EC325 Public Economics Notes

Public Economics Notes LT
Anya Shah
2023

,1 Tax Incidence
Incidence is a term defining who the tax actually affects.
 Who receives the money is actually meaningless when it comes to
measuring who actually benefits the most.
o When low-income people receive housing benefits, landlords
have an incentive to increase rent so they capture some of
these benefits.
o Earned Income Tax Credit (EITC); subsides poor people who
work – but this may cause employers to keep wages low.

1.1 Partial Equilibrium Incidence
Model Set-Up:
 Two goods: x and y (background good – numeraire = 1)
 Per-unit tax on good x q= p+t
o q  effective buyer’s price (price consumers pay)
o p  seller’s price (price sellers receive and keep)
o Ad Valorem (q=( 1+τ ) p ¿ and excise tax (q= p+t ¿ are equiva-
lent in a competitive case
 Market for x is small to minimize income effects affecting demand for
goods in other markets
 No close substitutes and complements  rule out spillovers into other
markets
 Tax revenue not spent on taxed good  to avoid feedback effects
 Perfect Competition
Demand:
 Consumer have Quasi-Linear Utility  u ( x , y )=u ( x ) + y and
wealth z
 Total demand  D ( q ) = ∑ x (q)
 Price elasticity of demand: ε D = ( dDdq )( Dq( q ) )
o q  after-tax price for buyers
o Percentage change in total demand when there is a 1%
change in price.
o Change in total demand with respect to after-tax buyers price
o q= p+t  tax levied on consumers
Supply:
 Firms maximize profits  π= pS−c ( S )
 Total Supply  p S

, dS p
 Price elasticity of supply: ε S=( )( )
dp S ( p )
o Percentage change in total supply when there is a 1% change
in price.
o p=q−t  tax levied on producers


Equilibrium:
 D ( q ) =pS  demand = supply
 D ( p ( t ) +t )=S( p ( t ) )
o Total demand is based on buyer’s price q
dD
dp dp ε
 = = D
dt dS dD ε S −ε D
−
dp dp
dp
o  effect of tax on seller’s price: incidence on sellers (if this
dt
is -1 then producers bear the full burden of the tax)
o Gives us demands share of the overall response
o If demand is inelastic, ε D would be small so incidence on sell-
ers is small as we would expect.
o Being elastic as an agent is beneficial as you can avoid
the burden.

dD
dp dp ε
Prove that = = D holds:
dt dS dD ε S −ε D
−
dp dp
 We start with demand = supply  Qd =Qs
 Qd=D ( q )=D (p +t) and Qs=S (p)
 Therefore, D ( q ) =S ( p)
dD dq dS dp
 Derivative of this expression  · = ·
dq dt dp dt
dq
o  total derivative of argument with respect to the tax
dt
dq dp dt dp
 Now take the derivative of q= p+t  = + = +1 and sub that
dt dt dt dt
into the above.

 · (
dD dp
dq dt )
dS dp
+1 = ·
dp dt

dD dp dD dS dp
· + = ·
dq dt dq dp dt
dD dS dp dD dp dD dp dS dD
 = · − ·  = ( − )
dq dp dt dq dt dq dt dp dq

, dD
dp dq q
 =  now multiply all parts by
dt dS dD D(q)
−
dp dq
dD q
·
dp dq D ( q )
 =
dt dS q dD q
· − ·
dp D ( q ) dq D ( q )
εD
 Then split up the bottom left into p+t  dS p dS t
· + · −ε
dp D ( q ) dp D ( q ) D
dS p
 As D ( q ) =S ( p ), · =ε
dp D ( q ) S
dS t
 ·  increase in tax proportional to the price change is a very
dp D ( q )
small number, we start at t=0 ; so, we approximate that to 0.
dp εD
 Therefore, we get =
dt ε S−ε D
 We have made two important assumptions:
1. Market is in equilibrium at the beginning: Qd=Qs
2. t=0 is the starting point




Now let’s consider the case where the tax is levied on producers:
 We start with demand = supply  Qd =Qs
 Qd=D ( q ) and Qs=S (p)
 However, now p=q−t

 Derivative of the above 
(
dD dq dS dq dt
· = − 
dq dt dp dt dt )
dD dq dS dq dS
· = · −
dq dt dp dt dp
dS

(
dS dq dS dD
=
dp dt dp dq
− 
) dq
=
dt dS dD
dp
−
dp dq
p
 Multiply all parts by
S( p)