Distortionary taxation and the free-rider problem

Abstract

This paper derives a version of the Samuelson rule which takes into account that a distortionary Ramsey-tax system is used to finance public-goods provision. Individuals have private information about their public-goods preferences. Moreover, individuals differ in their productive abilities. The incidence of taxation in the Ramsey model implies that more productive individuals have a lower willingness to pay for public goods than less productive individuals. They are therefore tempted to understate their valuation of public goods and less productive individuals are inclined to exaggerate theirs. The paper characterizes an optimal rule for taxation and public-goods provision that eliminates these biases.

Appendix

Proof of Lemma 1

Define

$$U^*(\tau,w_t) :=\mathrm{Y}\max_Y u\bigl(\bigl(1-\tau(s)Y\bigr)\bigr) - v\biggl(\frac {Y}{w_t} \biggr) ,$$

i.e., U ^∗ gives indirect utility as a function of the tax rate and the skill level. Straightforward calculations yield

$$\everymath{\displaystyle}\begin{array}{l} Y_t'(\tau) =\frac {u'((1-\tau)Y_t(\tau)) + (1-\tau)Y_t(\tau)u''((1-\tau)Y_t(\tau)) }{(1-\tau)^2u''((1-\tau)Y_t(\tau)) -w_t^{-2}v''(Y_t(\tau)/w_t ) }, \quad \mbox{and} \\\noalign{\vspace*{6pt}}\frac {\partial^2 U^*(\tau,w_t)}{\partial\tau\partial w_t} =\frac {Y_t'(\tau)}{w_t^{2}}\biggl[v'\biggl(\frac {Y_t(\tau)}{w_t}\biggr)+ \frac {Y_t(\tau)}{w_t}\; v''\biggl(\frac {Y_t(\tau)}{w_t}\biggr)\biggr]. \end{array}$$

Note that $Y_{t}'(\tau) < 0$ and v convex imply that the cross derivative is negative. This proves the lemma. □

Proof of Proposition 2

I first derive a concise statement of the planner’s problem. From substituting Q ^∗ into the incentive constraints, one finds that incentive compatibility for the low-skilled requires that

$$ \theta_H \geq V_1(s_{LL}) - V_1(s_{HL}) \geq\theta_L ,\qquad V_1(s_{LH}) = V_1(s_{HH}).$$

(8)

Similarly, the incentive constraints for the more productive are

$$ \theta_H \geq V_2(s_{LL}) - V_2(s_{LH}) \geq\theta_L , \qquad V_2(s_{HL}) = V_2(s_{HH}).$$

(9)

As the function U _t (τ) is strictly decreasing in τ, these incentive constraints imply that, whenever Q=1, the same tax rate has to be used to cover the cost of provision. This tax rate is henceforth called $\bar{\tau}$, i.e., $\bar{\tau} :=\tau(s_{HH}) =\tau(s_{LH})=\tau(s_{HL})$. Analogously, define $\underline{\tau}:=\tau(s_{LL})$. Using those tax rates, the incentive constraints for individuals with skill level w _t can be rewritten as: $\theta_{H} \geq U_{t}(\underline{\tau}) - U_{t}(\bar{\tau}) \geq\theta_{L}$. In addition, from the property of increasing differences established in Lemma 1, the private utility loss due to higher taxation is larger for high-skilled individuals, $U_{2}(\underline{\tau}) - U_{2}(\bar{\tau})> U_{1}(\underline{\tau}) - U_{1}(\bar{\tau})$. Consequently, the planner only has to take the constraints $\theta_{H} \geq U_{2}(\underline{\tau}) - U_{2}(\bar{\tau})$ and $U_{1}(\underline{\tau}) - U_{1}(\bar{\tau})\geq\theta_{L}$ into account.

The planner’s problem can now be stated in the following way: An optimal choice of $\bar{\tau}$ and $\bar{\tau}$ solves the following problem:

I denote by $\underline{\tau}^{**}$ and $\bar{\tau}^{**}$ the second best tax rates, which solve this problem.

Case 1. Suppose θ _H ≥U ₂(0)−U ₂(τ _1L ). Consider the relaxed problem, which ignores IC₂. At a solution, it has to be true that $\underline{\tau}^{**} = 0$. Otherwise, $\underline{\tau}^{**}$ could be reduced in a feasible and incentive compatible manner, thereby contradicting optimality. The optimal level of $\bar{\tau}$ then has to ensure that IC₁ is binding. Hence, $\bar{\tau}^{**} = \tau_{1L}$. The assumption θ _H ≥U ₂(0)−U ₂(τ _1L ) implies that IC₂ can indeed be ignored.

Case 2. Let θ _H <U ₂(0)−U ₂(τ _1L ), and suppose a solution exists. Let τ _2H be the tax rate that is implicitly defined by the equation θ _H =U ₂(0)−U ₂(τ _2H ). Note that τ _2H >τ _k . Also note that θ _H <U ₂(0)−U ₂(τ _1L ) implies τ _2H <τ _1L .

(i) It must be the case that $\bar{\tau}^{**} > \tau_{k}$. Suppose, to the contrary, that $\bar{\tau}^{**} = \tau_{k}$. Then IC₁ is violated. To see this, note that $\bar{\tau}^{**} =\tau_{k}$ implies

$$U_1\bigl(\underline{\tau}^{**}\bigr) - U_1\bigl(\bar{\tau}^{**}\bigr) \leq U_1(0) -U_1(\tau_k) < \theta_L .$$

(ii) It must be the case that $\underline{\tau}^{**}>0$. Suppose, to the contrary, that $\underline{\tau}^{**}=0$. Then IC₁ implies $\bar{\tau}^{**}\geq\tau_{1L}$, and IC₂ implies $\bar{\tau}^{**} \leq \tau_{2H}$, contradicting τ _2H <τ _1L .

(iii) At least one (IC) constraint has to be binding. Otherwise—with $\underline{\tau}^{**}> 0$ and $\bar{\tau}^{**} > \tau_{k}$—both tax rates could be reduced in a feasible and incentive compatible manner. To see that both (IC) constraints have to be binding, suppose, for instance that at an optimum, IC₂ binds and IC₁ does not. Then both tax rates could be reduced in a feasible and incentive compatible manner—keeping the equality in the constraint for high-skilled individuals, while not violating the one for the low-skilled—thereby increasing utilitarian welfare. □

Proof of Lemma 2

The monotonicity of admissible provision rules is derived as follows: consider, for example, the two incentive compatibility constraints for the low-skilled, given that θ ₂=θ _L : θ _L Q(s _LL )+V ₁(s _LL )≥θ _L Q(s _HL )+V ₁(s _HL ) and θ _H Q(s _HL )+V ₁(s _HL )≥θ _H Q(s _LL )+V ₁(s _LL ). Adding up these inequalities gives Q(s _HL )≥Q(s _LL ). Similarly, one derives the constraints Q(s _LH )≥Q(s _LL ), Q(s _HH )≥Q(s _HL ) and Q(s _HH )≥Q(s _LH ). □

Proof of Proposition 3

First, the maximal level of welfare for each of the six candidate provision rules satisfying the monotonicity conditions in Lemma 2 is derived. In the second step, these welfare levels are compared to determine the optimal provision rule.

Obviously, the expected welfare under the constant provision rules Q≡0 and Q≡1, respectively, is given by EW ^Q≡0=U ₁(0)+U ₂(0), and

(10)

Rule Q ^∗: The solution to this problem has been characterized in Lemma 2, i.e., if θ _H ≥U ₂(0)−U ₂(τ _1L ), expected utilitarian equals:

(11)

If, to the contrary, θ _H <U ₂(0)−U ₂(τ _1L ), then, if a solution to the planner’s problem exists, expected utilitarian welfare equals:

(12)

where τ _LL and τ _HH is a pair of tax rates so that both incentive constraints are binding, i.e., so that

$$U_1(\tau_{LL}) - U_1(\tau_{HH}) = \theta_L \quad \mbox{and}\quad \theta _H = U_2(\tau_{LL}) - U_2(\tau_{HH}) .$$

Rule Q′: Along the same lines as in the proof of Proposition 2, one derives that the planner has to solve the following problem if Q′ is chosen:

The solution to this problem has been characterized in Proposition 2. If the incentive problem is modest, then expected welfare is given as

$$ EW' =(1-p_{HH})\bigl(U_1(0) + U_2(0)\bigr) +p_{HH}\bigl(2\theta_H+U_1(\tau_{1L}) + U_2(\tau_{1L})\bigr) .$$

(13)

If the incentive problem is severe, either rule Q′ cannot be implemented or the optimal combination of $\underline{\tau}$ and $\bar{\tau}$, for which both IC constraints are binding, is chosen. Denote these as τ _LL and τ _HH , respectively. Then

(14)

Rule Q ¹: Under provision rule Q ¹, the incentive compatibility constraints imply V ₂(s _LH )=V ₂(s _LL ) and V ₂(s _HH )=V ₂(s _HL ), or equivalently $\underline{\tau}:=\tau(s_{LL}) = \tau(s_{LH})$ and $\bar{\tau}:=\tau(s_{HH}) = \tau(s_{HL})$. The planner’s problem becomes

It is easily verified that at an optimum, only the constraint $U_{1}(\underline{\tau}) - U_{1}(\bar{\tau}) \geq\theta_{L}$ is binding. Optimal taxes are given as $\underline{\tau}^{**} = 0$ and $\bar{\tau}^{**} = \tau_{1L}$. Expected utilitarian welfare under rule Q ¹ equals:

(15)

Rule Q ²: It is easily verified that, under Scenario 2, provision rule Q ² can be implemented such that for all s, τ(s) is chosen such that the budget constraint holds as an equality. This implies

(16)

The proof of Proposition 3 follows from Observations 1–6 below.

1. Q≡0 and Q′ are strictly dominated by Q ². To see this, note that by Assumption 1, EW ² exceeds EW ^Q≡0. To see that it also exceeds EW′, note that Assumption 1 also implies that EW ² exceeds (1−p _HH )(U ₁(0)+U ₂(0))+p _HH (U ₁(τ _k )+U ₂(τ _k )+2θ _H ). The right-hand side of this inequality is an upper bound on the welfare that can be achieved with provision rule Q′.

2. If θ _H <U ₂(0)−U ₂(τ _1L ), then EW′=EW ^∗. This follows from using θ _H =U ₂(τ _LL )−U ₂(τ _HH ) and U ₁(τ _LL )−U ₁(τ _HH )=θ _L to substitute for θ _L and θ _H in the expressions for EW′ and EW ^∗ in (12) and (14).

3. If θ _H <U ₂(0)−U ₂(τ _1L ), then EW ^∗<EW ². This is a direct consequence of Observations 1 and 2.

4. If θ _H ≥U ₂(0)−U ₂(τ _1L ), then EW ^∗≥EW ¹ with equality if and only if θ _H =U ₂(0)−U ₂(τ _1L ): To see this, use (11) and (15), as well as the definition of τ _1L , to derive:

$$ EW^* - EW^1 =p_{LH}\bigl(U_2(\tau_{1L}) + \theta_H - U_2(0)\bigr).$$

(17)

5. If θ _H <U ₂(0)−U ₂(τ _1L ), then EW ¹<EW ²: Equations (15) and (16) imply that EW ²−EW ¹ equals

All terms in this sum are strictly positive under Scenario 2.

6. EW ²−EW ^Q≡1 may become positive or negative, depending on the prior probabilities: From (10) and (16), EW ²−EW ^Q≡1 equals:

Under Scenario 2, the first term is positive and the second is negative. Assuming that incentive problems are modest, one can use similar arguments to verify that

and

are also convex combinations of a strictly positive term and a strictly negative term. Moreover, for any one of the rules Q ², Q ⁱ, or Q≡1 one can find probability weights such that it yields the maximal level of welfare. □

Proof of Proposition 4

The implementability constraints in (5) imply that for any τ(s) and α(s), C(s,w _t )=w _t (1−τ(s)) and $Y(s,w_{t}) =w_{t} - \frac{\alpha(s)}{1-\tau(s)}$. Substituting these expressions into the constraints and the objective function, and dropping constant terms that are irrelevant for the maximization problem, yields the following optimization problem: Choose τ(s _LL ), α(s _LL ), τ(s _LH ), and α(s _LH ) in order to maximize

$$\ln \bigl(1-\tau(s_{LL})\bigr) + \lambda\frac{\alpha(s_{LL})}{1-\tau(s_{LL})} +\ln \bigl(1-\tau(s_{LH})\bigr) + \lambda\frac{\alpha(s_{LH})}{1-\tau(s_{LH})}$$

subject to the feasibility constraints,

$$\bar{w}\tau(s)= \frac{\alpha(s)}{1-\tau(s)} + kq(s) ,$$

for each s, and the incentive compatibility constraint,

$$\theta_L + \ln \bigl(1-\tau(s_{LH})\bigr) + \frac{1}{w_2} \frac{\alpha (s_{LH})}{1-\tau(s_{LH})}\leq\ln \bigl(1-\tau(s_{LL})\bigr) + \frac{1}{w_2} \frac{\alpha(s_{LL})}{1-\tau (s_{LL})}.$$

Upon using the feasibility constraints to substitute for $\frac{\alpha (s_{LL})}{1-\tau(s_{LL})}$ and $\frac{\alpha(s_{LH})}{1-\tau(s_{LH})}$, this optimization problem can be rewritten as: Choose τ(s _LL ) and τ(s _LH ) in order to maximize

$$\ln \bigl(1-\tau(s_{LL})\bigr) + \lambda\bar{w}\tau(s_{LL}) + \ln \bigl(1-\tau(s_{LH})\bigr)+ \lambda\bar{w}\tau(s_{LH})$$

subject to the incentive compatibility constraint,

$$k- \theta_L \geq \ln \bigl(1-\tau(s_{LL})\bigr) - \ln \bigl(1-\tau(s_{LH})\bigr) + \frac{\bar {w}}{w_2}\bigl(\tau(s_{LH}) - \tau(s_{LL})\bigr).$$

The proof that τ(s _LL ) is lower and that τ(s _LH ) is higher, as compared to a model without incentive constraints, involves the following steps: First, use a Lagrangian approach, where μ is the non-negative multiplier on the incentive constraint. Second, derive first-order conditions and observe that τ(s _LL ) decreases in μ and that τ(s _LH ) increases in μ. Hence, if the incentive constraint is binding, μ is strictly positive and τ(s _LL ) is therefore larger and τ(s _LH ) smaller than otherwise. Finally, it follows from the feasibility constraints that this also implies that α(s _LL ) is smaller and that α(s _LH ) is larger than otherwise if the incentive constraint is binding. □