Who should be regulated: Genuine producers or third parties?

Abstract

This study develops a model in which a “genuine” producer supplying genuine products competes with competitive “third-party” producers supplying compatible third-party products. We use this model to examine (i) how the strategic behavior of the genuine producer to drive out third parties (running comparative advertising, establishing technical barriers) affects the market equilibrium, (ii) whether the government should regulate such behavior by the genuine producer, and (iii) whether the government should regulate the entry of firms that supply third-party products. We find that a small amount of spending on advertising and creating technical barriers improves social welfare. However, their amounts in market equilibrium are socially excessive because the negative effects (e.g., the cost of advertising and creating barriers and an increase in production cost for third parties) outweigh the positive effects (e.g., an increase in the consumption of the genuine product and mitigation of the distortion of insufficient supply). Furthermore, we find that prohibitive measures (e.g., prohibition of advertising and technical barriers, entry prohibition of third-party producers) may improve welfare.

Appendices

Appendix

A: On functional forms

1.1 A.1. General distribution function

Although we assume a uniform distribution for the consumers’ love of genuineness ($\theta$), the density may be generalized. As (1) holds even for general distributions, the demand for each type of product (2) can be rewritten as

$$\begin{aligned} x_1 = \int ^{\bar{\theta }}_{\hat{\theta }} f(\theta ) d\theta , \qquad x_2 = \int ^{\hat{\theta }}_0 f(\theta ) d\theta , \end{aligned}$$

where $f(\cdot )$ is a probability density function. Then, the profits of firm 1 is

$$\begin{aligned} \pi _1 = p_1 \int ^{\bar{\theta }}_{\hat{\theta }} f(\theta )d\theta - \frac{1}{2} \mu _s s^2 - \frac{1}{2} \mu _b b^2. \end{aligned}$$

Similar to the case of a uniform distribution, the FOC for firm 1 in the second stage is given by

$$\begin{aligned} \frac{\partial \pi _1}{\partial p_1} =\int ^{\bar{\theta }}_{\hat{\theta }} f(\theta ) d\theta - p_1 f(\hat{\theta }) =0. \end{aligned}$$

(20)

From (20), we obtain the second partial derivatives:

$$\begin{aligned} \frac{\partial ^2 \pi _1}{\partial p^2_1} = -2f(\hat{\theta }) - p_1 f'(\hat{\theta }) \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 \pi _1}{\partial s \partial p_1} = f(\hat{\theta }) + p_1 f'(\hat{\theta }). \end{aligned}$$

Thus, if $-2f(\theta ) - p_1 f'(\theta )<0$, the SOC is satisfied. Moreover, if $f(\theta ) + p_1 f'(\theta )>0$ holds, the positive relationship between advertising and the price of the genuine product holds as follows:

$$\begin{aligned} \frac{\partial p^*_1(s,b)}{\partial s}= \frac{f(\hat{\theta }^*) + p^*_1 f'}{2f(\hat{\theta }^*)+ p^*_1 f'(\hat{\theta }^*)} >0. \end{aligned}$$

In the case of uniform distribution, these conditions hold because $f'=0$. Moreover, it holds that

$$\begin{aligned} \frac{\partial \hat{\theta }^*(p^*_1(s,b), s,b)}{\partial s}= -\frac{f(\hat{\theta }^*)}{2f(\hat{\theta }^*+ p^*_1 f')} \end{aligned}$$

which is negative if the SOC is satisfied.

Using the envelope theorem, we also obtain the FOCs for firm 1 in the first stage:

$$\begin{aligned} \frac{\partial \pi ^*_1}{\partial j}= p^*_1 f(\theta ) - \mu _j j =0, \quad j=s,b, \end{aligned}$$

which is similar to the FOC in the case of uniform distribution. The second partial derivatives are given by

$$\begin{aligned} \frac{\partial ^2 \pi ^*_1}{\partial j^2} = \frac{\partial p^*_1}{\partial j} \cdot f(\hat{\theta }^*) + p^*_1 f' \frac{\partial \hat{\theta }^*}{\partial j} - \mu _j = \frac{f(\hat{\theta }^*)^2}{2f(\hat{\theta }^*) + p^*_1 f'} - \mu _j, \quad j=s,b. \end{aligned}$$

(21)

$$\begin{aligned} \frac{\partial ^2 \pi ^*_1}{\partial k \partial s} = \frac{f(\hat{\theta }^*)^2}{2f(\hat{\theta }^*) + p^*_1 f'}, \quad j,k=s,b, \quad j \ne k. \end{aligned}$$

(22)

Similar to the case of a uniform distribution, assuming that the size of the cost parameters ($\mu _j$) is larger than a certain level, the SOC in the first stage (21) is assured to hold as long as the SOC in the second stage is satisfied. Moreover, if the SOC in the second stage is satisfied, the stability condition is also satisfied.

An increase in the level of advertising (1) increases the price of the genuine product, and (2) shifts the demand from non-genuine to genuine products. Thus, consumers lose from the increase in advertising. However, firm 1 does not consider this loss of consumers when choosing its level of advertising. In addition to these effects, an increase in the level of technical barriers increases the marginal cost of the non-genuine product and its price. Thus, similar to the case of uniform distribution, the difference of the levels of advertising and technical barriers between those at the market equilibrium and the second-best optimum can be obtained.

Thus, the important factor for general distribution to generate the same result as a uniform distribution is the condition for the SOC in the second stage to hold. The same results are obtained if $f'>0$, while the same results may not be generated if $f'<0$. Considering any level of love of genuineness, $\tilde{\theta }(0 \le \tilde{\theta }< \bar{\theta }$). If the number of consumers with $\tilde{\theta }$ increases as $\tilde{\theta }$ becomes larger, the generalization of distribution does not change the results. Moreover, even if the number of consumers with $\tilde{\theta }$ decreases as $\tilde{\theta }$ becomes larger, the same results are obtained if the degree of the decrease is small.

1.2 A.2. Increasing marginal cost of non-genuine products

We assume a constant marginal cost of producing non-genuine products in the main text. However, even if we consider an upward sloping supply curve for third-party products, the same results are obtained. For simplicity, we consider a uniform distribution of the love of genuineness.

Let us consider a simple upward sloping curve as follows:

$$\begin{aligned} c_2 = b + \psi \theta , \end{aligned}$$

where $c_2$ and $\psi$ are the marginal cost of production for non-genuine products and a positive constant, respectively. Then, the marginal consumer who is indifferent between purchasing goods 1 and 2 is defined as:

$$\begin{aligned} v_1 - p_1 + \hat{\theta }+ s = v_2 - b - \psi \hat{\theta }\Leftrightarrow \hat{\theta }= \frac{p_1 - G-s-b}{1+\psi }. \end{aligned}$$

The profits of firm 1 can be defined in the same way as the case of constant marginal cost of non-genuine products. Then, the FOC for firm 1 in the second stage is given by

$$\begin{aligned} \frac{\partial \pi _1}{\partial p_1} = x_1 - \frac{p_1}{1+\psi } =0. \end{aligned}$$

Note that the SOC is satisfied: $\partial ^2 \pi _1/\partial p^2_1 = -2/(1+\psi ) <0$. From the FOC above, the equilibrium values in the second stage are obtained as follows:

$$\begin{aligned} p^*_1 = \frac{(1+\psi ) \bar{\theta }+G+s+b}{2}, \quad x^*_1 = \frac{(1+\psi ) \bar{\theta }+G+s+b}{2(1+\psi )}. \end{aligned}$$

(23)

Thus, similar to the case of constant marginal cost of non-genuine products, the analyses for the first stage at the market equilibrium and for the second-best optimum can be conducted.

There is one additional effect from introducing an upward sloping supply curve. It is obtained from (23) that

$$\begin{aligned} \frac{\partial x^*_1}{\partial \psi } = -\frac{G+s+b}{(1+\psi )^2}<0, \end{aligned}$$

which implies that the steeper the slope of the upward sloping supply curve of non-genuine products, the smaller is the equilibrium output of firm 1. When the slope is steep, the cost of the marginal third-party (the marginal seller of the non-genuine product) decreases greatly as the market share of firm 1 increases. In other words, firm 1 must decrease the price of its product greatly to gain additional consumers (to shift consumers from purchasing non-genuine to genuine products). However, the price decrease leads to the loss of profits from its existing customers. Thus, the steeper the slope of the upward sloping supply curve of non-genuine products, the weaker the incentive firm 1 has to increase its share. The same mechanism works when firm 1 chooses the levels of advertising and technical barriers in the first stage.

However, Eqs. (10), (11) reveal that the difference of the levels of advertising and technical barriers between the market equilibrium and the second-best optimum is generated by (1) the loss consumers incur caused by an increase in the price of genuine products; (2) the loss incurred by the marginal consumers who change their purchased products; and (3) the loss that consumers of non-genuine products face caused by an increase in the marginal cost of non-genuine products. These three factors exist even in the case of an upward sloping supply curve. Thus, the same results are obtained even in the case of upward sloping supply curves of non-genuine products.

1.3 A.3. Shapes of the cost functions of advertising and technical barriers

We assume a specific cost function of advertising and technical barriers, $C_j (j) = \mu _j j^2/2 \, (j=s,b)$, which means that these cost functions are convex. To obtain interior solutions, the convexity of these cost functions is crucial, that is, $C'_j (j) >0$ and $C^{''}_j(j) >0$. To verify this point, consider the FOC for firm 1 in the first stage. In the case of general functional forms, (8) and (9) can be rewritten as:

$$\begin{aligned} \frac{\partial \pi ^*_1}{\partial j} = \frac{\bar{\theta }+G+s+b}{2} - C'(j) =0, \quad j=s,b. \end{aligned}$$

Then, the second partial derivative is given by

$$\begin{aligned} \frac{\partial ^2 \pi ^*_1}{\partial j^2} = \frac{1}{2} - C^{''}(j), \quad j=s,b. \end{aligned}$$

Thus, when $C^{''}_j \le 0$, firm 1 chooses zero or the maximum value for j.

For the government, the SOCs may be satisfied if the cost functions are linear or convex because the government not only considers the profits of firm 1 but also the consumer surplus (see 10 and 11). Moreover, even if the case with corner solutions is considered, we can examine the difference between the levels of advertising and technical barriers at the market equilibrium and the second-best optimum.

B: Proof of Propositions 1 and 2

First, we compare the advertising and technical barriers in the market equilibrium with those of the second-best optimum. From Table 1, we obtain:

$$\begin{aligned} s^E - s^O = \frac{ \left( 2\mu _s \mu ^2_b +2\mu ^2_b -\mu _s -2\mu _b \right) \bar{\theta }+ \left( 2 \mu _s \mu ^2_b -2\mu _s \mu _b +2 \mu ^2_b -\mu _b \right) G}{\left( 2\mu _s \mu _b -\mu _s-\mu _b \right) \left( 4\mu _s\mu _b -3\mu _s +\mu _b-1 \right) }. \end{aligned}$$

Given that $\mu _s>1$ and $\mu _b>1$ hold, from Assumption 1, $s^E-s^O>0$ holds. Moreover,

$$\begin{aligned} s^E-s^o = \frac{ \left( 2\mu _s \mu _b + \mu _s + \mu _b+1 \right) \left( \bar{\theta }+G \right) }{ \left( 4\mu _s+1 \right) \left( 2\mu _s \mu _b -\mu _s -\mu _b \right) } >0 \end{aligned}$$

holds.

We also obtain

$$\begin{aligned} b^E - b^O = \frac{ \mu _s \left( 6\mu _s \mu _b -4\mu _s -1 \right) \bar{\theta }+ \mu _b \left( -2\mu ^2_s +2\mu _s+1 \right) G}{ \left( 2\mu _s \mu _b -\mu _s-\mu _b \right) \left( 4\mu _s\mu _b -3\mu _s +\mu _b-1 \right) }. \end{aligned}$$

$\mu _s>1$ and $\mu _b>1$ hold from the second inequality of Assumption 1. $\bar{\theta }>G$ also holds from the third inequality from Assumption 1. Then, as $\mu _s (6\mu _s \mu _b -4\mu _s -1) -2\mu ^2_s \mu _b >0$, $b^E-b^O>0$ holds. Moreover, as $b^E>0$, it is obvious that $b^E-b^o >0$.

Next, we examine the denominator of the equilibrium advertising and technical barriers in partial regulation (L).

$$\begin{aligned} \Gamma _L= & {} (2\mu _s-1)^2 \mu _b - 3\mu _s (\mu _s-1) \nonumber \\= & {} 4 \mu ^2_s \mu _b -4\mu _s\mu _b + \mu _b -3\mu ^2_s + 3\mu _s \nonumber \\= & {} \mu _s (\mu _s -1)(4\mu _b-3) +\mu _b. \nonumber \end{aligned}$$

As $\mu _s>1$ and $\mu _b>1$ hold from Assumption 1, the denominator is positive.

Next, we compare the advertising and technical barriers at the market equilibrium with those under partial regulation. From Table 1, we obtain:

$$\begin{aligned} s^E - s^L = \frac{(3\mu ^2_s \mu _b -2 \mu ^2_s - 2\mu _s \mu _b +\mu _s + \mu _b)\bar{\theta }+ \mu _s \mu _b (2-\mu _s) G}{(2\mu _s \mu _b - \mu _s -\mu _b) \left\{ (2\mu _s-1)^2 \mu _b - 3\mu _s (\mu _s-1)\right\} } \end{aligned}$$

Note that

$$\begin{aligned} 3\mu ^2_s \mu _b -2 \mu ^2_s - 2\mu _s \mu _b +\mu _s + \mu _b = \mu _s^2 \mu _b + 2\mu _s(\mu _b -1)(\mu _s-1) -\mu _s +\mu _b >0 \end{aligned}$$

holds. Recall that $\mu _s>1$, $\mu _b>1$, and $\bar{\theta }>G$. If $G \ge \bar{\theta }/2$, then $2\mu _s\mu _b G > \mu _s \bar{\theta }$ holds. If $G < \bar{\theta }/2$ and $\mu _s>2$, then $\mu _s(\mu _s \mu _b -1) \bar{\theta }> \mu ^2_s \mu _b G$ holds. Moreover, if $\mu _s \le 2$, then $s^E>s^L$ necessarily holds. Thus, $s^E>s^L$ necessarily holds. We also obtain that

$$\begin{aligned} s^E - s^l = \frac{\mu _s (\bar{\theta }+G)}{(2\mu _s -1)(2\mu _s \mu _b -\mu _s - \mu _b)} \end{aligned}$$

Thus, it is clear that $s^E>s^l$.

Moreover, we obtain

$$\begin{aligned}{} & {} b^E - b^L \\{} & {} \quad = \frac{ \left( 6\mu ^3_s \mu _b -7\mu ^2_s \mu _b - 4\mu ^3_s + 4 \mu ^2_s + 4\mu _s \mu _b -\mu _s - \mu _b \right) \bar{\theta }- \left( 2\mu ^3_s \mu _b - 5 \mu ^2_s \mu _b + 2\mu _s \mu _b \right) G}{ \left( 2\mu _s \mu _b - \mu _s -\mu _b \right) \left\{ (2\mu _s-1)^2 \mu _b - 3\mu _s (\mu _s-1)\right\} }. \end{aligned}$$

The numerator is rewritten as:

$$\begin{aligned}{} & {} \left( 2\mu ^3_s \mu _b - 5\mu ^2 \mu _b + 2\mu _s \mu _b \right) \left( \bar{\theta }-G \right) + (4\mu ^2_s (\mu _s-1)(\mu _b-1) \\{} & {} \quad + 2\mu ^2_s \mu _b +4\mu _s \mu _b -\mu _s - \mu _b) \bar{\theta }. \end{aligned}$$

The third inequality of Assumption 1 reveals that $\bar{\theta }- G > 0$. Thus, the numerator is positive. $b^E > b^L$ necessarily holds. As $b^l =0$, it is obvious that $b^E > b^l$.

C: The details for the welfare effect of complete strategy prohibition

We first investigate the sign of (17), which can be rewritten as:

$$\begin{aligned}{} & {} -\frac{1}{8(2\mu _s \mu _b -\mu _s - \mu _b)^2} \cdot \left\{ 4(2\mu _s \mu _b - \mu _s - \mu _b)(\mu _s + \mu _b)(\bar{\theta }+G)G \right. \nonumber \\{} & {} \qquad \left. +(2\mu _s \mu _b - \mu _s - \mu _b)^2 (\bar{\theta }-G)^2 - 4((\mu _s \mu _b -\mu _s -\mu _b)\bar{\theta }- \mu _s \mu _b G)^2\right\} . \nonumber \end{aligned}$$

The coefficient of $\bar{\theta }^2$ is given by

$$\begin{aligned} - 4\mu ^2_s \mu _b - 4\mu _s \mu ^2_b + 3\mu ^2_s + 3\mu ^2_s + 6\mu _s \mu _b= & {} -3(\mu _s+\mu _b)(\mu _s \mu _b- \mu _s - \mu _b) \nonumber \\{} & {} - \mu _s \mu _b (\mu _s + \mu _b) <0. \nonumber \end{aligned}$$

It is easily verified that the coefficients of $\bar{\theta }G$ and $G^2$ are negative. Thus, the third effect is necessarily negative.

We next examine the overall effect. Summing up (15) through (17), we obtain the difference between welfare under the prohibition of both advertising and technical barriers and welfare at market equilibrium:

$$\begin{aligned} W^{C} - W^{E}= & {} \frac{1}{8(2\mu _s \mu _b - \mu _s - \mu _b)^2} \cdot \left\{ \left( 12\mu ^2_s \mu _b + 4 \mu _s \mu ^2_b - 5\mu ^2_s -2\mu _s \mu _b+ 3\mu ^2_b \right) \bar{\theta }^2 \right. \nonumber \\{} & {} \ \left. - \left( 2\mu ^2_s - 4\mu _s \mu _b- 6 \mu ^2_b \right) \bar{\theta }G - \left( 4 \mu ^2_s \mu _b -3\mu ^2_s - 6\mu _s \mu _b - 3\mu ^2_b \right) G^2\right\} . \nonumber \end{aligned}$$

The third inequality of Assumption 1 reveals that $\theta > G$. Moreover, the second inequality of Assumption 1 implies that $\mu _b>1$. Thus, $12\mu ^2_s \mu _b \bar{\theta }^2 - 2\mu ^2_s \bar{\theta }G - 4\mu ^2_s \mu _b G^2 >0$ holds. Consequently, $W^C - W^E$ is positive.

D: Partial prohibition for the genuine producer

Let superscript $J$ denote the case of partial prohibition. Firm 1 can use one of two strategies, $j \, (j=s \, or \, b)$. The equilibrium price, consumption quantities, and levels of advertising/technical barriers are given as follows:

$$\begin{aligned}{} & {} p^{J}_1 = \frac{\mu _j (\bar{\theta }+ G)}{2\mu _j -1} +c_1, \quad x^{J}_1 = \frac{\mu _j (\bar{\theta }+ G)}{2\mu _j -1}, \nonumber \\{} & {} x^{J}_2 = \bar{\theta }- x^{J}_1, \quad j^{J}_1 = \frac{\bar{\theta }+G}{2\mu _j-1}. \nonumber \end{aligned}$$

The total cost of running advertising and setting technical barriers is smaller under complete strategy prohibition than under partial prohibition. Thus, the benefit (cost saving) caused by the change from partial prohibition to complete strategy prohibition is

$$\begin{aligned} \frac{\mu _j (\bar{\theta }+G)^2}{2(2\mu _j-1)^2}. \end{aligned}$$

(24)

Having said that, the quantity of genuine products consumed under partial prohibition is greater than that under complete prohibition. Therefore, the difference between the gross surplus in the latter case and that in the former case may be negative:

$$\begin{aligned} -\frac{(\bar{\theta }+G)G}{2(2\mu _j-1)} - \frac{(\bar{\theta }-G)^2}{8} + \frac{ \left\{ (\mu _j-1 ) \bar{\theta }- \mu _j G \right\} ^2}{2(2\mu _j-1)^2}. \end{aligned}$$

(25)

However, summing both effects, (24) and (25), reveals that the difference is necessarily positive: $3 (\bar{\theta }+ G)^2/\left\{ 8(2\mu _j-1)\right\} ^2>0$. Moreover, in the case of a prohibition on advertising, there is an additional cost to the third-party producers ($b$), which disappears under the complete strategy prohibition. Thus, it is concluded that welfare in complete prohibition is necessarily greater than that in partial prohibition.

E: Choice of partial coverage by firm 1 under entry prohibition of third parties

Let us first consider that firm 1 chooses full coverage in the case where (1) entry of third parties is prohibited and (2) firm 1 may run advertising. The consumer whose love of genuineness is zero purchases the genuine product if the price is equal to or less than $v_1+s$. Thus, when choosing full coverage, the profit function of firm 1 is given by

$$\begin{aligned} \pi _{1,fu} = (v_1 +s )\bar{\theta }- \frac{\mu _s s^2}{2}, \end{aligned}$$

where subscript fu denotes full coverage. Solving the profit maximization problem, we obtain the equilibrium level of advertising, which is $\bar{\theta }/\mu _s$. Then, the equilibrium profit is given by

$$\begin{aligned} \pi ^{MS}_{1,fu} = \bar{\theta }\left(v_1 + \frac{\bar{\theta }}{2 \mu _s}\right). \end{aligned}$$

(26)

Next, suppose that firm 1 chooses partial coverage in the same case as above. In this case, the FOCs for firm 1 in the second stage are given by (6) and (8) given $b=0$. The equilibrium profit is given by

$$\begin{aligned} \pi ^{MS}_{1, pa} = \frac{\mu _s (\bar{\theta }+ v_1)^2}{2(2\mu _s-1)}, \end{aligned}$$

(27)

where subscript pa denotes partial coverage. From (26) and (27), we obtain that

$$\begin{aligned} \pi ^{MS}_{1, pa} - \pi ^{MS}_{1,fu} = \frac{((\mu _s -1)\bar{\theta }- \mu _s v_1)^2}{2\mu _s (2\mu _s-1)} > 0. \end{aligned}$$

Thus, firm 1 necessarily chooses partial coverage in the case that (1) entry of third parties is prohibited and (2) firm 1 may run advertising.

Let us turn to the case in which (1) the entry of third parties is prohibited and (2) firm 1 does not run advertising. When firm 1 aims at full coverage, it must set the price equal to $v_1$. If the price is higher than this value, consumers with a low degree of love of genuineness make no purchases. The total supply is $\bar{\theta }$, and the profit is $v_1 \bar{\theta }$. Next, suppose that firm $1$ chooses partial coverage. Then, the surplus of consumer $\theta$ is $u = v_1 - p_1 +\theta$. Therefore, the demand for the genuine product is given by $x^M_1 = \bar{\theta }+v_1 -p_1$, and the profit of firm $1$ is given by $\pi ^M_1 = (p_1 -c_1) \cdot (\bar{\theta }+v_1 -p_1)$. Solving the profit maximization problem, we obtain the equilibrium price and output (See Table 2). Then, the profit at equilibrium is

$$\begin{aligned} \pi ^{M}_1 = \frac{(\bar{\theta }+ v_1)^2}{4}. \end{aligned}$$

A comparison of the profit under partial coverage and under full coverage reveals that the former is greater than the latter:

$$\begin{aligned} \frac{(\bar{\theta }+ v_1)^2}{4} - v_1 \bar{\theta }= \frac{(\bar{\theta }- v_1)^2}{4} >0. \end{aligned}$$

Thus, firm $1$ necessarily chooses partial coverage in both cases.

F: Details for Proposition 5

From the second-stage market equilibrium values and (18), we obtain that

$$\begin{aligned}{} & {} -\left( \frac{s^{V*}_1}{2} + \int ^{\bar{\theta }}_{\hat{\theta }^{V*}} \frac{1}{2} d\theta \right) \cdot \frac{ds^{V*}_1}{dV_1} - \left( \hat{\theta }^{V*} + \frac{s^{V*}_1}{2} + \int ^{\bar{\theta }}_{\hat{\theta }^{V*}} \frac{1}{2} d\theta \right) \cdot \frac{db^{V*}_2}{dV_1} \nonumber \\{} & {} \quad = -\frac{(1-\alpha )\mu _s}{2\mu _s \mu _b - \mu _s - \mu _b} \cdot \hat{\theta }^{V*} \nonumber \\{} & {} \quad \quad - \frac{(1-\alpha ) \mu _b (\mu _s+\mu _b)(\mu _s+1)}{2(2\mu _s \mu _b - \mu _s - \mu _b)^2} \cdot \left\{ \bar{\theta }+ (v^{V*}_1-c_1)-(v^{V*}_2-c_2) \right\} \nonumber \end{aligned}$$

However,

$$\begin{aligned} \alpha \hat{\theta }^{V*} - \frac{(1-\alpha )s^{V*}_1}{2} + \int ^{\bar{\theta }}_{\hat{\theta }^{V*}} \frac{1+\alpha }{2} d\theta= & {} \alpha \hat{\theta }^{V*} \nonumber \\{} & {} + \frac{\mu _b ((1+\alpha )\mu _s - (1-\alpha ))}{2(2\mu _s \mu _b - \mu _s - \mu _b)} \nonumber \\{} & {} \cdot \left\{ \bar{\theta }+ (v^{V*}_1-c_1)-(v^{V*}_2-c_2)\right\} . \nonumber \end{aligned}$$

Thus, the smaller (larger) the value of $\alpha$, the more likely it is that (19) is negative (positive).