Optimal Taxation
Old Riddles
Neoclassical Answers
Copyright 2008 by Peter Berck
Questions
• Optimal Tax
• Deadweight Loss
• Tax the Rich
• A compromise formula
• Government Efficiency
• Social Discount Rate
• Border Pricing
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Review of
Graphical Robinson Crusoe
Graphical Derivation: Offer
Offer Curve
E is the
consumer’s
endowment of
time. It is
allocated to leisure
or sold, called
work.
S
t
u
f
f
Leisure
E
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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’
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S
t
u
f
f
L*
•P S* = L* + profit
•(def. of profit)
S*
•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
0
x
L* + x
work
x is Profit
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S
t
u
f
f
S*
L*
0
profit
L* + profit
work
Profit Max Choice of a Firm
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Robinson Crusoe: A Firm
•The price is P = 1/-F’
•Pareto Optimal
•Competitive Equilibrium
S
t
u
f
f
Work
Leisure
E
Consumer spends endowment
plus all profits
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On to
Graphical Diamond and Mirrlees
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
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S
t
u
f
f
S*
•Gov’t buys labor to build
project
•There is a price line for
any point on f
Profits
L*
0 Work on project
Work for firm, L*
PPF with Project
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Optimal Outcome with Project
•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.!
Offer Curve
Leisure
E
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Consumer Prices
The slope of this
budget line is -1/q, q
is the price charged
to consumers.
Offer Curve
L(q)
E
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Producer Prices
The slope of this
tangent line is -1/p,
p is the price
charged to
producers.
Offer Curve
L(q)
E
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Optimal Tax
S*(P)
Intersects Offer Curve
-Slope is consumer
price, 1/q.
Tangent to PPF: -Slope is 1/P
As drawn, q > p
L(q)
Consumer’s Labor supply at q
L*
Firm’s Labor Demand at P
L(q) - L* = Gov’t Labor Demand =Project
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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))
•
•
•
•
• z1=g(z2,…zn)
x(q) is demand
indirect utility
Welfare(V1(q),..Vm(q))
Also any other function of q
• public output
• x(q) = y + z
• market clearing
• y1=f(y2,…yn)
• private output
• p’y = profit = 0
• by assumption of CRTS
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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.
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Firms Foc
• pn=- p1 fn
• price times marginal product = wage
• 1 = p1
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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 = g k
Government and Private have same MP!
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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
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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
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L  V (q)   ( f ( x2 (q)  z2 ,...xn (q)  zn )  g ( z2 ,...zn )  x1 (q))
xi
Vk    pi
qk
i 1, n
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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
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Tax Rule
x / q  x / t the latter with p held constant!

Vk  
tk

pi xi  

tk
i 1, n
t x
i 1, n
i i
since q ' x  0 Using Roy's identity Vk   x
xk 
 t ' x
 tk
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Tax Rule with Extern..
• V=U – D
• Vk = -axk - Dk
 t ' x Dk
xk  

 tk

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Conclusions
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!
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