# Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008, 2010 by Peter Berck.

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Optimal Taxation Old Riddles Neoclassical Answers Copyright 2008, 2010 by Peter Berck

P. Berck2 Questions Optimal Tax Deadweight Loss Tax the Rich A compromise formula Government Efficiency Social Discount Rate Border Pricing

Review of Graphical Robinson Crusoe

P. Berck4 Graphical Derivation: Offer Leisure StuffStuff E Offer Curve E is the consumer’s endowment of time. It is allocated to leisure or sold, called work.

P. Berck5 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. Berck6 StuffStuff 0 work L* S* L* + x x 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. Berck7 StuffStuff 0 work L* S* L* + profit profit Profit Max Choice of a Firm

P. Berck8 Robinson Crusoe: A Firm StuffStuff E The price is P = 1/-F’ Pareto Optimal Competitive Equilibrium Leisure Work Consumer spends endowment plus all profits

On to Graphical Diamond and Mirrlees

P. Berck10 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. Berck11 StuffStuff 0 L*+project 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

P. Berck12 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.! E Leisure

P. Berck13 Consumer Prices Offer Curve E The slope of this budget line is -1/q, q is the price charged to consumers. L(q)

P. Berck14 Producer Prices Offer Curve E The slope of this tangent line is -1/p, p is the price charged to producers. L(q)

P. Berck15 Optimal Tax S*(P) Tangent to PPF: -Slope is 1/P Intersects Offer Curve -Slope is consumer price, 1/q. L(q) L* Consumer’s Labor supply at q Firm’s Labor Demand at P L(q) - L* = Gov’t Labor Demand =Project As drawn, q > p

P. Berck16 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

P. Berck17 Conclusion From Graph Production is on PPF Tax induced equilibrium is not P.O. Optimal tax can be found

P. Berck18 D-M Algebra V(q) = U(X(q)) x(q) is demand indirect utility Welfare(V 1 (q),..V m (q)) Also any other function of q y 1 =f(y 2,…y n ) private output p’y = profit = 0 by assumption of CRTS z 1 =g(z 2,…z n ) public output x(q) = y + z market clearing

P. Berck19 Normalization Since p’y = 0 so does any multiple of p and there is a normalization of p 1 =1. The budget constraint is q’x = 0 and so one can normalize on q 1 =1. This makes the tax on good 1 zero.

Firms Foc Chose y to max p’y s.t. y 1 - f(y 2,…y n )=0 FOC for lagrange: p 1 + = 0; p i - f i =0 p i =- p 1 f i (is DM eq 9) price times marginal product = wage 1 = p 1 p i = - f i P. Berck20

P. Berck21 DM Maximization Problem Max z,q V(q) s.t. x 1 (q) = f(x 2 (q)-z 2,…x n (q)-z n ) + g(z 2 …z n ) Derivs wrt q lead to optimal tax rule Deriv wrt z f k = gkgk Government and Private have same MP!

P. Berck22 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(z 2,…z n ) is given by w’z= 0 or z 1 =-(w 2 z 2 +…+w n z n )/w 1 Then domestic producer prices are world prices

Lagrange for optimal tax P. Berck23 DM 17,18,19 P 1 = 1; p i =- f i

P. Berck24 Optimal Tax Max z,q V(q) s.t. x 1 (q) = f(x 2 (q)- z 2,…x n (q)-z n ) + g(z 2 …z n ) Lambda is the utility value of a free unit of good 1 V k could include an externality

P. Berck25 VkVk One consumer (or representative consumer) with externality caused by consumption. V” = U(x(q)) – D(x(q))= V(q) –D(q) Consumer max’s only U(x); D(x) external D k is total derivative of D w.r.t q k V” k = -x k a -D k Roy’s identity: V k =-a x k Where a is marg. Util of income

P. Berck26 Tax Rule

Another version of rule Consume goods in proportion to how the tax take changes with taxes. P. Berck27 Since the t on labor is big, much attention on how labor changes with tax. Even change in labor supply from a gas tax.

P. Berck28 Tax Rule with Extern.. V”=U – D V” k = -ax k - D k

Conclusions

P. Berck30 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

P. Berck31 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!