lecture 1: tax avoidance and excess burden · i “real” v. “accounting” responses...

Lecture 1: Tax avoidance and excess burden

Michael Smart

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

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 avoidancesufficient statistics for policy analysis

I “real” v. “accounting” responses

alternative approaches to estimation

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 avoidancesufficient statistics for policy analysis

I “real” v. “accounting” responses

alternative approaches to estimation

Today’s lecture: Tax avoidance and theory of excess burden

Outline:

1 consumer surplus and excess burden2 equivalent variation and excess burden3 analytical results4 approximation formulas

Consumer surplus

Consider a single consumer with demand for a single good x(q, I). Weseek a monetary measure of the change in consumer welfare resultingfrom a price increase from q0 to q1.

Define

∆CS =

q0x(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 topurchase at p.

Consumer surplus

Consider a single consumer with demand for a single good x(q, I). Weseek a monetary measure of the change in consumer welfare resultingfrom a price increase from q0 to q1.

Define

∆CS =

q0x(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 topurchase at p.

Consumer surplus and excess burden

��

XXX1 0

Revenue

Excess burden1 + t

Notice that ∆CS includes the additional revenue generated by the priceincrease, which is a transfer.

A better measure is therefore the excess burden of the price change:

EBm = ∆CS − (q1 − q0)x(q1, I)

Consumer surplus and excess burden

��

XXX1 0

Revenue

Excess burden1 + t

Notice that ∆CS includes the additional revenue generated by the priceincrease, which is a transfer.

A better measure is therefore the excess burden of the price change:

EBm = ∆CS − (q1 − q0)x(q1, I)

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 toincrease revenue by $50 million, reduce ridership by 11.5 million, andreduce operating costs by $9 million. What is EB?

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 toincrease revenue by $50 million, reduce ridership by 11.5 million, andreduce operating costs by $9 million. What is EB?

Solving for EB

∆EB ≈ −∆X (2.50 − c)

where c is marginal cost, estimated to be $0.78 per trip. So

∆EB ≈ 1.72 × 11.5 = 19.78

To gain $50 million in revenue, the TTC created $20 million in excess burden, or40 cents per dollar of marginal revenue.

Solving for EB

∆EB ≈ −∆X (2.50 − c)

where c is marginal cost, estimated to be $0.78 per trip. So

∆EB ≈ 1.72 × 11.5 = 19.78

To gain $50 million in revenue, the TTC created $20 million in excess burden, or40 cents per dollar of marginal revenue.

Marshallian EB: Pro and ConAdvantages of the Marshallian approach:

1 Requires minimal preference information x(q, I)2 Aggregates across consumers with only market demand information

X (q) =∑

h xh(q, Ih)

Disadvantages of the Marshallian approach:

1 Includes income effects of price changes – which even lump-sumtaxes would have

2 What if many prices are changing? Line integrals like

EBm(π) =∑π(i)

∫p1i

x∗i (p, y)dpi − R

are generally path-dependent3 Distributional insensitivity

Marshallian EB: Pro and ConAdvantages of the Marshallian approach:

1 Requires minimal preference information x(q, I)2 Aggregates across consumers with only market demand information

X (q) =∑

h xh(q, Ih)

Disadvantages of the Marshallian approach:

1 Includes income effects of price changes – which even lump-sumtaxes would have

2 What if many prices are changing? Line integrals like

EBm(π) =∑π(i)

∫p1i

x∗i (p, y)dpi − R

are generally path-dependent3 Distributional insensitivity

Equivalent variationEV measures consumer’s willingness to pay (as a lump-sum tax) for theright to purchase x at q0 instead of q1.

Let U1 be utility at prices (q1, 1), and let (x , y) be the cheapest way toattain U1 if prices are (q0, 1). Then

EV = (p0x0 + y0) − (p0x + y) = I − (p0x + y)

Let U1 be utility at prices (q1, 1), and let (x , y) be the cheapest way toattain U1 if prices are (q0, 1).

EV = (p0x0 + y0) − (p0x + y) = I − (p0x + y)

EV and the expenditure functionWe 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-changeutility at pre-change prices. So

EV = e(p0, u0) − e(p0, u1)

EV is a money-metric index of the utility change.

But if I is (fixed) lump-sum income then

I = e(p0, u0) = e(p1, u1)

SoEV = I − e(p0, u1)

= e(p1, u1) − e(p0, u1)

EV is an exact price index for the price change, at post-change utility u1.

e(p, u) = min{p · x : U(x) > u}

EV = e(p0, u0) − e(p0, u1)

I = e(p0, u0) = e(p1, u1)

= e(p1, u1) − e(p0, u1)

EV is an exact price index for the price change, at post-change utility u1.

e(p, u) = min{p · x : U(x) > u}

EV = e(p0, u0) − e(p0, u1)

I = e(p0, u0) = e(p1, u1)

= e(p1, u1) − e(p0, u1)

EV is an exact price index for the price change, at post-change utility u1.Michael Smart (UToronto) Lecture 1: Tax avoidance and excess burden 10 / 14

Measuring EB from compensated demands

The indifference curve analysis makes clear that distortionary effects ofthe tax result from substitution effects on demands alone: Any taxincluding 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(q1, u1) − e(q0, u1) =

∫L(q0,q1)

xi(p, u)dp

where L(q0, q1) is any line integral.

Measuring EB from compensated demands

The indifference curve analysis makes clear that distortionary effects ofthe tax result from substitution effects on demands alone: Any taxincluding 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(q1, u1) − e(q0, u1) =

∫L(q0,q1)

xi(p, u)dp

where L(q0, q1) is any line integral.

In one dimension, we can see how EV traces out the area under acompensated demand curve:

δ X C

≅∆ t

Slope = P > P01

Slope = P

X (P, U )C 1

0=P +tP

EB=EV−R

Approximating EB

Consider a change in prices from q0 = 1 to q1 = 1 + t. Excess burden:

EB(t, u) = e(1 + t, u) − e(1, u) −∑

tixci (1 + t, u)

Marginal excess burden is

∂EB(t, u)∂ti

=∂e∂ti

− xci −

tj∂xc

j (1 + t, u)

∂ti= −

tj∂xc

j (1 + t, u)

A useful approximation:

EB(t, u) ≈∑

ti∂EB(1

2 t, u)∂ti

= −∑

∂xcj

∂ti≡ −

t ′St

Approximating EB

EB(t, u) = e(1 + t, u) − e(1, u) −∑

tixci (1 + t, u)

∂EB(t, u)∂ti

=∂e∂ti

− xci −

tj∂xc

j (1 + t, u)

∂ti= −

tj∂xc

j (1 + t, u)

EB(t, u) ≈∑

ti∂EB(1

2 t, u)∂ti

= −∑

∂xcj

∂ti≡ −

t ′St

Approximating EB

EB(t, u) = e(1 + t, u) − e(1, u) −∑

tixci (1 + t, u)

∂EB(t, u)∂ti

=∂e∂ti

− xci −

tj∂xc

j (1 + t, u)

∂ti= −

tj∂xc

j (1 + t, u)

EB(t, u) ≈∑

ti∂EB(1

2 t, u)∂ti

= −∑

∂xcj

∂ti≡ −

t ′St

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

Figure : EB in multiple markets

∆EB ≈ −12(tg)2 ∂X c

∂pg− tr tg

∂X cr

∂pg< 0 for tg small

∆EB ≈ −12(tg)2 ∂X c

∂pg− tr tg

∂X cr

∆EB ≈ −12(tg)2 ∂X c

∂pg− tr tg

∂X cr

lecture 1: tax avoidance and excess burden · i “real” v. “accounting” responses...

Documents