optimal taxation old riddles neoclassical answers copyright 2008 by peter berck
TRANSCRIPT
![Page 1: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/1.jpg)
Optimal Taxation
Old Riddles
Neoclassical Answers
Copyright 2008 by Peter Berck
![Page 2: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/2.jpg)
P. Berck 2
Questions
• Optimal Tax• Deadweight Loss• Tax the Rich• A compromise formula
• Government Efficiency• Social Discount Rate• Border Pricing
![Page 3: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/3.jpg)
Review of
Graphical Robinson Crusoe
![Page 4: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/4.jpg)
P. Berck 4
Graphical Derivation: Offer
Leisure
Stuff
E
Offer CurveE is the consumer’s endowment of time. It is allocated to leisure or sold, called work.
![Page 5: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/5.jpg)
P. Berck 5
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’
![Page 6: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/6.jpg)
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
![Page 7: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/7.jpg)
P. Berck 7
Stuff
0
work
L*
S*
L* + profitprofit
Profit Max Choice of a Firm
![Page 8: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/8.jpg)
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
![Page 9: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/9.jpg)
On to
Graphical Diamond and Mirrlees
![Page 10: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/10.jpg)
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
![Page 11: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/11.jpg)
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
![Page 12: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/12.jpg)
P. Berck 12
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
![Page 13: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/13.jpg)
P. Berck 13
Consumer Prices
Offer Curve
E
The slope of this budget line is -1/q, q is the price charged to consumers.
L(q)
![Page 14: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/14.jpg)
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)
![Page 15: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/15.jpg)
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
![Page 16: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/16.jpg)
P. Berck 16
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
![Page 17: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/17.jpg)
P. Berck 17
Conclusion From Graph
• Production is on PPF
• Tax induced equilibrium is not P.O.
• Optimal tax can be found
![Page 18: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/18.jpg)
P. Berck 18
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
![Page 19: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/19.jpg)
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.
![Page 20: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/20.jpg)
Firms Foc
• pn=- p1 fn
• price times marginal product = wage
• 1 = p1
P. Berck 20
![Page 21: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/21.jpg)
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!
![Page 22: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/22.jpg)
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
![Page 23: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/23.jpg)
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
![Page 24: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/24.jpg)
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
![Page 25: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/25.jpg)
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
![Page 26: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/26.jpg)
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
![Page 27: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/27.jpg)
P. Berck 27
Tax Rule with Extern..
• V=U – D
• Vk = -axk - Dk
' kk
k
t x Dx
t
![Page 28: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/28.jpg)
Conclusions
![Page 29: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/29.jpg)
P. Berck 29
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
![Page 30: Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008 by Peter Berck](https://reader035.vdocuments.us/reader035/viewer/2022062713/56649f505503460f94c72b19/html5/thumbnails/30.jpg)
P. Berck 30
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!