Ad valorem versus per unit taxation: a perspective from price signaling

Abstract

This paper compares ad valorem and per unit taxation in the context of price signaling. In the model, a taxation designer chooses between ad valorem and per unit taxation to maximize tax revenues, and a monopoly firm, whose product quality can be either high or low, uses price as a quality signal. The analysis shows that, compared to per unit taxation, ad valorem taxation raises the low-quality firm’s mimicking cost and lowers the high-quality firm’s signaling cost. This leads to higher transaction volumes and tax revenues.

Appendices

Appendix A Proof of Lemma 1

The proof is similar to Bagwell and Riordan (1991). First, consider per unit taxation. Suppose there is a pooling equilibrium in which both types of firms charge the price $p_F^{**}$. The profit earned by the high-quality firm is

$$\begin{aligned} {\pi _F}(H,s,p_F^{**}) = (p_F^{**} - {c_H} - f)D(p_F^{**},s), \end{aligned}$$

and the profit of the low-quality firm is

$$\begin{aligned} {\pi _F}(L,s,p_F^{**}) = (p_F^{**} - {c_L} - f)D(p_F^{**},s). \end{aligned}$$

Clearly, for a pooling price that each type of firm can accept, $\pi _F (H,s,p_F^{**}) \le \pi _F (L,s,p_F^{**})$.

Since $\pi _F (L,s,p_F^{**}) < \pi _F (L,1,p_F^{**}) \le \pi _F (L,1,p_F^m(L))$, there exists a ${p'_F} > \max \{ p_F^{**},p_F^m(L)\}$ such that $\pi _F (L,1,{p'_F}) = \pi _F (L,s,p_F^{**})$. That is,

$$\begin{aligned} ({p'_F} - {c_L} - f)D({p'_F},1) = (p_F^{**} - {c_L} - f)D(p_F^{**},s), \end{aligned}$$

which further implies that $D({p'_F},1)< D(p_F^{**},s)$ as $p'_F > p_F^{**}$. This equation is also equivalent to

$$\begin{aligned}&({p'_F} - {c_H} - f)D({p'_F},1)+({c_H}- {c_L}) D({p'_F},1)\\&\quad = (p_F^{**} - {c_H} - f)D(p_F^{**},s)+({c_H}- {c_L})D(p_F^{**},s). \end{aligned}$$

This implies that

$$\begin{aligned} {\pi _F}(H,1,{p'_F}) - {\pi _F}(H,s,p_F^{**}) = ({c_H} - {c_L})[D(p_F^{**},s) - D({p'_F},1)] > 0. \end{aligned}$$

Moreover, as $\pi _F (L,1,p_F^{})$ decreases in $p_F$ at ${p'_F}$, ${p''_F} = {p'_F} + \varepsilon$ violates the Intuitive Criterion for a positive and sufficiently small $\varepsilon$.

The proof is similar in the case of ad valorem taxation. Suppose there is a pooling price $p^{**}_A$. Following the same logic as above, there exists a ${p'_A} > \max \{ p_A^{**},p_A^m(L)\}$ such that

$$\begin{aligned}{}[{p'_A}(1-r) - {c_L}]D({p'_A},1) = [p_A^{**}(1-r) - {c_L} ]D(p_A^{**},s). \end{aligned}$$

As ${p'_F}>p_F^{**}$ and $r<1$, we should have $D({p'_A},1)< D(p_A^{**},s)$. Besides, the equation can be equivalently written as

$$\begin{aligned}&[{p'_A}(1-r)- {c_H} ])D({p'_F},1)+({c_H}- {c_L}) D({p'_F},1) \\&\quad = [p_A^{**}(1-r) - {c_H} ]D(p_F^{**},s)+({c_H}- {c_L})D(p_F^{**},s). \end{aligned}$$

The rest of the proof is the same as above.

Appendix B Proof of Proposition 1

Part 1 The equilibrium prices under per unit taxation

First, consider per unit taxation. Inequality (6) can be explicitly written as

$$\begin{aligned} \max \{u-f-c_L, 0\} \ge (p_{F}^{*}(H)-f-c_L)D(H,1,p_{F}^{*}(H)). \end{aligned}$$

(A1)

Depending on the tax rate, there are two cases to be discussed: (i) $f< u-c_L$; (ii)$f\ge u-c_L$. In each case, the analysis consists of two parts. First, we display all prices satisfying inequality (6). Then, we examine whether the high-quality firm has an incentive to deviate and whether these prices meet the requirement of the Intuitive Criterion.

Case (i): $f< u-c_L$

In this case, inequality (A1) becomes

$$\begin{aligned} u-f-c_L \ge \pi _{F}(L,1,p_{F}^{*}(H))=(p_{F}^{*}(H)-f-c_L)D(H,1,p_{F}^{*}(H)). \end{aligned}$$

(A2)

According to Assumption 1, the $\pi _{F}(L,1,p_{F}^{*}(H))$ is concave in $p_{F}^{*}(H)$, and is maximized when $p_{F}^{*}(H)$ is equal to the low-quality firm’s monopoly price $p_F^m(L)$.

To discuss the solutions to the inequality, we divide the value range of $p_{F}^{*}(H)$ into three complementary cases: (i)$p_{F}^{*}(H) \le u$; (ii) $p_{F}^{*}(H) \in (u, p_F^m(H)]$ ; and (iii)$p_{F}^{*}(H)\in (p_F^m(H),1+u]$. We ignore all prices that are strictly higher than $1+u$, since no consumer would make a purchase.

Since $D(H,1,p_{F}^{*}(H))=1$ when $p_{F}^{*}(H) \le u$, it is straightforward that inequality (A2) holds if $p_{F}^{*}(H) \le u$.

Another solution lies in the interval $(p_F^m(H),1+u]$. As the monopoly price is strictly increasing in the marginal cost (i.e. $p_F^m(H)>p_F^m(L)$), the concavity of $\pi _{F}(L,1,p_{F}^{*}(H))$ implies that the profit is monotonically decreasing when $p_{F}^{*}(H)\ge p_F^m(H)$. Moreover, $\pi _{F}(L,1,p_{F}^{*}(H))=0$ when $p_{F}^{*}(H)=1+u$, and Assumption 2 states that $u-f-c_L < \pi _{F}(L,1,p_F^m(H))$. Therefore, there exist a unique ${\bar{p}}_{F}$ in $(p_F^m(H), 1+u]$ such that

$$\begin{aligned} \pi _{F}(L,0,u) = \pi _{F}(L,1,{\bar{p}}_{F}). \end{aligned}$$

(A3)

Clearly, inequality (A2) also holds if $p_{F}^{*}(H) \ge {\bar{p}}_{F}$.

The only interval left undiscussed is $(u, p_F^m(H)]$. Note that $\pi _{F}(L,1,u)=u-f-c_L$, $\pi _{F}(L,1,p_F^m(H))>u-f-c_L$ and $p_F^m(L)\in (u, p_F^m(H)]$. Since $p_F^m(L)$ maximizes $\pi _{F}(L,1,p_{F}^{*}(H))$, the concavity of the profit implies that $\pi _{F}(L,1,p_{F}^{*}(H))>u-f-c_L$ if $p_{F}^{*}(H)\in (u, p_F^m(H)]$. Thus, any price in the interval $(u, p_F^m(H)]$ cannot be the solution of inequality (A2).

The analysis above yields two solutions of inequality (A2): $p_{F}^{*}(H) \le u$ and $p_{F}^{*}(H) \ge {\bar{p}}_{F}$. Now, we check whether the high-quality firm has no incentive to deviate from these prices. The deviation that can enhance profit lies within $(u, {\bar{p}}_{F})$, and the Intuition Criterion has no restrictions on beliefs for such prices. Assuming that such a deviation would induce the belief that the product quality is low, the high-quality firm has no incentive to deviate if it earns a non-negative profit at the equilibrium. That is

$$\begin{aligned} \pi _{F}(H,1,p_{F}^{*}(H) ) = (p_{F}^{*}(H) - c_{H} - f)[1-G(p_{F}^{*}(H)-u)] \ge 0. \end{aligned}$$

(A4)

Solving inequality (A4) yields

$$\begin{aligned} p_{F}^{*}(H)\ge c_{H} + f. \end{aligned}$$

(A5)

This condition rules out any prices that are strictly lower than $c_{H} + f$. Moveover, as $p_F^m(H)>c_{H} + f$, it is straightforward that $p_{F}^{*}(H) \ge {\bar{p}}_{F}$ satisfies the requirement of inequality (A5).

However, only $p_{F}^{*}(H) = {\bar{p}}_{F}$ satisfies the Intuition Criterion. The Intuitive Criterion is satisfied if there does not exist a price ${\tilde{p}}$ such that $\pi _{F}(H,1,{\tilde{p}})>\pi _{F}(H,1,p_{F}^{*}(H))$ and $\pi _{F}(L,1,{\tilde{p}})<\pi _{F}(L,0,p_{F}^{*}(L))$. If $c_{H} + f\le p_{F}^{*}(H)<u$, then the Intuitive Criterion fails for ${\tilde{p}}\in (p_{F}^{*}(H), u)$; if $p_{F}^{*}(H)>{\bar{p}}_{F}$, then the Intuitive Criterion fails for ${\tilde{p}}\in ({\bar{p}}_{F},p_{F}^{*}(H))$. This leaves the possibility that $p_{F}^{*}(H)=u$ or $p_{F}^{*}(H)={\bar{p}}_{F}$. However, the former one can be further excluded. Based on equation (A3), we have

$$\begin{aligned} u-f-c_H+c_H-c_L = ({\bar{p}}_{F}-f-c_H)D(H,1,{\bar{p}}_{F})+(c_H-c_L)D(H,1,{\bar{p}}_{F}). \end{aligned}$$

Rearranging yields

$$\begin{aligned} \pi _{F}(H,1,u)-\pi _{F}(H,1,{\bar{p}}_{F})=(c_H-c_L)[D(H,1,{\bar{p}}_{F})-1]<0, \end{aligned}$$

from which it follows that ${\tilde{p}}= {\bar{p}}_{F}-\epsilon$ violates the Intuitive Criterion for sufficiently small $\epsilon$.

Case (ii): $f\ge u-c_L$

Under the separating equilibrium, consumers correctly infer the firm’s type. Given that $f\ge u-c_L$, the low-quality firm could only earn a zero profit.

Note that the tax designer will not raise the tax rate to $f\ge 1+u-c_H$. Otherwise, the high-quality firm cannot attract any customers at a price that covers its average cost $f+c_H$; the low-quality firm also has no sales no matter whether it mimics its high-quality counterpart or not. Given that $u-c_L \le f< 1+u-c_H$, the left-hand-side of inequality (A1) becomes zero. The right-hand-side would be strictly positive unless $p_{F}^{*}(H)\le f+c_L$ or $p_{F}^{*}(H)\ge 1+u$. Since we have postulated that the high-quality firm will not further raise the price once the demand drops to zero, the solutions of (A1) in this case are $p_{F}^{*}(H)\le f+c_L$ and $p_{F}^{*}(H)= 1+u$.

It is straightforward that the high-quality firm has no incentive to stick to the price $p_{F}^{*}(H)\le f+c_L$, as it causes a negative profit. Deviating to some prices lies in $(f+c_L, 1+u)$ would at worst lead to the belief that the product quality is low and yield zero profit, which makes the firm better.

The high-quality firm has no incentive to deviate from the price $p_{F}^{*}(H)= 1+u$. A potential attractive deviation must satisfy $p_{F}^{*}(H)< 1+u$. The Intuition Criterion does not restrict beliefs for such prices, and at worst such a deviation would induce the belief that the product quality is low. Thus, the “no-defect” condition for the high-quality firm is equivalent to non-negative profit at the separating equilibrium, which requires that $p_{F}^{*}(H)\ge c_{H} + f$. Since $f< 1+u-c_H$, it is straightforward that $p_{F}^{*}(H)= 1+u>c_{H} + f$.

When $p_{F}^{*}(H)= 1+u$, both sides of inequality (A1) are equal to zero. Based on the definition of ${{\bar{p}}}_F$ (see Eq. (A3)), it can be inferred that ${{\bar{p}}}_F=1+u$ in the case that $f\ge u-c_L$.

Part 2 The equilibrium prices under ad valorem taxation

The proof follows exactly the same approach as above. So, we go through some key points briefly. Under ad valorem taxation, inequality (6) becomes

$$\begin{aligned} \max \{u(1-r)-c_L, 0\} \ge [p_{A}^{*}(H)(1-r)-c_L]D(H,1,p_{A}^{*}(H)). \end{aligned}$$

(A6)

When $r< 1-\frac{c_L}{u}$, there exists a unique ${\bar{p}}_{A}$ in $(p_A^m(H), 1+u]$ such that

$$\begin{aligned} \pi _{A}(L,0,u) = \pi _{A}(L,1,{\bar{p}}_{A}). \end{aligned}$$

(A7)

Both $p_{A}^{*}(H)\le u$ and $p_{A}^{*}(H)\ge {\bar{p}}_{A}$ are solutions of inequality (A6). However, only $p_{A}^{*}(H)={\bar{p}}_{A}$ satisfies the Intuitive Criterion. The existence of the separating equilibrium also requires that $p_{A}^{*}(H)\ge \frac{c_{H}}{1-r}$ so that the high-quality firm earns non-negative profit (and thus will not deviate from the signaling price). As $p_A^m(H)\ge \frac{c_{H}}{1-r}$, it is straightforward that ${\bar{p}}_{A}$ meets this requirement.

When $r\ge 1-\frac{c_L}{u}$, the left-hand-side of inequality (A6) is equal to zero. Since the tax designer will not choose a tax rate $r>1-\frac{c_L}{1+u}$ to drive the firm out of the market regardless of its type, the inequality holds only if $p_{A}^{*}(H)= 1+u$ so that $D(H,1,p_{A}^{*}(H))=0$. In this case, ${{\bar{p}}}_{A}= 1+u$.

Appendix C Proof of Proposition 2

The key point behind Proposition 2 is it is more costly for the low-quality firm to mimic its counterpart under ad valorem taxation.

According to Proposition 1, a per-unit tax f that is strictly higher than $u-c_L$ would deter the entry of the low-quality firm and leads to a zero transaction volume when the product quality is high. However, by lowering the tax rate to $u-c_L$, the tax designer can at least obtain a revenue $u-c_L$ when the firm is of low quality. Thus, for per unit taxation, the tax designer will not choose a tax rate $f>u-c_L$. Similarly, we would have $r\le 1-\frac{c_L}{u}$ under ad valorem taxation.

Therefore, under ad valorem taxation, inequality (6) can be explicitly written as

$$\begin{aligned} u-c_L-ru\ge [{{\bar{p}}}_A(1-r)-c_L]D(H,1, {{\bar{p}}}_A), \end{aligned}$$

(A8)

while under per unit taxation, its explicit form is

$$\begin{aligned} u-c_L-f \ge ({{\bar{p}}}_F-f-c_L)D(H,1, {{\bar{p}}}_F). \end{aligned}$$

(A9)

Comparing left-hand-sides in (A8) and (A9), it is straightforward that when the low-quality firm chooses the separating price u, it earns a higher profit if taxation is proportional to price: provided that $r{{\bar{p}}}_A=f$, we should have $ru=\frac{fu}{{{\bar{p}}}_A}<f$. Therefore, the opportunity cost of mimicking is higher under ad valorem taxation.

Now, suppose ${{\bar{p}}}_A={{\bar{p}}}_F$. In the case that $f=u-c_L$, ${{\bar{p}}}_F= 1+u$ (see Appendix B), and $D(H,1, {{\bar{p}}}_A)=D(H,1, {{\bar{p}}}_F)=0$; when $f<u-c_L$, ${{\bar{p}}}_A={{\bar{p}}}_F$ and $r{{\bar{p}}}_A=f$ still suggest that the right-hand-sides of (A8) and (A9) would be equal. Therefore, in both cases, inequality (A9) would be tight, while (A8) would be slack, since its left-hand-side is larger. This implies that the high-quality firm does not need to raise its price to ${{\bar{p}}}_F$ to deter mimicking under ad valorem taxation. So, given that $r{{\bar{p}}}_A=f$, we should have ${{\bar{p}}}_A<{{\bar{p}}}_F$.

Appendix D Proof of Proposition 3

Let $f^*$ denote the optimal per unit tax, and suppose $r'u = f^*$. That is, the ad valorem tax is set to $r'$ so that the tax revenue from the low-quality firm equals that under the optimal per unit taxation. Note that $r'$ exists and is feasible as $0<f^{*}\le u - c_{L}$, and $r' = \frac{f^{*}}{u} \le \frac{u - c_{L}}{u} = 1 - \frac{c_{L}}{u}$.

Proposition 1 suggests that $p_A^*(H)={{\bar{p}}}_A>u$. This further implies $r'p_A^*(H)>r'u = f^{*}$. Thus, the per-transaction tax revenue is strictly higher under ad valorem taxation when the product is of high quality.

The last step is to check the transaction volumes. It is straightforward that transaction volumes under both taxation forms are equal to 1 when the firm is of low-quality type.

Now, consider the transaction volume of the high-quality product. In the case that $f^*=u-c_L$, $r'u = f^*$ implies that the low-quality firm’s profit at the separating equilibrium is constantly zero under both tax regimes. Thus, we would have ${{\bar{p}}}_A={{\bar{p}}}_F=1+u$, and $D(H,1,p_A^*(H)|_{r=r'}) =D(H,1,p_F^*(H)|_{f=f^{*}})=0$. However, when $f^*<u-c_L$, recall that we have shown $r'u = f^{*}$ implies that ${{\bar{p}}}_A<{{\bar{p}}}_F$. Therefore, in this case, $D(H,1,p_A^*(H)|_{r=r'}) \ge D(H,1,p_F^*(H)|_{f=f^{*}})>0$ and $R_A|_{r=r'} > R_F|_{f=f^{*}}$.

Note that $r'$ is only a potential solution in the feasible set. Suppose $r^{*}$ maximizes $R_A$. We have

$$\begin{aligned} R_A|_{r=r^{*}} \ge R_A|_{r=r'} \ge R_F|_{f=f^{*}}. \end{aligned}$$

This suggests that the optimized tax revenue under ad valorem taxation is higher than that under per unit taxation.

Appendix E Proof of Proposition 4

Recall that, under per unit taxation, the optimal tax rate might either be an inner solution or a corner solution. We first consider the case of the corner solution. Given $f^*=u-c_L$, under per unit taxation, only low-quality products can be traded. Under ad valorem taxation, as $r^{*}\le 1-\frac{u}{c_L}$, social welfare would be weakly higher than that under per unit taxation.

Now, consider the case of the inner solution. In this case, both the high- and the low-quality firm would enter under either type of taxation. When the product quality is low, the welfare under two forms of taxation is the same: at the equilibrium, the firm charges u in both cases to extract all consumers’ surplus, and the demand is constantly equal to 1. Therefore, regardless of the taxation form, social welfare is $u - c_{L}$ when the firm is of low-quality type. When the product quality is high, it can be shown that social welfare is strictly lower under per unit taxation by proving that ${\bar{p}}_A|_{r=r^{*}} < {\bar{p}}_F|_{f=f^{*}}$. Suppose that the ad valorem tax rate is set to $r''$ such that ${\bar{p}}_A|_{r=r^{''}} = {\bar{p}}_F|_{f=f^{*}}$. Then, transaction volumes under both forms of taxation should be equal, i.e. $D(H,1,{\bar{p}}_A|_{r=r^{''}}) = D(H,1,{\bar{p}}_F|_{f=f^{*}})$. Differentiating $R_A$ with respect to r evaluated at $r''$, we have

$$\begin{aligned} \frac{\partial R_A}{\partial r}\bigg |_{r=r''} &=s{\bar{p}}_{A}D(H,1,{\bar{p}}_{A})+(1-s)uD(L,0,u) \\&+ r s \frac{\partial {\bar{p}}_{A}}{\partial r}D(H,1,{\bar{p}}_{A}) + rs{\bar{p}}_{A} \frac{\partial D(H,1,{\bar{p}}_{A})}{\partial {\bar{p}}_{A}} \frac{\partial {\bar{p}}_{A}}{\partial r} \\ &={\bar{p}}_{A}sD(H,1,{\bar{p}}_{A}) +rs(1+u- 2{\bar{p}}_{A})\frac{\partial {\bar{p}}_{A}}{\partial r} +(1-s)u \\ &={\bar{p}}_{A}(s+sf^{*}-1)+ rs(u-2c_{L}-1 - 2f^{*}) \frac{\partial {\bar{p}}_{A}}{\partial r} +(1-s)u \\ &={\bar{p}}_{A}(s{\bar{p}}_{A} - s c_{L} -1)-r(s c_{L}+1)\frac{\partial {\bar{p}}_{A}}{\partial r}+(1-s)u \\ &=(s-1)({\bar{p}}_{A}-u) + s(1+c_{L}+f^{*} )f^{*}-\frac{r c_{L} (s c_{L} + 1) }{(1-r)^2} \\ &=(s-1)({\bar{p}}_{A}-u) + s(1+c_{L}+f^{*} )f^{*}- \frac{f^{*} (c_{L}+f^{*}) (s c_{L} + 1) }{c_{L}} \\ &=(s-1)({\bar{p}}_{A}-u) +f^{*} (s-1-\frac{f^{*}}{c_{L}}) \\ &<0, \end{aligned}$$

where the third line is obtained by substituting the first-order condition under per unit taxation (i.e., $sD(H,1,{\bar{p}}_{A}) + 1-s-sf^{*} =0$) into the first term and by substituting ${\bar{p}}_A|_{r=r^{''}} = {\bar{p}}_F|_{f=f^{*}} = 1+c_{L}+f^{*}$ into the second term; the fourth line is reached by substituting ${\bar{p}}_A|_{r=r^{''}} = {\bar{p}}_F|_{f=f^{*}} = 1+c_{L}+f^{*}$ into the first term and by substituting the inner solution $f^{*} = \frac{u - c_{L}}{2} + \frac{1-s}{2s}$ into the second term; the fifth line is obtained by substituting $\frac{\partial {\bar{p}}_{A}}{\partial r}$ at $r''$, by substituting $r''$ solved from ${\bar{p}}_A|_{r=r^{''}} = {\bar{p}}_F|_{f=f^{*}}$, and by reorganizing.

Moreover, we can prove that under ad valorem taxation, the total tax revenue $R_A$ is globally concave (please see Appendix F for the proof). As a result, the optimal tax rate $r^{*}$ should be strictly smaller than $r^{''}$, and we can conclude that ${\bar{p}}_A|_{r=r^{*}} < {\bar{p}}_A|_{r=r^{''}} = {\bar{p}}_F|_{f=f^{*}}$.

Appendix F The concavity of $R_A$

The second derivative of $R_A$ is

$$\begin{aligned} \frac{\partial ^2 R_A}{\partial r^2} &=s[D(H,1,{\bar{p}}_{A})+{\bar{p}}_{A}D'(H,1,{\bar{p}}_{A})] \left(2\frac{\partial {\bar{p}}_{A}}{\partial r}+r\frac{\partial ^2 {\bar{p}}_{A}}{\partial r^2}\right)\\ \nonumber&\quad+rs[2D'(H,1,{\bar{p}}_{A})+{\bar{p}}_{A}D''(H,1,{\bar{p}}_{A})] \left(\frac{\partial {\bar{p}}_{A}}{\partial r}\right)^2 . \end{aligned}$$

(A10)

The concavity of D guarantees that the second term is negative. However, the sign of the first term is difficult to identify. For instance, from equation (A7) it can be calculated that

$$\begin{aligned} \frac{\partial {{\bar{p}}}_A}{\partial r}=\frac{{{\bar{p}}}_AD(H,1,{\bar{p}}_{A}) -u}{(1-r)D(H,1,{\bar{p}}_{A})+[{{\bar{p}}}_A(1-r)-c_L]D'(H,1,{\bar{p}}_{A})}, \end{aligned}$$

(A11)

which is positive if and only if the numerator is negative. The sign of $\frac{\partial ^2 {\bar{p}}_{A}}{\partial r^2}$ is also ambiguous.

However, provided that G is a uniform distribution on [0, 1], we have $D'(H,1,{\bar{p}}_{A})=-1$, $D''(H,1,{\bar{p}}_{A})=0$, $\frac{\partial {\bar{p}}_{A}}{\partial r}=\frac{c_{L} }{(1-r)^2}$ and $\frac{\partial ^2 {\bar{p}}_{A}}{\partial r^2}=\frac{2rc_{L} }{(1-r)^3}$. Equation (A10) is thus simplified to

$$\begin{aligned} \frac{\partial ^2 R_A}{\partial r^2} =&s(D(H,1,{\bar{p}}_{A})-{\bar{p}}_{A})( 2 \frac{\partial {\bar{p}}_{A}}{\partial r}+r\frac{\partial ^2 {\bar{p}}_{A}}{\partial r^2}) -2s r (\frac{\partial {\bar{p}}_{A}}{\partial r})^2 \\ =&2s(D(H,1,{\bar{p}}_{A})-{\bar{p}}_{A}) \left[ \frac{c_{L} }{(1-r)^2}+\frac{rc_{L} }{(1-r)^3}\right] -2s r (\frac{\partial {\bar{p}}_{A}}{\partial r})^2 \\ =&2s(1+u-2{\bar{p}}_{A}) \left[ \frac{c_{L} }{(1-r)^2}+\frac{rc_{L} }{(1-r)^3} \right] -2s r (\frac{\partial {\bar{p}}_{A}}{\partial r})^2\\ <&0. \end{aligned}$$

The inequality sign holds because ${\bar{p}}_{A}>p_A^m(H)$ and $p_A^m(H)=\frac{1+u}{2}+\frac{c_H}{2(1-r)}>\frac{1+u}{2}$ under the uniform distribution. Therefore, $R_A$ is globally concave when $\theta$ is uniformly distributed.