Lecture 4

Tax structures: production efficiency andconsumption efficiency

April 4, 2013

Practical implementation and the interaction betweenvarious 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 modellingproduction.

Production efficiency: in an economy with linear commodity taxes wherefirst-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 firstbest.

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 singledeviation from first best need not do so in second-best.

Ex: tariffs can improve welfare by reducing distortions in other part ofeconomy.

Destructive result for welfare economics.

Diamond and Mirrlees: Production Efficiency

Diamond and Mirrlees: a general policy lesson even in second-bestenvironment.

Example: Suppose government can tax consumption goods and alsoproduce some goods on its own (e.g. postal services).

One might have the intuition that the government should try to generateprofits in postal services by increasing the price of stamps.

This intuition is wrong: it is optimal to have no distortions in productionof goods.

Bottom line: only tax goods when they appear in agents’ utilityfunctions.

One should not distort production decisions via taxes on intermediategoods, tariffs, etc.

The framework

Many consumers, many goods.

Important assumption: either constant returns to scale in production (noprofits) or pure profits can be fully taxed.

With this assumption, profits do not enter the social welfare function.

Model

Let zh be the net consumption plan of household h.

When she faces consumer prices q, she chooses the net trade ζh(q)which maximizes utility Uh(zh) subject to the budget constraint

qzh ≤ 0.

Indirect utility is V h(q) = Uh(ζh(q)).

There are j = 1, . . . , J firms with a production plan y j in a production setY j (inputs are counted negatively, outputs positively).

The government wants to implement a public expenditure vector g . Itcan use linear taxes on the various commodities (extension to nonlineartaxes 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 wouldface if it had direct control of production:

max W (V 1(q), . . . ,V H(q))

H∑h=1

ζh(q) + g = y =J∑

j=1

y j for y j ∈ Y j .

Suppose by contradiction that y belongs to the interior of the aggregateproduction set

∑Ji=1 Y j .

Suppose that there is a good i that is taxed linearly and is in positive netdemand by all households, so that a decrease in qi increases the utility ofall by Roy’s identity. For a small enough change, this is feasible, so thatthe initial allocation was not an optimum.

Implementation in a decentralized economy

With convex production sets, any point at the frontier of the set can bereached by profit maximization, with a well chosen production pricevector p: for every firm j , py j maximizes py over Y j .

Furthermore, by Walras’ law, with full taxation of profits, the governmentbudget constraint is satisfied:

pg = py − pH∑

h=1

zh

= profits + (q − p)H∑

h=1

zh

= profits + tH∑

h=1

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, notgenerating 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 toindiv. 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 ispossible to trade at linear prices (for a small country) with othercountries;

I Diamond-Mirrlees result: small open economy should be on thefrontier of the extended production set

I Implies that no tariffs should be imposed on goods and inputsimported 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, butshould 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 Ramseymany-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 actualp’s that arise in equilibrium.

I Key point: Incidence in the production sector and GE responses canbe completely ignored in formulas.

Diamond and Mirrlees Model: discussion of theassumptions

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 isconstant-returns-to-scale).

The latter implies that there is no way to improve welfare by taxingprofitable industries at expense of efficiency.

These assumptions effectively separate the production and consumptionproblems.

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 tohave a shadow price for skilled labor in public production that is higherthan the market price: this reduces aggregate demand for skilled labourand 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 protectdomestic 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 toindividual characteristics of the taxpayer and so they should be used forequity 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∑ni=1(1 + ti )Xi = wL− T (wL).

Given a government policy (t1, . . . , tn,T (·)), this gives the indirect utilityfunction U(w) of the household of productivity w .

The government maximizes a social welfare function∫w

Ψ(U(w))dF (w)

under the budget constraint∫w

[n∑

i=1

tiξi (w) + T (wξL(w))]dF (w) ≥ R.

Results obtained so far

First order conditions for linear commodity taxes and nonlinear incometaxes.

Ramsey: denote bw = Ψ′(U(w))u′(C (w))/λ+∑n

j=1 tj∂ξwj /∂Mw be the

net social marginal utility of income of consumer w .

ri =

∫w

(bw

b− 1

)(ξwiξi− 1

)dF (w) =

∫w

bw

b

ξwiξi− 1

∑nj=1 tj

∫w

Swij

ξi= −1 + b + rib.

Mirrlees:

S(w) =1

1− F (w)

∫ ∞w

Ψ′(U(x))u′(C (x))dF (x)

T ′

1− T ′=

(1 +

1

εL

)1− F (w)

wf (w)

(1− S(w)

S(0)

)

Uniform transfers and Ramsey

Suppose that it is possible to undertake uniform transfers in the Ramseyworld, i.e. taxes or subsidies ds equal for everyone (this is of courseincentive compatible, and therefore allowed in the Mirrlees setup).

The change in the Lagrangian is

∂L∂s

=

∫w

Ψ′(U(w))u′(C (w)) + λ(n∑

j=1

tj∂ξwj∂Mw

− 1)

dF (w).

This derivative is equal to zero if s is chosen optimally which implies

b = 1

Implications for indirect taxes

∑nj=1 tj

∫w

Swij

ξi= ri .

If all consumers are alike (ri = 0) , the discouragement index is equal tozero 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 equityconcerns through lump sum transfers without indirect taxes, theSamuelson rule applies to the provision of public goods, and Pigoviantaxes are used to correct externalities.

In a second best environment, the social planner faces additionalconstraints, but second best rules may sometimes have a ‘first best’flavor.

(Most of) these examples involve separability coupled with specificinformational 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∫h

xhdF (h) ≤ z .

Definition 1. A feasible allocation which satisfies all second bestconstraints is non-satiated when an increase dz, dz ≥ 0, dz 6= 0, in theresources of private goods, leading to the feasibility constraint∫

h

xhdF (h) ≤ z + dz ,

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∫w

pxwdF (w) ≤∫w

w`wdF (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 onconsumption 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, (yw ,R(yw )) which maximizes a socialwelfare function subject to the feasibility constraint and the incentiveconstraints (IC),

U

(V (γ(q,R(yw ))),

yw

w,w

)≥ U

(V (γ(q,R(yw ′

))),yw ′

w,w

)

for every (w ,w ′).

Lemma 1. Consider a non-satiated second best allocation in which agentw has before tax income yw∗ and consumes xw∗. Given (yw∗), (xw∗) is afirst best allocation of the economy where all the agents have the samequasi-concave and increasing utility function V (·) and the aggregateproduction set is ∫

z

pxwdF (w) ≤∫w

yw∗dF (w).

The proof is simple!

1. With V w∗ = V (xw∗), (IC) rewrite

U

(V w∗,

yw∗

w,w

)≥ U

(V w ′∗,

yw ′∗

w,w

)for every (w ,w ′).

Hence, given (yw∗), any (xw ) that yields (V w∗) satisfies (IC).

2. From the second welfare theorem, any first-best optimum (of theeconomy described in the lemma) can be decentralized with anappropriate choice of (q, (Rw )). Note that Rw can be written asR(yw ): two agents w and w ′ with the same y must have the sameV (by (IC)), and thus the same R.

3. If the reference allocation is not a first best optimum, then one canachieve (V w∗) with less than

∫w

yw∗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 madedependent on the observable characteristics a. Consider a non-satiatedsecond best allocation in which agent h chooses ah∗, has a before taxincome yh∗ and consumes xh∗. Given (ah∗, yh∗), (xh∗) is a first bestallocation of the economy where, for all h, agent h has preferences forgoods given by the quasi-concave and increasing utility function V (·, ah∗)and the aggregate production set is∫

h

pxhdF (h) ≤∫h

yh∗dF (h).

Public good provision

Agent h has utility U(V (x , g), `), where g is a public good (chosen bythe government).

The aggregate feasibility constraint is∫h

xhdF (h) + g =

∫h

yhdF (h).

Lemma 3. Consider a non-satiated second best allocation((yh∗, xh∗), g∗). Then, given (yh∗), ((xh∗), g∗) is a first best allocationof the economy with utility functions V (·) and production set∫

h

xhdF (h) + g =

∫h

yh∗dF (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. Competitionpolicy.

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 includingtax on capital).

Empirical evidence: work related expenses.