• On the demand side of the market income and substitution effects predicts the consequences of a price rise • On the supply side the tax is a cost increase and firms respond accordingly • The central question is the choice of the best set of taxes – Usually interpreted as the taxes that raise at given level of revenue with least efficiency cost • This introduces the analysis of optimal taxation Deadweight Loss • Lump-sum taxation does not cause any distortions • Commodity taxation does cause distortions – Demand shifts from goods with high taxes to goods with low taxes.

• These substitution effects are the tax-induced distortions • A commodity tax raises revenue but reduces welfare • The deadweight loss is the amount the welfare reduction exceeds revenue raised Deadweight Loss • Fig. 14. 1 illustrates deadweight loss • Without taxation the price is p, consumption X0, and consumer surplus abc • With a tax of t the price is q = p + t, consumption X1, and consumer surplus aef • The tax raises revenue tX1 which is area cdef • The deadweight loss (DWL) is the triangle bde Price af q ? p? t p c e b DWL d X1 X0 Quantity Figure 14.

1: Deadweight loss Deadweight Loss • • • • Triangle bde is equal to [1/2]t[X0 – X1] The elasticity of demand is ? d = pdX/Xdp Let dX = X0 – X1 and dp = t Using these definitions the deadweight loss is approximately DWL = [1/2]|? d|[X0/p]t2 • DWL is proportional to the square of the tax rate – It rises rapidly with increases in taxation • DWL is proportional to the elasticity of demand – For a given tax less deadweight loss is produced when demand is inelastic.

Deadweight Loss • In Fig. 14. 2 a is the initial choice with no taxation • Point b is the choice with a lump-sum tax • A commodity tax on good 1 raising the same revenue as the lump-sum tax leads to c • U1 – U2 is the deadweight loss in utility terms • The shift in budget from b to d is a monetary measure of loss Good 2 c b d a U0 U1 U2 Good 1 Figure 14. 2: Income and substitution effects Deadweight Loss • Deadweight loss is a consequence of substitution between commodities.

• For the indifference curves in Fig. 14. 3 there is no substitution • The initial choice is at a • The choice with a lumpsum tax and with a commodity tax on good 1 is at b • There is no deadweight loss Good 2 a b U1 U0 Good 1 Figure 14. 3: Absence of deadweight loss Optimal Taxation • Optimal commodity taxes attain the highest level of welfare possible whilst: – Raising the revenue required by the government – Allowing consumers to maximize utility – Letting firms maximize profit.

• Welfare is measured using the government’s objective function • With a single consumer the optimal taxes describe an efficient tax system • With many consumers the optimal taxes describe an equitable tax system Optimal Taxation • Consider a two-good economy with a single consumer and a single firm – This is the Robinson Crusoe economy • Labor is used as an input by the firm to produce an output with constant returns to scale – Constant returns implies zero profit • The output is sold by the firm to the consumer • The consumer supplies labor and demands the output • The revenue requirement of the government is R units of labor.

Optimal Taxation • Fig. 14. 4 shows the production possibilities of the economy • Y is the production set of the firm • This is displaced from the origin by the government use of labor • The normalised wage of 1 and output price p leads to zero profit • The firm produces on the boundary of the production set Good 2 (Consumption) Y p 1 R Good 1 (Labour) Figure 14. 4: Revenue and production possibilities Optimal Taxation • Consumer choices are shown in Fig. 14. 5 • The consumer price is q so the tax is t = q – p • The consumer’s budget constraint is qx = ? where x is consumption and ?labor supply • The locus of optimal choices as q varies traces out the offer curve • Utility rises as a move is made up the offer curve.

Good 2 (Consumption) Offer curve q p 1 Good 1 (Labour) Figure 14. 5: Consumer choice Optimal Taxation • Fig. 14. 6 superimposes consumer choice on the production set • The highest achievable utility is at the intersection of the offer curve and the production frontier • The optimum is denoted by point e and the optimal tax t* • The first-best is at e* and is achieved by a lumpsum tax of – R • Point e is the second-best Good 2 (consumption).

I1 e* e Y q I0 -R t* p 1 Good 1 (labor) Figure 14. 6: Optimal commodity taxation Optimal Taxation • Fig. 14. 6 also shows why labour can remained untaxed without affecting the outcome • The choices of the consumer and the firm are determined by the ratio of prices they face – This appears as the direction of the price vector in the figure • Changing the length but not the direction of the price vector introduces a tax on labour but does not alter the fact that e is the optimum • The zero tax on labour is a normalisation not a real restriction on the system.

Production Efficiency • The analysis of optimal taxation in the oneconsumer economy provides an important efficiency result • The Diamond-Mirrlees Production Efficiency Lemma states the production must be efficient when the optimal taxes are employed • Production efficiency occurs when an economy is maximising the output attainable from its given set of resources • This can only happen when the economy is on the boundary of its production set Production Efficiency.

• Production efficiency occurs when the marginal rate of substitution (MRS) between any two inputs is the same for all firms • This is attained in the absence of taxation by profit maximisation of firms in competitive markets – Each firm sets the marginal rate of substitution equal to the ratio of factor prices – Since factor prices are the same for all firms this equalizes the MRSs. • This is also true with taxation provided all firms face the same post-tax prices for inputs Production Efficiency • The optimum with commodity taxation must be on the boundary of the production set • Consider interior point f in Fig.

14. 7 • From f utility is raised by reducing input use keeping output constant • This is feasible so f cannot be an optimum • This reasoning can be applied to any interior point Good 2 (consumption) e f q I0 p -R 1 Good 1 (labor) Figure 14. 7: Production efficiency Production Efficiency • The reasoning can be extended to an economy with many consumers • With a single-consumer a reduction in labor use increases utility • This is true with many consumers if all supply labour or prefer to have more, rather than less, of a consumption good – This holds if there is agreement in the tastes of the consumers.

• If so moving from an interior point to a boundary point is unanimously preferred • The optimum must then be on the boundary Production Efficiency • Production efficiency implies no distortion in input prices • The Diamond-Mirrlees lemma provides a persuasive argument for: – The non-taxation of intermediate goods – The non-differentiation of input taxes between firms • The result is of immediate practical importance – It provides a basic property that an optimal tax system must possess • Value Added Taxation satisfies this property – Taxes paid on inputs can be reclaimed – Only final consumers pay tax.

Tax Rules • With more than one commodity the issue is how to distribute the tax burden • Assume that goods are produced using labor as the only input • The before-tax price of good i is pi = ci where ci is the constant marginal cost • The after-tax price is qi = pi + ti • The level of government revenue is R = ? itixi where xi is demand • Labor is good 0 with supply x0 and tax t0 = 0 – The tax of zero is a normalization Tax Rules • The inverse elasticity rule is derived assuming: 1. A single consumer 2. The demand for each good depends only on its own price and the wage rate.

• • • The commodity taxes are chosen to maximize welfare subject to a revenue requirement The assumption of a single consumer implies the optimal tax system is efficient The second assumption ensures that are no cross-price effects between the taxed goods – This independence of demands is a strong assumption Tax Rules • Assume there are two taxed goods and labor • The preferences of the consumer are represented by U(x0, x1, x2) • Consumption and labor supply satisfy the budget constraint q1x1 +q2x2 = x0 • Utility maximisation gives the first-order conditions U0 = – ????

Ui = ? qi, i = 1, 2 – Ui is the marginal utility of good i – ? the marginal utility of income Tax Rules • The government revenue constraint is R = t1x1 +t2x2 • But ti = qi – pi so q1x1 +q2x2 = R + p1x1 +p2x2 • The Langrangean of the optimization is L = U(x0, x1, x2) + ? [q1x1 +q2x2 - R - p1x1 -p2x2] • The assumption of independent demands implies qi = qi(xi) so the first-order condition is ? ? ? ? qi ? ?qi U i ? U 0 ? qi ? xi ? pi ? ? 0 ? ? ? ?qi ? xi ? xi ? ?xi ? ? ? Tax Rules • Using the facts on marginal utilities ? ? ?qi ? qi ? ? xi ? ? ? q i ? xi ? pi ? ? 0 ? xi ? xi ? ?

• Define the elasticity of demand for good i by ? i = qidxi/xidqi • The first-order condition can be solved to write ti ?? ? ? ? 1 . ? pi ? ti ? ? ? ?? ? i • This is the inverse elasticity rule Tax Rules • To interpret the rule it must be noted that – The marginal utility of another unit of income for the consumer is ? – The utility cost of another unit of government revenue is ? – Since taxes are distortionary ? > ? • The rule states that the proportional rate of tax on good i should be inversely related to its elasticity of demand • Furthermore the constant of proportionality is the same for all goods Tax Rules.

• These observations imply that necessities which by definition have low elasticities of demand should be highly taxed • Luxuries with a high elasticity of demand should have a low rate of tax • This efficient tax system minimizes the distortion in demand • The system described is efficient but is not equitable – If implemented low-income households would bear a disproportionately large share of the tax burden Tax Rules • Accounting for the cross-price effects in demand leads to the Ramsey rule • Now let the demand for good i be xi = xi(q1, q2) • The Lagrangean for the optimisation is L = U(x0, x1, x2) + ?

[t1x1 +t2x2 - R] • The first-order condition now has cross-price effects 2 2 ? ?xi ? xi ? ? ? ? xk ? ? t i ? Ui ?? 0 ? qk i ? 1 ? qk ? i ? 0 ? Tax Rules • From the budget constraint of the consumer • The Slutsky equation gives ? xi ? qk ? Sik ? xk ? xi ? I where I is lump-sum income • Using these the first-order condition becomes 2 ? ?xi ? ?? ?? ? ? ti ? Sik ? xk ? ? ? ? ? ? xk ? I ? ? ? i ? 1 ? • This solves for the Ramsey rule 2 ? ? 2 ? xi ? ? ? ?1 ? ? ? ti ? ti S ki ? ?? xk , k ? 1,2 ? i ? 1 ? I ? ? ? i ? 1 ? x0 ?

x1 ? x2 q1 ? q2 ? xk ? ?qk ? qk ? qk Tax Rules • To a first approximation the Ramsey rule implies that optimal taxes cause the same proportional reduction in compensated demand for each good relative to the before-tax position • This emphasises that it is the distortion caused by the tax system in terms of quantities that should be minimised – Consumption determines utility – What happens to prices is secondary • The tax rates remain implicit in the Ramsey rule since it focuses on what happens to demand Tax Rules.

• The Ramsey rule suggests goods whose demand is unresponsive to price changes must bear higher taxes – Goods that are unresponsive to price changes are typically necessities such as food and housing – This tax system would bear most heavily on necessities • The inequitable nature of this is simply a reflection of the single consumer assumption – The optimization does not involve equity and the solution reflects only efficiency Equity Considerations • The focus of taxes on necessities will be moderated by equity considerations • Assume there are two consumers and that the government has a social welfare function 1 1 W ?

W U 1 x1 , x1 , x1 , U 2 x1 , x1 , x1 0 2 0 2 ? ? ? ? ? ?? • The key term is the social marginal utility of income ? W h h ? ? • The value of ? h is determined by the social weight of h (? W/? Uh) and private valuation of income (? h) ?U h Equity Considerations • Maximizing welfare subject to the revenue constraint gives the necessary condition h h h h h h ? h ? i ti S ki 1 ? h ? xk ? h ? i ti ? xi ? I xk ? ?1 ? h h h ? ? h xk ? h xk ? h xk ? ? • The left-hand side is the proportional change in aggregate compensated demand for good k • The reduction differs across goods • It is smaller if ?

h and demand are positively correlated (equity) and if tax payment varies little with income (efficiency) Applications • Practical policy requires a method for numerically implementing the optimal tax rules • Two basic pieces of information are needed in order to calculate the tax rates – The demand functions of the consumers to find the levels of demand and the demand derivatives – The social valuations of each consumer • The demand functions are obtained by estimation • The valuations should be calculated from a specified social welfare function and individual utility functions for the consumers Applications.

• One common procedure is to employ the utility function U ? K M h ? ? h 1?? / ? 1 ? ? ? • With a utilitarian welfare function the social marginal utility is h h ?? ? ? KM • K is fixed by setting ? h = 1 for the lowest income consumer • If ? > 0 then ? h declines as income rises • It decreases faster as ? rises Applications • The first application is to reform of taxaxation • Define the welfare effect of the tax on good k by • Define the revenue effect by ? W h ? ? ? h ? h xk ? t k ? h ? xih ? ?R ? ? h ? xk ? ? i t i ? ?t k ? qk ? ? ? ? • The marginal revenue benefit (the extra revenue relative to the welfare change) for good k is ?

R ? t k MRBk ? ? ? W ? t k Applications • At the optimum MRBk should be the same for all goods • The tax on a good with ahigh MRBk should be raised and lowered on low MRBk • Tab. 14. 1 presents the MRB for several categories of commodity • Tax reform should lower the tax on tobacco and raise the tax on “other goods” Good Other goods Services Petrol Food Alcohol Transport and equipment Fuel and power Clothing and footwear Durables Tobacco ? =2 2. 316 2. 258 1. 785 1. 633 1. 566 1. 509 1. 379 1. 341 1. 234 0. 420 ? =5 4. 349 5. 064 3. 763 3. 291 3. 153 3. 291 2. 221 2. 837 2. 514 0. 683 Table 14.

1: Tax reform Source: Madden (1995) Applications • Calculations of optimal taxes have used data from the Indian National Sample Survey • Define ? to be the wage as a proportion of expenditure • Tab. 14. 2 reports optimal taxes for ? = 2 and two value of ? • Redistribution is observed in the subsidies to cereals and milk products Item Cereals Milk and milk products Edible oils Meat, fish and eggs Sugar and tea Other food Clothing Fuel and light Other non-food ? =2 -0. 015 -0. 042 0. 359 0. 071 0. 013 0. 226 0. 038 0. 038 0. 083 ? =5 -0. 089 -0. 011 0. 342 0. 083 0. 003 0. 231 0. 014 0. 014 0. 014 Table 14.

2: Optimal tax rates Source: Ray (1986a) Applications • The extent of redistribution of Indian commodity taxes has been calculated explicitly • Th is commodity tax payment by consumer h • Expenditure of consumer h is ? h • The net gain from the tax system is – Th/? h • Tab. 14. 3 shows the redistribution achieved by the actual tax system Rural Urban Expenditure ? T h / ? h ? T h / ? h level Rs. 20 Rs. 50 . 105 . 004 . 220 . 037 Table 14. 3: Redistribution of Indian commodity taxes Source: Ray (1986b) Applications • Tab.

14. 4 reports the redistribution achieved by the optimal tax system • A higher value of ? represents more social concern for equity • The value of – T/? is calculated for a household with ? of mean expenditure • The potential redistribution is substantial ? ? 0. 1? ? 1. 5 ? ? 5 ?T / ? .07 .343 . 447 Table 14. 4: Optimal redistribution Source: Ray (1986b) Efficient Taxation • The tax rules have only analyzed competition • Imperfect competition provides an additional motivation for tax intervention • An imperfectly competitive firm produces less then the efficient level • Providing a subsidy will raise output but imposes a tax cost elsewhere • Whether this is efficient depends on the extent of tax-shifting.

Efficient Taxation • Consider an economy with two goods – Good 1 is produced by a competitive firm and good 2 by a monopoly • There is a single consumer and no revenue is to be raised – Taxation is used only to correct the imperfect competition • The profit-maximizing price of the monopoly is represented by q2 = q2 (q1, t2) • There is undershifting if ? q2/? t2 < 1 and overshifting if ? q2/? t2 > 1 Efficient Taxation • The utility effect of variations in the two taxes is ?q1 ? q2 ? q2 dt1 ? ?x2 dt 2 dU ? ?? x1 dt1 ? ?x2 ? t1 ? t1 ? t 2 • The taxes raise zero revenue so dR ? 0 ? x1dt1 ? x2 dt 2 • Eliminating dt1 between these equations shows • The monopoly should be subsidized if ? q2/? t2 is large and ? q2/? t1 is negative ? ?q2 ? ?q2 x1 ? 1 ? ? 0 ? dt 2 ? 0 ? ? x2 ? t1 ? ?t 2 ?

Public Sector Pricing • The theory of commodity taxation has a second application • A public sector firm should set a price that maximizes welfare subject to a revenue target • This is the same problem that has already been analyzed • Optimal public sector prices are generally known as Ramsey prices • The optimal tax rate becomes the optimal markup over marginal cost.