Tax Distortions

February 3, 2016

Tax distortion

I Economic efficiency requires that an extra unit is produced and traded if theconsumers incremental willingness to pay (MRS) > incremental (true) cost ofproducing (MRT ) the unit: Efficiency (maximum economic surplus) =)tradeuntil MRS = MRT .

I If taxes varies with economic behaviiour they will typically distort choices suchthat MRS 6= MRT or MRS differs between different group of consumers,because they are taxed differently (or MRT differs between different group ofproducers).

I This distortion makes it costly for the government to collect a tax revenue.Collecting a tax revue of T NOK costs (1 + �)T with � > 1. Another way toput it: Suppose consumers requires a cash transfer of CV to attain the samewelfare as they had before the tax: Excess burden EB =) CV � T > 0. Thereis a deadweight loss associated with taxation, an excess burden of taxation.

I The excess burden (EB) of taxation = the sum (integral) of the value (differencebetween MRS and MRT ) of all transactions that are not realized due to the tax.

Excess burden

I A tax affects behaviour (what we consume, how much wework, how much we save..) through two channels; by alteringrelative prices (substitution effect) and by reducing thepurchasing power of private income and wealth (incomeeffect).

I It is the substitution effect that creates an EB of taxation(even a lump sum tax creates a tax burden and implies lowerwelfare for the consumers)

I To measure EB we need to get rid of the income effects. Twoways to proceed.

I Assume quasi linear utilityI More general utility fn but work with Hicksian (compensated)

demand functions

EB in a simple case with quasi linear utility

I Two goods x and y .

Iu(x , y) = h(x) + y with h(x) increasing and concave

I Per unit tax on good x , the tax inclusive price is q = p + ⌧

Iy is untaxed and the price is normalized to 1.

I Consumers maximize u(x , y) s.t. y + (p + ⌧)x = z

I Demand function implicitly defined by h

0(x) = (p + ⌧)I Producers use c(s) units of good y to produce s units of

x ; c 0 > 0, c 00 > 0I profits ⇡ = ps � c(s) f.o.c. p = c

0(s(p)) with elasticity"S = @S

@ppS

EB in a simple case with quasi linear utilityillustration of the excess burden of taxation

When a tax is imposed the supply curve shifts upwards and there is a new after taxequilibirum. This shift creates a deadweight loss equal to ABC: Equal to the value(marginal benefit minus marginal cost) of all “trades” that do not take place.

EB in a simple case with quasilinear utilitymeasure excess burden

I EB = �12d⌧dQ

IdQ = S

0(p)dp

Idp =

⇣⌘D

⌘S�⌘D

⌘d⌧ to get

=)EB = �12

⇣⌘D⌘S⌘S�⌘D

⌘pQ(d⌧p )2

I Triangle ) EB ! 0 if d⌧ ! 0. There is no first order EBeffect of a very very small tax starting from a zero tax (theeconomic value of the transaction the tax preempt is ⇡ 0)

Two important lessons

1 EB increases with the square of the excise tax (more thanproportional), argument for a broad tax base.

Two important lessons

2 EB increases in the drop in demand caused by the tax, and thisdrop depends on the demand and supply elasticity. Higher taxes oninelastic tax bases.

Excess burden with income effectsI Use the Hicksian demand functions (demand for a product

when prices change but utility is fixed) and calculate sum ofthe economic surplus that is lost due to taxes.

The expenditure functionI Introduces a tax on good x , good y is the untaxed numeraire.

Suppose pretax price is fixed (totally elastic supply of good x).

I The increase in expenditures (income) that is needed to attainthe same level of satisfaction (utility) after a tax is imposed isgiven by

CV = E (p + ⌧, 1, u0)� E (p, 1, u0) = E (p + ⌧, 1, u0)� I

.

I After the tax (and wtih no compensation) the consumerobtains utility u

1. We can then measure the amount of incomewe can take away from the consumer to leave her on u

1 if weabolished the tax (the maximum amount the conusmer iswilling to pay in order to get rid of the tax)

EV = E (p + ⌧, 1, u1)� E (p, 1, u1) = I � E (p, 1, u1)

Excess burden and Hicksian demand functions

I EB based on CV: EBcv = CV � ⌧h(p + ⌧, 1, u0) =´ q1

q0@E(q,1,u0)

@q dq � ⌧h(p + ⌧, 1, u0)

I EB based on EV: EBEV = EV � ⌧h(p + ⌧, 1, u1)...

Excess burden and Hicksian demand functions

p"+"τ""

p""

h(p"+"τ,"1,"u0)"h(p"+"τ,"1,"u1)"

EBEV" EBCV"

x"x0(p,I)"x1(p+τ,CV)"

Three ways to represent the EB of taxation

Excess burden of an increase in the tax rateAn approximation

I Suppose a tax is levied on good 1, no other taxes in the systemI We have

EB(⌧) = E (p + ⌧, 1,U)� E (p, 1,U)� ⌧h1(p + ⌧, 1,U)

=) MEB = EB(⌧ +4⌧)� EB(⌧) ⇡ dEBd⌧ 4⌧ + 1

2d2EBd⌧2 (4⌧)2

I dEBd⌧ = h1 � h1 � ⌧ @h1

@⌧

I d2EBd⌧2 = �@h1

@⌧ � ⌧ @h21

@⌧2

I Assume linear hicksian =)MEB = �⌧ (4⌧) @h1@⌧ � 1

2@h1@⌧ (4⌧)2

I if the preexisting tax is zero =) MEB is the HarbergerTriangle.

I If there is a tax before the additional tax (4⌧) is imposed =)4⌧ has a first order effect on EB (MEB does not ! 0 as4⌧ ! 0.)

Excess burden of labour income taxationI

Can analyze the EB of labour taxation in a two goods economy

(consumption and leisure), leisure is the untaxed good

I u(l , c) where l is leisure and c is consumption of goods and service: (and

H = time that can be divided between leisure and work)

max u(l , c) s.t. w(1 � ⌧)(H � l) = c

Ithe EB associated with a labour income tax is again the sum of the value

that is lost by the fact that a labour income tax leads to a substitution

towards leisure even though the value of the lost production is larger than

the value of the extra leisure (the surplus of a labour transaction is the

difference between the marginal product of labour and the marginal value

of leisure).

Iwe can represent EB in the same three ways as above (one of the seminar

questions ask you to do this).

Inote that in a one period model a tax on income is just like a

general consumption tax (value added tax for example)

Ia labour income tax that does not change the number of hours

individuals want to work (substitution effect + income affect =)unaltered labour supply) will have an EB associated with it: it is the

substitution effect that creates an efficiency loss.

Excess burden of a tax on returns to savings (interest rates)

IAgain the EB of a tax on savings can be analyzed in a two good model,

where consumption in this period and in the next period are the two

goods.

max u(c1, c2) s.t. (z1 � c1) (1 + r)(1 � ⌧) + z2 = c2

IThere is an EB of this tax if a tax on savings induce consumers to

substitute towards more consumption today.

IAgain, even if a saving tax does not change savings behavior, there is still

an excess burden associated with this tax since a tax drives a wedge

between MRT (interest rate) and the consumers marginal rate of

substitution between the periods. (Very well discussed in Stiglitz

(Taxation and economic efficiency)

Excess burden when several goods are taxedWhat is the effect of introducing a tax on good i if a number of other goods are already

taxed?

I Change in Excess burden of increasing a tax on good i when other goods are taxedMEB = EB(⌧i + 4⌧i , ⌧�i ) � EB(⌧i , ⌧�i ) is given (approximately) by

@EB(⌧ )

@⌧i4⌧i +

1

2

@2EB(⌧ )

@⌧2i

4⌧2i

)@EB(⌧ )

@⌧i=

@E(p0 + ⌧ ,U)

@⌧i�

@⇣Pn

j=1 ⌧j hj (p0 + ⌧ ,U)⌘

@⌧i

= hi �

0

@hi +nX

j=1⌧j

@hj (p0 + ⌧ ,U)

@⌧i

1

A

= �⌧i@hi (p0 + ⌧ ,U)

@⌧i�

nX

j=16=i

⌧j@hj (p0 + ⌧ ,U)

@⌧i

)@2EB(⌧ )

@⌧2i

= �@hi (p0 + ⌧ ,U)

@⌧i� ⌧i

@2hi (p0 + ⌧ ,U)

@⌧2i

�nX

j=16=i

⌧j@2hj (p0 + ⌧ ,U)

@⌧2i

Excess burden when several goods are taxed

I Assuming no tax on good i initially

@EB

@⌧i

��⌧i=0 = �nX

j=16=i

⌧j@hj (p0 + ⌧ ,U)

@⌧i

@EB

@⌧2i

��⌧i=0 = �@hi (p0 + ⌧ ,U)

@⌧i�

nX

j=16=i

⌧j@2hj (p0 + ⌧ ,U)

@⌧2i

I We can now approximate the change EB if a tax on good i is introduced byusing a Taylor approximation of the EB function around ⌧i = 0.

Excess burden when several goods are taxed

EB(⌧i , ⌧�i ) � EB(0, ⌧�i ) ⇡@EB

@⌧i

���⌧i=0 (⌧i � 0) +1

2

@EB

@⌧2i

���⌧i=0 (⌧i � 0)2

⇡ �nX

j=16=i

⌧i⌧j@hj (p0 + ⌧ ,U)

@⌧i+

1

2

2

6664�

@hi (p0 + ⌧ ,U)

@⌧i�

nX

j=16=i

⌧j@2hj (p0 + ⌧ ,U)

@⌧2i

3

7775⌧2i

⇡ �1

2

@hi (p0 + ⌧ ,U)

@⌧i

!⌧2i �

nX

j=16=i

⌧i⌧j@hj (p0 + ⌧ ,U)

@⌧i

I Key observation; introducing a tax on good i has a second order effect on EB in “own market”but a first order effect on the EB in the other markets: If good j is a substitute for good iconsumer will substitute towards good j as good i gets taxed and this will have a first ordereffect of the taxes collected from the market for commodity j

I If there is a labour tax present EB burden of taxing a consumer good i will be minimized if thegovernment imposes the tax on a good that is complementary to leisure (since it will induceagents to work more). (Corrlet & Hague (1953))