Optimal taxation under a consumption target

Abstract

While in the familiar problem of optimal commodity taxation the government faces a constraint on tax revenue, we consider the case of a consumption target on a group of commodities (i.e., a weak constraint on total consumption), instead. This optimal commodity tax problem with a consumption target brings about taxation rules that are mainly at variance with the standard results of commodity taxation. In our main theorem, we derive a general, though quite simple, rule of optimal commodity taxation under a target on total consumption: in particular, we establish that higher consumer prices should be charged for commodities with (1) high price elasticities of total demand and (2) low consumption shares in total demand. From this theorem we deduce three important corollaries: an anti-inverse elasticity result, an anti-Corlett–Hague result and a uniform-pricing result. All of these results are (generically) at variance with well-known rules of commodity taxation.

Appendix: Proof of the generalised uniform-pricing result

We first prove the “only if.” For equal weighted consumer prices, $q_{i} / k_{i} = r~\forall i=1,\ldots ,n$, the right-hand side of Eq. (9) becomes $(D^{\prime } / r) \sum _{i=1}^{n} \varepsilon _{ji}^{h}$. Using Hicks’ (1939) “third law,” we get $(D^{\prime } / r) \sum _{i=1}^{n} \varepsilon _{ji}^{h} = (D^{\prime } / r) (- \varepsilon _{j0}^{h})$. Then, all $\varepsilon _{j0}^{h}$ ($j \ne 0$) must be equal. This implies that all $\varepsilon _{i0}^{h}$ and $\varepsilon _{0i}^{h}$ ($\forall i \ne 0$) have the same sign, since $\partial x_{i}^{h} / \partial q_{j} = \partial x_{j}^{h} / \partial q_{i}$. Also, using Hicks’ “third law” and the fact that $\varepsilon _{00}^{h} \leqq 0$, we obtain $\sum _{i=1}^{n} \varepsilon _{0i}^{h} = - \varepsilon _{00}^{h} \geqq 0$. Therefore, all $\varepsilon _{0i}^{h}$ and $\varepsilon _{i0}^{h}$ ($\forall i \ne 0$) must be non-negative.

Next, we prove the “if.”¹⁶ Suppose that all commodities are equally weakly substitutable with respect to the composite good, $\varepsilon _{j0}^{h} = \alpha \geqq 0, \; \forall j \ne 0$. Then, by definition, $\partial x_{j}^{h} / \partial q_{0} = \alpha x_{j} / q_{0}$. Thus, by symmetry, $\partial x_{0}^{h} / \partial q_{j} = \alpha x_{j} / q_{0}$. Substituting this relation into the Slutsky equation yields

$$\begin{aligned} \frac{\partial x_{0}^{m}}{\partial q_{j}} = \frac{\partial x_{0}^{h}}{\partial q_{j}} - \frac{\partial x_{0}^{m}}{\partial I} x_{j} = \left( \alpha - \frac{\partial x_{0}^{m}}{\partial I} q_{0} \right) \frac{x_{j}}{q_{0}}, \quad j = 1, 2, \dots , n. \end{aligned}$$

(A-1)

The term within the parentheses in Eq. (A-1) is independent of j, and we denote it by $\beta$, that is, $\partial x_{0}^{m} / \partial q_{j} = \beta x_{j} / q_{0}$. Now, differentiating both sides of the identity $\sum _{i=0}^{n} q_{i} x_{i}^{m} (q_{0}, \mathbf {q}, I) \equiv I$ with respect to $q_{j}$ and using the above relation, we obtain

$$\begin{aligned} x_{j} = - \frac{1}{1 + \beta } \sum _{i=1}^{n} q_{i} \frac{\partial x_{i}^{m}}{\partial q_{j}}. \end{aligned}$$

(A-2)

Substituting this into Eq. (2) yields

$$\begin{aligned} \sum _{i=1}^{n} \left[ \frac{\mu }{(1 + \beta )} q_{i} - D^{\prime } k_{i} \right] \frac{\partial x_{i}^{m}}{\partial q_{j}} = 0, \quad j = 1, 2, \dots , n. \end{aligned}$$

(A-3)

Then, $[ \mu / (1 + \beta ) ] q_{i} - D^{\prime } k_{i} = 0, \; \forall i \ne 0$ is a solution of Eq. (A-3).¹⁷ That is, $q_{i} / k_{i} = (1 + \beta ) D^{\prime } / \mu , \; \forall i \ne 0$. The solution is independent of i and may be denoted by r.