Optimal Taxation

Old Riddles

Neoclassical Answers

Copyright 2008 by Peter Berck

P. Berck 2

Questions

• Optimal Tax• Deadweight Loss• Tax the Rich• A compromise formula

• Government Efficiency• Social Discount Rate• Border Pricing

Review of

Graphical Robinson Crusoe

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Graphical Derivation: Offer

Leisure

Stuff

E

Offer CurveE is the consumer’s endowment of time. It is allocated to leisure or sold, called work.

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Profit Maximization

• Stuff = F(L) (work is L; we measure inputs as negative quantities; -F’ is marginal product!)

• w = 1 (wage)

• P is price of stuff

• Profit Max• -P F’ = w• P = -1/F’

P. Berck 6

Stuff

0

work

L*

S*

L* + xx

•P S* = L* + profit •(def. of profit)

•slope of the tangent line is• -S*/ (L* +x)• = F’ = -1/P

•F.O.C. for a profit max•P*S* = L + x

•x = profit

x is Profit

P. Berck 7

Stuff

0

work

L*

S*

L* + profitprofit

Profit Max Choice of a Firm

P. Berck 8

Robinson Crusoe: A Firm

Stuff

E

•The price is P = 1/-F’•Pareto Optimal•Competitive Equilibrium

LeisureWork

Consumer spends endowmentplus all profits

On to

Graphical Diamond and Mirrlees

P. Berck 10

D-M Graphic Setup

• Consumer owns only labor

• Sells labor; buys stuff at price q

• Firm receives p for stuff

• Gov’t collects tax on Stuff, q-p

• Gov’t gets profits from firm

• Gov’t buys labor and builds project with tax and profits

• No or separable utility from project

P. Berck 11

Stuff

0L*

S*

Work for firm, L*

Work on project

Profits

•Gov’t buys labor to build project•There is a price line for any point on f

PPF with Project

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Optimal Outcome with Project

Offer Curve

•Price Lines and Indifference Curves are used to find Offer Curve•PPF and Offer intersect at best allocation consumer can get using prices•But, that is not a P.O.!

ELeisure

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Consumer Prices

Offer Curve

E

The slope of this budget line is -1/q, q is the price charged to consumers.

L(q)

P. Berck 14

Producer Prices

Offer Curve

E

The slope of this tangent line is -1/p, p is the price charged to producers.

L(q)

P. Berck 15

Optimal Tax

S*(P)

Tangent to PPF: -Slope is 1/P

Intersects Offer Curve-Slope is consumerprice, 1/q.

L(q)

L*

Consumer’s Labor supply at q

Firm’s Labor Demand at PL(q) - L* = Gov’t Labor Demand =Project

As drawn, q > p

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Adding Up

• Gov’t gets (q - p) S* (the tax take)• q S* = L* + government labor = E (budget

constraint)• P S* = L* + profit• Taxes = government labor - profit• Government budget constraint requires:

• profits to go to government

• no profits (constant returns to scale)

• inframarginal taxes to raise extra money

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Conclusion From Graph

• Production is on PPF

• Tax induced equilibrium is not P.O.

• Optimal tax can be found

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D-M Algebra

• V(q) = U(X(q))• x(q) is demand

• indirect utility

• Welfare(V1(q),..Vm(q))

• Also any other function of q

• y1=f(y2,…yn)• private output

• p’y = profit = 0

• by assumption of CRTS

• z1=g(z2,…zn)• public output

• x(q) = y + z• market clearing

P. Berck 19

Normalization

• Since p’y = 0 so does any multiple of p and there is a normalization of p1=1.

• The budget constraint is q’x = 0 and so one can normalize on q1=1.

• This makes the tax on good 1 zero.

Firms Foc

• pn=- p1 fn

• price times marginal product = wage

• 1 = p1

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P. Berck 21

DM Maximization Problem

• Maxz,q V(q)

• s.t. x1(q) = f(x2(q)-z2,…xn(q)-zn) + g(z2…zn)

• Derivs wrt q lead to optimal tax rule

• Deriv wrt z

• fk = gk

• Government and Private have same MP!

P. Berck 22

G and Trade

• Instead of G being government, let it be an international trade sector. (Or add a new sector)• Let w be the vector of exogenous international

prices

• suppose g(z2,…zn) is given by

• w’z= 0 or z1 =-(w2 z2 +…+wn zn)/w1

• Then domestic producer prices are world prices

P. Berck 23

Optimal Tax

• Maxz,q V(q)• s.t. x1(q) = f(x2(q)-z2,

…xn(q)-zn) + g(z2…zn)• Lambda is the utility

value of a free unit of good 1 which is also $

• Vk could include an externality

P. Berck 24

2 2 2 1( ) ( ( ( ) ,... ( ) ) ( ,... ) ( ))n n nL V q f x q z x q z g z z x q

1,

ik i

i n k

xV p

q

P. Berck 25

Vk

• One consumer (or representative consumer) with externality caused by consumption.

• V = U(x) – D(x)

• Consumer max’s only U(x); D(x) external

• Vk = -xk a +Dk

P. Berck 26

Tax Rule

1, 1,

k

/ / the latter with p held constant!

since ' 0 Using Roy's identity V

'

k i i i ii n i nk k

kk

x q x t

V p x t xt t

q x x

t xx

t

P. Berck 27

Tax Rule with Extern..

• V=U – D

• Vk = -axk - Dk

' kk

k

t x Dx

t

Conclusions

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Efficiency Consequences

• Gov’t and Private Use Same Prices to guide decisions

• If g() is opportunities from trade, algebra and conclusion is same: economy operates efficiently w.r.t. border prices

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Social Rate of Discount

• No “social rate of discount”: MRP of gov’t investment = MRP of private investment

• Yes “social rate:” investments that favor poor (possible future generations) could have subsidy (p>q) over projects that favor rich (us.) But, it is true for both gov’t and private projects!