Characterization of a K-L equilibrium

With the functional form of labor income taxes that we assume, we cannot directly apply the method of characterizing a K-L equilibrium that we used for the linear tax case because marginal rates of substitution will be different across agents in this case. We need to modify the planner’s static problem so that its solution can be equated with a K-L equilibrium allocation. Also, we need to assume a functional form of utility from the beginning, since a modification of the planner’s static problem will be different for different utility forms. Here, we will assume an isoelastic preference as defined in section 4.3.

Even though marginal rates of substitution are not equated across agents, we can find modified terms of marginal rates of substitution that are equated across agents using first-order conditions:

· v⁰(lt(θ^t))lt(θ^t)^τp

θ^1−τ_t ^pu⁰(c_t(θ^t)) = τ_l,t(1 − τ_p)w_t^1−τp

· β^tπ(θ^t)M_t(θ^t)u⁰(c_t(θ^t))

u⁰(c₀(θ₀)) = Q(θ^t)

Now, we can define a modified planner’s static problem for the separable isoelastic utility.

The following modified static problem is constructed so that a K-L equilibrium allocation solves the planner’s static problem given an aggregate K-L equilibrium allocation (C_t, L_t), Pareto-Negish weights {M_t(θ^t)}, and a progressivity of labor income tax, τ_p.

U^m(C_t, L_t, τ_p; M ) = max

ct(θ^t),lt(θ^t)

X

θ^t

π(θ^t)M_t(θ^t)

"

u(c_t(θ^t)) − γ

τ_p+ γθ^τ_t^pl_t(θ^t)^τpv(l_t(θ^t))

#

s.t. (λ^m_t ) ^X

θ^t

π(θ^t)c_t(θ^t) = C_t (µ^m_t ) ^X

θ^t

π(θ^t)θ_tl_t(θ^t) = L_t

Using the envelope condition, we can express prices only with the aggregate allocations, Pareto-Negish weights, and τ_p.

· τ_l,t(1 − τ_p)w_t^1−τp = Mt(θ^t)v⁰(l_t(θ^t))l_t(θ^t)^τp

M_t(θ^t)u⁰(c_t(θ^t))θ_t^1−τp = −U_L^m(C_t, Lt, τp; M ) U_c^m(C_t, L_t, τ_p; M )

· Q(θ^t) = β^tπ(θ^t)M_t(θ^t)u⁰(c_t(θ^t))

u⁰(c₀(θ₀)) = β^tπ(θ^t)U_c^m(C_t, L_t, τ_p; M ) U_c^m(C₀, L₀, τ_p; M )

By substituting out these prices in the budget constraint of the household, we obtain an implementability constraint¹⁹ of the form:

X

Among five conditions that characterize a K-L equilibrium, only the implementability condition is changed as we change the tax system from linear to non-linear. So we can use the same relaxed Ramsey problem with the modified implementability constraint to analyze optimal taxes. See the Appendix C for a complete description of the relaxed Ramsey problem.

For a fixed τ_p, the Ramsey government still has two conflicting goals: 1. minimizing distortions when financing government expenditures, and 2. internalizing the externality to improve risk sharing. The next proposition discusses the optimal structure of the labor income tax and the capital income tax.

Proposition 13. Assume an isoelastic utility of the form (24). If the tax system takes the functional form specified in (25), and ^∂w_∂L^autt

t < 0, ^∂w_∂K^autt

t > 0 for all t, then an optimal tax system for t ≥ 1 satisfies

τ_k,t+1 = 1

With this specific functional form of a non-linear tax system and isoelastic preference, proposition 13 shows that capital taxes are levied only to remove the externality of capital, and labor income taxes are responsible for all distortions due to remaining budgetary needs.

In particular, if we let τ_p = 0, we can see that the result reduces to the previous finding with linear labor income taxes.

19Notice that the derivation of an implementability constraint exploits the connection between government revenues and agents’ marginal rates of substitution for this specific functional form. With a more general non-linear tax system, however, this connection might be eliminated. Then, we cannot derive an implementability constraint and we cannot apply the same analysis with a more general non-linear tax system.

In summary, we here demonstrated that the optimal taxation results - the optimal structure between optimal and labor income tax with linear taxation - are robust to some class of progressive taxation functions.

6 Conclusion

In this paper, we have studied optimal Ramsey taxation in a limited commitment model.

The goal of the Ramsey government in this economy is to maximize welfare subject to an allocation being a K-L equilibrium. To achieve this goal, the government faces two conflicting objectives: 1. minimizing distortions when financing government expenditures, and 2. internalizing the externality of labor and capital on the value of autarky and thus the degree of risk-sharing. Balancing these two objectives, the steady-state optimal capital taxes are levied only to remove the external cost of capital, even though its level could depend on the elasticities and the shadow cost of distortion because of normalization. All the remaining budgetary needs of the government will be financed using labor income taxes in the long run, even though labor has a positive externality. This main result can be extended to non-linear labor income taxes if we take specific functional forms of labor income taxes.

Therefore, our analysis shows that there is a good reason to tax capital income in a limited commitment economy - the externality of capital. Our result, however, can also be interpreted as a version of the famous zero capital tax result of Chamley-Judd. If there is no capital externality in a limited commitment economy, then the result reverts to Chamley-Judd’s result. If there is an external cost of capital, then the result is a generalization of the Chamley-Judd’s result because capital taxes are levied only to remove the externality.

This result, however, implies that when thinking about the real world implications of optimal Ramsey tax analysis, the market structure matters crucially for the optimal level of capital income taxes and labor income taxes. Thus, a policy prescription should be based on a thorough investigation of the relevant financial market structure and its friction associated.

We see this paper as the first attempt to consider the risk-sharing effects endogenous to a tax code in the Ramsey literature. If private insurance markets are not perfect, a tax system implemented by the government affects private risk sharing. Thus, the government has to consider how the tax rate endogenously changes risk sharing in private markets when it optimally sets a tax system to finance government expenditures. This paper provides an analysis of optimal Ramsey taxation focusing on one source of imperfect private risk sharing,

limited commitment. Analyzing optimal Ramsey taxation with other sources of imperfect risk sharing would be important and complementary future work.

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Appendix