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Legalizing Harmful Drugs: Government Participation and Optimal policies

  • Luis Rodrigo Arnabal Rocca ORCID logo EMAIL logo
Published/Copyright: November 14, 2022
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The B.E. Journal of Economic Analysis & Policy
From the journal The B.E. Journal of Economic Analysis & Policy Volume 23 Issue 1

Abstract

We are currently witnessing a shift in the approach to combat traffic and consumption of illegal harmful drugs, being cannabis legalization a prominent example. In this paper, we study how to optimally regulate the market for cannabis, in a setting where consumers differ in their utility from consumption of the psychoactive component of cannabis, THC, and suffer from misperception of the health damage it causes. We analyze this problem through a vertical differentiation model, where a black market firm and a public firm compete in prices and qualities (THC content). A paternalistic government would like to correct for the misperceived health damage caused by cannabis consumption, as well as to reduce the size of the black market. It is the undesirability of black market profits what explains that the first-best allocation cannot be decentralized. We find two possible equilibria, depending on whether the public firm serves those consumers with the highest or lowest willingness to pay for quality. Paradoxically, when the public firm serves those consumers with higher taste for THC, a lower average health damage is achieved together with a better economic result for the public firm.

Keywords: legalization; cannabis; marijuana; optimal policies; government participation
JEL Classification: H44; I18; L13; L51

1 Introduction

There is an increasing consensus that the so called “War on Drugs” has failed. As a consequence, policymakers are exploring other alternatives to discourage consumption of illicit drugs.[1] Cannabis is today the most consumed illicit substance, with around 200 million people having used it at least once in 2019 according to the World Drug Report of 2021. As a natural alternative to its prohibition, several jurisdictions have recently decided to legalize cannabis for recreational purposes, bringing to the table the discussion on how to optimally regulate the market for harmful drugs.[2]

The arguments for cannabis legalization are many, standing out the reduction on enforcement costs and the increase in tax revenue. Perhaps more importantly, legalization is to diminish black market revenues, reducing the strength of the mafias and their criminal activity, improving thus welfare for the whole population.[3] This argument was very relevant for pursuing cannabis legalization in Uruguay, where it has been framed as a policy to combat crime, rather than a public health or consumers rights issue (see Queirolo et al. 2019). In any case, cannabis legalization has not been able so far to wipe out the black market. Indeed, according to the Canadian Cannabis Survey 2021, only 43% of cannabis consumers report to have always acquired cannabis from a legal source in the past twelve months, while the public agency that regulates cannabis in Uruguay, IRCCA, acknowledges that only 30% of those who consumed cannabis in 2018 did so through the legal channel. Differences in prices and the amount of the psychoactive component Δ9-tetrahydrocannabinol (THC) are the main explanations for the coexistence of legal an illegal suppliers. This last dimension is particularly relevant in Uruguay, where THC levels of publicly provided cannabis are surprisingly low (initially fixed at 2%, to later being increased up to 9%).

On the other hand, few restrictions exist regarding THC content in the US states and Canada, where legal cannabis is mostly privately provided, reaching levels of 32%.[4] These differences raise the question of what should be the optimal levels of THC available for consumers upon legalization.[5]

Achieving higher levels of THC content implies increasing marginal production costs, as a detailed care on environmental conditions, such as humidity, temperature and light is required throughout the whole production process.[6] While for most consumers the amount of THC is perceived as the main measure of quality for recreational cannabis, this psychoactive component is also associated with potential brain damage.[7] Furthermore, evidence from the medical literature suggests that health damage associated to cannabis consumption is convex in THC levels, e.g. Di Forti et al. (2014), Rigucci et al. (2016), Di Forti et al. (2019) and Arterberry et al. (2019).

The damage caused by cannabis is in general not fully acknowledged by consumers, raising public health issues that governments cannot ignore.[8] The increase in cannabis consumption among high school US students is indeed mainly due to changes in the disapproval of its use and in the risk perception of the harm it causes (Bachman, Johnson, and O’Malley 1998). This problem is becoming more relevant today, as recent evidence suggests that health damage perception from cannabis consumption is declining (Johnston et al. 2019).

While the debate on cannabis legalization revolves around whether the benefits outweigh the costs aforementioned (see, for instance, Galenianos, Pacula, and Persico 2012; Caulkins et al. 2012; Jacobi and Sovinsky 2016), our study focuses on how to regulate the market for cannabis once the decision to legalize has already been made. We thus want to contribute to the debate on how to regulate this market, e.g. Pacula et al. (2014) and Caulkins and Kilmer (2016). In this line, the study of Caputo and Ostrom (1996) discusses the optimal combination of tax and enforcement rate when the government is a dominant firm and black market firms are considered as a competitive fringe. More recently, the analysis of Auriol, Mesnard, and Perrault (2019) determines the optimal combination of enforcement and taxes necessary to eradicate the black market, while keeping demand for cannabis at desirable levels.

This paper aims to contribute to this discussion by formally analyzing the question of what are the optimal policies for regulating an imperfectly competitive market for cannabis, in a setting characterized by the presence of a black market firm, where consumers differ in their taste for THC and suffer from misperception of the health damage caused by its consumption.[9]

In our framework, cannabis legalization translates into a direct participation of the government through a public firm that aims to maximize welfare, as it has been the case in Uruguay.[10] While our study is motivated by this particular example, we believe the lessons extracted from our analysis contribute to a better understanding on the issue of how to optimally regulate the market for cannabis. We consider that legal and illegal provision of cannabis coexist after legalization. Profits generated in the black market are deemed socially undesirable since they foster criminal activity. Consequently, for a paternalistic government the motivation to intervene in the cannabis market is twofold: reduce black market activity and correct for the overconsumption of THC that arises due to health damage misperception suffered by consumers.

Rather than to focus on repressive measures to control demand or to incorporate a revenue objective, our focus is on the optimal level of THC content to be offered, as well as on the pricing strategy to be followed by the public firm, in order to reduce black market activity and to correct for the misperceived health damage caused by cannabis consumption. We incorporate therefore a new dimension to the discussion related to the optimal amount of THC to be offered upon the decision to legalize cannabis.

Regarding the timing, we depart from a situation where firms compete first in qualities and then simultaneously in prices, in order to account for the fact that THC content is more difficult to adjust, since, for instance, the crop cycle of cannabis takes at least six months from germination to its drying and curing. We focus on pure strategies, where the concept of equilibrium is subgame perfect Nash equilibrium. We assume that the market is fully covered and restrict our attention to situations where both firms are active in the market and constrained to offer only one quality.[11] To take into account that the government has a stronger commitment and credibility than the black market, we also model the quality selection stage à la Stackelberg.

We find that as long as black market profits are deemed socially undesirable, the first-best allocation cannot be decentralized. While the presence of a public firm allows to correct for the health damage misperception through Pigouvian prices, splitting consumers optimally between the two available qualities, the undesirability of black market profits drives the public firm to reduce its price in an effort to attract more demand. For the same reason, the public firm distorts its quality to increase its market share, what in turn triggers a strategic reaction on the quality chosen by the black market firm, who seeks to make more profits through a higher product differentiation.

There are two possible equilibrium configurations that arise as a result of the competition between the black market firm and the public firm, depending on whether the latter serves the consumers with higher or lower willingness to pay for quality. When the government supplies the product with more THC content, a better economic result for the public firm is achieved together with a lower average health damage. Extensions of our model indicate that this equilibrium may also yield higher welfare levels and a lower black market share. We also find that adding a first-mover advantage to the public firm in the quality selection stage of the game does not modify the equilibria that result from simultaneous quality choice, as long as the public firm is allowed to make negative profits.

The remainder of the paper is organized as follows. Section 2 describes the economic environment and the normative benchmark. Section 3 studies the competition between a black market firm and a public firm in prices and qualities. The main policy implications of our analysis are discussed in Section 4. Section 5 concludes.

2 Economic Environment and Normative Benchmark

Consider a unit mass of consumers that differ in their marginal valuation for quality (THC) denoted by θ, positive and uniformly distributed between [ θ ̲ , θ ̄ ] , with θ ̄ = θ ̲ + 1 .[12] Individuals incur on a monetary cost p i > 0, to purchase at most one unit of cannabis from firm i with THC content q i > 0. Cannabis consumption entails health damage that, in accordance to the medical literature, is considered to be convex in the amount of THC, captured through the following harm function: h ( q i ) = q i 2 / 2 . Consumers are only able to perceive a fraction β < 1 of this damage. We consider a model of vertical differentiation, where to the utility from consumption described in Mussa and Rosen (1978), we add a health cost related to the perceived harm caused by cannabis consumption.[13] The perceived utility for a consumer of type θ and with a health damage perception of β from buying one unit of cannabis from firm i is then given by:

U β ( p i , q i ; β ) = θ q i − p i − β q i 2 2 .

The black market firm and the public firm are assumed to have the same marginal production costs, given by the following expression: C ( q i ) = c q i 2 / 2 , with c > 0.[14] This specification aims to capture the fact that to achieve higher THC levels, a more detailed care is required throughout the whole production process. The only difference between both firms in our setting is the social value assigned to the profits made by each type of firm.

To focus on the strategic interaction between qualities offered by both firms, we make the assumption that each firm is restricted to offer a unique quality. A firm i will then set a level of THC q i and a price p i for its product, in order to maximize profits Π i . For a consumer to participate in the market, it must be that his taste for quality is higher than the following participation threshold:

θ P ( p j , q j ; β ) = p j q j + β q j 2 ,

where here j corresponds to the firm offering the lowest quality product.

From the expression above we have that the socially desirable participation threshold is the following:

θ PS ( p j , q j ) = p j q j + q j 2 .

We make the assumption that the utility of the consumer with the lowest marginal willingness to pay for THC θ ̲ is positive, even when fully perceiving the health damage caused by cannabis consumption:

(1) θ ̲ > θ PS ( p j , q j ) .

This assumption implies that the demand is fixed and that participation of all consumers is socially desirable. The impact of relaxing this assumption is analyzed in Appendix A.1.

Two sources of inefficiency motivate the intervention of our paternalistic government: the misperception of the health damage that THC causes and the presence of a black market firm, whose profits are deemed socially undesirable. The tools available for the government are the price p G and quality q G of the product to be supplied through the public firm. Additionally, the government may also make use of a lump sum transfer T to redistribute profits or losses made by the public firm. The budget constraint faced by the government is therefore:

T ≤ Π G = p G − c q G 2 2 D G ,

where D G corresponds to the demand faced by the public firm.

2.1 First-Best Allocation and Optimal Splitting

Our first-best scenario is a situation where consumers do not suffer from misperception of the health damage caused by THC, and where profits are just transfers between agents who have the same value from a welfare perspective. In other words, consumers fully perceive the health damage caused by the quality content of the product they purchase and there is no black market.

As a normative benchmark and for comparative purposes, let’s consider the situation where the government is constrained to offer two different qualities, q L and q H, with q H > q L.[15] In order to derive the optimal allocation, we must first determine the demands for the low and high quality products. For the consumer that is indifferent between both varieties it must verify that:

θ q L − β q L 2 2 − p L = θ q H − β q H 2 2 − p H ,

so that the indifferent consumer is given by:

(2) θ ̂ ( p L , p H , q L , q H ; β ) = p H − p L q H − q L + β q H + q L 2 .

For both qualities to have positive demand in equilibrium, besides the participation constraint in (1), we further assume that there is enough consumer heterogeneity such that the market is covered. In particular, we will make the following assumption:

(3) θ ̄ > θ ̂ ( p L , p H , q L , q H ; β ) > θ ̲ .

A paternalistic government maximizes social welfare, considering the true health damage caused by cannabis consumption.[16] The (constrained) first-best allocation would then correspond to offer a low and a high quality product such that the true utility of consumers is maximized, i.e. with β = 1. The welfare function can be then expressed in the following way:

(4) W = ∫ θ ̲ θ ̂ θ q L − ( 1 + c ) q L 2 2 d θ + ∫ θ ̂ θ ̄ θ q H − ( 1 + c ) q H 2 2 d θ .

The government will set qualities q L and q H, and how consumers should be split among these two qualities. To derive how consumers should be optimally split, let’s take the first-order condition (FOC) of the welfare function in (4) with respect to the indifferent consumer:

∂ W ∂ θ ̂ = θ ̂ q L − ( 1 + c ) q L 2 2 − θ ̂ q H + ( 1 + c ) q H 2 2 = 0 .

Solving for θ ̂ yields the optimal splitting condition:

(5) θ ̂ ̃ = ( 1 + c ) q L + q H 2 .

Optimal splitting is achieved when the indifferent consumer is such that his marginal willingness to pay for quality equals the marginal social cost of the average quality.

Let’s now determine the first-best qualities. Taking the FOCs of the welfare function with respect to qualities yields:

∂ W ∂ q L = θ ̂ + θ ̲ 2 − ( 1 + c ) q L ( θ ̂ − θ ̲ ) = 0 , ∂ W ∂ q H = θ ̂ + θ ̄ 2 − ( 1 + c ) q H ( θ ̄ − θ ̂ ) = 0 .

Conditions above together with the optimal splitting condition in (5) determine the (constrained) first-best qualities:

(6) q L FB = 4 θ ̲ + 1 4 ( 1 + c ) ,

(7) q H FB = 4 θ ̲ + 3 4 ( 1 + c ) .

The expressions above correspond to the qualities that maximize welfare, once the true health damage is taken into account.[17] Plugging in the expressions for the first-best qualities in (6) and (7) into the expression that characterizes the optimal splitting in (5), we have that:

θ ̂ FB = θ ̲ + 1 2 .

The optimal splitting between first-best qualities is then such that the indifferent consumer lays at the middle of the distribution. These first-best qualities will be used as a benchmark to address the case where a black market firm and a public firm engage in price and quality competition in the following section. The comparable laissez-faire situation, where two profit-maximizing firms are constrained to offer each a unique quality, results in a higher degree of quality differentiation, as they seek to further differentiate under this dimension in order to soften price competition. The formal analysis of the equilibrium outcome under a black market duopoly is presented in Appendix A.3.

2.2 First-Best Decentralization

Decentralization of the optimal allocation will require individualized prices that induce consumers to make the “right” quality choices. This amounts to choose the prices p L and p H in such a way that the condition for optimal splitting in (5) is satisfied. Combining the expressions for the indifferent consumer in (2) and the optimal splitting condition in (5), we have that:

θ ̂ = ( 1 + c ) q L + q H 2 ,

while the optimal prices are characterized by the following condition:

p H FB − p L FB = ( 1 − β + c ) q H 2 − q L 2 2 .

From the condition above we see that what matters for the optimal splitting between consumers is the price difference between the two available qualities. The optimal allocation can be decentralized by making this price difference equal to the sum of the differences in marginal production costs and in the fraction of health damage that is misperceived. This can be achieved, for instance, by setting prices:

p L = ( 1 − β + c ) q L 2 2   and   p H = ( 1 − β + c ) q H 2 2 ,

together with the following individualized transfers to have budget balance:

T L = ( 1 − β ) ( θ ̂ − θ ̲ ) q L 2 2   and   T H = ( 1 − β ) ( θ ̄ − θ ̂ ) q H 2 2 .

If the government were to be restricted to uniform lump sum transfers, budget balance could be achieved instead by setting:

T = ( 1 − β ) ( θ ̂ − θ ̲ ) q L 2 + ( θ ̄ − θ ̂ ) q H 2 2 ,

what would imply a monetary transfer from consumers with low to those with high taste for THC.

3 Mixed Duopoly

Consider now that the government participates in the market for cannabis through a public firm, competing in prices and qualities with a black market firm. In our framework, we define a black market firm as a profit-maximizing firm who does not account for the fact that consumers suffer from misperception of the health damage caused by THC and whose profits have a negative value for society. Regarding the timing, we begin by considering a situation where firms first choose simultaneously which quality to provide and then compete in prices à la Bertrand. Finally, we introduce a first-mover advantage to the public firm in the quality selection stage. We solve the games by backward induction, where the solution concept is subgame perfect Nash equilibrium.

3.1 Simultaneous Decisions

Since the black market firm and the government have different objectives, whether a firm serves the lower or upper-end of the demand will be of relevance for our analysis. There are two possible scenarios: either the public firm supplies a product with higher or lower quality than the black market firm.[18] We will present both cases separately and then proceed to compare the equilibrium outcomes.

3.1.1 Public Firm Supplies the Low Quality Product

Let’s begin by considering that the government offers a lower quality than the black market. This scenario may arise, for instance, when the government is faced with some ethical, legal or political restriction, that constraints the amount of THC to be offered in the legal market.[19]

Assuming that the whole market is covered and restricting our attention to the situation where both firms are active, the government will serve consumers with a lower valuation for THC than the indifferent consumer, while the black market firm will serve the rest. Consequently, demands for the black market firm and the public firm have respectively the following expressions:

D B ( p B , p G , q B , q G ; β ) = θ ̄ − p B − p G q B − q G + β q B + q G 2 , D G ( p B , p G , q B , q G ; β ) = p B − p G q B − q G + β q B + q G 2 − θ ̲ .

The objective of the black market firm is to maximize profits:

(8) Π B = p B − c q B 2 2 D B .

For our analysis, we define welfare as the sum of consumer surplus and profits made by the public firm, while we assign a negative social valuation to profits made by the black market firm that will be captured by the parameter λ > 0, in order to emphasize the damage that the presence of a black market causes to society through crime and violence.[20]

The objective of the government is to maximize welfare, subject to its resource constraint:

max p G , q G , T W = ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q B − ( 1 + c ) q B 2 2 d θ − ( 1 + λ ) p B − c q B 2 2 D B   s . t . p G − c q G 2 2 D G − T ≥ 0 .

The welfare function W has been expressed as the sum of the social surplus that corresponds to the trade of the low and high quality product, accounting for the “true” health damage caused by THC (i.e. with β = 1), while accounting for the welfare loss associated to the profits generated in the black market (weighted by −λ). Notice that due to the fact that prices are not anymore just transfers between agents the government cares for equally, black market profits appear in our welfare expression weighted by −(1 + λ). The resource constraint considered allows for the public firm to make a loss, as long as it can be covered through a lump sum tax to be imposed on consumers.[21]

In the second stage of the game, the black market firm and the government choose simultaneously their prices to maximize profits and social welfare respectively, for any given qualities q B and q G. The FOCs associated to the objective of the black market firm and the government are given respectively by:

∂ Π B ∂ p B = 2 ( p G − 2 p B ) + 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 + c q B 2 2 ( q B − q G ) = 0 , ∂ W ∂ p G = − 2 ( λ p B + p G ) − ( 1 − β ) ( q B 2 − q G 2 ) + c q G 2 + λ q B 2 2 ( q B − q G ) = 0 .

Solving for p B and p G, we get the following best reply correspondences:

(9) p ̄ B = c q B 2 + 2 p G + 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 4 ,

(10) p ̄ G = c q G 2 − ( 1 − β ) ( q B 2 − q G 2 ) − λ 2 p B − c q B 2 2 .

The first term in the expressions above refer to the marginal production costs per unit of THC. The second term of expression (9) reflects the strategic interaction at the price setting stage. For given qualities, an increase in the price of the public firm will result in a higher price set by the black market firm, thus exhibiting a pattern of complementarity. The third term shows how the quality differentiation allows the black market firm to charge a higher price. The last term accounts for the fact that consumers are less willing to pay for a higher quality, as they acknowledge part of the health damage caused by higher THC levels.

The second term in (10) is a Pigouvian component, which is negative (q B > q G), so that as long as there is some misperception of the health damage caused by cannabis consumption in the population, the government would like to lower its price in order for consumers to internalize the misperceived difference in the health damage caused by the THC content of both products. The last component tell us that the government will reduce its price proportionally to the degree of undesirability of black market profits. These last two terms imply pricing below marginal costs, what requires for a lump sum tax to achieve budget balance. This is a remarkable difference with respect to the price that would decentralize the first-best allocation in our normative setting, as in that case the government increased its price above marginal cost in order for consumers to internalize the health damage caused by cannabis consumption and split them optimally between the two available qualities. The price set by the government appears as a useful tool to attract demand towards the public firm and fight black market activity, as well as to correct for the misperception of the health damage caused by cannabis consumption.

Combining the best reply correspondences for the black market firm and the public firm in the price subgame given by (9) and (10), results in the following Nash equilibrium prices:

(11) p B * ( q B , q G ) = c ( q G 2 + ( 1 + λ ) q B 2 ) + 2 θ ̄ ( q B − q G ) − q B 2 − q G 2 2 ( 2 + λ ) ,

(12) p G * ( q B , q G ) = 2 c q G 2 − 2 ( 1 − β ) ( q B 2 − q G 2 ) − λ 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 − c q B 2 2 ( 2 + λ ) .

Before moving to the first stage of the game and derive the optimal qualities, let’s analyze the implications for the utility of consumers of the pricing strategy followed by both firms. For a consumer of type θ, we have that the utility difference between buying from the black market firm or from the public firm is the following:

(13) U B β p B * , q B − U G β p G * , q G = θ ( q B − q G ) − ( 1 + c ) q B 2 − q G 2 2 − ( 1 + λ ) ( θ ̄ − θ ) ( q B − q G ) 2 + λ .

The first two terms of the numerator capture the utility difference from buying the high versus the low quality product without misperception, that is, considering the differences in the true health damage and in marginal production costs. The last term accounts for the additional distortion that arises due to the undesirability of black market profits. The public firm distorts its price downwards in order to steal more consumers from the black market. This reduction is decreasing in the taste for quality and it becomes zero for consumers with the highest marginal willingness to pay for quality θ ̄ . Notice also from expression (13) that the equilibrium prices are such that the misperception of health damage is corrected, as individuals take into account the true (rather than the perceived) difference in health damage caused by the two products. This is due to the fact that what matters for consumers’ choice is the price difference. Optimal splitting is not possible to achieve anymore due to the fact that black market profits are socially undesirable. Indeed, plugging in the equilibrium prices in (11) and (12) into the expression for the indifferent consumer in (2) and rearranging terms we get:

(14) θ ̂ * = ( 1 + c ) q B + q G 2 + 1 + λ 2 + λ θ ̄ − ( 1 + c ) q B + q G 2 .

Taking the difference of the indifferent consumer above in (14) with respect to the optimal splitting condition in (5), we have that:

θ ̂ * − θ ̂ ̃ = 1 + λ 2 + λ θ ̄ − ( 1 + c ) q B + q G 2 > 0 .

The expression above captures the deviation from the optimal splitting that arises due to the fact that black market profits entail a welfare loss. The higher it is the undesirability of black market profits, the more the government will distort its price to attract more consumers to the public firm. This deviation is positive when the government supplies the low quality product, thus attracting more consumers with higher willingness to pay for quality than what it would be optimal under our first-best scenario.[22] We then have the following result:

Proposition 3.1

As long as black market profits entail a welfare loss, the first-best allocation cannot be decentralized by the direct participation of the government through a public firm in the market for cannabis.

For the result above, it is irrelevant whether the public firm serves the consumers with higher or lower willingness to pay for quality, as it is driven by the inefficiency caused by the undesirability of black market profits.

Let’s now proceed to analyze the second stage of the game. We first plug in the equilibrium prices in (11) and (12) into the objective functions of the black market firm and the government, to then solve for their respective qualities. The FOCs are as follows:

∂ Π B ∂ q B = [ 2 θ ̄ + ( 1 + c ) ( q G − 3 q B ) ] [ 2 θ ̄ − ( 1 + c ) ( q B + q G ) ] 4 ( 2 + λ ) 2 = 0 , ∂ W ∂ q G = 1 8 ( 2 + λ ) 2 × − 4 θ ̄ 2 + 8 θ ̄ q G ( 1 + c ) + ( 1 + c ) 2 q B 2 − 2 q B q G − 3 q G 2   + 4 ( 2 + λ ) 2 ( 1 + 2 ( θ ̲ − q G ( 1 + c ) ) ) = 0 .

Solving the system of equations above yields the following equilibrium qualities:

(15) q B * = 8 θ ̄ − 3 ( 2 + λ ) 2 + ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) ,

(16) q G * = 8 θ ̄ − 9 ( 2 + λ ) 2 + 3 ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) .

The equilibrium when both firms compete simultaneously, first in qualities and then in prices, is characterized by expressions (11), (12), (15) and (16). From the expression in (16) we see that the quality of the public firm is increasing in the degree of undesirability of black market profits. The public firm increases its market participation at the expense of deviating from the social optimum qualities. By doing so, it triggers a strategic increase in the quality of the black market firm, in order to extract higher rents through more differentiation.[23] As a result, as long as black market profits entail a welfare loss, the resulting equilibrium qualities are higher than our first-best qualities.

While the inefficiency related to health damage misperception is only addressed with one instrument, the government makes use of both prices and qualities to fight black market activity. Once the optimal splitting is distorted due to the undesirability of black market profits, so are the qualities that maximize welfare. In comparison with a black market duopoly, the presence of a public firm results in less quality differentiation, as now only the black firm differentiates its quality with the goal to soften competition.

3.1.2 Public Firm Supplies the High Quality Product

We now turn to the case where the public firm serves the consumers with higher willingness to pay for quality. In order to determine the optimal prices and qualities we follow the same procedure as for the previous case, presenting here the main results.[24]

For the price subgame, the best reply correspondences are now the following:

p ̄ B = c q B 2 + 2 p G − 2 θ ̲ ( q G − q B ) + β q G 2 − q B 2 4 , p ̄ G = c q G 2 + ( 1 − β ) ( q G 2 − q B 2 ) − λ 2 p B − c q B 2 2 .

From the expressions above we see that while the government has the same best reply function than when it offers the low-quality product, the black market firm now adapts its pricing strategy to the fact that it serves consumers with lower willingness to pay for quality. From the best reply correspondences above we get the following Nash equilibrium prices:

(17) p B * * ( q B , q G ) = c ( q G 2 + ( 1 + λ ) q B 2 ) − 2 θ ̲ ( q G − q B ) + q G 2 − q B 2 2 ( 2 + λ ) ,

(18) p G * * ( q B , q G ) = 2 c q G 2 + 2 ( 1 − β ) ( q G 2 − q B 2 ) − λ − 2 θ ̲ ( q G − q B ) + β q G 2 − q B 2 − c q B 2 2 ( 2 + λ ) .

Plugging in the prices above into the objective functions of the black market firm and the government and solving for the optimal qualities, leads to the following equilibrium qualities:

(19) q B * * = 8 θ ̲ + 3 ( 2 + λ ) 2 − ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) ,

(20) q G * * = 8 θ ̲ + 9 ( 2 + λ ) 2 − 3 ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) .

When the government supplies the high quality product, the resulting equilibrium is characterized by expressions (17)–(20).[25]

The main findings from the characterization of the two equilibria that arise when the black market firm and the public firm engage in simultaneous competition, first in qualities and then in prices, can be summarized in the following proposition:

Proposition 3.2

By reducing its price, the government is able to successfully reduce black market participation. When the public firm supplies the low (high) quality product the resulting allocation is one where both qualities are higher (lower) than our first-best allocation. These quality deviations increase with the degree of undesirability of black market profits.

Given that for the black market firm prices are strategic complements, a price reduction by the government not only brings more consumers to the legal market, but also triggers a price reduction of the black market firm, reducing its markup for given qualities. With respect to the first-best allocation, unlike the previous case where the government also increased its quality in order to attract more demand, now in turn it reduces its quality, triggering a reduction in the quality offered by the black market firm, given the strategic complementarity.

3.2 Discussion of Equilibrium Outcomes

Table 1 presents the equilibrium outcomes, where for ease of presentation we have denoted Δ = 9(2 + λ)2 − 8.

Table 1:

Equilibrium outcomes.

Public firm supplies low quality Public firm supplies high quality
q B 8 θ ̄ − 3 ( 2 + λ ) 2 + ( 2 + λ ) Δ 8 ( 1 + c ) 8 θ ̲ + 3 ( 2 + λ ) 2 − ( 2 + λ ) Δ 8 ( 1 + c )
q G 8 θ ̄ − 9 ( 2 + λ ) 2 + 3 ( 2 + λ ) Δ 8 ( 1 + c ) 8 θ ̲ + 9 ( 2 + λ ) 2 − 3 ( 2 + λ ) Δ 8 ( 1 + c )
|q G − q B| 3 ( 2 + λ ) 2 − ( 2 + λ ) Δ 4 ( 1 + c ) 3 ( 2 + λ ) 2 − ( 2 + λ ) Δ 4 ( 1 + c )
D B 3 ( 2 + λ ) − Δ 4 3 ( 2 + λ ) − Δ 4
D G − 3 λ − 2 + Δ 4 − 3 λ − 2 + Δ 4
ΠB ( 2 + λ ) [ 3 ( 2 + λ ) − Δ ] 3 64 ( 1 + c ) ( 2 + λ ) [ 3 ( 2 + λ ) − Δ ] 3 64 ( 1 + c )
ΠG ( 2 + λ ) ( − 3 λ − 2 + Δ ) 32 ( 1 + c ) 2 − 2 θ ̄ ( 1 − β ) ( 6 + 3 λ − Δ ) ( 2 + λ ) ( − 3 λ − 2 + Δ ) 32 ( 1 + c ) 2 2 θ ̲ ( 1 − β ) ( 6 + 3 λ − Δ )
+ ( 2 ( 1 − β ) − λ ( c + β ) ) ( 9 λ 2 + 36 λ + 32 − ( 6 + 3 λ ) Δ ) + ( 2 ( 1 − β ) − λ ( c + β ) ) ( 9 λ 2 + 36 λ + 32 − ( 6 + 3 λ ) Δ )
W 32 θ ̄ θ ̲ + ( 2 + λ ) ( − 9 ( 3 λ 3 + 18 λ 2 + 32 λ + 16 ) + Δ 3 / 2 ) 64 ( 1 + c ) 32 θ ̄ θ ̲ + ( 2 + λ ) ( − 9 ( 3 λ 3 + 18 λ 2 + 32 λ + 16 ) + Δ 3 / 2 ) 64 ( 1 + c )

We observe that the two equilibria are symmetric, in the sense that they both yield the same market shares, black market profits and welfare levels. This symmetry breaks if one were to remove the assumption that demand is fixed or that marginal production costs are equal among firms.

Intuitively, any policy that increases production costs of the black market firm relative to the public firm improves welfare, provided that the costs of implementing this policy are not too high. More interestingly, if the legal and illegal supplier face different production costs, efficiency gains call for allocating the production of the high quality product to the firm with the lowest production costs. A thorough analysis of the impacts of considering asymmetric costs is presented Appendix A.2.

When relaxing the assumption that demand is fixed, the government faces an additional trade-off: by reducing its price not only it attracts more consumers to the public firm, but it may also attract new consumers to the market. This new dilemma favo urs the equilibrium where the government supplies the high quality product, resulting in a lower participation of consumers than what would be optimum, but in higher welfare levels. A more detailed analysis of the consequences of relaxing the assumption that the demand is fixed is presented in Appendix A.1.

Notice that while the quality difference between both equilibria in Table 1 is the same, when the government offers the low quality product, the resulting equilibrium qualities are higher. The economic result of the public firm also differs and together with the resulting health damage to the population, they are two relevant features that policy makers may take into account to decide between the two possible equilibrium configurations. We discuss these two outcomes next.

3.2.1 Profits of the Public Firm

Taking the difference of the profits of the public firm when it supplies the low and high quality product we have that:

Π G * − Π G * * = ( 1 − β ) ( 1 + 2 θ ̲ ) ( 2 + λ ) ( 40 + 6 λ ( 10 + 3 λ ) − 2 ( 4 + 3 λ ) Δ ) 16 ( 1 + c ) 2 < 0 .

From the expression above we observe that the public firm achieves a better economic result when supplying the product with higher THC content. The difference between the economic result of the public firm when supplying the low and quality product is increasing in the degree of misperception in the population. Given that consumers misperceive the health damage caused by THC, when supplying the low quality product the government finds optimal to reduce its price in order to deter consumers from purchasing the higher THC content product. For the same reason, when supplying the high quality product the government finds optimal to increase its price, improving its markup. From the comparison above, we have the following result:

Proposition 3.3

As long as there is some misperception in the population, the public firm has a better economic result when it supplies the high quality product. This difference becomes more relevant, the higher the degree of misperception in the population.

It is worth mentioning that if at least one of the two inefficiencies that motivate government intervention is present, the public firm will set its price below marginal costs when supplying the low quality product, thus resulting in a loss for the public firm. On the other hand, the public firm will make positive profits when it supplies the high quality product, as long as the undesirability of black market profits is not too high.

3.2.2 Health Damage

Equilibrium qualities are higher when the government supplies the low quality product, i.e. q G * > q B * * and q B * > q G * * . Since it is always the public firm who serves more consumers, a priori it is not clear which equilibrium is more beneficial in terms of the average health damage caused to the population.

The average health damage H is given by:

H ( q L , q H ) = D L × h ( q L ) + D H × h ( q H ) .

Comparing the average health damage resulting from the two equilibria derived, we have that:

(21) H q G * , q B * − H q B * * , q G * * = ( 1 + λ ) ( 1 + 2 θ ̲ ) 9 ( 2 + λ ) 2 − 4 − 3 ( 2 + λ ) Δ 8 ( 1 + c ) 2 > 0 .

We then have the following result:

Proposition 3.4

The equilibrium in which the government supplies the high THC content product results in a lower average health damage with respect to the situation where it supplies the low THC content product.

Notice from expression (21) that when λ = −1, the total health damage is the same in both equilibria. This is because when black market firm profits do not imply a welfare loss, we are back to the scenario where the government is able to decentralize the first-best allocation.

From a welfare point of view, the government should be indifferent between either of the two equilibria described in this section. In practice, the one where the public firm serves consumers with higher willingness to pay for quality results in a lower average health damage caused to the population, together with a better economic result for the public firm, making it from a political point of view more appealing. On the other hand, for the government to supply cannabis with high THC content may raise some ethical issues or face some political constraints. Which equilibrium would be more feasible will depend, among other things, on how policy makers resolve these trade-offs.

3.3 Public Firm has a First-Mover Advantage on Quality Selection

In reality, the black market firm is already present when the government takes the decision to enter in the market for cannabis. Nevertheless, the higher commitment and credibility that the government has over the black market ought to be interpreted as if the government were to have a first-mover advantage in the quality selection stage of the game. In other words, the government is strong enough such that once it has announced that it will participate in the market with a certain quality for its product, the black market will take this information into account and adapt its strategy to the new legalized scenario. Accordingly, in this section we modify the timing of the game by introducing a first-mover advantage on the public firm in the quality selection stage of the game. Since prices can be easily adjusted, in the second stage we consider that both firms compete simultaneously in prices as before.

Let’s consider that the public firm supplies the low quality product. As nothing changes in the second stage of the game, the optimal prices conditional on qualities are given as before by expressions (11) and (12). In the second stage of the game nothing changes either for the black market firm. On the other hand, the government now observes the best reply function of the black market firm and taking into account this new information maximizes its objective function given now by:

W p G * , p B * , q ̄ B , q G = ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q ̄ B − ( 1 + c ) q ̄ B 2 2 d θ − ( 1 + λ ) p B * − c q ̄ B 2 2 D B ,

where q ̄ B = q G 3 + 2 θ ̄ 3 ( 1 + c ) .

From FOC:

∂ W ∂ q G = 9 ( 2 + λ ) 2 [ 1 + 2 θ ̲ − 2 q G ( 1 + c ) ] − 8 [ q G ( 1 + c ) − θ ̄ ] 2 18 ( 2 + λ ) 2 = 0 .

Solving for q G yields the same equilibrium quality for the public firm than when qualities were chosen simultaneously, given by expression (16). Since the best reply function of the black market firm and the equilibrium prices are also the same, we arrive to the same equilibrium than when we had simultaneous competition in qualities. The same holds true for the case where the government supplies the high quality product.[26] We then have the following result:

Proposition 3.5

Adding a first-mover advantage to the public firm at the quality selection stage does not improve welfare with respect to a situation where qualities were chosen simultaneously.

In the context of a mixed duopoly under vertical differentiation, the analysis of Grilo (1994) shows that direct public intervention in the market suffices to decentralize the social optimum allocation. In our framework, the fact that black market profits generate a welfare loss for society makes the decentralization of the first-best allocation unfeasible. However, direct public intervention is sufficient to achieve the welfare-maximizing allocation constrained to the fact that black market profits generate a welfare loss and thus adding a first-mover advantage to the public firm does not help to improve welfare. The intuition for this result lies in the fact that the public firm already internalizes how its quality choice ends up determining both equilibrium qualities. Nonetheless, a first-mover advantage for the public firm will guarantee that out of the two possible equilibria, the most desired one is attained.

4 Policy Implications

An important lesson from our analysis is that if the government is really concerned on the negative effects that the presence of a black market may have on society, it should sell its product at a relatively low price, and even at a loss if the amount of THC on the legal product is below what consumers may find by resorting to illegal suppliers. This becomes particularly relevant, when for some political or ethical reasons, the legal product faces a cap on its THC content. Regarding the amount of THC to be offered, when the public firm supplies the high quality product, it will find optimal to reduce the amount of THC offered with respect to a first-best situation, what in turn will reduce the amount supplied by the black market, given the strategic complementarity among THC levels. On the other hand, when supplying the low quality product, it will find optimal to increase the amount of THC content with respect to a first-best situation, in order to steal more consumers from the black market. Paradoxically, introducing a cap on the THC offered in the legal market may result in a higher health damage to the population, even when its price is relatively low.

In terms of the equilibria derived from our analysis, the one where the public firm supplies a product with low psychoactive content resembles the strategy followed upon legalization in Uruguay, where the public supplier offers cannabis with relatively low THC content and where cannabis is sold at a very competitive price (1.9 USD per gram). This has led to an estimated reduction of black market profits of 30 million USD since cannabis has been legalized in 2014, according to the IRCCA. Still, revenues generated from the legal sale of cannabis are not enough to cover for the costs incurred by this public agency, resulting in a net loss for tax payers. While the low amount of psychoactive content supplied by the legal source is the main driver for the blooming of the black market in Uruguay, the main factor driving the purchases of cannabis from an illegal source in Canada is its price, which is estimated to be almost half than the legal alternatives (Donnan et al. 2022).[27] Although there are no legal limits for the THC content on legal cannabis in Canada, recent evidence suggest that consumers may find higher potency cannabis in the black market (see Mahamad et al. 2020). The case of Canada seems to indicate that the tax revenue motive may be also playing an important role in the design of the optimal policies towards harmful drugs, regardless whether cannabis is provided through a provincial monopoly as in Quebec or through licensed stores like in Alberta.

To keep the problem tractable, in our exposition we have considered the demand for cannabis to be fixed. Preliminary evidence suggests that cannabis legalization in Uruguay was not associated with changes in the prevalence of adolescent cannabis use or self-reported frequency of use (Laqueur et al. 2020), while the study of Cerdá et al. (2020) find a possible increase in the risk for cannabis use disorder among adolescent and adult users after legalization of recreational cannabis in the US. In Canada, the prevalence of cannabis use after legalization remained unchanged for those between 18 and 24 years old, while it increased for those aged 25 and older (Rotermann 2021). Relaxing this assumption brings a new trade-off to the table, as government intervention to reduce black market participation and to correct for the misperceived difference in the health damage, may come at the cost of attracting consumers whose participation may be socially undesirable. If this is indeed the case, the idea that for the government to provide the high quality product seems the best strategy to follow is strengthened, as now not only a lower average health damage together with a better economic outcome for the public firm are achieved, but also higher welfare levels. If instead cannabis legalization attracts additional consumers whose presence is socially desirable, and who are no different than those who previously bought from the black market, then the government should keep the same pricing policy as if demand was fixed, but it ought to adjust its quality towards the taste of the average consumer.

Our analysis focused on a situation where both firms had the same production costs. In reality, the black market firm faces additional costs associated to operating in an illegal environment such as sanctions and seizures, while legal production must incur in idiosyncratic costs such as quality testing and traceability. Relaxing this assumption, we find that any policy that increases marginal production costs of one firm with respect to its rival will translate in higher equilibrium prices, due to the strategic interaction between firms, and in a higher demand for the product offered by the firm who is now faced with relative lower marginal production costs. In this sense, increasing enforcement activities that target black market supply or reducing the costs associated to quality and traceability controls for legal production will increase the amount of consumers in the legal market and reduce black market profits. Another interesting implication of considering asymmetric costs steams from the fact that, since the objective of the government is to maximize welfare, the welfare-maximizing allocation will be such that the most efficient firm supplies the high quality product, given our assumption that marginal production costs are increasing in quality.

5 Conclusion

Motivated by the recent wave of cannabis legalization in several jurisdictions, we have analyzed the optimal policies for the government to undertake when directly participating in a vertically differentiated market, characterized by the undesirable presence of a black market firm and with the peculiarity that a higher quality of the product increases both the joy from consumption and the health damage it causes, where the latter is misperceived by consumers.

We find two possible equilibrium configurations depending on whether the government supplies a product with lower or higher psychoactive content than the one supplied by the black market. These equilibria are symmetric, in the sense that they yield the same market shares, black market profits and welfare levels. However, they differ in the profits made by the public firm and in the resulting average health damage. Paradoxically, the average health damage will be lower if the public firm supplies the high quality product. We also find that as long as there are no constrains on the economic result of the public firm, introducing a first-mover advantage for the public firm at the quality selection stage does not help to improve welfare, as simultaneous competition already achieves the constrained welfare-maximizing allocation.

Our analysis allowed us to understand the role of the price and quality choices made by the public firm in order to correct for the misperception of health damage and to fight black market activity. With these instruments, as long as black market profits entail a welfare loss, the first-best allocation cannot be restored. Provided that the public firm faces no constraints on the price it can set, this instrument is sufficient to effectively correct for the health damage misperception. It is the strategic interaction between the black market firm and the public firm what results in the government being able to correct for the health damage misperception by only setting its own price. The government will distort its price in order to attract more demand, departing from the optimal splitting between qualities as a consequence of the undesirability of black market profits. Moreover, it will also deviate from first-best qualities in an effort to steal more demand from the black market.

There are several policy implications that arise from our analysis. First, the price of the public firm should be set in accordance to how undesirable is the presence of black market activity and it should also be adjusted to take into account that consumers misperceive the health damage caused by the amount of THC they consume. This seems to be in line with the approach followed by the Uruguayan government, who sells its product at a very competitive price but in contrast with introducing high taxes on legal cannabis sales, as it is the case in the US or Canada. Second, the THC content offered by the public firm should also be adjusted in accordance to how important it is for the government to diminish the presence of the black market. Finally, we find that when the public firm provides the high THC content product, a lower average health damage is achieved, together with a better economic result for the public firm. Furthermore, it may also result in higher welfare levels if legalization happens to attract consumers whose participation is socially undesirable. This approach is in sharp contrast with the quality strategy followed by the Uruguayan government, and more in line with the high potency cannabis offered by legal suppliers in Canada or in the US States where cannabis is legal.

While our analysis considered the case where the government competes directly with the black market, alternatively, one could address the case of a regulated supplier and focus on the optimal tax policy that the government should set to maximize welfare. In this aspect, the analysis of Cremer, Goulão, and Lozachmeur (2019) shows how market power and misperception of the health damage caused by a harmful good interact in the design of the optimal tax policy. Regarding the market for cannabis, an additional tool for the regulator would be to set, for instance, a cap (or a floor) on the THC content for the regulated firm.

Our main results were derived restricting our attention to situations where the heterogeneity in the taste for quality is such that both firms are active in equilibrium. We have focused on THC as a proxy for quality and it has been considered as the main driver for health damage, though the quantity consumed of this harmful drug is also relevant from a public health perspective. Relaxing the assumption that consumers only buy one unit of the harmful drug would enrich the analysis, as it would allow us to understand how variations in quality may affect the total quantity demanded, though this is beyond the scope of this paper. We have also abstracted from redistributional considerations, that may arise, for instance, due to differences in health damage perception, what may constitute an interesting avenue for future research.

Corresponding author: Luis Rodrigo Arnabal Rocca, Banco Central del Uruguay, J.P. Fabini 777, 11100 Montevideo, Uruguay; and Toulouse School of Economics (TSE), Toulouse, France, E-mail:

Funding source: H2020 European Research Council

Award Identifier / Grant number: 669217

Acknowledgments

I would like to thank L. Abreu, F. Barigozzi, N. Bonneton, P. Bontemps, C. Canta, H. Cremer, C. Goulão, A. Grillo, J.M. Lozachmeur, P. Pestieau, as well as attendants to the 25th Annual ENTER Jamboree (Tilburg) and the 4th Annual Workshop of the LACEA’s Health Economics Network for helpful comments and discussions.

  1. Research funding: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 669217).

  2. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

  3. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Appendix A

A.1 Demand Expansion

We first consider the case where legalization attracts a new batch of consumers who purchase exclusively from the public firm. Then, we analyze the impact of relaxing the assumption that the marginal willingness to pay for quality is such that participation of all consumers is socially desirable.

A.1.1 A New Batch of Consumers

Consider now that the black market firm and the government compete for a mass γ ∈ (0, 1) of consumers uniformly distributed between [ θ ̲ , θ ̄ ] ∈ R + , with θ ̄ = θ ̲ + 1 . Moreover, there is an additional mass of consumers 1 − γ with the same characteristics, that will exclusively buy from the public firm as a consequence of cannabis legalization. This parameter γ can be therefore interpreted as a measure of how easy it is to acquire cannabis. A situation where cannabis would be easy to come by, would depict a situation where γ is close to one, while if this parameter is close to zero, legalization makes cannabis available for a large number of consumers, that before found it very difficult to get access to.

We then consider a situation where demand for cannabis is divided in two groups: a group of size 1 − γ, that exclusively buy from the public firm (or otherwise do not buy the product at all) and a market of size γ, for which the black market firm and the public firm compete. Except for this new batch of consumers that demand exclusively from the public firm after legalization, the assumptions considered in the main text remain the same, in particular, that their participation is socially desirable.

Consider the case where the public firm supplies the low quality product. Demands faced by the black market firm and the public firm are given respectively by:

D ̂ B ( p B , p G , q B , q G ; β ) = γ ( θ ̄ − θ ̂ ) , D ̂ G ( p B , p G , q B , q G ; β ) = γ ( θ ̂ − θ ̲ ) + ( 1 − γ ) ( θ ̄ − θ ̲ ) .

where θ ̂ is the indifferent consumer in (2).

There is then a contestable market of size γ and a non-contestable market of size 1 − γ served by the government. The objective functions of the black market firm and the government are now given respectively by:

Π ̂ B = p B − c q B 2 2 D ̂ B , W ̂ = γ ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q B − ( 1 + c ) q B 2 2 d θ − ( 1 + λ ) p B − c q B 2 2 D ̂ B + ( 1 − γ ) ∫ θ ̲ θ ̄ θ q G − ( 1 + c ) q G 2 2 d θ .

In the second stage of the game, the black market firm and the public firm set their prices, for given qualities. Solving the problem results in the Nash equilibrium prices that correspond to expressions (11) and (12) in the main text. Let’s plug in these prices into the objective functions of the black market firm and the government to then solve for their respective qualities. The FOCs for the black market firm and the public firm are as follows:

∂ Π ̂ B ∂ q B = γ [ 2 θ ̄ + ( 1 + c ) ( q G − 3 q B ) ] [ 2 θ ̄ − ( 1 + c ) ( q B + q G ) ] 4 ( 2 + λ ) 2 = 0 , ∂ W ̂ ∂ q G = 1 8 ( 2 + λ ) 2 × γ − 4 θ ̄ 2 + 8 θ ̄ q G ( 1 + c ) + ( 1 + c ) 2 q B 2 − 2 q B q G − 3 q G 2   + 4 ( 2 + λ ) 2 ( 1 + 2 ( θ ̲ − q G ( 1 + c ) ) ) = 0 .

Solving the system of equations above yields the following equilibrium qualities:

q ̂ B * = 8 γ θ ̄ − 3 ( 2 + λ ) 2 + ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 γ 8 γ ( 1 + c ) , q ̂ G * = 8 γ θ ̄ − 9 ( 2 + λ ) 2 + 3 ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 γ 8 γ ( 1 + c ) .

The fact that legalization brings along a new batch of consumers to the public firm has an impact on the quality it offers, with the consequent strategic reaction of the black market firm. The public firm will now offer a higher quality than when there is no additional demand accruing from legalization. This is due to the fact that since the new batch of consumers have only access to the product offered by the government, the public firm must adjust its quality upwards, in order to better serve their taste. In response, the black market firm will strategically increase its quality.

For the alternative case where the government supplies the high quality product, the Nash equilibrium prices correspond to the expressions (17) and (18) in the main text.

Following the same procedure than before yields the following equilibrium qualities:

q ̃ B * * = 8 γ θ ̲ + 3 ( 2 + λ ) 2 − ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 γ 8 γ ( 1 + c ) , q ̃ G * * = 8 γ θ ̲ + 9 ( 2 + λ ) 2 − 3 ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 γ 8 γ ( 1 + c ) .

The main results of the analysis above can be summarized in the following proposition:

Proposition A.1

When legalization attracts a new batch of consumers with the same characteristics as those who were already in the market and who exclusively buy from the public firm:

  1. The pricing strategy of the public firm is not modified.

  2. The quality offered by the public firm is adjusted towards the one that maximizes the utility of the average consumer.

All in all, the impact of this new non-contestable consumers that enter the market upon legalization is on the quality set by the public firm, that is adjusted towards the quality that maximizes the utility of the average consumer: q M = θ ̄ + θ ̲ 2 ( 1 + c ) , what in turn triggers a strategic interaction on the quality offered by the black market firm.

A.1.2 Endogenous Participation Threshold

Let’s now relax the assumption that the marginal willingness to pay for quality is such that participation of all consumers is socially desirable, given by condition (1). Consider now that θ is uniformly distributed between [ 0 , θ ̄ ] ∈ R + . The consumers that participate in the market are those who get a (perceived) positive utility from consuming the low quality product and consists on those who have a marginal willingness to pay for quality higher than the participation threshold:

θ P ( p L , p H , q L , q H ; β ) = p L q L + β q L 2 ,

while participation will be socially desirable when above:

θ PS ( p L , p H , q L , q H ) = p L q L + q L 2 .

Consequently, demands for the low and high quality product are now given by:

D L ( p L , p H , q L , q H ; β ) = θ ̂ − θ P θ ̄ , D H ( p L , p H , q L , q H ; β ) = θ ̄ − θ ̂ θ ̄ .

The first-best allocation with endogenous participation constraint is given by:

q G FB = 2 θ ̄ 5 ( 1 + c ) , q B FB = 4 θ ̄ 5 ( 1 + c ) ;

while the optimal participation threshold is given by:

(22) θ ̃ P = ( 1 + c ) q L 2 .

The optimal participation threshold and the optimal splitting condition under our first-best scenario are now given respectively by: θ P FB = θ ̄ 5 and θ ̂ FB = 3 θ ̄ 5 .

Consider the case when the government supplies the low quality product. The problem faced by the government is now the following:

max p G , q G , T W P = ∫ θ P θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q B − ( 1 + c ) q B 2 2 d θ − ( 1 + λ ) p B − c q B 2 2 D B   s . t . p G − c q G 2 2 D G − T ≥ 0 .

In the second stage of the game, the black market firm and the government set their prices to maximize their objective functions.

The FOCs are given by:

∂ Π B ∂ p B = 2 ( p G − 2 p B ) + 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 + c q B 2 2 θ ̄ ( q B − q G ) = 0 , ∂ W P ∂ p G = − 2 ( λ p B q G + p G q B ) − ( 1 − β ) q G q B 2 − q B q G + c q G 2 q B + λ q B 2 q G 2 θ ̄ q G ( q B − q G ) = 0 .

Solving for p B and p G, we get the following best reply correspondences:

(23) p ̄ B = c q B 2 + 2 p G + 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 4 ,

(24) p ̄ G = c q G 2 − ( 1 − β ) ( q B q G − q G 2 ) 2 − λ q G q B p B − c q B 2 2 .

While the black market firm has the same best reply function than in our previous analysis, the public firm must now take into account that its pricing decision will have an impact not only on those consumers who have to decide whether to purchase the low or high quality product, but also on those who may decide not to buy any product at all. Indeed, the participation constraint affects directly the decisions of the firm that is serving the consumers with lower marginal willingness to pay for quality, what via the strategic interaction between firms affects the equilibrium outcomes. The new trade-off introduced by the endogenous participation constraint is that by increasing its quality and reducing its price not only does the government steal consumers from the black market, but it also attracts new consumers, whose participation may not be socially desirable. This trade-off becomes more relevant the higher the degree of misperception of the health damage caused by THC in the population.

Comparing the expression for the price setting of the public firm when supplying the low quality product in (24) with the one where demand was fixed in (10), it becomes clear that having an endogenous participation constraint mitigates the price reduction that the two motives for government intervention call for.

Solving the system given by conditions (23) and (24), we get the following Nash equilibrium prices when there is an endogenous participation constraint:

(25) p B * ( q B , q G ) = q B c ( q G 2 + q B ( q B + λ q G ) ) + 2 θ ̄ ( q B − q G ) − β q B 2 − q B q G − q B q G − q G 2 2 ( 2 q B + λ q G ) ,

(26) p G * ( q B , q G ) = 2 q B c q G 2 − ( 1 − β ) q B q G − q G 2 − λ q G 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 − c q B 2 2 ( 2 q B + λ q G ) .

Optimal splitting between qualities is not possible to achieve when considering an endogenous participation constraint, even when the motives that call for government intervention are not present. Moreover, the price set by the government is not able to correct for the misperception suffered by agents anymore. When the participation constraint was not binding, setting its price to correct for the misperceived difference in the health damage caused by the two different qualities was sufficient for consumers to internalize the true health damage caused by THC content. This is not anymore the case, as the government would also like to correct for the health damage misperception of those consumers who face the decision of whether to buy the low THC content product or not buy anything at all.

For prices (25) and (26) the resulting participation threshold is given by:

θ P * ( q B , q G ) = 2 ( 1 + c ) q B q G − 2 ( 1 − β ) q B 2 + λ q B 2 ( β + c ) − 2 θ ̄ ( q B − q G ) 4 q B + 2 λ q G .

Taking the difference of the expression above with the one given by the optimal participation constraint we have that:

θ P * − θ ̃ P = − 2 ( 1 − β ) q B 2 + λ q B 2 ( β + c ) − q G 2 ( 1 + c ) − 2 θ ̄ ( q B − q G ) 4 q B + 2 λ q G < 0 .

We then conclude that with an endogenous participation constraint, when the government supplies the low quality product, there will always be more consumers participating in the market than what would be optimum.

When the government supplies the high quality product, the best reply correspondences of the second stage of the game are given by:

p ̄ B = c q B 2 + β q G q B − q B 2 4 + p G 2 q B q G , p ̄ G = c q G 2 + ( 1 − β ) ( q G 2 − q B 2 ) − λ 2 p B − c q B 2 2 .

We observe that now the government has the same best reply function than in our previous analysis, while the black market firm now has an incentive to lower its price in order to attract more consumers into the market. Equilibrium prices when government supplies the high quality product are given by:

p B * * ( q B , q G ) = q B c q G 2 + q B q G + λ q B 2 + β q B 2 − q B q G + q G 2 − q B 2 2 ( 2 q G + λ q B ) , p G * * ( q B , q G ) = q G 2 c q G 2 + 2 ( 1 − β ) q G 2 − q B 2 − λ ( β q B q G − q B 2 − c q B 2 ) 2 ( 2 q G + λ q B ) ;

while the expression for the resulting participation threshold is given by:

θ P * * ( q B , q G ) = − ( 1 − β ) q B 2 + q G 2 ( 1 + c ) + q G q B ( β + c ) + λ q B 2 ( β + c ) 4 q G + 2 λ q B .

Taking the difference with the optimal participation threshold in (22) results in:

θ P * * − θ ̃ P = q G 2 − q B q G ( 1 + c ) − ( 1 − β ) ( 1 + λ ) q B 2 + q B q G 4 q G + 2 λ q B .

For reasonable values of the parameters considered in our analysis, the expression above is positive.

The results of the analysis above can be summarized in the following proposition:

Proposition A.2

When the government supplies the low (high) quality product, participation of consumers in the market for cannabis will be above (below) what would be optimum.

This implies that if the government supplies the low quality product due, for instance, to some political constraint, there will be more consumers than what would be optimum. The intuition behind this result lies in the fact that, for given qualities, the government will set its price below marginal costs, and therefore, below the optimal participation threshold.

When supplying the low quality product the government is now faced with the dilemma of whether to reduce its price, in order for consumers to internalize the difference in health damage caused by the high versus the low quality product, or rather to increase it, in order to deter consumers whose marginal willingness to pay for quality is sufficiently low to participate in the market. From the equilibrium price expression of the government in (26) it becomes clear that this dilemma is resolved in favour of internalizing the difference in health damage caused by the two available qualities.

Considering an endogenous participation constraint comes at the cost of not having tractable analytical solutions anymore. The equilibrium qualities are then solved numerically, for a given set of parameters. The equilibrium outcomes with and without an endogenous participation constraint are presented in Table 2.[28]

Table 2:

Numerical simulations of equilibrium outcomes with endogenous participation. Results expressed for parameters: λ = 1, θ ̄ = 1.35 , c = 1.

Public firm supplies low quality Public firm supplies high quality
Demand fixed Endogenous Demand fixed Endogenous
participation participation
β 0.25 0.75 0.25 0.75 0.25 0.75 0.25 0.75
p B 0.1932 0.1932 0.3745 0.2435 0.0534 0.0534 0.0444 0.0386
p G 0.0034 0.0465 0.0465 0.0822 0.1180 0.0883 0.1275 0.0967
q B 0.5895 0.5895 0.8194 0.6655 0.2605 0.2605 0.2088 0.1892
q G 0.4185 0.4185 0.5095 0.4674 0.4315 0.4315 0.4362 0.4417
θ p 0.35 0.35 0.1549 0.3511 0.35 0.35 0.2387 0.2751
θ ̂ 1.236 1.236 1.2247 1.2389 0.4640 0.4640 0.4463 0.4667
D B 11.40 11.40 10.48 11.12 11.40 11.40  18.68 17.83
D G 88.60 88.60 89.52 88.87 88.60 88.60 81.32 82.17
ΠB 0.0022 0.0022 0.0036 0.0018 0.0022 0.0022 0.0035 0.0029
ΠG −0.0745 −0.0364 −0.0660 −0.0178 −0.0220 −0.0042 0.0217 −0.0005
H 0.0974 0.0974 0.1340 0.0900 0.0863 0.0863 0.0670 0.0664
W 0.1817 0.1817 0.1012 0.1315 0.1817 0.1817 0.1322 0.1338

The equilibrium qualities that result from the numerical simulations in Table 2 do not vary significantly with the degree of undesirability of black market profits, while it does have a considerable impact on the price set by the public firm and consequently on its economic result. In the limit, it would be welfare improving to even subsidize the product offered by the public firm. On the other hand, if the motives for government intervention are of little significance, a higher willingness to pay for quality is required to have both firms active in equilibrium, otherwise, the public firm would find optimal to serve all consumers.

Notice also that with an endogenous participation constraint, as the government is unable to effectively correct for the misperception of health damage with its price, the degree of misperception now plays a more important role in the quality choice made by the public firm, affecting the resulting market shares, and more in general, having an impact on all equilibrium outcomes.[29] Introducing an endogenous participation constraint breaks the symmetry of the two equilibria described in the main text in terms of welfare, black market profits and market shares, due to the fact that the government is now unable to correct for the misperception of health damage in the population by setting its own price. Indeed, from our numerical simulations we observe that welfare and black market profits are higher when the government supplies the high quality product. In turn, the share of the public firm is lower than when it supplies the low quality product, while the public firm has a better economic result. The resulting average health damage is lower when the public firm supplies the high quality product, despite the fact that this equilibrium may result in more cannabis consumers, what crucially depends of the degree of misperception in the population. We then have the following proposition:

Proposition A.3

When considering an endogenous participation constraint, a higher welfare is achieved when the public firm supplies the high quality than when it supplies the low quality product, also resulting in a better economic outcome for the public firm, together with a lower average health damage. The difference in this last two outcomes becomes more relevant, the higher the degree of misperception in the population.

We conclude that the main results of our analysis with fixed demand hold true when considering an endogenous participation constraint. In particular, equilibrium qualities are higher (lower) than what it would be optimal when the government supplies the low (high) quality product. The equilibrium where the government supplies the high quality product still results in a lower average health damage for the population and in a better economic result for the public firm. On the other hand, introducing a participation constraint breaks the symmetry in terms of the resulting welfare levels of both equilibria, favouring the equilibrium where the government supplies the high quality product.

A.2 Asymmetric Production Costs

Throughout our analysis we have considered that the legal and illegal supplier had the same production costs. In reality, the black market firm faces additional costs associated to operating in an illegal environment, such as sanctions and seizures, while legal production must incur in idiosyncratic costs, such as quality testing and traceability. In order to take this into account, consider now that the marginal production costs of the public firm are a fraction α > 0 of those faced by the black market firm. Consequently, the profits of the public firm are now given by:

Π G α = p G − α c q G 2 2 D G .

Let’s proceed to study the implications of relaxing the assumption of symmetric costs when firms compete simultaneously, first in qualities and then in prices.

Consider the case when the public firm supplies the low quality product. In the second stage of the game, the black market firm and the public firm set their prices, for given qualities. The FOCs for their objective functions with respect to prices are given respectively by:

∂ Π B α ∂ p B = 2 ( p G − 2 p B ) + 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 + c q B 2 2 ( q B − q G ) = 0 , ∂ W α ∂ p G = − 2 ( λ p B + p G ) − ( 1 − β ) ( q B 2 − q G 2 ) + c α q G 2 + λ q B 2 2 ( q B − q G ) = 0 .

Solving the system above for p B and p G results in the following Nash equilibrium prices when firms have asymmetric production costs:

(27) p B α ( q B , q G ) = c ( α q G 2 + ( 1 + λ ) q B 2 ) + 2 θ ̄ ( q B − q G ) − q B 2 − q G 2 2 ( 2 + λ ) ,

(28) p G α ( q B , q G ) = 2 α c q G 2 − 2 ( 1 − β ) ( q B 2 − q G 2 ) − λ 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 − c q B 2 2 ( 2 + λ ) .

Taking the difference with the expressions of the Nash equilibrium prices with symmetric costs given by respectively by (11) and (12), for given qualities, we have that:

p B * − p B α = ( 1 − α ) × c q G 2 2 ( 2 + λ ) , p G * − p G α = ( 1 − α ) × c q G 2 ( 2 + λ ) .

We conclude that, for given qualities, if the public firm has higher (lower) marginal production costs relative to the black market firm, then both prices would be higher (lower) than those that result when costs are symmetric.

For prices in (27) and (28), the indifferent consumer is such that:

θ ̂ α ( q B , q G ) = ( 1 + c ) q B + ( 1 + α c ) q G 2 + 1 + λ 2 + λ θ ̄ − ( 1 + c ) q B + ( 1 + α c ) q G 2 + ( 1 − α ) × c q B q G 2 ( 2 + λ ) ( q B − q G ) .

Comparing with the expression for the indifferent consumer when considering symmetric costs in (2), for given qualities, we have that:

θ ̂ * − θ ̂ α = − ( 1 − α ) × c q G 2 2 ( 2 + λ ) ( q B − q G ) .

From the expression above we observe that, for given qualities, the firm who has higher marginal production costs will face a lower demand with respect to a situation where marginal production costs are equal. We have an analogous situation for the alternative case where the government supplies the high quality product. We then have the following result:

Proposition A.4

For any given qualities supplied by the black market firm and the public firm, a policy that results in reducing (increasing) the marginal production cost of the public firm relative to those of the black market firm results in:

  1. Lower (higher) prices.

  2. A higher (lower) market share of the public firm.

  3. A reduction (an increase) of black market profits.

As when we introduced an endogenous participation constraint, asymmetry in production costs results in the problem not having a tractable analytical solution. We therefore solve our model using numerical simulations for a given set of parameters, considering that participation of all consumers is socially desirable and restricting our analysis to situations where both firms are active in equilibrium. The equilibrium outcomes are presented in the table below, for specific values of the parameters considered in our model.[30]

The resulting equilibrium qualities from our simulation do not vary significantly neither for different values of the degree of misperception of the health damage caused by THC nor for the degree of undesirability of black market profits, while they both have an important impact on the profits of the public firm. When the importance of the aforementioned motives for government intervention are of great magnitude, the public firm may find optimal to subsidize its product.

For both equilibria described in Table 3, when the public firm has lower marginal production costs a higher welfare is achieved, together with a lower demand and profits for the black market firm. We observe that when the public firm has lower marginal production costs relative to those of the black market firm, it will always offer a higher quality than what it would under symmetric production costs. In turn, the black market firm will offer a higher (lower) quality when it is supplying the high (low) quality product.

Table 3:

Numerical simulations of equilibrium outcomes with asymmetric production costs. Results are expressed for parameters: λ = 1, β = 0.75, θ ̄ = 1.35 .

Public firm supplies low quality Public firm supplies high quality
α = 0.9 α = 1 α = 1.1 α = 0.9 α = 1 α = 1.1
c 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1
p B 0.1846 0.2034 0.1804 0.1932 0.1769 0.1851 0.0556 0.0520 0.0561 0.0534 0.0570 0.0558
p G 0.0143 0.0512 0.0135 0.0465 0.0127 0.0424 0.0825 0.0899 0.0830 0.0884 0.0832 0.0867
q B 0.8089 0.6160 0.7860 0.5895 0.7655 0.5670 0.3385 0.2510 0.3473 0.2605 0.3587 0.2744
q G 0.5784 0.4420 0.5580 0.4185 0.5393 0.3979 0.5948 0.4538 0.5753 0.4315 0.5572 0.4114
θ ̂ 1.2589 1.2713 1.236 1.236 1.2155 1.2060 0.4551 0.4513 0.4640 0.4640 0.4753 0.4825
D B 9.11 7.86 11.40 11.40 13.45 14.39 10.51 10.13 11.40 11.40 12.53 13.25
D G 90.89 92.14 88.60 88.60 86.55 85.61 89.49 89.87 88.60 88.60 87.47 86.75
ΠB 0.0019 0.0011 0.0030 0.0022 0.0041 0.0035 0.0028 0.0021 0.0030 0.0022 0.0031 0.0024
ΠG −0.0554 −0.0339 −0.0570 −0.0364 −0.0582 −0.0383 −0.0026 −0.0024 0.0003 −0.0042 −0.0019 −0.0056
H 0.1818 0.1049 0.1731 0.0974 0.1653 0.0909 0.1643 0.0957 0.1535 0.0863 0.1438 0.0784
W 0.2500 0.1906 0.2423 0.1817 0.2350 0.1737 0.2505 0.1911 0.2423 0.1817 0.2346 0.1732

The main results from our numerical simulations can be summarized in the following proposition:

Proposition A.5

Any policy that results in reducing (increasing) the marginal production cost of the public firm relative to the black market firm results in:

  1. A higher (lower) quality offered by the public firm.

  2. A higher (lower) market share of the public firm.

  3. A reduction (an increase) of black market profits.

  4. An ambiguous effect on the economic result of the public firm.

  5. A welfare increase (decrease).

While with symmetric costs both equilibria resulted in the same welfare levels, market shares and black market profits, having asymmetric production costs introduces an asymmetry in these outcomes. The government can achieve a higher welfare outcome by supplying the high quality product when it faces lower marginal production costs than the black market, otherwise it will be better off by supplying the low quality product. Comparing equilibrium outcomes in Table 3, leads to the following result:

Proposition A.6

When the public firm has lower (higher) marginal production costs relative to the black market firm, the equilibrium where it supplies the high (low) quality product results in higher welfare levels.

The intuition for this result is that there are now gains of efficiency if production of the high quality product is assigned to the firm with lower marginal production costs. Our analysis suggest that any policy that reduces marginal production costs of the public firm relative to those of the black market firm is welfare enhancing. The reader must bear in mind however that this analysis does not take into account the costs associated to such policy.

A.3 Black Market Duopoly

In a laissez-faire situation, the market for cannabis would be in the hands of the black market. In order to compare with our normative benchmark in Section 2.1, we consider that there are two black market firms competing for the market of cannabis.

Each firm maximizes profits, where without loss of generality, one firm offers the low quality product q L and the other one the high quality product q H. The profit function for a firm who chooses quality i is given by:

Π i ( p i , q i ; β ) = p i − c q i 2 2 D i ( p i , q i ; β )   with i = L,H .

where demands have the following expressions:

D L ( p L , p H , q L , q H ; β ) = p H − p L q H − q L + β q L + q H 2 − θ ̲ , D H ( p L , p H , q L , q H ; β ) = θ ̄ − p H − p L q H − q L + β q L + q H 2 .

The FOCs are given by:

∂ Π H p H = 2 p L − 4 p H + 2 ( 1 + θ ̲ ) ( q H − q L ) − β q H 2 − q L 2 + c q H 2 2 ( q H − q L ) = 0 , ∂ Π L p L = 2 p H − 4 p L − 2 θ ̲ ( q H − q L ) + β q H 2 − q L 2 + c q L 2 2 ( q H − q L ) = 0 .

Solving the system above for p L and p H, we get the following equilibrium prices of the second stage of the game:

p L PD ( q L , q H ) = 2 ( 1 − θ ̲ ) ( q H − q L ) + β q H 2 − q L 2 ) + c q H 2 + 2 q L 2 6 , p H PD ( q L , q H ) = 2 ( 2 + θ ̲ ) ( q H − q L ) − β q H 2 − q L 2 ) + c 2 q H 2 + q L 2 6 .

Let’s plug the equilibrium prices of the second stage game above into the profit functions and take the FOCs with respect to qualities:

∂ Π H q H = [ 4 + 2 θ ̲ + ( β + c ) ( q L − 3 q H ) ] [ 4 + 2 θ ̲ − ( β + c ) ( q L + q H ) ] 36 = 0 , ∂ Π L q L = [ 2 − 2 θ ̲ + ( β + c ) ( 3 q L − q H ) ] [ 2 − 2 θ ̲ + ( β + c ) ( q L + q H ) ] 36 = 0 .

Solving for the system above for q L and q H leads to the following equilibrium qualities:

q L PD = 4 θ ̲ − 1 4 ( β + c ) , q H PD = 4 θ ̲ + 5 4 ( β + c ) .

While the indifferent consumer is such that optimal splitting is achieved, under a black market duopoly the quality differentiation is higher. The respective distances between qualities are given by the following expressions:

q H FB − q L FB = 4 θ ̲ + 3 4 ( 1 + c ) − 4 θ ̲ + 1 4 ( 1 + c ) = 1 2 ( 1 + c ) , q H PD − q L PD = 4 θ ̲ + 5 4 ( β + c ) − 4 θ ̲ − 1 4 ( β + c ) = 3 2 ( β + c ) .

Consequently, compared with our normative benchmark, with a black market duopoly there is too much quality differentiation.

A.4 Restriction on the Profitability of the Public Firm

When the public firm supplies the low quality product, it finds optimal to set its price below marginal cost, thus making a loss. This may not be feasible, for instance, due to political reasons. Here we consider a situation where the public firm is constrained to make non-negative profits and we study the impact of this restriction on equilibrium qualities.[31]

The price restriction will be binding under the scenario where the public firm supplies the low quality product. The public firm will then price at marginal costs, while the black market firm will set its price according to (9) as before. We then have the following Nash equilibrium prices:

(29) p B r * ( q B , q G ) = c q B 2 + q G 2 + 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 4 ,

(30) p G r * ( q B , q G ) = c q G 2 2 .

Due to the fact that the government is constrained to set prices at marginal cost, the equilibrium prices of the second stage of the game do not depend anymore neither on the undesirability of black market profits nor on the misperception of health damage. The profitability constraint forces the public firm to set a higher price than what it would find optimal, what in turn triggers a strategic price increase of the black market firm, resulting in higher prices for both products, for any given qualities.

We will now proceed to the quality selection stage of the game. As in the main text, we first address the case where quality choice is made simultaneously and then when the public firm has a first-mover advantage.

A.4.1 Simultaneous Competition

In the first stage of the game the government solves the following problem:

max q G W = ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q B − ( 1 + c ) q B 2 2 d θ − ( 1 + λ ) p B − c q B 2 2 D B ,

while the black market firm maximizes profits as before.

Let’s plug in the expressions (29) and (30) into the profit function of the black firm given by (8) and in the expression for social welfare above and take FOCs with respect to their respective qualities:

∂ Π B ∂ q B = [ 2 θ ̄ + ( β + c ) ( q G − 3 q B ) ] [ 2 θ ̄ − ( β + c ) ( q B + q G ) ] 16 = 0 , ∂ W ∂ q G = 1 32 × 16 ( 1 + 2 θ ̲ ) − 4 θ ̄ 2 ( 1 − 2 λ ) + ( β + c ) q B 2 − 2 q B q G − 3 q G 2 ( − β ( 3 + 2 λ ) + c ( 1 − 2 λ ) + 4 ) + 8 θ ̲ q G   × ( − β ( 1 + 2 λ ) + c ( 1 − 2 λ ) + 2 ) − 2 q G ( 8 ( 1 + λ ( β + c ) ) + 4 β + 12 c ) = 0 .

Solving the system above yields the optimal qualities when the government is constrained to make non-negative profits and supplies the low quality product. The optimal qualities are given by the following expressions:

(31) q B r * = 1 4 ( β + c ) ( 4 − β ( 3 + 2 λ ) + c ( 1 − 2 λ ) θ ̄ ( 7 − 9 β + 4 c − 8 λ ( β + c ) ) + 6 θ ̲ − 6 c + 17 + 9 θ ̲ 2 ( 1 − β ) 2 − β 2 ( 16 λ + 15 ) + β ( 26 + 32 λ − 4 c )   − 2 θ ̲ ( 1 − β ) 5 − β ( 16 λ + 15 ) − 2 c ( 8 λ + 5 ) + 4 c ( 15 + 7 c + 4 λ ( 2 + c ) ) 1 / 2 ,

(32) q G r * = 1 4 ( β + c ) ( 4 − β ( 3 + 2 λ ) + c ( 1 − 2 λ ) θ ̄ ( 7 − 3 β + 4 c − 8 λ ( β + c ) ) − 18 ( 1 + c )   + 3 17 + 9 θ ̲ 2 ( 1 − β ) 2 − β 2 ( 16 λ + 15 ) + β ( 26 + 32 λ − 4 c )   − 2 θ ̲ ( 1 − β ) 5 − β ( 16 λ + 15 ) − 2 c ( 8 λ + 5 ) + 4 c ( 15 + 7 c + 4 λ ( 2 + c ) ) 1 / 2 .

The equilibrium when the public firm supplies the low quality product and is constrained not to make non-negative profits is characterized by Eqs. (29)–(32).

Both a higher degree of misperception of the health damage and a higher undesirability of black market profits increase the quality offered by the public firm. Compared with the unconstrained case presented in the main text, the public firm increases its quality, in an effort to correct both for the health damage misperception and the negative welfare impact of black market profits, what triggers a strategic increase of the quality offered by the black market firm, resulting in higher equilibrium qualities.

A.4.2 Stackelberg Competition

The second stage of the game remains the same, so that p B r * and p G r * are the optimal prices, conditional on quality choices. The black market firm solves the following problem:

max q B Π B p B r * , p G r * , q B , q G = p B r * − c q B 2 2 [ θ ̄ − θ ̂ ] .

For the black market firm nothing has changed, having the following best reply function:

q ̄ B r = q G 3 + 2 θ ̄ 3 ( β + c ) .

In the first stage of the game the government solves the following problem:

max q G W p B r * , p G r * , q ̄ B r , q G = ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q ̄ B r − ( 1 + c ) q ̄ B r 2 2 d θ − ( 1 + λ ) p B r * − c q ̄ B r 2 2 D B .

The FOCs are given by:

∂ W ∂ q G = 1 18 × 9 − 2 θ ̄ 2 ( − 2 λ + 1 ) + 2 θ ̲ ( 2 q G ( − β ( 2 λ + 1 ) + c ( 1 − 2 λ ) + 2 ) + 9 )   − 2 q G 2 ( β + c ) ( − β ( 2 λ + 3 ) + c ( 1 − 2 λ ) + 4 )   + 2 q G ( − β ( 4 λ + 2 ) − c ( 4 λ + 7 ) − 5 ) = 0 .

Solving condition above for q G yields the optimal quality for the public firm:

q G S * = 1 2 ( β + c ) ( 4 − β ( 3 + 2 λ ) + ( 1 − 2 λ ) c ) × − 2 θ ̄ ( 1 + ( β + c ) ( 1 + 2 λ ) ) − 3 − 5 c + 2 θ ̲ ( 3 + 2 c ) + 16 θ ̲ 2 ( 1 − β ) 2 + 4 θ ̲ ( 1 − β ) ( β ( 19 + 18 λ ) + 9 c ( 1 + 2 λ ) − 10 )   − 2 β 2 ( 19 + 18 λ ) + β ( 76 + 72 λ ) + 9 c ( 7 + 4 λ ) ( 2 + c ) + 25 1 / 2 .

The quality offered by the public firm when having a first-mover advantage will be lower than when quality choices were made simultaneously, and so will be the quality offered by the black market firm, given the strategic complementarity. This is true as long as consumers suffer from misperception of the health damage caused by cannabis consumption. If consumers fully perceive the damage caused by THC, that is, if β = 1, then adding a first-mover advantage does not help to improve welfare.

Comparing the outcomes with simultaneous and Stackelberg competition in qualities we have the following proposition:

Proposition A.7

If the public firm supplies the low quality product when constrained to make non-negative profits, then adding a first-mover advantage in the quality selection stage improves welfare, provided that consumers suffer from misperception of the health damage caused by THC.

The intuition behind proposition above is that as long as the government is not able to use its price to correct for the inefficiency related to the misperception of health damage, adding a first-mover advantage to the public firm helps to improve welfare by reducing both equilibrium qualities. By having a first-mover advantage, the public firm can commit to a lower quality, what in turn triggers a strategic reduction on the quality supplied by the black market firm.

A.5 Characterization of the Best Reply Correspondences in a Mixed Duopoly

Consider first that the public firm supplies the low quality product, that is, q G < q B. To decide how to optimally split consumers between both qualities, the government solves the following problem:

max θ ̂ W = ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q B − ( 1 + c ) q B 2 2 d θ − ( 1 + λ ) p B − c q B 2 2 [ θ ̄ − θ ̂ ] .

FOC yields:

∂ W ∂ θ ̂ = θ ̂ q G − ( 1 + c ) q G 2 2 − θ ̂ q B + ( 1 + c ) q B 2 2 + ( 1 + λ ) p B − c q B 2 2 = 0 ,

so that for any given price of the black market firm, the indifferent consumer must be such that:

(33) θ ̂ ( q B , q G ) = ( 1 + c ) q B + q G 2 + 1 + λ q B − q G p B − c q B 2 2 .

For the case where the public firm supplies the high quality product, the expression for the indifferent consumer that splits consumers optimally between both qualities is the same one. The only difference is that the second term is positive when the government supplies the low quality product and negative in the other case.

Recall from condition (2) that for q H > q L, the indifferent consumer θ ̂ is such that:

θ ̂ ( p L , p H , q L , q H ; β ) = p H − p L q H − q L + β q H + q L 2 .

If q G < q B, the black market firm solves the following problem:

max p B Π B = p B − c q B 2 2 [ θ ̄ − θ ̂ ] .

FOC yields:

(34) ∂ Π B ∂ p B = θ ̄ − θ ̂ − p B − c q B 2 2 q B − q G = 0 .

Using condition (34), we have that:

Π B = p B − c q B 2 2 [ θ ̄ − θ ̂ ] = ( q B − q G ) [ θ ̄ − θ ̂ ] 2 .

From conditions (33) and (34) we get the following equilibrium splitting condition when q G < q B:

(35) θ ̂ * ( q B , q G ) = 1 2 + λ ( 1 + c ) q B + q G 2 + ( 1 + λ ) θ ̄ .

When q G > q B, the black market firm solves the following problem:

max p B Π B = p B − c q B 2 / 2 [ θ ̂ − θ ̲ ] .

FOC yields:

(36) ∂ Π B ∂ p B = θ ̂ − θ ̲ + p B − c q B 2 2 q B − q G = 0 .

Using condition (36), we have that:

Π B = p B − c q B 2 2 [ θ ̂ − θ ̲ ] = ( q G − q B ) [ θ ̂ − θ ̲ ] 2 .

From conditions (33) and (36) we get the following equilibrium splitting condition when q G > q B:

(37) θ ̂ * * ( q B , q G ) = 1 2 + λ ( 1 + c ) q B + q G 2 + ( 1 + λ ) θ ̲ .

The objective of the black market firm is then given by:

Π B = ( q B − q G ) ( θ ̄ − θ ̂ ) 2   if  q G ≤ q B ; ( q G − q B ) ( θ ̂ − θ ̲ ) 2   if  q G > q B .

where for the first case the expression for θ ̂ is given by (35) and for the second by (37).

To find the quality supplied by the public firm that makes the black market firm indifferent between offering the low or the high quality product, we must first determine the best reply function of the black market firm. So that maximizing black market profits with respect to its own quality, for any given quality offered by the public firm, we have that:

q ̄ B = q G 3 + 2 θ ̄ 3 ( 1 + c )   if  q G ≤ q B ; q G 3 + 2 θ ̲ 3 ( 1 + c )   if  q G > q B .

We can now substitute the best reply correspondences into the expressions for the black market profits and find the quality of the public firm that makes the black market firm indifferent between offering the low or the high quality product. Following this procedure we find that this quality is given by:

q ̃ G = θ ̄ + θ ̲ 2 ( 1 + c ) .

The best reply function of the black market firm is then given by:

q ̄ B = q G 3 + 2 θ ̄ 3 ( 1 + c )   if  q G ≤ θ ̄ + θ ̲ 2 ( 1 + c ) , q G 3 + 2 θ ̲ 3 ( 1 + c )   if  q G > θ ̄ + θ ̲ 2 ( 1 + c ) .

From the expression above, it becomes clear that both qualities will never be equal in equilibrium, as the black market profits depend crucially on quality differentiation.

The same procedure can be followed to characterize the best reply correspondence of the public firm, the resulting expressions are quite cumbersome and are therefore not presented here.

A.6 Equilibrium Characterization When Public Firm Supplies the High Quality Product

When the public firm supplies the high quality product, the demands faced by the black market firm and the public firm are given respectively by:

D B ( p B , p G , q B , q G ; β ) = θ ̂ − θ ̲ , D G ( p B , p G , q B , q G ; β ) = θ ̄ − θ ̂ .

The public firm now serves the consumers with higher taste for quality, so that the objective functions of the black market firm and the government are now given respectively by:

Π B = p B − c q B 2 2 [ θ ̂ − θ ̲ ] , W = ∫ θ ̲ θ ̂ θ q B − ( 1 + c ) q B 2 2 d θ + ∫ θ ̂ θ ̄ θ q G − ( 1 + c ) q G 2 2 d θ − ( 1 + λ ) p B − c q B 2 2 [ θ ̂ − θ ̲ ] .

In the second stage of the game, the black market firm and the public firm set their prices, for given qualities. The FOCs for their respective objective functions with respect to prices are given respectively by:

∂ Π B ∂ p B = 4 p B − 2 p G + 2 θ ̲ ( q G − q B ) + β q B 2 − q G 2 − c q B 2 2 ( q B − q G ) = 0 , ∂ W ∂ p G = 2 ( p G + λ p B ) + ( 1 − β ) ( q B 2 − q G 2 ) − c λ q B 2 + q G 2 2 ( q B − q G ) = 0 .

Solving the system of equations above for p B and p G we get the Nash equilibrium prices that correspond to expressions (17) and (18) in the main text:

p B * * ( q B , q G ) = c ( q G 2 + ( 1 + λ ) q B 2 ) − 2 θ ̲ ( q G − q B ) + q G 2 − q B 2 2 ( 2 + λ ) , p G * * ( q B , q G ) = 2 c q G 2 + 2 ( 1 − β ) ( q G 2 − q B 2 ) − λ − 2 θ ̲ ( q G − q B ) + β q G 2 − q B 2 − c q B 2 2 ( 2 + λ ) .

Following the same procedure as before, we plug in the prices above into the objective function of the black market firm and the government, to then solve for their respective qualities. The FOCs for the black market firm and the public firm are as follows:

∂ Π B ∂ q B = [ 2 θ ̲ + ( 1 + c ) ( q G − 3 q B ) ] [ 2 θ ̲ − ( 1 + c ) ( q B + q G ) ] 4 ( 2 + λ ) 2 = 0 , ∂ W ∂ q G = 1 8 ( 2 + λ ) 2 × 4 θ ̲ 2 − ( 1 + c ) 2 q B 2 − 2 q B q G − 3 q G 2   + 4 ( 2 + λ ) 2 ( 1 + 2 ( θ ̲ − q G ( 1 + c ) ) ) − 8 θ ̲ q G ( 1 + c ) = 0 .

Solving the system above for q B and q G we get the equilibrium qualities when the public firm supplies the high quality product:

q B * * = 8 θ ̲ + 3 ( 2 + λ ) 2 − ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) , q G * * = 8 θ ̲ + 9 ( 2 + λ ) 2 − 3 ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) .

For the qualities above, that corresponds to the expressions (19) and (20) in the main text, no firm has incentive to deviate.

A.7 Second Order Properties

We want to verify that in the first stage of the game, the objective functions for the black market firm and the government are concave in their respective qualities, for the given price equilibrium of the second stage of the game.

A.7.1 Public Firm Supplies the Low Quality Product

The second order properties for the black market firm are as follows:

∂ 2 Π B ( q B , q G ) ∂ q B 2 = 1 + c 2 ( 2 + λ ) 2 − 4 θ ̄ + ( 1 + c ) ( 3 q B + q G ) , ∂ 2 Π B ( q B , q G ) ∂ q B ∂ q G = ( 1 + c ) 2 2 ( 2 + λ ) 2 ( q B − q G ) > 0 .

For the black market profits to be concave in its own quality it must verify that:

q B < 4 θ ̄ 3 ( 1 + c ) − q G 3 .

The second order properties for government are as follows:

∂ 2 W ( q B , q G ) ∂ q G 2 = 1 + c 4 ( 2 + λ ) 2 4 ( θ ̄ − ( 2 + λ ) 2 ) − ( 1 + c ) ( q B + 3 q G ) , ∂ 2 W ( q B , q G ) ∂ q G ∂ q B = ( 1 + c ) 2 4 ( 2 + λ ) 2 ( q B − q G ) > 0 .

So that the welfare function is concave in the quality chosen by the government as long as it verifies that:

q G > 4 ( θ ̄ − ( 2 + λ ) 2 ) 3 ( 1 + c ) − q B 3 .

There exists then a pattern of complementarity between qualities offered by the black market firm and the public firm. For the equilibrium qualities (15) and (16), both objective functions satisfy the conditions for concavity.

A.7.2 Public Firm Supplies the High Quality Product

The second order properties associated to the profit function of the black market firm are as follows:

∂ 2 Π B ( q B , q G ) ∂ q B 2 = 1 + c 2 ( 2 + λ ) 2 4 θ ̲ − ( 1 + c ) ( 3 q B + q G ) , ∂ 2 Π B ( q B , q G ) ∂ q B ∂ q G = ( 1 + c ) 2 2 ( 2 + λ ) 2 ( q G − q B ) > 0 .

For the black market profits to be concave in its own quality it must hold that:

q B > 4 θ ̲ 3 ( 1 + c ) − q G 3 .

The second order properties associated to the welfare function are as follows:

∂ 2 W ( q B , q G ) ∂ q G 2 = 1 + c 4 ( 2 + λ ) 2 − 4 ( θ ̲ + ( 2 + λ ) 2 ) + ( 1 + c ) ( q B + 3 q G ) , ∂ 2 W ( q B , q G ) ∂ q G ∂ q B = ( 1 + c ) 2 4 ( 2 + λ ) 2 ( q G − q B ) > 0 .

The welfare function is then concave in the quality chosen by the government as long as it holds that:

q G < 4 ( θ ̲ + ( 2 + λ ) 2 ) 3 ( 1 + c ) − q B 3 .

There exists then a complementarity pattern between the qualities offered by the black market firm and the public firm. For the equilibrium qualities (19) and (20), both objective functions satisfy the conditions for concavity.

A.8 Equilibrium Characterization When the Public Firm has a First-Mover Advantage

Consider the case where the public firm supplies the low quality product. Solving for the second stage of the game we have the following Nash equilibrium prices that correspond to expressions (11) and (12) in the main text:

p B * ( q B , q G ) = c ( q G 2 + ( 1 + λ ) q B 2 ) + 2 θ ̄ ( q B − q G ) − q B 2 − q G 2 2 ( 2 + λ ) , p G * ( q B , q G ) = 2 c q G 2 − 2 ( 1 − β ) ( q B 2 − q G 2 ) − λ 2 θ ̄ ( q B − q G ) − β q B 2 − q G 2 − c q B 2 2 ( 2 + λ ) .

Moving now to the first stage of the game, let’s plug in the prices above into the objective functions of the black market firm and the government. Since nothing has changed for the black market firm, the best reply function is the same as with simultaneous competition and given by:

q ̄ B = q G 3 + 2 θ ̄ 3 ( 1 + c ) .

The government now chooses its quality taking into account the best reply function above of the black market firm:

W p B * , p G * , q ̄ B , q G = ∫ θ ̲ θ ̂ θ q G − ( 1 + c ) q G 2 2 d θ + ∫ θ ̂ θ ̄ θ q ̄ B − ( 1 + c ) q ̄ B 2 2 d θ − ( 1 + λ ) p B * − c q ̄ B 2 2 D B .

FOC yields:

∂ W ∂ q G = 9 ( 2 + λ ) 2 [ 1 + 2 θ ̲ − 2 q G ( 1 + c ) ] − 8 [ q G ( 1 + c ) − θ ̄ ] 2 18 ( 2 + λ ) 2 = 0 .

Solving for q G we get the following equilibrium quality for the public firm:

q G * = 8 θ ̄ − 9 ( 2 + λ ) 2 + 3 ( 2 + λ ) 9 ( 2 + λ ) 2 − 8 8 ( 1 + c ) ,

which is the same quality offered when the black market firm and the public firm engage in simultaneous competition in (16).

The equilibrium when the public firm has a first mover advantage in the quality selection stage is the same than when engaging in simultaneous competition, and therefore characterized by expressions in (11), (12), (15) and (16). The outcome regarding the case where the government supplies the high quality product is analogous to the one just presented.

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Received: 2021-08-25
Revised: 2022-09-19
Accepted: 2022-11-02
Published Online: 2022-11-14

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Political Commitment, Policy Consequences, and Moral Beliefs: Survey Evidence on the Minimum Wage
  4. Managerial Delegation, Product R&D and Subsidies on R&D Investment Costs
  5. Counterfactual Decomposition of Health Care Expenditures: From Cost-Based Payment to a Global Budget System
  6. Legalizing Harmful Drugs: Government Participation and Optimal policies
  7. Subsidized Crop Insurance under Limited Access to Incomplete Financial Markets
  8. Network Effect, Business Dynamism and Wage Inequality in a Sharing Economy
  9. Letters
  10. Global Politico-Economic Dynamics during a Local Conflict: Reciprocal Patterns of Cooperation and Sanctions
  11. Welfare Effects of Socially Conscious Platforms in Two-Sided Markets
  12. Private Benefits of Control and the Core
  13. What is the Short-Term Impact of Decreasing the Speed Limit for Motor Vehicles on Bicycle Commuting? Evidence from French Cities during Covid-19 Pandemic
  14. Gone with the Flood: Natural Disasters and Children’s Schooling in Pakistan
  15. Major-Occupation Match Quality: An Empirical Measure Based on Relative Productivity
https://doi.org/10.1515/bejeap-2021-0309
Keywords for this article
legalization; cannabis; marijuana; optimal policies; government participation

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Political Commitment, Policy Consequences, and Moral Beliefs: Survey Evidence on the Minimum Wage
  4. Managerial Delegation, Product R&D and Subsidies on R&D Investment Costs
  5. Counterfactual Decomposition of Health Care Expenditures: From Cost-Based Payment to a Global Budget System
  6. Legalizing Harmful Drugs: Government Participation and Optimal policies
  7. Subsidized Crop Insurance under Limited Access to Incomplete Financial Markets
  8. Network Effect, Business Dynamism and Wage Inequality in a Sharing Economy
  9. Letters
  10. Global Politico-Economic Dynamics during a Local Conflict: Reciprocal Patterns of Cooperation and Sanctions
  11. Welfare Effects of Socially Conscious Platforms in Two-Sided Markets
  12. Private Benefits of Control and the Core
  13. What is the Short-Term Impact of Decreasing the Speed Limit for Motor Vehicles on Bicycle Commuting? Evidence from French Cities during Covid-19 Pandemic
  14. Gone with the Flood: Natural Disasters and Children’s Schooling in Pakistan
  15. Major-Occupation Match Quality: An Empirical Measure Based on Relative Productivity
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