Income taxes and endogenous fluctuations: a generalization

Appendix A (Proof of proposition 1)

When (20) implies that the trace is negative for any tax rate on capital income if:

Noting assumption 1, (A.1) is rewritten as:

As the left hand (the right hand) in (A-2) is negative (positive), (A-2) is satisfied. For any tax rates on capita income, therefore, the steady state is always a sink, when and σⁱ=σ(i=k, l).

Appendix B (The reason why the required externalities are lower in the continuous-time case than in the discrete-time case)

If we express the local dynamics of one-sector growth model in the plane (k_t, c_t)_,(17) becomes:

where

and

In the discrete-time one-sector Ramsey model, the counterpart of (B-1) is expressed as:

The trace (T) and determinant (D) can be derived as:

These equations are almost the same as in Guo and Lansing (2002) except that we set σ^k≠σ^l and τ_r≠τ_w. The Euler equation is the equation that determines the growth rate of consumption, but the timing of the interest rate in the Euler equation is different between the continuous-time and discrete-time cases. The interest rate in this period (i.e., the marginal product of capital in this period) determines the growth rate in the continuous case, while the interest rate in the next period (i.e., the marginal product of capital in the next period) determines the growth rate in the discrete case. As the next period capital stock k_t+1 is a function of current period consumption c_t and current period capital stock k_t, equation (B-2) is more complicated than (B-1).

In the numerical simulations in Guo and Lansing (2002), the parameters were chosen such that T<0 and 0<D<1 are satisfied for 0≤τ_r<1. They considered the combination of the tax rates and the sizes of externalities that satisfy D+T+1=0. Noting that indeterminacy arises in the range D+T+1>0, they concluded that larger externalities are needed (i.e., D+T+1>0 is more likely to hold), as τ_r is higher. We should note that the externalities of capital and labor are not separated in their numerical simulations (i.e., σ^k=σ^l is assumed).

Then, let us consider why we can obtain Proposition 2 in contrast with Guo and Lansing (2002). We examine the continuous-time growth model. Suppose that the sizes of labor externalities are fixed in the range σ^l>(1+γ)/(1–α). Then, the determinant is positive. If the trace is negative, indeterminacy arises. We consider the case of τ_r=0. The trace is positive (negative) if σ^k=0 (σ^k=σ^l). Noting that the trace is negatively related to σ^k and the trace is zero at we can easily verify As the trace is negative for indeterminacy emerges at ¹⁹ Suppose that τ_r is increased from 0 to some value within the range (0,1). From the trace is negative at smaller sizes of σ^k. Thus, indeterminacy always emerges for σ^k=σ^l∈((1+γ)/(1–α), 1/α–1 and τ_r∈0, 1).

We consider the discrete-time growth model (B-2) investigated in Guo and Lansing (2002). Note that they focused only on the case of σ^k=σ^l(=σ). Defining the size of the following is satisfied in their numerical simulations. When τ_r=0, the steady state is a saddle for and is sink (locally indeterminate) for Moreover, is positively related to τ_r (see Figure 1 in their paper). Unlike the continuous case, the steady state might not be locally indeterminate in the discrete-time growth model even if σ>(1+γ)/(1–α) and τ_r∈[0, 1). Larger externalities are needed in the discrete-time model. Phrased differently, indeterminacy is more likely to arise in the continuous-time growth model than in the discrete-time one.

Appendix C (The case of σⁱ=0 and ϕ≠1 )

Appendix C proves that the trace T is negative for only if the sector-specific externalities is slightly above the minimum sizes of externalities θ_min(τ_r). In the case of σⁱ=0 and ϕ≠1, let us write the trace T as

where and (As for the signs of Θ and Ω, see footnote 17).

(34) is rewritten as

Noting the second row in (35) and we can obtain

only if θ_I is slightly higher than θ_min(τ_r), and equivalently Γ is sufficiently close to zero.