p 1, u 1 ) Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 10 / 14

21 EV and the expenditure function We can generalize this using the consumer expenditure function e(p, u) = min{p x : U(x) u} EV is the change in lump-sum income required to attain the post-change utility at pre-change prices. So EV = e(p 0, u 0 ) e(p 0, u 1 ) EV is a money-metric index of the utility change. But if I is (fixed) lump-sum income then I = e(p 0, u 0 ) = e(p 1, u 1 ) So EV = I e(p 0, u 1 ) = e(p 1, u 1 ) e(p 0, u 1 ) EV is an exact price index for the price change, at post-change utility u 1. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 10 / 14

22 Measuring EB from compensated demands The indifference curve analysis makes clear that distortionary effects of the tax result from substitution effects on demands alone: Any tax including a lump-sum tax would have comparable income effects. Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 11 / 14

23 Measuring EB from compensated demands The indifference curve analysis makes clear that distortionary effects of the tax result from substitution effects on demands alone: Any tax including a lump-sum tax would have comparable income effects. So EV measure is integral under the (Hicksian) compensated demand, instead of the regular Marshallian demand. EV = e(q 1, u 1 ) e(q 0, u 1 ) = ˆx i (p, u)dp where L(q 0, q 1 ) is any line integral. L(q 0,q 1 ) i Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 11 / 14

24 In one dimension, we can see how EV traces out the area under a compensated demand curve: Y EV Slope = P 1 > P 0 Slope = P 0 U 1 1 P 0 P =P +t EB=EV R X P 0 X 1 X c t δ X C δ P X^ C 1 X (P, U ) X Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 12 / 14

25 Approximating EB Consider a change in prices from q 0 = 1 to q 1 = 1 + t. Excess burden: EB(t, u) = e(1 + t, u) e(1, u) t i x c i (1 + t, u) Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 13 / 14

26 Approximating EB Consider a change in prices from q 0 = 1 to q 1 = 1 + t. Excess burden: EB(t, u) = e(1 + t, u) e(1, u) t i x c i (1 + t, u) Marginal excess burden is EB(t, u) t i = e xi c t i j xj c (1 + t, u) t j = t i j t j x c j (1 + t, u) t i Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 13 / 14

27 Approximating EB Consider a change in prices from q 0 = 1 to q 1 = 1 + t. Excess burden: EB(t, u) = e(1 + t, u) e(1, u) t i x c i (1 + t, u) Marginal excess burden is EB(t, u) t i = e xi c t i j xj c (1 + t, u) t j = t i j t j x c j (1 + t, u) t i A useful approximation: EB(t, u) i EB( 1 2t, u) t i = t i i j t j xj c t i 2 t i 1 2 t St Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 13 / 14

28 Exercise: EB with pre-existing taxes A consumer supplies labour and buys gin and rum. Show that, if t r > 0, then introducing a small tax on gin causes excess burden to fall.

29 Exercise: EB with pre-existing taxes A consumer supplies labour and buys gin and rum. Show that, if t r > 0, then introducing a small tax on gin causes excess burden to fall. 1+tr 1+tg 1 c1 X r 1 c X g c0 X r 0 1 X X g g 1 X g 0 Xg Figure : EB in multiple markets

30 Exercise: EB with pre-existing taxes A consumer supplies labour and buys gin and rum. Show that, if t r > 0, then introducing a small tax on gin causes excess burden to fall. 1+tr 1+tg 1 c1 X r 1 c X g c0 X r 0 1 X X g g 1 X g 0 Xg Figure : EB in multiple markets EB 1 2 (t g) 2 X c g p g t r t g X c r p g < 0 for t g small