The Ramsey rule revisited.

I. Introduction

In pursuit of an optimum tax scheme in terms of minimizing welfare loss, economists have been searching for a better solution since the seminal contribution of Ramsey [17]. The literature is voluminous: they include the works by Hotelling [9], Hicks [8], Harberger [7], Dimond and Mirrlees [2], Dixit [3; 4], and Sandmo [19]. The Ramsey optimum tax rule, that is, the percentage reduction in quantity demanded of each commodity be the same was interpreted by Kahn [11] as the inverse elasticity rule. The inverse elasticity rule states that the optimum tax rates and price elasticities of demand should be inversely related. The merit of the inverse elasticity rule is that the optimum tax rate does not require the explicit knowledge on prices and quantities of the taxed commodities. To evaluate policy implications, one needs only to examine the published price elasticities of demand to arrive at optimum tax rates. However, the inverse elasticity rule normally applies only to a constant cost case in which average cost (hence marginal cost) is horizontal [10; 18]. In a competitive market, while the constant cost case is plausible, it assumes away the producer's surplus. In this paper, we prove that the inverse elasticity rule with the ad valorem tax on the gross demand price is equivalent to that with the supply ad valorem tax developed by Ramsey.(1) However, the inverse elasticity formula with the unit tax as shown in this paper is different. Furthermore, we propose a sufficient condition for optimum tax ratio with a positively sloped supply schedule. In addition, we explore a sufficient condition for an optimum equiproportional tax across all commodities. Last, we employ some empirical estimates of price elasticities of both demand and supply to calculate the optimum tax ratios.

II. Inverse Elasticity Rule with Increasing Costs

Given a set of linear demand and supply functions an ad valorem tax v based on the gross demand price [P.sub.G] is equivalent to pivoting down the demand curve by v[P.sub.G] [14]. In Figure 1, the excess burden is measured by the well-known welfare triangle whose size is determined by the tax rate [P.sub.G] - [P.sub.N] = v[P.sub.G] (difference between gross and net demand prices) and loss in output [Delta]Q. We first measure the excess burden via the following price elasticities of demand ([Alpha]) and supply ([Beta])

[Alpha] = ([Delta]Q/Q*)(P*/([P.sub.G] - P*)) [is less than] 0 (1)

[Beta] = ([Delta]Q/Q*)(P*/([P.sub.N] - P*)) [is greater than] 0 (2)

where P* and Q* are pretax equilibrium price and quantity. Hence it follows immediately from (1) and (2) that

[P.sub.G] - P* = ([Delta]Q/Q*)(P*/[Alpha]) (3)

[P.sub.N] - P* = ([Delta]Q/Q*)(P*/[Beta]) (4)

[P.sub.G] - [P.sub.N] = (([Delta]QP*)/Q*)((1/[Alpha]) - (1/[Beta])) (5.A)

or

v[P.sub.G] = ([Delta]QP*/Q*)([Beta] - [Alpha]/[Alpha][Beta]) (5.B)

where v is the demand ad valorem tax rate. Solving for [Delta]Q from (5.B) yields

[Delta]Q = (Q*[Alpha][Beta]v[P.sub.G])/(P*([Beta] - [Alpha])). (6)

The size of the excess burden W is therefore

[Mathematical Expression Omitted]

For a group of commodities (i) whose income effect is negligible such that the compensated demand curve is a good approximation for the ordinary demand curve and (ii) that are insignificantly related, i.e., commodities are not close substitutes or complements,(2) the Ramsey rule can be cast into the following minimization problem:

[Mathematical Expression Omitted]

subject to [summation over i] [v.sub.i][P.sub.iG][Q.sub.i] = R (9)

where subscript i denotes the ith commodity and R denotes the tax revenue constraint. The Lagrangian equation and its corresponding first-order conditions are:

L = [Phi] + [Lambda](R - [summation over i] [v.sub.i][P.sub.iG][Q.sub.i]) (10)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

and from (11) and (12) we have

[Lambda] = (

.sub.i] - [[Alpha].sub.i])[Q.sub.i]) (13)

[Lambda] = (

.sub.j] - [[Alpha].sub.j])[Q.sub.j]). (14)

Note that the changes in both prices and quantities must be small enough for the elasticity formulas to be valid. Besides, such a small change is needed to free us from the theoretical difficulty of income effect and interrelatedness between a pair of commodities. As a result of such a small change, i.e., [Q*.sub.i] [is equivalent to] [Q.sub.i] and [P*.sub.i] [is equivalent to] [P*.sub.iG], the inverse elasticity rule is readily

[v.sub.i]/[v.sub.j] = [[[Alpha].sub.j][[Beta].sub.j]([[Beta].sub.i] - [[Alpha].sub.i])]/[[[Alpha].sub.i][[Beta].sub.i]([[Beta].sub.j] - [[Alpha].sub.j])]. (15)

Equation (15) is exactly the same as that developed by Ramsey [17] in the case of the supply ad valorem tax. Consequently, the Ramsey rule applies to both demand and supply ad valorem taxes: that is, it is valid for an ad valorem tax be it levied on suppliers or consumers.(3) However, for a unit tax on the demand price, i.e., [P.sub.G] - [P.sub.N] = u, the inverse elasticity rule does not have its convenient forms. Following exactly the same procedure, the optimum taxation ratio can be shown to be

[u.sub.i]/[u.sub.j] = [[[Alpha].sub.j][[Beta].sub.j]([[Beta].sub.i] - [[Alpha].sub.i])[P*.sub.i]]/[[[Alpha].sub.i][[Beta].sub.i]([[Beta].sub.j] - [[Alpha].sub.j])[P*.sub.j]]. (16)

Note that the pretax prices are included in (16); hence, one must know these prices in order to calculate the optimum taxes. In such a case, one can still employ the Ramsey Rule simply by multiplying it by the pretax price ratio. Since many taxes are levied in terms of the demand ad valorem taxes, we shall limit our analysis to the optimum ad valorem tax ratio of equation (15).

Since [[Alpha].sub.i] [is less than] 0 and for ease of presentation, we denote [a.sub.i] as the absolute value of [[Alpha].sub.i]. Unfortunately, neither [a.sub.i] + [[Beta].sub.i] [is greater than] [a.sub.j] + [[Beta].sub.j] implies [a.sub.i][[Beta].sub.i] [is less than] [a.sub.j][[Beta].sub.j] nor [a.sub.i][[Beta].sub.i] [is less than] [a.sub.j][[Beta].sub.j] implies [a.sub.i] + [[Beta].sub.i] [is greater than] [a.sub.j] + [[Beta].sub.j]. Consequently, the knowledge on the sum or product of two price elasticities for a pair of commodities is not adequate to draw a useful conclusion about the optimum tax ratio.

In order to preserve the result of the inverse elasticity rule, we set forth the following proposition:

PROPOSITION 1. With a set of well-behaved linear demand and supply functions, the optimum ad valorem tax rate of commodity i is lower than that of commodity j if (i) the price elasticity of demand for commodity i (absolute value) exceeds that for commodity j and (ii) the price elasticity of supply for commodity i exceeds that for commodity j.

Proof. Let [a.sub.i] [is greater than] [a.sub.j] such that [a.sub.i] = [a.sub.j] + [Delta] for some [Delta] [is greater than] 0 and [[Beta].sub.i] [is greater than] [[Beta].sub.j] such that [[Beta].sub.i] = [[Beta].sub.j] + [Epsilon] for some [Epsilon] [is greater than] 0. The optimum tax ratio of (15) can be rewritten as

[v.sub.i]/[v.sub.j] = [[a.sub.j][[Beta].sub.j]([[Beta].sub.j] + [Epsilon] + [a.sub.j] + [Delta])]/[([a.sub.j] + [Delta])([[Beta].sub.j] + [Epsilon])([[Beta].sub.j] + [a.sub.j])] = [[a.sub.j][[Beta].sub.j]([[Beta].sub.j] + [a.sub.j]) + [Delta][[Beta].sub.j][a.sub.j] + [Epsilon][[Beta].sub.j][a.sub.j]] /[[a.sub.j][[Beta].sub.j]([[Beta].sub.j] + [a.sub.j]) + [Delta][[Beta].sub.j]([[Beta].sub.j] + [a.sub.j]) + [Epsilon][a.sub.j]([[Beta].sub.j] + [a.sub.j]) + [Delta][Epsilon]([[Beta].sub.j] + [a.sub.j])]. (17)

The value of (17) must be less than 1 because (i) the first term of both denominator and numerator is identical, (ii) the second and third terms of the denominator are greater than that of the numerator respectively, and (iii) the fourth term of the denominator is positive.

The proof of Proposition 1 preserves the result of the original inverse elasticity rule with constant costs if both price elasticities of the commodity are greater than that of the other. By utilizing increasing costs, the result generalizes the original inverse elasticity rule. However, sales taxes are normally identical for a group of commodities within a state. To justify the equiproportional sales tax across a group of commodities, it does not require that price elasticities be identical for all the commodities as suggested by the inverse elasticity rule under a constant cost case. Evidently, it is sufficient from (15) that if [[Alpha].sub.i] = [[Alpha].sub.j] and [[Beta].sub.i] = [[Beta].sub.j], the optimum ad valorem tax rate on commodity i must be the same as that on commodity j. Unfortunately it is highly unlikely that the price elasticity of demand (supply) for corn, for example, is identical to that of the price elasticity of demand (supply) for wheat. Given such an unlikeliness of equal price elasticities of demand and supply, we propose a sufficient condition which justifies an across-the-board equiproportional sales tax.

PROPOSITION 2. An equiproportional ad valorem tax between a pair of commodities i and j is optimum if the sum of the price elasticities of both demand and supply equals their product.

Proof. An examination of (15) indicates that [v.sub.i] = [v.sub.j] at optimality if [a.sub.j][[Beta].sub.j] = [[Beta].sub.j] + [a.sub.j] and [a.sub.i][[Beta].sub.i] = [[Beta].sub.i] + [a.sub.i] for all i and j.

Clearly, the solution to [a.sub.j][[Beta].sub.j] = [[Beta].sub.j] + [a.sub.j] can be traced out by the locus of the elasticity equation: [a.sub.j] = [[Beta].sub.j]/([[Beta].sub.j] - 1) for [[Beta].sub.j] [is greater than] 1. Hence, there is an infinite number of possibilities along the elasticity equation. For example, the pair of [[Beta].sub.j] = 1.5 and [a.sub.j] = 3 and [a.sub.i] = 2.25 and [[Beta].sub.i] = 1.8 would justify an equiproportional tax while the conventional wisdom suggests [[Alpha].sub.i] = [[Alpha].sub.j] for constant costs. Hence, the result of Proposition 2 provides a more realistic condition for imposing an equiproportional tax.

The sufficient condition (Proposition 2) is quite restrictive. The price elasticity of supply being greater than one ensures that [a.sub.j] is positive. Thus, it seems almost certain that different tax rates need to be applied to different commodities to minimize excess burden. What is true in theory may not be workable in the real world. Applying different tax rates would be politically difficult. One can imagine lobbyists exerting great pressure to prevent a relatively higher tax on their commodity. In addition, the use of different tax rates would necessitate an extra computational burden.

The results of both propositions can be applied to the general set of nonlinear demand and supply equations. A linear approximation at the pretax equilibrium price and quantity P* and Q* via the Taylor expansion would suffice. For instance, a tangency at P* and Q* should in general provide a reasonable approximation in a small neighborhood. Note that the "small neighborhood" is also required in proving the two propositions, as was true for the original inverse elasticity rule. In addition, to alleviate the income effect due to price changes and to avoid a strong interrelatedness among commodities in summing up the excess burden, a small change in P* and Q* is assumed. As a result, nonlinearity does not impose a serious problem to our results. In the case of unstable price elasticities manifested in great curvatures, one needs to further reduce the range within which P* and Q* can vary.

III. Applications of the Ramsey Rule

Note that the Ramsey rule in the case of a unit tax or (16) requires the information on pretax equilibrium prices. Hence, it does not lend itself conveniently toward policy implications. We shall calculate optimum tax ratios based on (15) which is exactly the same as that developed by Ramsey with a supply ad valorem tax. Since an ad valorem tax on gross demand prices is often practiced, it is appropriate to calculate the optimum tax ratios so that the excess burden is minimized. Price elasticities are reported in Table I; and the Ramsey optimum tax ratios are reported in Table II. An examination of Table II indicates that the optimum tax ratios of green peas to corn or cotton or wheat are very small since both price elasticities are greater for green peas. For example, from Table I, the price elasticity of demand is -2.8 and the price elasticity of supply is 4.4 for green peas. For corn, the price elasticity of demand is -0.49 and the price elasticity of supply is 0.18. The optimum tax ratio of green peas to corn can be readily calculated, using equation (15), to be 0.077 as shown in Table II. In contrast, the optimum tax ratio of natural gas to oil is higher because the price elasticities of natural gas are relatively inelastic. The validity of Proposition 1 is verified in the case of the demand ad valorem tax with a general upward sloping supply curve.

It is to be noted that only seven of twenty-eight optimum tax ratios (8[C.sub.2]) need to be calculated; i.e., [v.sub.i]/[v.sub.i + 1] for all i are calculated based on equation (15). For other [v.sub.i]/[v.sub.k] for k [is not equal to] i + 1 can be derived via

[v.sub.i]/[v.sub.k] = ([v.sub.i]/[v.sub.i + 1])([v.sub.i + 1)/(v.sub.i + 2]) ... ([v.sub.i + (k - i - 1)]/[v.sub.k]). (18)

Consequently, the Ramsey rule for n taxable commodities involves only n - 1 pieces of computations instead of n[C.sub.2]. The substitution formula of (18) can reduce significantly the burden of the computation (i.e., only 999 instead of 499500 optimum tax ratios need to be calculated in a case of 1000 commodities in a state).

[TABULAR DATA OMITTED]

If a tax is levied in the form of a per unit tax, the Ramsey rule takes the form of equation (16) in which pretax equilibrium prices need to be known. In the case of two similar pretax equilibrium prices, the Ramsey rule would serve as a good approximation to the optimum taxation rule. Finally, none of the optimum tax ratios from our simulation suggests the need for an equiproportional tax between a pair of commodities. That is, to justify such a tax it is sufficient that the sum of price elasticities be the same as their product as stated in Proposition 2. Hence, equiproportional taxes on a group of energy or agricultural products may not be justified for efficiency purposes as is witnessed in our numerical simulation.

[TABULAR DATA OMITTED]

IV. Concluding Remarks

In this paper, we prove that the Ramsey optimum taxation rule under the demand ad valorem tax is exactly the same as that under the supply ad valorem tax proved by Ramsey in 1927. Hence, it applies to all cases of the ad valorem taxes; be it on the demand price or on supply prices in a competitive market. However, we also show that the same formula can be applied in the case of a per unit tax. To evaluate policy implications for a unit tax, the optimum tax ratios requires the information of pretax equilibrium prices. Furthermore, we refine the inverse elasticity rule with a general upward sloping supply curve. It is sufficient that both price elasticity values be higher for a lower tax rate. To justify an equiproportional tax across a group of commodities, as is usually seen in reality, we derive a sufficient condition. Finally, applications of the Ramsey rule are made for eight agricultural or energy products. It is found that only seven of twenty-eight optimum tax ratios need to be calculated. None of the calculated optimum tax ratios calls for an equiproportional tax to minimize the excess burden.

2. If commodities are close substitutes or complements, the summation of excess burden may not be exact. For a large number of commodities, the underestimation and overestimation of excess burden may cancel out.

3. The relationship between demand and cost ad valorem taxes (the Musgravian transformation) is discussed by Musgrave [14] and Yang [21, 22].

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