Regulation on coexisting legal and illegal markets with quality differentiation

Abstract

This paper considers coexisting legal and illegal markets for goods with undesirable consumption externalities, such as marijuana markets. This paper introduces a quality differentiation between legal and illegal markets, e.g., product differentiation and its consumption externality, and investigates the effects of regulations such as taxation and quality control on legal markets and enforcement against illegal producers and consumers. This paper shows that these regulations of legal and illegal markets are likely to cause divergence between legal and illegal prices and may fail to reduce social costs.

Appendix

1.1 Appendix A: The effect of regulation costs for the licensed producer

Proof

We provide the explanation for $\sigma ^{*}_t-m^{*}_t>0$, where

$$\begin{aligned} \sigma ^{*}_t-m^{*}_t=\frac{-\pi ^{l}_{\sigma t}\left( \pi ^{ul}_{m m}+\pi ^{ul}_{m \sigma }\right) }{T}. \end{aligned}$$

(58)

Since $i_{\sigma } =1/(1-k)>0$ and $i_{m} =-1/(1-k)<0$, we have

$$\begin{aligned} \pi ^{ul}_{m\sigma }= & {} i_{\sigma } -c^{ul}_{ii}i_\sigma i_m >0 \end{aligned}$$

(59)

$$\begin{aligned} \pi ^{ul}_{m m}= & {} 2i_{m}-c^{ul}_{ii}(i_m)^2<0. \end{aligned}$$

(60)

Since $|i_\sigma |=|i_m|$ according to (15) and (19), we have $\pi ^{ul}_{m m}+\pi ^{ul}_{m \sigma }<0$ and $\sigma ^{*}_t-m^{*}_t>0$. $\square$

1.2 Appendix B: The effect of the substitution of licensed products

Proof

The sign of $\sigma ^{*}_{v_1}$ is determined by

$$\begin{aligned} \underbrace{-\underbrace{\pi ^{l}_{\sigma v_1}}_{+} \underbrace{\pi ^{ul}_{m m}}_{-}}_{+}+\underbrace{\underbrace{\pi ^{ul}_{m v_1}}_{-}\underbrace{\pi ^{l}_{\sigma m}}_{+}}_{-}. \end{aligned}$$

(61)

Furthermore, since

$$\begin{aligned} \pi ^{l}_{\sigma v_1}= & {} q_{v_1}-c^{l}_{qq}q_{\sigma }q_{v_1}>0 \end{aligned}$$

(62)

$$\begin{aligned} \pi ^{l}_{\sigma m}= & {} q_m -c^{l}_{qq}q_\sigma q_m >0 \end{aligned}$$

(63)

$$\begin{aligned} \pi ^{ul}_{m v_1}= & {} i_{v_1}-c^{ul}_{ii}i_m i_{v_1}<0 \end{aligned}$$

(64)

$$\begin{aligned} \pi ^{ul}_{m m}= & {} 2i_{m}-c^{ul}_{ii}(i_m)^2<0, \end{aligned}$$

(65)

and $|q_{v_1}|>|q_m|$ and $|i_m|=|i_{v_1}|$ according to (33), (15), and (13). Then, it is likely that $-\pi ^{l}_{\sigma v_1}\pi ^{ul}_{m m}>0$ is more important than $\pi ^{ul}_{m v_1}\pi ^{l}_{\sigma m}<0$. Therefore, we are likely to have $\sigma ^{*}_{v_1}>0$.

The sign of $m^{*}_{v_1}$ is determined by

$$\begin{aligned} \underbrace{-\underbrace{\pi ^{l}_{\sigma \sigma }}_{-} \underbrace{\pi ^{ul}_{m v_1}}_{-}}_{-}+\underbrace{\underbrace{\pi ^{ul}_{m \sigma }}_{+}\underbrace{\pi ^{l}_{\sigma v_1}}_{+}}_{+}. \end{aligned}$$

(66)

Furthermore, since

$$\begin{aligned} \pi ^{l}_{\sigma v_1}= & {} q_{v_1}-c^{l}_{qq}q_{\sigma }q_{v_1}>0 \end{aligned}$$

(67)

$$\begin{aligned} \pi ^{l}_{\sigma \sigma }= & {} 2q_{\sigma }-c^{l}_{qq}(q_\sigma )^2<0 \end{aligned}$$

(68)

$$\begin{aligned} \pi ^{ul}_{m v_1}= & {} i_{v_1}-c^{ul}_{ii}i_m i_{v_1}<0 \end{aligned}$$

(69)

$$\begin{aligned} \pi ^{ul}_{m\sigma }= & {} i_{\sigma } -c^{ul}_{ii}i_\sigma i_m >0 \end{aligned}$$

(70)

and$|q_{v_1}|=|q_{\sigma }|$ and $|i_{\sigma }|=|i_{v_1}|$ according to (33), (19), and (9). Then, it is likely that $-\pi ^{l}_{\sigma \sigma }\pi ^{ul}_{m v_1}<0$ is more important than $\pi ^{ul}_{m \sigma }\pi ^{l}_{\sigma v_1}>0$. Therefore, we are likely to have $m^{*}_{v_1}<0$. As a result, we are likely to have $\sigma ^{*}_{v_1}-m^{*}_{v_1}>0$. $\square$

1.3 Appendix C: The effect of sanctions against the unlicensed producer

Proof

First, we have

$$\begin{aligned} \sigma ^{*}_{e_p}-m^{*}_{e_p}=\frac{\pi ^{ul}_{m e_p}( \pi ^{l}_{\sigma \sigma }+\pi ^{l}_{\sigma m})}{T}. \end{aligned}$$

(71)

Then, in addition to $\pi ^{ul}_{m e_p}>0$,

$$\begin{aligned} \pi ^{l}_{\sigma m}= & {} q_{m} -c^{l}_{qq}q_\sigma q_m >0 \end{aligned}$$

(72)

$$\begin{aligned} \pi ^{l}_{\sigma \sigma }= & {} 2q_{\sigma }-c^{l}_{qq}(q_\sigma )^2<0. \end{aligned}$$

(73)

Additionally, although we have $|q_\sigma |>|q_m|$, according to (9) and (13), which indicates $\pi ^{l}_{\sigma m}+\pi ^{l}_{\sigma \sigma }<0$ and $\sigma ^{*}_{e_p}-m^{*}_{e_p}<0$. $\square$

1.4 Appendix D: The effect of sanctions against illicit consumers

Proof

The sign of $\sigma ^{*}_{e_d}$ is determined by

$$\begin{aligned} \underbrace{-\underbrace{\pi ^{l}_{\sigma e_d}}_{+} \underbrace{\pi ^{ul}_{m m}}_{-}}_{+}+\underbrace{\underbrace{\pi ^{ul}_{m e_d}}_{-}\underbrace{\pi ^{l}_{\sigma m}}_{+}}_{-}. \end{aligned}$$

(74)

Furthermore, since

$$\begin{aligned} \pi ^{l}_{\sigma e_d}= & {} q_{e_d}-c^{l}_{qq}q_{\sigma }q_{e_d}>0 \end{aligned}$$

(75)

$$\begin{aligned} \pi ^{l}_{\sigma m}= & {} q_m -c^{l}_{qq}q_\sigma q_m >0 \end{aligned}$$

(76)

$$\begin{aligned} \pi ^{ul}_{m e_d}= & {} i_{e_d}-c^{ul}_{ii}i_m i_{e_d}<0 \end{aligned}$$

(77)

$$\begin{aligned} \pi ^{ul}_{m m}= & {} 2i_{m}-c^{ul}_{ii}(i_m)^2<0, \end{aligned}$$

(78)

and $|q_{e_d}|=|q_m|$ and $|i_m|=|i_{e_d}|$ according to (50), (15), and (13). Then, it is likely that $-\pi ^{l}_{\sigma e_d}\pi ^{ul}_{m m}>0$ is more important than $\pi ^{ul}_{m e_d}\pi ^{l}_{\sigma m}<0$. Therefore, we are likely to have $\sigma ^{*}_{e_d}>0$.

The sign of $m^{*}_{e_d}$ is determined by

$$\begin{aligned} \underbrace{-\underbrace{\pi ^{l}_{\sigma \sigma }}_{-} \underbrace{\pi ^{ul}_{m e_d}}_{-}}_{-}+\underbrace{\underbrace{\pi ^{ul}_{m \sigma }}_{+}\underbrace{\pi ^{l}_{\sigma e_d}}_{+}}_{+}. \end{aligned}$$

(79)

Furthermore, since

$$\begin{aligned} \pi ^{l}_{\sigma e_d}= & {} q_{e_d}-c^{l}_{qq}q_{\sigma }q_{e_d}>0 \end{aligned}$$

(80)

$$\begin{aligned} \pi ^{l}_{\sigma \sigma }= & {} 2q_{\sigma }-c^{l}_{qq}(q_\sigma )^2<0 \end{aligned}$$

(81)

$$\begin{aligned} \pi ^{ul}_{m e_d}= & {} i_{e_d}-c^{ul}_{ii}i_m i_{e_d}<0 \end{aligned}$$

(82)

$$\begin{aligned} \pi ^{ul}_{m\sigma }= & {} i_{\sigma } -c^{ul}_{ii}i_\sigma i_m >0 \end{aligned}$$

(83)

and $|q_{e_d}|<|q_{\sigma }|$ and $|i_{\sigma }|=|i_{e_d}|$ according to (50), (19), and (9). Then, it is likely that $-\pi ^{l}_{\sigma \sigma }\pi ^{ul}_{m e_d}<0$ is more important than $\pi ^{ul}_{m \sigma }\pi ^{l}_{\sigma e_d}>0$. Therefore, we are likely to have $m^{*}_{e_d}<0$. As a result, we are likely to have $\sigma ^{*}_{e_d}-m^{*}_{e_d}>0$. $\square$