Lecture 3: Taxes, Tariffs and Quota
(Chapter 5)
•
•
•
•
•
Relation to “work horses”
Government and taxation
Taxes and quotas in general equilibrium
Welfare implications of taxes and tax reforms
Practices in applied modeling
Aim of lecture 3
• Highlighting welfare effects of different taxes
• Illustrating implementation of taxes and quotas in general
equilibrium
Mapping from theorems to “working
horses”
5.1: existence of eq with cons tax
Slater
Constraint set of optimization program
Non-negativity of
incomes
Assumptions on T(.)
Maximum
theorem
Upper semicontinuity , compactness and
convex-valued ness of price correspondence
in consumption of individuals.
Kakutani
Fixed point in p, driven by p
5.2 Welfare loss from cons tax
5.4 Eventual welfare gains
from reducing taxes in the
closed ec.
consumer demand is continuous
function of wedge. (strict
concavity of utility is used)
Continuity of social welfare
in "policy scalar".
Government and taxation
• Direct taxes
– Take away (give) a part of income and profits from (to) the agents.
Can be used for redistribution purposes and to distribute the
proceeds from:
• Indirect taxes
– Create a wedge between prices at various stages of supply and
utilization of a commodity
Price wedges
Production
Indirect taxes
Consumption
Indirect taxes
Intermediate demand
Indirect taxes
Tariffs
Import
Market clearing
Tariffs
Export
Government and taxation:
tax on consumption
• Recall from lecture 1 that a tax on consumption of goods
implies that the relation between the market clearing price
c
c
and the consumer price is given as: pk  (1   k ) pk .
• Types of consumption taxes:
– Ad valorem tax:  kc is fixed
– Nominal tax:  kc pk is fixed
– Variable levy: pkc is fixed,  kc adjusts
• Homogeneity of demand in market prices is preserved only
under ad valorem tax
Example (continued)
• Proceeds of indirect taxes
– Public consumption
– Indirect subsidies
– Direct subsidies (lump-sum transfers)
• Budget of the agent now reads:
c
0
0
0
(1


)
p
x

p




(
p
)

T
(
p
,
T
,
h
,
h
,...,
h
k k k k
 j ij j
i
i
1
2
m)
• All consumers (including government) take lump-sum transfers
as given!
Taxes and quotas in general equilibrium:
consumption tax
(a) Optimization with given consumptions x̂i :
maxm,e0,y j ,all j p e e  p m m
subject to
 j y j  m  e   i ˆxi   i i
( p)
y j Yj
(b) A feedback relation that sets x̂i by solving the consumer problem:


0
x̂i  arg max ui ( xi ) p c xi  hi0  Ti ( p,T ,h10 ,h20 ,...,hm
),xi  0 ,all i
with hi0  pi   j ih  j ( p )
Taxes and quotas in general equilibrium:
taxes on intermediate demand
(a) Optimization with given consumptions x̂i :
maxm,e0,y+ ,y  0,y
j
j
j
all j
p e e  p m m     j y j     j y j
subject to


 j ( y j  y j )  m  e  i ˆxi   i i
( p)
y j  y j  y j
( pf )
y j Yj
(b) Feedback on consumption as before and tax feedback relations:
 k   k pk ,  k   k pk
Taxes and quotas in general equilibrium:
tariffs in the international model
• Under small country assumption, import and export prices are given and
tariffs are included as:
maxm,e0,0,y j all j p ee  p m m   ee   m m
subject to
 j y j  m  e  i ˆxi   i i
( p)
y j Y j
• Quota can be included as additional restrictions
ee
( e )
m  m ( m )
• With  e ,  m (the shadow prices) being the quota rents.
Taxes and quotas in general equilibrium:
tariffs in the international model (continued)
• International model can also be seen as set of national models
linked through trade
– World prices (import and export prices) are now endogenous and clear world
markets
– Tariffs now are the difference between clearing prices at world level and
domestic clearing prices
– We may assume that tariff revenue is distributed only among agents in the
own country
Taxes and quotas in general equilibrium: tariffs
in the international model (continued)
• Optimization with given consumption
maxm,e0,mc ,ec 0, all c, y j ,c all j ,c p ee  p m m   c  ceec   c cm mc
subject to
m  e   c ( mc  ec )
 j y j ,c  mc  ec  i ˆxi,c  i i,c
y j ,c  Y j ,c
• and feedback relations as before
( pw )
( pc )
Government activity: a summary
• Government can be:
– Social planner
– Consumer (distortion, since public demand does not enter in private utility
function, see Chapter 9)
– Owner of resources (that can be used for transfers to private agents)
• Activities and transactions in which government is involved include
–
–
–
–
–
–
–
Public consumption
Income from endowment and production and remittances from abroad
Redistributed income
Transfers to private consumers
Proceeds from indirect taxation and tariffs
Budget balancing
Trade balancing
Welfare implications: consumption tax
• Shift to Negishi format
• As in open economy format, taxes are represented in the
objective
• To recover consumer prices, an additional variable (total
consumption) and an additional constraint (equating the
sum of individual consumption to total consumption) are
inserted
• Feedback on welfare weights is adjusted to reflect taxincluded budget constraint
Welfare implications: consumption tax (continued)
max x , xi  0, all i , y j , all j  i  i ui ( xi )   i  c x
subject to
x   j y j   i i
x   i xi
( p)
( pc )
y j Yj
With welfare weights set such that budget constraints hold:
pc xi  hi0  Ti (), with hi0  pi   j  ij  j ( p),
T   c  i xi ,  kc   kc pk
Note: homogeneity of the objective in welfare weights is lost for given
 c  0!
How to reform?
• Proposition 5.2 (welfare loss of a consumption tax) “removing
taxes cannot reduce total welfare”. No gain if:
– All commodities are taxed in the same way and
– redistribution of tax gains is such that every consumer receives a
lump-sum transfer equal to the amount of taxation paid
• Proposition 5.3 (welfare gains from reform)
– Recall from lecture 1 that in general, subsidies, tariffs, monopoly
premiums and wage subsidies can be represented by separate agent
terms in objective with a weight factor.
– Consumer welfare rises as weight factor is reduced
• Proposition 5.4 (eventual welfare gains from reducing taxes)
– If indirect taxes are close to zero, there can be no welfare loss from
reduction of taxes
How to reform? (continued)
• Proposition 5.2 suggest that all indirect taxes should be removed in one
reform
• Since this is rarely feasible, it is important to look at effects of gradual
reforms
• Phasing of reform, starting from a situation with high indirect taxes
– Proposition 5.3 tells us that all wedges should first be decreased
simultaneously and proportional
– If taxes are decreased to low levels, then Proposition 5.4 tells us that
reform could be stepwise: no tax should be raised, but some can be set to
zero in a series of piecemeal reforms
– Throughout the process, compensating transfers should be given
– Agents should not be able to anticipate the reform process
How to reform? Why compensation is important
• (Weakly) Pareto-superior tax reforms can be designed, if
they are supported by compensating transfers
• However, if there is no compensation, even full elimination
of taxes may lead to welfare losses:
–
–
–
–
–
Closed economy, 2 consumers, 1 commodity, no production
Consumer 1 owns all endowments
100% tax on consumption
Tax receipts given to consumer 2
Consumers have identical, strictly concave, increasing, continuously
differentiable utility functions
– It follows that both have the same income
– It also follows that they have the same welfare weight
How to reform? Why compensation is important
max x1 , x2  0  u ( x1 )   u ( x2 )   c ( x1  x2 )
subject to
x1  x2  
( p),
where  c  1
• Abolishing tax with no compensation leads to zero income
for consumer 2, and by strict concavity this implies:
  1 
 1  
  u    u       u 1   u  0  
 2 
  2 
Practices in application:
taxes and tariffs in CGE
• Introduction of tariffs and taxes does not change
commodity balances, but only budgets and price formation
• Special price equations:
m
m
m
– Trade: tariff-ridden import prices are exogenous: pˆ k  (1   k ) pk
– Trade: exports are assumed not to be perfect substitutes in the rest
of the world, so prices are endogenous, and the price adjusts
according to: pke  pkg (1   ke ) . This price then enters the export
demand function and the balance of payments
• Budget equations:
– Export demand
– Tax functions, household budgets, government budget
Practices in application:
Specification of wedges in CGE
• Tax wedges can be fixed, observed parameters, which may be
varied in scenario analyses
• Tax wedges may be specified as explicit functions of
endogenous variables, the parameters of which can be
estimated.
• Tax wedges can be specified as variable levy, to maintain
relative prices with certain range (e.g. relation domestic price
and world market price)
• Tax wedges may be fully endogenous and result from social
welfare optimization
• Taxes may be used to stabilize quantities: tariff quota, with
voluntary export restraint as special case where tariff revenue
accrues to the exporter
Practices in applications: a SAM


Goods
Firms
Consumers
Foreign
sector
Government
j

j
g

j,g
g
pg y
f
p f y j , f
j
Factors
Foreign sector
Consumers
Firms
Factors
Goods

i
g

p g xi , g
Government
p g eg
g
p g y j , g

i

f
f
p f i, f
Sf
  pgj y j , g 

 g

 j   g pg y j , g 
  p y  
f j, f 
f

T
pf mf
 j  g  j , g pg y j , g 
 j  g  j , g pg y j , g 

j
f
 j , f p f y j , f

i
g
 ic, g p g xi , g
 pe
  p m
e
g
g
f
w
g g
m
f
w
f
f
Practices in applications:
properties of income tax schedule
• Income tax schedule Ti () should satisfy following properties
– Continuity
– Homogeneity of degree 1 in ( p, T , h10 , h20 ,..., hm0 )
– Adding up: T   i hi0  0 , with equality whenever hi0  Ti  0
– Positiveness:
 T ()  T
i i
– There exists a constant   (0,1) such that if  [0, ] all consumers
are taxed at the same proportional rate so that for all i, hi  hi0 for
T  (1   ) i hi0  0
Practices in applications:
Tax functions
• Tax functions can be specified in an ad-hoc manner, where
properties of the tax schedule should be satisfied
• However, optimal taxation schedules may theoretically be
derived from maximization of social welfare, according to
some social welfare function that is agreed upon by society
Practices in applications:
Tax functions (continued)
• Consider social welfare optimization program for an
exchange economy:
max xi  0,ui 0 V (u )
subject to
ui  ui ( xi )
 x  
i
i
i
( i )
i
( p)
With V (u) being strictly concave increasing
Practices in applications:
Tax functions (continued)
Substituting ui ( xi )  ui leads to a composite mapping W. Note that we have to
assume concavity of the individual utility functions to preserve concavity of the
sum of these functions, and thus concavity of W in x (see proposition A.1.4)
max xi ,all i W [u1 ( x1 ), u2 ( x2 ),..., ui ( xi ),...um ( xm )]
subject to

i
pxi   i pi
Which equivalent to the previous problem for p  1  p
This can be decomposed as a two-level problem:
1.Optimal expenditure allocation
2.Consumer i’s utility maximization
Practices in applications:
Tax functions (continued)
Level 1 maximization:
max ei 0,all i W * v1 ( p e1 ), v2 ( p e2 ),..., vi ( p ei ),..., vm ( p em )
subject to
 i ei   i pi
Individual maximization:
vi  p ei   max ui ( xi ) pxi  ei , xi  0 , where his expenditure ei is
obtained from (1)
From this, we can obtain expenditure allocation functions ei and
derive direct tax functions Ti  pi  ei
Practices in applications:
Tax functions (continued)
• The optimal allocation that follows is unique
• This implies that tax schedules are not only useful to reach
specific distributional goals, but also to stabilize the
economy
• This also holds if there are distortions already in the
economy that must be taken as given by the modeler
Practices in applications:
Tax functions (continued)
• Individual optimization
– Concave utility function
– Increasing in all commodities
– Slutsky equation
– Preferences over known goods
• Social welfare optimization
– Concave composite social
welfare function
– Increasing in all utilities
– “Slutsky equation”
– Anonymity
Practices in applications:
Tax functions (continued)
• We did not impose restrictions on transfers that relate to property
rights of individuals.
• If all endowments are divided over individuals and all rights need to
be respected, then we seem to be back in the Negishi format, where
there is no room from redistribution and uniqueness is lost
• However, even under full respect of rights tax revenue needed for
redistribution can be mobilized without creating distortions:
– VAT: given tax rate that does not differentiate between commodities. This
simply raises the overall price level and does not affect relative prices
– Profits of state-owned companies can be used
– Government could establish markets for commodities that went unpriced
before (Pigouvian taxes; double dividend discussion)