# Mathematical Formulation

The Ramsey rule is a basic criterion for any optimal taxation problem. See Diamond and Mirrlees (1971). We derive this rule using a dual approach. Suppose in the economy that there is only one consumer. This individual consumes leisure and two goods. Producers produce two consumption goods and a public good, g, by applying leisure (labor). The variable indexed by 3 is associated with leisure and the variables indexed by 1 and 2 are associated with the consumption goods. The prices that the consumer faces are called the consumer’s prices and are denoted by the vector q = (q_{b} q_{2}, q3). The consumer’s net demand vector is x = (x_{b} x_{2}, x_{3}). The consumer’s utility function is given by u = u((x_{b} x_{2}, x_{3}).

Then, the consumer’s budget equation is given as

Note that the consumer’s net demand for leisure, x_{3}, is negative and her or his demand for other goods is positive on the relevant domain of the prices. Namely, net leisure is the difference between leisure Z — L, and available time, Z. Thus, x_{3} = (Z — L) — Z. Alternatively, it is minus labor supply, —L. Equation (9.7) is the same as the standard expression of budget constraint:

where the left-hand side denotes consumption spending and the right-hand side denotes after-tax wage income.

The production possibility frontier is the constant cost type. The production possibility frontier is given as

where producer’s prices, p = (p_{b} p_{2}, p_{3}), are constants. This constraint is also the same as

where the left-hand side corresponds to total output and the right-hand side corresponds to labor input.

Specific excise taxes and a wage tax are imposed. Thus, we have

When a positive wage tax is imposed, the consumer’s after-tax pay is less than the amount that her or his employer pays. This implies q_{3} < p_{3} and t_{3} < 0. An increase in t_{3} implies a decrease in the wage tax. For example, p_{3} = w and q_{3} = (1 — t_{w})w. Then, t_{3} = —t_{w}w. Thus, we have t_{3}x_{3} = t_{w}wL > 0.

The tax revenue collected is spent on the public good. The government budget constraint is given as

Equation (9.10) may be derived from Eqs. (9.7), (9.8), and (9.9). Thus, this equation will not explicitly be considered below as a constraint.

Using the dual approach, the consumer’s optimizing behavior may be summarized in terms of the expenditure function:

Equation (9.11) summarizes the optimizing behavior of the consumer and the budget constraint. The production possibility frontier, Eq. (9.8), may be rewritten as

where E_{;} = I^{е} = x_{;}(q, и) (i = 1, 2, 3), is the compensated demand function for good i.

The maximization problem is to maximize utility, u, subject to Eqs. (9.11) and (9.12). The associated Lagrange function is given as

where A_{;} (i = 1,2) is a Lagrange multiplier.

The first-order conditions are given as

*dip*

where Ец *= *_{dq} p_{q}. denotes the substitution effect. Considering Eq. (9.9), Eq. (9.14) may be rewritten as

Considering the homogeneity condition, q_{;}Ey *=* 0, and the symmetrical

cross effects, Ец *=* E_{j;}, we obtain the Ramsey rule as

Alternatively, in the elasticity term, we have

where = f_{;}/q_{;} is the effective tax rate and *вц = qjEy/Ei* is the compensated elasticity. The Ramsey rule means that under an optimal tax structure, the marginal deadweight burden of a unit increase in each tax rate is proportional to the demand for that good. Alternatively, the marginal excess burden is proportional to the marginal tax revenue for that good.

From the Ramsey rule (9.15) or (9.15'), we may derive some special propositions, as explained in Sect. 2.2.

*The Inverse Elasticity Proposition* Assume that the cross-substitution terms among the commodities are all zero (*E _{i}j =* 0

*fori = j*). Then, the intrinsic tax rate of a commodity is inversely related to its demand elasticity. Thus,

where *X = ХфХ _{2}* takes a common value for i = 1, 2, 3.

*The Uniform Tax Rate Proposition* A uniform tax structure is optimal if and only if wage elasticities of demand are equal for all commodities; namely, ff_{13} = ff_{23}. Thus,

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