Minimum Tax Anticipated

A. Appendix

A.4. Minimum Tax Anticipated

In the following, we show that even when the minimum tax is anticipated, rents for the respective governments have the same qualitative characterization as in Section 5.1 where public good provision was fixed. Specifically, rB(0, x^µ_B(ε)) is monotonically increasing in ε while rA(0, x^µ_B(ε)) is concave in ε with a unique optimum that defines the lower bound of the non-renegotiable minimum tax frontier.

We assume that the imposition of the minimum tax is anticipated before the start of Stage 1, so each government takes the minimum tax into account when determining the level of public good provision. Thus, the minimum tax is agreed upon after which the sequence of events is exactly as in Section 4. Best response taxes with the minimum tax are as follows:

if xB > xA then τ^µ_A = ¹₃k

x_B^θ−x_A^θ

+ε and τ^µ_B = ²₃k

x_B^θ−x_A^θ

+ ¹₂ε; on the other hand if xA > xB then τ^µ_A = ²₃k

x_A^θ−x_B^θ

+ ¹₂ε and τ^µ_B = ¹₃k

x_A^θ−x_B^θ

+ε. If xA = xB then τ^µ_A = τ^µ_B = 0. But now the levels of public good provision xA and xB are determined optimally in Stage 1. Using these expressions for τ^µ_A and τ^µ_B, Government A’s rent function is defined as follows for ε∈

0,²₃kx_B^θ−x_A^θ: r^µ_A(xA, xB;ε) =

⎧⎪

⎨

⎪⎩

19k

x_B^θ−x_A^θ

−xA+¹₆ε− _2k(x_B^θ1−x_A^θ)ε² if 0≤xA < xB;

−xA if 0≤xA=xB;

49k

x_A^θ−x_B^θ

−xA+²₃ε+ ¹

4k(x_A^θ−x_B^θ)ε² if 0≤xB < xA.

(A.7)

For Government B,

r_B^µ(xA, xB;ε) =

⎧⎪

⎨

⎪⎩

49k

x_B^θ−x_A^θ

−xB+ ²₃ε+ ¹

4k(x_B^θ−x_A^θ)ε² if 0≤xA< xB;

−xB if 0≤xA=xB;

19k

x_A^θ−x_B^θ

−xB+ ¹₆ε− _2k(x_A^θ1−x_B^θ)ε² if 0≤xB < xA.

(A.8)

As was the case for when there was no minimum tax, when it maximizesr_A^µ(xA, xB;ε) a level of public good provision x^µ_A(ε) of Government A is a best response against a level of public good provision BRA(xB;ε).⁴⁷ A Nash equilibrium in levels of public good provision is a pair (x^µ_A(ε), x^µ_B(ε)) where x^µ_A(ε) is a best response against x^µ_B(ε) and vice-versa.

The characterization of equilibrium is technically the same as discussed in Section 4.2 for the case with no minimum tax; see the appendix for details. The equilibrium is asymmetric, with one government providing no public good and the other providing the public good at a positive level. As before, w.o.l.o.g. we let A be the country with no public good provision in equilibrium; x^µ_A(ε) = 0. The top panel of Figure 3 shows a plot of x^µ_B(ε) while the bottom panel shows 1−sˆ(0, x^µ_B(ε)) for k = 1 (andθ = ¹₂) as ε is varied.⁴⁸ Note from the bottom panel that 1−sˆ(0, x^µ_B(ε)) is increasing in ε and 1−ˆs(0, x^µ_B(ε)) = 1 for ε= ²₃k

x_B^θ −x_A^θ

= ₃₆⁵ . Also note that all values forε= 0 correspond to equilibrium values given in Proposition 3. Thus, atε= 0,x^µ_B(0) =x^µ_B =₂

9

₂

. The top panel shows thatx^µ_B(ε) decreases monotonically with ε until the point where 1−sˆ(0, x^µ_B(ε)) = 1. Government B’s incentive to compete in public good provision (by offering the public good at a higher level than Government A) is reduced by the fact that Government A is limited in the extent to which it is allowed to set its tax lower than B’s.

Turning to Figure 4, we see that for k= 1, θ= ¹₂, rents for the respective governments have the same qualitative characterization as in Section 5.1 where public good provision was fixed. Thus, as claimed, rB(0, x^µ_B(ε)) is monotonically increasing in ε while rA(0, x^µ_B(ε)) is concave in ε. The non-renegotiable minimum tax frontier is shown in Figure 4 as the interval ε ∈ [ε, ε). The upper and lower bounds, ε and ε, are defined in the same way as in Proposition 4. For ε < εboth governments would agree to implement a higher minimum tax. But for ε > ε, Government A makes higher rent with no minimum tax.

47Note the distinction we make between the best response function and rent function with and without the minimum tax; the functions are shown to be dependent on the parameterεin the former case.

48We have written ˆs(τ_A, τ_B, x_A, x_B) in the form ˆs(0, x^µ_B(ε)) to represent the fact that taxesτ_A=τ^µ_A and τ_B =τ^µ_B have been determined as functions ofx^µ_A(ε) = 0 andx^µ_B(ε).

Minimum Tax Anticipated: Characterization of Equilibrium. To determine Gov- ernment A’s set of best responses with the minimum tax, we investigate the properties of r^µ_A(xA, xB;ε). First we check the range 0 ≤ xA < xB, over which r_A^µ(xA, xB;ε) =

19k

x_B^θ−x_A^θ

−xA+¹₆ε−ε²/2k

x_B^θ−x_A^θ

. Taking the first derivative, we have drA/dxA=

−1−θx^θ−1_A

2k²+ 9ε²/

x_B^θ−x_A^θ₂

/18k <0;r_A^µ(xA, xB;ε) is everywhere downward slop- ing over the range 0 ≤ xA < xB. Now note that r_A^µ(xA, xB;ε) >−xA at xA =xB > 0 for ε ∈

0,²₃k

x_B^θ−x_A^θ

, and r^µ_A(xA, xB;ε) = −xA at xA = xB > 0 for ε = ²₃k

x_B^θ−x_A^θ . We can conclude that r_A^µ(xA, xB;ε)≥ −xA for all ε asxA →xB from below. Consequently, xA= 0 maximizes r_A^µ(xA, xB;ε) for the range 0≤xA< xB and xA= 0 dominates xA=xB. Thus x^µ_A(ε) = 0 is the best response over the range 0≤xA < xB.

If on the other hand 0 ≤ xB < xA, then r_A^µ(xA, xB;ε) = ⁴₉k

x_A^θ−x_B^θ

−xA+ ²₃ε+ ε²/4k

x_A^θ−x_B^θ

. Taking the first derivative, we have drA/dxA=−1 + 4

9kθx^θ−1_A

1−9ε²/16

x_A^θ−x_B^θ₂ .

We cannot solve explicitly for x^µ_A(ε) over the range 0 ≤ xB < xA without specifying θ. However, by specifying values of ε we can obtain a characterization of x^µ_A(ε). To illustrate, fixεat its maximum admissible valueε =ε= ²₃k

x_B^θ−x_A^θ

, and substitute this into the first derivative. We have drA/dxA=−1 +₁₂⁵kθx^θ−1_A . Setting the result equal to zero and solving, we have x^µ_A(ε) = ₅

12θk_1−θ¹

. Then, following the same logic as in Section 4.2 preceding Proposition 3, and using the assumption that xA < xB, we have that (x^µ_A(ε), x^µ_B(ε)) =

0, ₅

12θk_1−θ¹

is the unique Nash equilibrium. Taxes are obviously the same across countries for ε=ε, at τ^µ_A=τ^µ_B =k ₅

12θk ^θ

1−θ and ˆs= 1.

Notice that x^µ_A(ε) = ₅

12θk_1−θ¹

< x^µ_A(0) = ₄

9θk_1−θ¹

. More generally, by the implicit function theorem we know thatx^µ_A(ε) may be treated as a continuous function of ε. It can be established that x^µ_A(ε) is inversely related to ε as ε is varied over the interval ε ∈ [0, ε] for 0 ≤ xB < xA. Following, once again the, same logic as in Section 4.2 we have that (x^µ_A(ε), x^µ_B(ε)) = (0, BRB(0;ε)) is the unique Nash equilibrium (given thatxA < xB).⁴⁹

We want to go one step further, and investigate the behavior of rA(0, BRB(0;ε) ;ε) and rB(0, BRB(0;ε) ;ε) as ε is varied in order to determine the non-renegotiable minimum tax frontier. While this cannot be done at a general level, it can be done for the specific

49By the same arguments as in Section 4.2,BR_B(0;ε)= 0.

value θ = ¹₂, which we believe to be generally illustrative. We maintain the assumption that xA < xB and solve for x^µ_B(ε). This root is very cumbersome to write down, and since x^µ_A(ε) = 0 is a dominant strategy for Government A we jump straight to the equilibrium value:

x^∗_B(ε) = 512 (−2)^1/3k⁸−(−2)^2/3φ^2/3+ 16k⁴

−2187 (−2)^1/3ε² + 2φ^1/3

/

1944φ^1/3

where

φ =−8192k¹²+ 839808k⁸ε²−14348907k⁴ε⁴+ 59049√

−768k¹²ε⁶+ 59049k⁸ε⁸.

This solution for x^µ_B(ε) is illustrated for k = 1 in Figure 3 and used to define the non- renegotiable minimum tax frontier illustrated in Figure 4.