Commodity taxes and welfare under endogenous market conduct

Abstract

We consider consumption taxes in a model of endogenous Cournot versus Bertrand competition. It is argued that when the choice of unit versus ad valorem taxes affects longer-term decisions beyond the customary price or quantity decisions, the mix of the two taxes co-determines market conduct. This gives ad valorem taxes an anti-competitive effect that harms ad valorem taxes’ efficiency in comparison with unit taxes. We show that a mix of the taxes—or a unit tax alone if we compare one or the other of the taxes—is sometimes welfare superior on account of consumer-price and tax revenue effects. A practical implication of our findings is that pass-through rates are only sometimes useful guides for policy. In fact, we show when the proper response to demand for higher revenue is a higher unit tax rate and a lower ad valorem tax rate.

Appendix

Proof of Proposition 1

The proof uses the following Lemma.

Lemma

To find the way firm i reacts to the price charged by firm j, define $q^{i}=a-bp^{j}-\hbox {X}^{i}\left( {p^{j},k^{i}} \right) ,$ with $q^{i}=k^{i}$. That is, ${\mathrm{X}}\left( {p^{j},k^{i}} \right)$ is $p^{i}$ so that firm i produces exactly at its capacity, $k^{i},$ given $p^{j}.$ Now, let $R^{i}\left( {p^{j},k^{i}} \right)$ be the reaction function of firm i. We have:

$$\begin{aligned} R^{i}\left( {p^{j},k^{i}} \right)= & {} r^{i}\left( {p^{j},x^{i}} \right) , \quad x^{i}=\left( {1-\tau } \right) ^{-1}\left( {z+t} \right) \;{\mathrm{where}}\;{\mathrm{X}}\left( {p^{j},k^{i}} \right) <r^{i}\left( {p^{j},z+t} \right) .\\ R^{i}\left( {p^{j},k^{i}} \right)= & {} r^{i}\left( {p^{j},x^{i}} \right) , \quad x^{i}=\left( {1-\tau } \right) ^{-1}\left( {z+t+\theta } \right) \;{\mathrm{where}}\;{\mathrm{X}}\left( {p^{j},k^{i}} \right)>r^{i}\left( {p^{j},z+\theta +t} \right) .\\ R^{i}\left( {p^{j},k^{i}} \right)= & {} a+bp^{j}-k^{i}\;{\mathrm{where }}\; r^{i}\left( {p^{j},z+\theta +t} \right)>{\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right) >r^{i}\left( {p^{j},z+t} \right) . \end{aligned}$$

Proof of Lemma

Now, ${\mathrm{X}}\left( {p^{j},k^{i}} \right) <r^{i}\left( {p^{j},z+t} \right)$ means that firm i sets a price where the marginal revenue curve cuts the marginal cost curve given by $z+t.$ In this situation Eq. (1) defines the reaction function:

$$\begin{aligned} R^{i}\left( {p^{j},k^{i}} \right) =1/2\left( {\left( {1-b} \right) a+bp^{j}+\left( {1-\tau } \right) ^{-1}\left( {z+t} \right) } \right) . \end{aligned}$$

(4)

By analogy, when ${\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right) <r^{i}\left( {p^{j},z+\theta +t} \right) ,$ firm $i^{\prime }$s marginal revenue curve cuts the marginal cost curve given by $z+\theta +t$ and the reaction function is:

$$\begin{aligned} R^{i}\left( {p^{j},k^{i}} \right) =1/2\left( {\left( {\left( {1-b} \right) a+bp^{j}} \right) +\left( {1-\tau } \right) ^{-1}\left( {z+t+\theta } \right) } \right) . \end{aligned}$$

(5)

Here firm i produces beyond capacity. Finally, in the intermediate case, firm $i^{\prime }$s marginal revenue curve cuts the marginal cost curve where marginal costs jump from $z+t$ to $z+\theta +t.$ That is, the firm ideally sets a price that results in production at the capacity limit. Clearly, it can charge a higher price and produce less, or it can charge a lower price and sell more. That is, $r^{i}\left( {p^{j},z+\theta +t} \right)>{\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right) >r^{i}\left( {p^{j},z+t} \right) .$ In this situation, the firm sets a price given by ${\mathrm{X}}\left( {p^{j},k^{i}} \right) ,$ hence the reaction function is:

$$\begin{aligned} R^{i}\left( {p^{j},k^{i}} \right) =\left( {a+bp^{j}-k^{i}} \right) . \end{aligned}$$

(6)

Proof of Proposition 1

In the second stage, firms’ capacities are given, and the Bertrand response functions are the solutions to $argmax_{p^{i}} \left( {\left( {1-\tau } \right) p^{i}-x^{i}} \right) q^{i}$, where the marginal cost is $x^{i}=z+t$ when $q^{i}\le k^{i}$ and $x^{i}=z+\theta +t$ when $q^{i}>k^{i}$. We have:

$$\begin{aligned} r^{i}\left( {p^{j},x^{i}} \right) =1/2\left( {\left( {\left( {1-b} \right) a+bp^{j}} \right) +\left( {1-\tau } \right) ^{-1}x^{i}} \right) ,i,\quad j=1,2,\quad i\ne j.\nonumber \\ \end{aligned}$$

(7)

Equation (7) defines firm i’s best response to the price charged by firm j. To determine the reaction function, define ${\mathrm{X}}\left( {p^{j},k^{i}} \right)$ as the combination of the price set by firm j and the capacity that happens to equalize the output of firm i to its capacity. Formally $k^{i}=a-bp^{j}-{\Phi }^{i}\left( {p^{j},k^{i}} \right)$. Under this definition firm i charges a price that implies production below capacity when ${\mathrm{X}}\left( {p^{j},k^{i}} \right) <r^{i}\left( {p^{j},z+t} \right)$. In this situation, the reaction function of firm i is given by Eq. (7) with $x^{i}=z+t$. Similarly, when ${\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right) >r^{i}\left( {p^{j},z+\theta +t} \right) ,$ the firm operates beyond capacity and the reaction function is defined by the best response function in Eq. (7) where $x^{i}=z+t+\theta$. Finally, when $r^{i}\left( {p^{j},z+\theta +t} \right)>{\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right) >r^{i}\left( {p^{j},z+t} \right)$ firm i produces at its capacity constraint and the reaction function is $a+bp^{j}-k^{i}$.¹⁶

Based on the firm’s reaction function we can see how taxes have a direct effect when the reaction function is given by $r^{i}\left( {p^{j},x^{i}} \right)$. It is easy to check that firm i sets a higher price in response to the price of firm j as tax rates go up. Equation (7) also shows that unit and ad valorem taxes are quantitatively different. When the unit tax rate increases at the margin, the effect on the price charged by firm i,  given the price set by firm j, is $\left. {\partial p^{i}/\partial t} \right| _{p^{j}} =1/2\left( {1-\tau } \right) ^{-1}$. For a marginal increase in the ad valorem tax rate, the effect is $\left. {\partial p^{i}/\partial \tau } \right| _{p^{j}} =1/2\left( {1-\tau } \right) ^{-2}x^{i}$. Hence, the ad valorem tax rate’s price effect at the margin becomes more pronounced when the ad valorem tax rate is already high. There is a second effect of taxes. Clearly, the ${\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right)$-function is not affected by the tax firms pay. Nevertheless, changes in the tax rates have an effect through changes in the reaction-function’s domain because taxes affect firm i’s best response function. That is, in the condition $r^{i}\left( {p^{j},z+\theta +t} \right)>{\mathrm{X}}^{i}\left( {p^{j},k^{i}} \right) >r^{i}\left( {p^{j},z+t} \right) ,$ changes in the taxes affect the $r^{i}$-functions.

Because the introduction of taxes does not change the fact that the reaction-functions’ slopes lie in the range $\left[ {0,1} \right]$, Proposition 1 in Maggi (1996) establishes an equilibrium in pure strategies. More precisely, let $p^{c}\left( \psi \right)$, $\psi =\left( {1-\tau } \right) ^{-1}\left( {z+c+t} \right)$ be the Cournot price under a marginal cost of $z+c+t$ and $q^{c}\left( \psi \right)$ the corresponding firm-output:

$$\begin{aligned} \left\{ {p^{c}\left( \psi \right) ;q^{c}\left( \psi \right) } \right\} =\left\{ {\left( {2+b} \right) ^{-1}\left( {a+\left( {1+b} \right) \psi } \right) ;\left( {2+b} \right) ^{-1}\left( {a-\psi } \right) } \right\} . \end{aligned}$$

(8)

Similarly, $p^{b}\left( \omega \right)$, $\omega =\left( {1-\tau } \right) ^{-1}\left( {z+\theta +t} \right)$, is the Bertrand price under a marginal cost of $z+t+\theta$ and $q^{b}\left( \omega \right)$ the corresponding firm-output:

$$\begin{aligned} \left\{ {p^{b}\left( \omega \right) ;q^{b}\left( \omega \right) } \right\} =\left\{ {\left( {2-b} \right) ^{-1}\left( {\left( {1-b} \right) a+\omega } \right) ;\left( {\left( {2-b} \right) \left( {1+b} \right) } \right) ^{-1}\left( {a-\omega } \right) } \right\} .\nonumber \\ \end{aligned}$$

(9)

If we define $\theta _c$ by $p^{b}\left( \omega \right) =p^{c}\left( \psi \right)$, the Cournot outcome survives when $\theta >\theta _c$. When $\theta \le \theta _c$, the outcome is that of a Bertrand duopoly where firms have a marginal cost equal to $z+\theta +t.$

Proof of Lemma 1

Because $\theta =\theta _c \left( {0,t_m } \right)$ defines $t_m$, it follows that any unit tax rate below $t_m$ sustains Bertrand equilibrium. Next, consider the solution of $\theta =\theta _c \left( {\tau _m, 0} \right)$. This defines the ad valorem tax rate where a slight increase changes market conduct from Bertrand to Cournot conduct. Thus, any $\tau \le \tau _m$ sustains the Bertrand equilibrium. Define $t_l$ by $z+\theta +t_l =\left( {1-\tau _m } \right) ^{-1}\left( {z+\theta } \right)$. Clearly, when the unit tax rate satisfies $z+\theta +t=\left( {1-\tau } \right) ^{-1}\left( {z+\theta } \right)$, $\tau <\tau _m,$ a pure unit tax system as well as a pure ad valorem tax system sustains Bertrand equilibrium. To the contrary, take any unit tax rate satisfying $t>t_l$ and define the ad valorem tax rate by $\left( {1-\tau } \right) ^{-1}\left( {z+\theta } \right) =z+\theta +t$. By construction, this defines ad valorem tax rates bounded from below by $\tau _m,$ proving the claim.

Proof of Proposition 2

By Lemma 1, the price is $p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right)$ under the ad valorem tax-only policy with a tax rate of $\tilde{\tau }$, and $p^{b}\left( {\omega _{0,\tilde{t} } } \right)$ under the unit tax-only policy. Thus, the tax change’s effect on price is unambiguously positive so that $p^{b}\left( {\omega _{0,\tilde{t} } } \right) <p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right)$.

The tax revenue changes from $\tilde{\tau } p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) q^{c}\left( {\psi _{\tilde{\tau }, 0} } \right)$ to $\tilde{t} q^{b}\left( {\omega _{0,\tilde{t} } } \right)$, and the effect on the tax revenue is beneficial when

$$\begin{aligned} \tilde{\tau } p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) q^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) <\tilde{t} q^{b}\left( {\omega _{0,\tilde{t} } } \right) . \end{aligned}$$

(10)

Notice that, by symmetry, it follows from the demand function that $q^{b}\left( {\omega _{0,\tilde{t} } } \right) =q^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) +\left( {1-b} \right) ^{-1}\left( {p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) -p^{b}\left( {\omega _{\omega _{0,\tilde{t} } } } \right) } \right)$. Therefore, Eq. (10) becomes:

$$\begin{aligned}&\tilde{\tau } p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) q^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) \nonumber \\&\quad < {\tilde{t} \left( {q^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) +\left( {1-b} \right) ^{-1}\left( {p^{c}\left( {\psi _{\tilde{\tau }, 0} } \right) -p^{b}\left( {\omega _{0,\tilde{t} } } \right) } \right) } \right) .} \end{aligned}$$

(11)

We have to prove that the inequality in Eq. (11) is satisfied. Using the expressions for prices and quantities in Proposition 1, we see that these are all finite as $b\rightarrow 1$. This also applies to $\tilde{t}$ and, in consequence for $\tilde{\tau }$. Hence, the right-hand side grows without an upper bound as $b\rightarrow 1$, while the left-hand side is finite showing the claim.

Proof of Proposition 3

When the pair of taxes is in ${\Phi }$ we have that:

$$\begin{aligned} t=\left( {1-\tau } \right) a-\left( {z+c} \right) -b^{-2}\left( {2+b} \right) \left( {\theta -c} \right) . \end{aligned}$$

(12)

When taxes are in $\tilde{\Phi }_{\tilde{t} }$ we know from $p^{b}\left( {\omega _{\tau , t} } \right) =p^{b}\left( {\omega _{0,\tilde{t} } } \right)$ that:

$$\begin{aligned} t=\left( {1-\tau } \right) \left( {z+\theta +\tilde{t} } \right) -\left( {z+\theta } \right) . \end{aligned}$$

(13)

Collecting (12) and (13) we have:

$$\begin{aligned} \left( {1-\tau } \right) \left( {a-\left( {z+\theta +\tilde{t} } \right) } \right) =b^{-2}\left( {1+b} \right) \left( {1-b} \right) \left( {\theta -c} \right) . \end{aligned}$$

(14)

Now, (14) shows that $1-\tau >0$ or $\tau <1$. Next, notice that $\tau$ is given by:

$$\begin{aligned} \tau =1-\left( {a-\left( {z+\theta +\tilde{t} } \right) } \right) ^{-1}b^{-2}\left( {1+b} \right) \left( {1-b} \right) \left( {\theta -c} \right) . \end{aligned}$$

(15)

Thus, $\tau >0$ when:

$$\begin{aligned} \left( {a-\left( {z+\theta +\tilde{t} } \right) } \right) >b^{-2}\left( {1+b} \right) \left( {2-b} \right) \left( {\theta -c} \right) . \end{aligned}$$

(16)

We know that $\tilde{t} <t_m =a-\left( {z+c} \right) -b^{-2}\left( {2+b} \right) \left( {\theta -c} \right)$. Replacing $t_m$ for $\tilde{t}$ in the left-hand side of (16), we see that $a-\left( {z+\theta +\tilde{t} } \right) >a-\left( {z+\theta +\tilde{t} } \right) t_m =b^{-2}\left( {1+b} \right) \left( {2-b} \right) \left( {\theta -c} \right)$ showing that (16) is satisfied.

Proof of Proposition 7

Define $t_0^*$ by

$$\begin{aligned} z+\theta +t_0^*=\left( {1-\tau ^{*}} \right) ^{-1}\left( {z+\theta +t^{*}} \right) . \end{aligned}$$

(17)

and $t_\epsilon ^*$ by:

$$\begin{aligned} z+\theta +t_\epsilon ^*=\left( {1-\tau ^{*}-\epsilon } \right) ^{-1}\left( {z+\theta +t^{*}} \right) . \end{aligned}$$

(18)

Lemma 6.1

$t_0^*$ is in ${\Phi }_-$,that is, $t_0^*<t_m$ and $t_\epsilon ^*<t_m$

Proof

Because $\left( {\tau ^{*},t^{*}} \right)$ and $\left( {0,t_m } \right)$, both are in ${\Phi }$ we have [using Eq. (3)] that $t_m =a\tau ^{*}+t^{*}$. Some re-arranging shows that $t_0^*<t_m$:

$$\begin{aligned} \left( {1-\tau ^{*}} \right) ^{-1}\left( {z+\theta +t^{*}} \right) -\left( {z+\theta } \right) <a\tau ^{*}+t^{*}, \end{aligned}$$

(19)

or

$$\begin{aligned} \left( {1-\tau ^{*}} \right) ^{-1}\left( {\tau ^{*}\left( {z+\theta } \right) +t^{*}} \right) <a\tau ^{*}+t^{*}, \end{aligned}$$

(20)

or

$$\begin{aligned} \left( {1-\tau ^{*}} \right) a-\left( {z+\theta +t^{*}} \right) >0, \end{aligned}$$

(21)

which is satisfied as Bertrand quantities, $\left( {1-\tau ^{*}} \right) a-\left( {z+\theta +t^{*}} \right)$, are positive.

Lemma 6.2

When $t_0^*$ is in ${\Phi }_-$, $t_\epsilon ^*$ is also in ${\Phi }_-$ for $\left( {z+c+t_m } \right) \epsilon <\left( {1-\tau ^{*}} \right) \left( {t_m -t_0^*} \right)$

Proof

Combining (17) and (18) we have:

$$\begin{aligned} z+c+t_0^*=\left( {1-\tau ^{*}} \right) ^{-1}\left( {1-\tau ^{*}-\epsilon } \right) \left( {z+\theta +t_\epsilon ^*} \right) . \end{aligned}$$

(22)

Solving for $t_\epsilon ^*$ gives:

$$\begin{aligned} t_\epsilon ^*=\left( {1-\tau ^{*}-\epsilon } \right) ^{-1}\left( {\left( {1-\tau ^{*}} \right) t_0^*-\left( {z+\theta } \right) \epsilon } \right) . \end{aligned}$$

(23)

Rearranging of $t_\epsilon ^*<t_m$ gives:

$$\begin{aligned} \left( {z+c+t_m } \right) \epsilon <\left( {1-\tau ^{*}} \right) \left( {t_m -t_0^*} \right) . \end{aligned}$$

(24)

Lemma 6.3

Comparing the tax policies $\left( {\tau ^{*}+\epsilon ,t^{*}} \right)$ and $\left( {0,t_\epsilon ^*} \right)$ we have $p^{b}\left( {\omega _{0,t_\epsilon ^*} } \right) <p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right)$ and tax revenue under $\left( {0,t_\epsilon ^*} \right)$ exceeds that under $\left( {\tau ^{*}+\epsilon ,t^{*}} \right)$ for b sufficiently large.

Proof

The tax policy $\left( {\tau ^{*}+\epsilon ,t^{*}} \right)$ implements Cournot prices and quantities. Thus the price under this policy is $p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right)$. When the tax policy $\left( {0,t_\epsilon ^*} \right)$ replaces $\left( {\tau ^{*}+\epsilon ,t^{*}} \right)$, Lemma 6.2 implies that the price is $p^{b}\left( {\omega _{0,t_\epsilon ^*} } \right) <p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right)$.

The effect on the tax revenue is beneficial when

$$\begin{aligned} \left( {\left( {\tau ^{*}+\epsilon } \right) p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) +t^{*}} \right) q^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) <t_\epsilon ^*q^{b}\left( {\omega _{0,t_\epsilon ^*} } \right) . \end{aligned}$$

(25)

Notice that by symmetry, it follows from the demand function that $q^{b}\left( {\omega _{0,t_\epsilon ^*} } \right) =q^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) +\left( {1-b} \right) ^{-1}\left( {p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) -p^{b}\left( {\omega _{0,t_\epsilon ^*} } \right) } \right)$. Using this in (25):

$$\begin{aligned}&\left( {\left( {\tau ^{*}+\epsilon } \right) p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) +t^{*}} \right) q^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) \nonumber \\&\quad < t_\epsilon ^*\left( {q^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) +\left( {1-b} \right) ^{-1}\left( {p^{c}\left( {\psi _{\tau ^{*}+\epsilon ,t^{*}} } \right) -p^{b}\left( {\omega _{0,t_\epsilon ^*} } \right) } \right) } \right) . \end{aligned}$$

(26)

Because prices, quantities and taxes are finite for all b-values, we see that the right-hand side grows without an upper bound as $b\rightarrow 1$ while the left-hand side is finite showing the claim.

Proof of Proposition 6

We first establish that $\tau _{\Phi }^*<\tau ^{*}+\epsilon$ and $t_{\Phi }^*>t^{*}$. Notice that $\left( {\tau ^{*}+\epsilon ,t^{*}} \right)$ as well as $\left( {\tau _{\Phi }^*, t_{\Phi }^*} \right)$ are in $\tilde{\Phi }_{t_\epsilon ^*}$ so that $p^{b}\left( {\omega _{\tau _{\Phi }^*, t_{\Phi }^*} } \right) =p^{b}\left( {\omega _{\tau ^{*}+\epsilon ,t^{*}} } \right)$, or:

$$\begin{aligned} \left( {1-\tau ^{*}-\epsilon } \right) ^{-1}\left( {z+\theta +t^{*}} \right) =\left( {1-\tau _{\Phi }^*} \right) ^{-1}\left( {z+\theta +t_{\Phi }^*} \right) . \end{aligned}$$

(27)

Next, because $\left( {\tau ^{*}+\epsilon ,t^{*}} \right)$ and $\left( {\tau _{\Phi }^*, t_{\Phi }^*} \right)$ are also in ${\Phi }$ we have:

$$\begin{aligned} \left( {1-\tau ^{*}} \right) a-t^{*}= & {} k, \end{aligned}$$

(28)

$$\begin{aligned} \left( {1-\tau _{\Phi }^*} \right) a-t_{\Phi }^*= & {} k. \end{aligned}$$

(29)

Using this in (27) we have:

$$\begin{aligned} \left( {a^{-1}\left( {k-t^{*}} \right) -\epsilon } \right) ^{-1}\left( {z+\theta +t^{*}} \right) =\left( {a^{-1}\left( {k-t_{\Phi }^*} \right) } \right) ^{-1}\left( {z+\theta +t_{\Phi }^*} \right) . \end{aligned}$$

(30)

If we set $t_{\Phi }^*=t^{*}$ the left-hand side is strictly greater than the right-hand side because of $\epsilon$. The right-hand side is increasing in $t_{\Phi }^*$ and it follows that $t_{\Phi }^*>t^{*}$. Because the ad valorem tax rate and the unit tax rate varies inversely along ${\Phi }$, it follows that $\tau _{\Phi }^*<\tau ^{*}$.

With respect to tax revenue we have $R_{\tau _{\Phi }^*, t_{\Phi }^*}>R_{0,t_\epsilon ^*}>R_{\tau ^{*}+\epsilon ,t^{*}} >R_{\tau ^{*},t^{*}}$. Therefore, by continuity, there are tax rates, called $\hat{\tau }$ and $\hat{t}$, that satisfy $\tau _{\Phi }^*<\hat{\tau }<\tau ^{*}$ and $t_{\Phi }^*>\hat{t}>t^{*}$ and $R_{\hat{\tau }, \hat{t}} =R_{\tau ^{*}+\epsilon ,t^{*}}$.

Evaluating tax changes that keep the Bertrand price intact at $p^{b}\left( {\omega _{\tau _{\Phi }^*, t_{\Phi }^*} } \right)$ we have $\left. {dt/d\tau } \right| _{\tilde{\Phi }_{t_\epsilon ^*} } =-\left( {1-\tau _{\Phi }^*} \right) ^{-1}\left( {z+\theta +t_{\Phi }^*} \right)$. On the other hand, the tax changes around $\left( {\tau _{\Phi }^*, t_{\Phi }^*} \right)$ are restricted to be in ${\Phi }$ so that $\left. {dt/d\tau } \right| _{\Phi } =-a$. At any meaningful solution we have $\left. {dt/d\tau } \right| _{\Phi } <\left. {dt/d\tau } \right| _{\tilde{\Phi }_{t_\epsilon ^*} }$ showing that the price goes down as we move from $\left( {\tau _{\Phi }^*, t_{\Phi }^*} \right)$ to $\left( {\hat{\tau }, \hat{t}} \right)$.