Lecture 4
Tax structures: production efficiency and
consumption efficiency
April 4, 2013
Practical implementation and the interaction between
various instruments
1. Production: should it be taxed?
2. Putting together direct and indirect taxes
3. Separability
I. Production: should it be taxed?
Previous analysis assumed fixed producer prices.
Diamond and Mirrlees (1971) relax this assumption by modelling
production.
Production efficiency: in an economy with linear commodity taxes where
first-best is unattainable, optimal policy maintains production efficiency.
Lipsey and Lancaster (1956): Theory of the Second Best
Standard optimal policy results only hold with single deviation from first
best.
Ex: Ramsey formulas invalid if there are pre-existing distortions,
imperfect competition, etc.
In second-best, anything is possible.
Policy changes that would increase welfare in a model with a single
deviation from first best need not do so in second-best.
Ex: tariffs can improve welfare by reducing distortions in other part of
economy.
Destructive result for welfare economics.
Diamond and Mirrlees: Production Efficiency
Diamond and Mirrlees: a general policy lesson even in second-best
environment.
Example: Suppose government can tax consumption goods and also
produce some goods on its own (e.g. postal services).
One might have the intuition that the government should try to generate
profits in postal services by increasing the price of stamps.
This intuition is wrong: it is optimal to have no distortions in production
of goods.
Bottom line: only tax goods when they appear in agents’ utility
functions.
One should not distort production decisions via taxes on intermediate
goods, tariffs, etc.
The framework
Many consumers, many goods.
Important assumption: either constant returns to scale in production (no
profits) or pure profits can be fully taxed.
With this assumption, profits do not enter the social welfare function.
Model
Let z h be the net consumption plan of household h.
When she faces consumer prices q, she chooses the net trade ζ h (q)
which maximizes utility U h (z h ) subject to the budget constraint
qz h ≤ 0.
Indirect utility is V h (q) = U h (ζ h (q)).
There are j = 1, . . . , J firms with a production plan y j in a production set
Y j (inputs are counted negatively, outputs positively).
The government wants to implement a public expenditure vector g . It
can use linear taxes on the various commodities (extension to nonlinear
taxes on labor?) so as to finance g , and wants to maximize
W (V 1 (q), . . . , V H (q)).
Proof of Production Efficiency Result
Consider the following maximization problem, that the government would
face if it had direct control of production:
max W (V 1 (q), . . . , V H (q))
H
X
h=1
ζ h (q) + g = y =
J
X
yj
for y j ∈ Y j .
j=1
Suppose by contradiction
that y belongs to the interior of the aggregate
PJ
production set i=1 Y j .
Suppose that there is a good i that is taxed linearly and is in positive net
demand by all households, so that a decrease in qi increases the utility of
all by Roy’s identity. For a small enough change, this is feasible, so that
the initial allocation was not an optimum.
Implementation in a decentralized economy
With convex production sets, any point at the frontier of the set can be
reached by profit maximization, with a well chosen production price
vector p: for every firm j, py j maximizes py over Y j .
Furthermore, by Walras’ law, with full taxation of profits, the government
budget constraint is satisfied:
pg = py − p
H
X
zh
h=1
= profits + (q − p)
H
X
h=1
= profits + t
H
X
h=1
zh
zh
Policy Consequences: Public Sector Production
Public sector production should be efficient.
If there is a public sector producing some goods, it should:
I
Face the same prices as the private sector;
I
Choose production with the unique goal of maximizing profits, not
generating government revenue.
Ex. postal services, electricity, health care, ...
Policy Consequences: No Taxation of Intermediate Goods
Intermediate goods: goods that are neither direct inputs or outputs to
indiv. consumption.
Taxes on transactions between firms would distort production.
Computers:
I
Sales to firms should be untaxed,
I
But sales to consumers should be taxed.
Policy Consequences: Tariffs
I
In open economy, the production set is extended because it is
possible to trade at linear prices (for a small country) with other
countries;
I
Diamond-Mirrlees result: small open economy should be on the
frontier of the extended production set
I
Implies that no tariffs should be imposed on goods and inputs
imported or exported by the production sector.
Ex. sales of IBM computers to other countries should be untaxed.
Ex. purchases of oil by oil companies should be untaxed.
Ex. should be no special tariff on imported clothes from China, but
should bear same commodity tax as clothes made in Europe.
Diamond and Mirrlees: Optimal Tax Rates
I
Optimal tax formulas take the same form as the solution to Ramsey
many-persons problem
I
Result holds even where producer prices are not constant.
I
Same formulas as in Ramsey just by replacing the p’s by the actual
p’s that arise in equilibrium.
I
Key point: Incidence in the production sector and GE responses can
be completely ignored in formulas.
Diamond and Mirrlees Model: discussion of the
assumptions
Result hinges on government’s ability to:
1. Set a full set of differentiated tax rates on each input and output;
2. Tax away fully pure profits (or production is
constant-returns-to-scale).
The latter implies that there is no way to improve welfare by taxing
profitable industries at expense of efficiency.
These assumptions effectively separate the production and consumption
problems.
Naito (1999)’s criticism
The Ramsey setup assumes that all goods can be taxed differently.
Naito supposes that two production inputs, skilled and unskilled labor,
cannot be taxed differently. Then he shows that it may be optimal to
have a shadow price for skilled labor in public production that is higher
than the market price: this reduces aggregate demand for skilled labour
and contributes to redistributing income.
Similar restrictions to the available tax instruments might justify
1. Subsidize low skilled intensive industries,
2. Set tariffs on low skilled intensive imported goods (to protect
domestic industry).
But nonlinear taxation of income work in the other direction.
II Putting together indirect and direct taxes
The usual (old) view is that direct taxes can be closely adapted to
individual characteristics of the taxpayer and so they should be used for
equity reasons. Indirect taxes would only matter for efficiency.
The ‘two attractive sisters’ of Gladstone.
We have to consider a model featuring both direct and indirect taxes.
A model
Households maximize U(X , L) subject to their budget constraint
P
n
i=1 (1 + ti )Xi = wL − T (wL).
Given a government policy (t1 , . . . , tn , T (·)), this gives the indirect utility
function U(w ) of the household of productivity w .
The government maximizes a social welfare function
Z
Ψ(U(w ))dF (w )
w
under the budget constraint
Z X
n
[
ti ξi (w ) + T (w ξL (w ))]dF (w ) ≥ R.
w i=1
Results obtained so far
First order conditions for linear commodity taxes and nonlinear income
taxes.
Pn
Ramsey: denote b w = Ψ0 (U(w ))u 0 (C (w ))/λ + j=1 tj ∂ξjw /∂M w be the
net social marginal utility of income of consumer w .
w
Z w
Z w w
b
ξi
b ξi
ri =
−1
− 1 dF (w ) =
−1
b
ξ
w
w b ξi
i
R w
Pn
j=1 tj w Sij
= −1 + b + ri b.
ξi
Mirrlees:
S(w ) =
1
1 − F (w )
T0
=
1 − T0
Z
∞
Ψ0 (U(x))u 0 (C (x))dF (x)
w
1 1 − F (w )
S(w )
1−
1+
εL
wf (w )
S(0)
Uniform transfers and Ramsey
Suppose that it is possible to undertake uniform transfers in the Ramsey
world, i.e. taxes or subsidies ds equal for everyone (this is of course
incentive compatible, and therefore allowed in the Mirrlees setup).
The change in the Lagrangian is


Z
n
w
X
∂ξ
∂L
j
Ψ0 (U(w ))u 0 (C (w )) + λ(
=
− 1) dF (w ).
tj
w
∂s
∂M
w
j=1
This derivative is equal to zero if s is chosen optimally which implies
b=1
Implications for indirect taxes
Pn
j=1 tj
ξi
R
w
Sijw
= ri .
If all consumers are alike (ri = 0) , the discouragement index is equal to
zero for all goods: elasticities do not play any role!
Public expenditures are financed by equal lump sum taxes on all citizens.
Otherwise taxes are justified by the ri ’s: positive taxes on luxuries,
negative for necessities.
III Weak separability of utility functions
Atkinson Stiglitz (1976), Gauthier Laroque (2009).
In a first best setting, efficiency can be achieved independently of equity
concerns through lump sum transfers without indirect taxes, the
Samuelson rule applies to the provision of public goods, and Pigovian
taxes are used to correct externalities.
In a second best environment, the social planner faces additional
constraints, but second best rules may sometimes have a ‘first best’
flavor.
(Most of) these examples involve separability coupled with specific
informational assumptions.
A non-satiation property
Agent h (h possibly multidimensional, with cdf F ) buys private goods x,
supplies labor `. Her utility is U(x, `, h).
For a net production z, the feasibility constraint on private goods is
Z
x h dF (h) ≤ z.
h
Definition 1. A feasible allocation which satisfies all second best
constraints is non-satiated when an increase dz, dz ≥ 0, dz 6= 0, in the
resources of private goods, leading to the feasibility constraint
Z
x h dF (h) ≤ z + dz,
h
allows a Pareto improvement while satisfying the second best constraints.
Indirect taxes: Atkinson Stiglitz setup
Agent w has separable utility U(V (x), `, w ) and her before tax income
(‘efficient labor’) is y = w `.
The technology is linear. The feasibility constraint is
Z
Z
px w dF (w ) ≤
w `w dF (w ),
w
w
where p is a fixed vector of (producer) prices.
The government observes y = w `, but neither w nor ` separately.
It announces an after tax income schedule R(·) and linear taxes q − p on
consumption goods.
Given y , an agent maximizes V (x) subject to qx = R(y ).
Her conditional demand function is γ(q, R(y )).
The government chooses (q, (y w , R(y w )) which maximizes a social
welfare function subject to the feasibility constraint and the incentive
constraints (IC),
!
0
yw
yw
w0
w
, w ≥ U V (γ(q, R(y ))),
,w
U V (γ(q, R(y ))),
w
w
for every (w , w 0 ).
Lemma 1. Consider a non-satiated second best allocation in which agent
w has before tax income y w ∗ and consumes x w ∗ . Given (y w ∗ ), (x w ∗ ) is a
first best allocation of the economy where all the agents have the same
quasi-concave and increasing utility function V (·) and the aggregate
production set is
Z
Z
px w dF (w ) ≤
y w ∗ dF (w ).
z
w
The proof is simple!
1. With V w ∗ = V (x w ∗ ), (IC) rewrite
w∗
w∗ y
,w ≥ U
U V ,
w
0
V
w 0∗
yw ∗
,
,w
w
!
for every (w , w 0 ).
Hence, given (y w ∗ ), any (x w ) that yields (V w ∗ ) satisfies (IC).
2. From the second welfare theorem, any first-best optimum (of the
economy described in the lemma) can be decentralized with an
appropriate choice of (q, (R w )). Note that R w can be written as
R(y w ): two agents w and w 0 with the same y must have the same
V (by (IC)), and thus the same R.
3. If the reference allocation is not
R a first best optimum, then one can
achieve (V w ∗ ) with less than w y w ∗ dF (w ). There is an extra dz:
the non-satiation property gives the desired contradiction.
Indirect taxes: heterogeneous preferences
Utility of agent h = (w , a) is U(V (x, a), `, a, w ).
Indirect taxes are useful when a is private information.
Indirect taxes are useless when a is publicly observable (given or chosen).
If chosen, we have:
Lemma 2. Suppose that the after tax income schedule can be made
dependent on the observable characteristics a. Consider a non-satiated
second best allocation in which agent h chooses ah∗ , has a before tax
income y h∗ and consumes x h∗ . Given (ah∗ , y h∗ ), (x h∗ ) is a first best
allocation of the economy where, for all h, agent h has preferences for
goods given by the quasi-concave and increasing utility function V (·, ah∗ )
and the aggregate production set is
Z
Z
h
px dF (h) ≤ y h∗ dF (h).
h
h
Public good provision
Agent h has utility U(V (x, g ), `), where g is a public good (chosen by
the government).
The aggregate feasibility constraint is
Z
Z
h
x dF (h) + g = y h dF (h).
h
h
Lemma 3. Consider a non-satiated second best allocation
((y h∗ , x h∗ ), g ∗ ). Then, given (y h∗ ), ((x h∗ ), g ∗ ) is a first best allocation
of the economy with utility functions V (·) and production set
Z
Z
x h dF (h) + g = y h∗ dF (h).
h
h
(A similar result holds in the presence of externalities.)
Conclusion
1. Production efficiency: value added tax is the best tool.
How to tax profits? Rents vs imperfect competition. Competition
policy.
2. Indirect taxes: not much of a role in the presence of income taxes.
Number of rates in VAT.
3. Separability: indirect taxes become superfluous (possibly including
tax on capital).
Empirical evidence: work related expenses.