Topic-Feature Lattices Construction and Visualization for Dynamic Topic Number

Kai WANG; Fuzhi WANG

doi:10.21078/JSSI-2021-558-17

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Journal of Systems Science and Information ›› 2021, Vol. 9 ›› Issue (5) : 558-574. DOI: 10.21078/JSSI-2021-558-17

Topic-Feature Lattices Construction and Visualization for Dynamic Topic Number

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Abstract

The topic recognition for dynamic topic number can realize the dynamic update of super parameters, and obtain the probability distribution of dynamic topics in time dimension, which helps to clear the understanding and tracking of convection text data. However, the current topic recognition model tends to be based on a fixed number of topics K and lacks multi-granularity analysis of subject knowledge. Therefore, it is impossible to deeply perceive the dynamic change of the topic in the time series. By introducing a novel approach on the basis of Infinite Latent Dirichlet allocation model, a topic feature lattice under the dynamic topic number is constructed. In the model, documents, topics and vocabularies are jointly modeled to generate two probability distribution matrices: Documentstopics and topic-feature words. Afterwards, the association intensity is computed between the topic and its feature vocabulary to establish the topic formal context matrix. Finally, the topic feature is induced according to the formal concept analysis (FCA) theory. The topic feature lattice under dynamic topic number (TFL DTN) model is validated on the real dataset by comparing with the mainstream methods. Experiments show that this model is more in line with actual needs, and achieves better results in semi-automatic modeling of topic visualization analysis.

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dynamic topic number / infinite latent Dirichlet allocation (ILDA) / formal concept analysis / topic feature lattice / topic feature lattice under dynamic topic number (TFL DTN) model

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Kai WANG , Fuzhi WANG. Topic-Feature Lattices Construction and Visualization for Dynamic Topic Number. Journal of Systems Science and Information, 2021, 9(5): 558-574 https://doi.org/10.21078/JSSI-2021-558-17

1 Introduction

Pharmaceutical industry is a typical innovation-driven industry. Efficient R&D innovation is an important source for pharmaceutical enterprises to obtain core competitiveness^[1]. However, owning to strong spillover effect, high failure rate and easy imitation ex-post, pharmaceutical enterprises often have little incentive to conduct innovation in the absence of policy intervention^[2]. To solve the above problems, regulators in various countries have put forward innovation incentive policies from the national strategic level. Generally, there are mainly two types of innovation incentive policies. One is the supply-side incentive policy and the other is the demand-side incentive policy. The innovation subsidy is viewed as a representative supply-side incentive policy widely used in the entire world. In the United State, for example, the FDA has subsidized more than ＄350 million for "orphan" drugs creation since 1983¹. European Commission and Federation of European Pharmaceutical Industries and Associations launched IMI2 (the second phase of Innovative Medicines Initiative) with a total budget of 3.3 billion EUR in the period of 2014–2020 to help discover the next generation of drugs². The government of China uses the "Creation and Development of Major New Drugs" fund to enhance the pharmaceutical firms' independent innovation capacity, by which the central government invested 23.3 billion RMB (About ＄350 million) in the period of 2008–2020 to support more than 3000 pharmaceutical innovation projects³.

¹http://www.evaluategroup.com.

²https://www.imi.europa.eu.

³http://www.gov.cn/xinwen/2021-02/02/content_5584285.htm.

Health insurance plan, as one of key demand-side incentives has been proved to influence pharmaceutical innovation^[3]. On the one hand, the inclusion of drugs in the health insurance plan can lower actual expenditure of patients for those drugs, and thereby expanding the market demand^[4]. On the other hand, such demand-side intervention may have spillover effects on supply side by inducing pharmaceutical innovation via expected market size expansion^[5]. Evidence from developed countries has shown that the inclusion of new drugs in the health insurance plan can incentivize pharmaceutical innovation^[6]. A typical example is the inclusion of new drugs for senile diseases in the Medicare Part D (prescription drug insurance) spurs pharmaceutical innovation in drugs used by seniors in the United State^[7]. The EU has started to incentivize pharmaceutical innovation and enhance the accessibility of new drugs by including them in the health insurance plan^[8]. The National Healthcare Security Administration of China has explicitly proposed that health insurance should play the role of encouraging the guiding the innovation of pharmaceutical industry, and encouraged new drugs with independent intellectual property rights to be included in health insurance plan so as to promote pharmaceutical innovation⁴. However, some studies have shown that regulators should be cautious when using health insurance fund to encourage pharmaceutical innovation, because this may trigger producers' opportunistic behavior and even lead to underinvest^[9].

⁴http://www.nhsa.gov.cn/art/2020/12/28/art_52_4222.html.

Although innovation subsidies and the inclusion of new drugs in the health insurance plan may both promote pharmaceutical innovation, their effects may vary. Thus, it is very necessary to explore policy design and policy selection decision problems for pharmaceutical innovation. Specifically, we attempt to address the following research questions: (ⅰ) Under what conditions should innovation subsidies or the inclusion of new drugs in the health insurance plan be selected? (ⅱ) Which type of policy allows the pharmaceutical enterprises to be more profitable, and generates higher consumer surplus as well as social welfare? (ⅲ) Whether the regulator can choose a hybrid policy? And when is the hybrid policy better than a single policy?

To answer these questions, we formulate a stylized model consisting of a regulator and two representative pharmaceutical producers, i.e., the new drug producer and the general drug producer. The regulator, acting as the leader, first choose and design the incentive scheme, namely innovation subsides, the inclusion of new drugs in the health insurance plan, or a hybrid policy (i.e., the combination of government subsidies and the inclusion of new drugs in the health insurance plan) to maximize social welfare. As a follower, the new drug producer decides its innovation level and then both the new drug producer and the general drug producer set their own prices simultaneously aiming to maximizing individual profits. We first analyze all players' optimal decisions under a given policy. Then we discuss the policy design of the regulator as well as the policy selection from innovation subsidies, the inclusion of new drugs in the health insurance plan, and the hybrid policy. The main insights of our paper are as follows.

First, in the competitive market, innovation subsidies and the inclusion of new drugs in the health insurance plan can both enhance innovation incentives and profits of the new drug producer, expand the accessibility of new drugs. Second, innovation subsidies perform better than the inclusion of new drugs in the health insurance plan from the perspective of social welfare improvement, but worsen in raising the innovation level, expanding the accessibility of new drugs and improving patient welfare, if the copayment level is sufficiently high. Third, a hybrid policy combined by innovation subsidies and the inclusion of new drugs in the health insurance plan benefits to lower fiscal subsidy expenditure, increase the innovation reward of the new drug producer, but will push up the new drug's price. This implies that the hybrid policy does not necessarily perform better than innovation subsidies in expanding the accessibility of new drugs, raising the new drug's innovation level, as well as improving patient welfare and social welfare.

Our study makes two-ford contributions. First, we examine the difference between innovation subsidies and the inclusion of new drugs in the health insurance plan in terms of their incentive effects of pharmaceutical innovation, which helps integrate health insurance policy and innovation incentive policy to promote pharmaceutical innovation, but to our best knowledge, has not been studied before. Second, we provide useful insights on how the regulator select incentive tools to encourage pharmaceutical innovation as well as improve social welfare and patient welfare. Our analysis shows that it is optimal for the regulator to adopt innovation subsidies if focuses on social welfare improvement, while it should include new drugs in the health insurance plan and raise the copayment level properly, if intended to expand the accessibility of new drugs, provide promising drugs and improve patient welfare. These endow our study with both theoretical and practical significance.

The rest of our study is organized as follows. Section 2 reviews the most relevant literature and highlights our contributions. In Section 3, we introduce the model setting of this paper. In Section 4, we evaluate the incentive effects of innovation subsidies and the inclusion of new drugs in the health insurance plan by analyzing the model and comparing the results. In Section 5, we investigate the implication of hybrid policy combined by innovation subsidies and the inclusion of new drugs in the health insurance plan. Section 6 considers a scenario of endogenous copayment level as an extension and we conclude this paper in Section 7. All proofs are provided in the Appendix.

2 Literature Review

Our paper is broadly relevant to the literature on incentive policies for pharmaceutical innovation. Generally speaking, innovation incentive policies in the pharmaceutical industry can mainly be categorized into two types, namely, the supply-side incentive policy and demand-side policy. In the literature on supply-side incentive policy, Rao^[10] examined impacts of greater exclusivity protections and a faster FDA approval process on drug development incentives, and shown that such policies lead to quicker realization of profits, but they also intensify competition thus reducing per-firm profits. Yin^[11] investigated how intellectual property regimes affect pharmaceutical innovation based on individual-level prescription data, and pointed that policy makers may need to revisit the provisions for granting market exclusivity to incremental innovations in the pharmaceutical industry. Altug and Sahin^[12] explored how parallel imports influence product launch and pricing strategies of newly developed drugs and further discussed impacts of insurance coverage, market size, quality perceptions of the parallel imported drug and valuations on the above strategies. More similar to this paper, some scholars focused on the roles of government subsidies on pharmaceutical innovation. For example, Yin^[13], Czamitzki and Lopesbento^[14], and Zhang, et al.^[15] respectively confirmed the value of government subsidies in pharmaceutical innovation from the perspectives of promoting innovation investment, reducing R&D risk and improving innovation outputs. Zhang, et al.^[16] designed the strategic incentive policy for pharmaceutical innovation and shown that such policy can achieve social optimal allocation of pharmaceutical innovation under certain conditions. However, Wallsten^[17] argued that government subsidies will crowd out enterprises' innovation investments, and thus resulting in inefficient or even ineffective subsidy policy. Montmartin and Herrera^[18] and Li, et al.^[19] demonstrated that the effect of government subsidies on enterprises' innovation incentives is not linear relationship, and there exists an optimal subsidy level.

In the literature of demand-side incentive policies, some studies (e.g., Bardey, et al.^[20]; Straume^[21]) emphasized effects of reference pricing on pharmaceutical innovation as well new drugs' entrance and pricing. Hu and Schwarz^[22] investigated the impact of group purchasing on pharmaceutical innovation and pointed out that such behavior stifled innovation of existing drugs. In recent years, as one of the most important demand-size incentive policies, the role of health insurance in pharmaceutical innovation has attracted considerable academic attention. Numerous studies showed that the expansion of health insurance may benefit pharmaceutical innovation. Representative studies are as follows. Blume-Kohout and Sood^[7] found that the passage of the Medicare Part D (prescription drug insurance) promotes pharmaceutical innovation in drugs for the elderly. The empirical analysis by Zhang and Nie^[6] showed that governments can create incentives for pharmaceutical innovation by offering public health insurance to low-income individual in developing countries. Gailey, et al.^[23] developed and empirically calibrated theoretical model to examine welfare effects of Medicare Part D, and showed that such the prescription drug insurance can improve incentives for pharmaceutical innovation. Hawkins, et al.^[24] showed that the structure of the health insurance market may influence expected profits, and hence innovation incentives for pharmaceutical enterprises. Garthwaite, et al.^[25] revealed that the drug prices by Medicaid (which accounted for most of the expansion in coverage) are relatively low in US, leading to seemingly no role of the expansion of health insurance in pulling in pharmaceutical innovation. Nakamura and Wakutsu^[26] investigated the reimbursement drug price risk of pharmaceutical firms as well as its impacts on pharmaceutical firms' R&D incentives.

We summarize the most relevant literature in Table 1 to highlight our main contributions. From Table 1, it is shown that pharmaceutical innovation incentives have been widely concerned by academia and the existing studies have conducted relatively sufficiently modelling and empirical research on this problem. Our work belongs to the category of modelling studies, but differs from previous literature at least in twofold: First, the extant literature on the incentive effects of innovation subsidy policy and health insurance plan for pharmaceutical innovation are independent of each other. We add to this stream literature by comparing the incentive effects of two types of policy tools and then provide insights for the choice of incentive policies from the perspectives of social welfare maximization and consumer welfare maximization. Second, we further analyze incentive effects of the combination of government subsidies and inclusion of new drugs in health insurance plan (i.e., the hybrid policy) and prove that it can be welcomed by the regulator, the new drug producer and consumers and makes them achieve a "win-win-win" outcome.

Table 1 Summary of literature

Representative paper	Pharmaceutical innovation	Innovation subsidies	The roles of health insurance	Modeling studies	Empirical studies
^[2]	$✓$			$✓$
^[6]	$✓$		$✓$		$✓$
^[7]	$✓$			$✓$	$✓$
^[12]	$✓$	$✓$		$✓$
^[13]	$✓$	$✓$		$✓$
^[26]	$✓$		$✓$		$✓$
This paper	$✓$	$✓$	$✓$	$✓$

3 Model Setup

We consider a stylized model comprising a regulator (he) and two representative pharmaceutical producers (she), namely, the new drug producer and the general drug producer. The two pharmaceutical producers produce and sell partially substitutable prescription drugs, such as different brands of drugs for the same diseases. Compared with the existing drug (i.e., general drug), the new drug has better efficacy and brings higher health gains to all patients^[27]. Note that the new drug producer incurs R&D costs because the new drug producer should adopt new processes or new technologies to improve therapeutic effects and/or reduce side effects. To capture this feature, we denote

θ

as the new drug's efficacy and normalize the efficacy of the general drug to 0 without loss of generality. Thus,

θ

measures the degree of therapeutic difference between two types of drugs. The higher value of

θ

is, the better therapeutic effects the new drug has. Clearly,

θ

can also be interpreted as the new drug's innovation level.

The new drug producer incurs two types of costs. The first one is the innovation cost. Without loss of generality, we use a quadratic form

κ θ^{2}

to capture the innovation cost, which is independent of the unit production cost.

κ

is the cost coefficient of innovation measuring the innovation efficiency of the new drug producer. The higher the value of

κ

is, the lower innovation efficiency the new drug producer has. The second one is the unit production cost, which can be normalized to 0 without affecting our qualitative results. The reasons are as follows. First, since the drugs are usually produced in accordance with GMP standards, the unit production is generally constant. Second, pharmaceutical innovation in our model is the improvement of therapeutic efficacy compared to the existing drugs but not the reduction of unit production cost.

The regulator can choose either innovation subsidies or the inclusion of new drugs in the health insurance plan to promote pharmaceutical innovation. Specifically, if the regulator chooses the innovation subsidy policy, he determines the subsidy rate

s

to subsidize the innovation cost for the new drug producer. Conversely, if the regulator chooses to incorporate new drugs into the health insurance plan, consumers can gain a portion of reimbursement rate (i.e., copayment level),

α

, when purchasing the new drug. In practice, the copayment level is a long-term decision and predetermined by the health insurance organization, thus is cannot be modified easily. Taking China as example, the National Healthcare Security Administration stipulates the copayment levels of different classes of drugs. Therefore, we assume that the copayment level is exogenous, which is a common setting in the previous studies^{[1, 27, 28]}. We relax this assumption and discuss the implication of endogenous copayment level in Section 6, though it is very rare in reality. Furthermore, we evaluate incentive effects of a hybrid policy which combines innovation subsidies and the inclusion of new drugs in the health insurance plan, and provide conditions under which the hybrid policy is preferred. For ease of exposition, we use superscripts

N

S

M

and

B

to represent the scenarios of without regulator's intervention, innovation subsidies, the inclusion of new drugs in the health insurance plan, and hybrid policy, respectively, and subscripts

n

and

g

to denote the new drug producer (or the new drug) and the general drug producer (or the general drug). The relationships among players are illustrated in Figure 1.

Figure 1 The relationships between decisions makers

Full size|PPT slide

Consistent with De Frutos, et al.^[29], we assume that the prescription drug market consists of a continuum of consumers (physician-patient pairs) who are heterogeneous when evaluating therapeutic effects of drugs. This is reasonable because, on the one hand, in the prescription drug market the choice of drug is generally made by physicians rather than patients. On the other hand, the abolition of the drug markup system has transformed hospitals from drug retailers into bridges between drug producers and patients. For simplicity, we breviate physician-patient pairs to patients. Moreover, due to heterogeneity among patients and differences in efficacy of different drugs, physicians have heterogenous preference for different drugs. To capture the above features, we follow Brekke, et al.^[30] and employ a model combined horizontal and vertical differentiation, where all patients are distributed uniformly on a linear market with its size normalized to one, and two available drugs (i.e., the new drug and the general drug) locate at two end points of this linear market respectively. Similar models are adopted commonly to capture competition in a therapeutic market (e.g., [21, 27, 28, 31]). Without loss of generality, the new drug producer locates at 0 and the general drug producer locates 1. Each patient needs one unit of drug treatment. The difference in patients reflects the heterogeneity of treatment response, i.e., the closer the patient is to the drug, the higher the therapeutic benefit of a particular drug treatment is. Specifically, a patient located at

x

incurs a therapeutic mismatch cost

t x

when choosing the new drug and a therapeutic mismatch cost

t (1 - x)

when choosing the general drug, where

t

measures the relative importance of therapeutic mismatch, which also reflects the degree of horizontal differentiation between the two drugs. Hence, the utility functions of patients for therapeutic treatment for new and general drugs are given as

\begin{array}{rcl} U_{n} = v + θ - (1 - I_{n} α) p_{n} - t x, \end{array}

(1)

\begin{array}{rcl} U_{g} = v - p_{g} - t (1 - x), \end{array}

(2)

where

v

is the consumers' evaluation of the therapeutic utility of general drug, and

θ

is the innovation level of the new drug, reflecting the vertical differentiation between the new drug and the general drug.

p_{n}

and

p_{g}

are prices of the new drug and the general drug, respectively. Without loss of generality, we assume that

p_{n} > p_{g}

, indicating that consumers are willing to pay premium for the new drug.

I_{n}

is an indicator.

I_{n} = 1

represents the inclusion of the new drug in the health insurance plan, and

I_{n} = 0

indicates that the new drug is not in the health insurance plan.

To rule out the uninteresting case with no market competition, we assume that

v

is sufficiently large so that the market is fully covered. If

v

is too low, the market is not covered and both producers will be localized monopolies. Such an assumption is widely used in the literature of horizontal differentiation competition^{[22, 27, 32]}. Notice that, we consider the case where the health insurance plan only comprises the new drug. The reasons are as follows. First, for each indication, the number of drugs in the health insurance play is limited, which means that when some drugs enter the list, others are moved out of the list. With increasing numbers of new drug in the health insurance plan, general drugs for the same indication are moved out. Moreover, such assumption helps focus our attentions on the incentive effects of inclusion of new drugs in the health insurance plan on pharmaceutical innovation.

Maximizing the above utility functions yields the following demand functions for the two drugs

d_{n} = \frac{1}{2} + \frac{(θ + p_{g} - (1 - I_{n} α) p_{n})}{2 t},

(3)

d_{g} = \frac{1}{2} - \frac{(θ + p_{g} - (1 - I_{n} α) p_{n})}{2 t} .

(4)

The sequence of events is as follows. First, the regulator acting as a Stackelberg leader selects between the innovation subsidies and the inclusion of new drugs in the health insurance plan to maximize social welfare. If the regulator chooses innovation subsidies, he decides the subsidy level

s

. After observing the regulator's choice, the new drug producer decides the innovation level of the new drug

θ

, then the new drug producer and the general drug producer set individual prices

p_{n}

and

p_{g}

simultaneously. On the opposite, if the regulator chooses to incorporate new drugs to the health insurance plan (i.e.,

I_{n} = 1

), the two drug producers decide individual prices simultaneously. Finally, consumers make purchase decisions based on therapeutic utility maximization, then market demand is realized and both the new and general drug producers gain profits. Table 2 summarizes the notations and definitions in our model.

Table 2 Notations and definitions

Notation	Definition
$α$	Copayment level
$κ$	Unit innovation cost measuring the innovation efficiency of the new drug producer
$θ$	Innovation level of the new drug
$v$	Consumer's valuation for the general drug
$t$	Travel cost or disutility per unit distance
$p_{n} (p_{g})$	Price of the new (general) drug
$d_{n} (d_{g})$	Market demand for the new (general) drug
$s$	Innovation subsidy level
$C S$	Consumer surplus reflecting consumer welfare
$S W$	Social welfare
$Π_{n} (Π_{g})$	Profit of the new (general) drug producer
$I_{n}$	Indicator parameters

4 Equilibrium Analysis

In this section, we first discuss the decisions of the regulator and two producers in equilibrium under different incentive policies. Then we compare these results and evaluate the incentive effect of different policies.

4.1 The Benchmark: Without Regulator Intervention

We first consider the case without regulator intervention as benchmark. The events sequence is described as follows: First, the new drug producer decides the innovation level

θ

. Second, the new drug producer and the general drug producer set individual prices simultaneously. Hence, the profit functions of the new drug and general drug producers are given by

\begin{array}{rcl} Π_{n} (p_{n}, θ) = p_{n} (\frac{1}{2} + \frac{θ + p_{g} - p_{n}}{2 t}) - k θ^{2}, \end{array}

(5)

\begin{array}{rcl} Π_{g} (p_{g}) = p_{g} (\frac{1}{2} - \frac{θ + p_{g} - p_{n}}{2 t}) . \end{array}

(6)

Accordingly, the consumer surplus and social welfare are given by

\begin{array}{rcl} C S = \int_{0}^{\frac{1}{2} + \frac{θ + p_{g} - p_{n}}{2 t}} U_{n} d x + \int_{\frac{1}{2} + \frac{θ + p_{g} - p_{n}}{2 t}}^{1} U_{g} d x, \end{array}

(7)

\begin{array}{rcl} S W = Π_{n} + Π_{g} + C S . \end{array}

(8)

By backward induction, we solve this game and obtain the following lemma, which summarizes the equilibrium outcomes.

Lemma 1 Without regulator intervention, $(i)$ if $κ > \frac{1}{9 t}$ , there exists a unique competitive equilibrium. The equilibrium innovation level of the new drug is $θ^{N *} = \frac{3 t}{18 κ t - 1}$ , the prices of new drug and general drug are $p_{n}^{N *} = \frac{18 κ t^{2}}{18 κ t - 1}$ , $p_{g}^{N *} = \frac{2 t (9 κ t - 1)}{18 κ t - 1}$ , the market share of new drug and general drug are $d_{n}^{N *} = \frac{9 κ t}{18 κ t - 1}$ , $d_{g}^{N *} = \frac{κ t - 1}{18 κ t - 1}$ , and the new drug producer's and the general drug producer's profits are $Π_{n}^{N *} = \frac{9 κ t^{2}}{18 κ t - 1}$ , $Π_{g}^{N *} = \frac{2 t {(9 κ t - 1)}^{2}}{{(18 κ t - 1)}^{2}}$ ; $(i i)$ if $κ < \frac{1}{9 t}$ , the new drug producer will monopoly the whole market. The equilibrium innovation level, price and market share of the new drug are $θ^{N *} = \frac{1}{2 κ}$ , $p_{n}^{N *} = v - t + \frac{1}{2 κ}$ , and $d_{n}^{N *} = 1$ . Accordingly, the profit of the new drug producer is $Π_{n}^{N *} = v - t + \frac{1}{4 κ}$ .

Lemma 1 shows that in a competitive market, the new drug producer always conduct innovation even if the regulator does not intervene in pharmaceutical innovation. More specifically, when the innovation efficiency is sufficiently high (i.e.,

κ \leq \frac{1}{9 t}

), the general drug producer will be squeezed out of the market, and the new drug producer monopolies the entire market. On the opposite, when the innovation efficiency is relatively low (i.e.,

κ > \frac{1}{9 t}

), the new drug and the general drug co-exist in the therapeutic market. Since patients prefer the new drug to the general drug, the price of the new drug is higher than that of the general drug, and thus the new drug producer gains more profits than her rival.

Since we focus on the evaluation of impacts of different incentive policies on pharmaceutical innovation in competitive market, in the rest of paper, we only consider the case of competition equilibrium, that is,

d_{n}

d_{g} > 0

. Furthermore, substituting the results in Lemma 1 into the functions of consumer surplus and social welfare shown in (7) and (8), we obtain the consumer surplus and social welfare intervention as

{C S}^{N *} = v - \frac{(810 κ^{2} t^{2} - 144 κ t + 5) t}{2 (18 κ t - 1)^{2}}

{S W}^{N *} = v - \frac{(162 κ^{2} t^{2} - 54 κ t + 1) t}{2 (18 κ t - 1)^{2}}

4.2 Innovation Subsidies

In this setting, the regulator first determines the subsidy level

s

for the new drug producer to maximize social welfare. Then, the new drug producer decides on the innovation level

θ

. Finally, the new drug producer and the general drug producer decide on individual prices simultaneously. The profit functions of the new drug producer and the general drug producer are

\begin{array}{rcl} Π_{n} (p_{n}, θ) = p_{n} (\frac{1}{2} + \frac{θ + p_{g} - p_{n}}{2 t}) - (1 - s) κ θ^{2}, \end{array}

(9)

\begin{array}{rcl} Π_{g} (p_{g}) = p_{g} (\frac{1}{2} - \frac{θ + p_{g} - p_{n}}{2 t}) . \end{array}

(10)

To find the equilibrium outcomes under innovation subsidy policy, we first determine the optimal decisions of the two drug producers. The results are summarized in Lemma 2.

Lemma 2 Given the subsidy level $s$ , the optimal innovation of the new drug is $θ^{*} (s) = \frac{3 t}{18 (1 - s) κ t - 1}$ , and the optimal prices of the new drug and the general drug are $p_{n}^{*} (s) = \frac{18 (1 - s) κ t^{2}}{18 (1 - s) κ t - 1}$ and $p_{g}^{*} (s) = \frac{2 (9 (1 - s) κ t - 1) t}{18 (1 - s) κ t - 1}$ . Accordingly, the market share of the new drug and the general drug are $d_{n}^{*} (s) = \frac{9 κ t (1 - s)}{18 (1 - s) κ t - 1}$ and $d_{g}^{*} (s) = \frac{9 κ t (1 - s) - 1}{18 (1 - s) κ t - 1}$ .

Intuitively, a higher subsidy level leads to a high innovation level, resulting in an increase in the new drug's price. This is understandable, because the new drug producer incurs a lower innovation cost. Recall that the higher the innovation level of the new drug, the greater the willingness of patients to choose the new drug, and thus expanding the market share of the new drug and encroaching on the market share of the general drug. To stimulate demand, the general drug producer has to reduce her own price.

Next, we consider the government's subsidy decision, which can be formulated as follows:

max_{s} S W (s) = Π_{n} (s) + Π_{g} (s) + C S (s) - s κ θ (s)^{2},

(11)

where

C S = \int_{0}^{\frac{1}{2} + \frac{θ + p_{g} - p_{n}}{2 t}} U_{n} d x + \int_{\frac{1}{2} + \frac{θ + p_{g} - p_{n}}{2 t}}^{1} U_{g} d x

By using backward induction, we characterize the equilibrium under innovation subsidy policy. The results are summarized in Lemma 3.

Lemma 3 Under innovation subsidy policy, when $κ > \frac{2}{9 t}$ holds, there exists a unique competitive equilibrium. The equilibrium subsidy level is $s^{S *} = \frac{9 κ t + 1}{27 κ t}$ ; the innovation level of the new drug is $θ^{S *} = \frac{9 t}{36 κ t - 5}$ ; the prices of new drug and general drug are $p_{n}^{S *} = \frac{2 t (18 κ t - 1)}{36 κ t - 5}$ , $p_{g}^{S *} = \frac{4 t (9 κ t - 2)}{36 κ t - 5}$ ; the market shares of new drug and general drug are $d_{n}^{S *} = \frac{18 κ t - 1}{36 κ t - 5}$ , $d_{g}^{S *} = \frac{2 (9 κ t - 2)}{36 κ t - 5}$ . Accordingly, the equilibrium profits of the new drug producer and the general drug producers are $Π_{n}^{S *} = \frac{(18 κ t - 1) t}{36 κ t - 5}$ and $Π_{g}^{N *} = \frac{8 t (9 κ t - 2)^{2}}{(36 κ t - 5)^{2}}$ , consumer surplus is ${C S}^{S *} = v - \frac{t (3240 κ^{2} t^{2} - 1224 κ t + 103)}{2 (36 κ t - 5)^{2}}$ , and the social welfare is ${S W}^{S *} = v - \frac{(18 κ t - 7) t}{2 (36 κ t - 5)}$ .

One may intuit that with the increasing innovation efficiency (smaller

κ

), the new drug producer is more willing to conduct innovation, thus the regulator will reduce his subsidy. However, Lemma 3 shows the opposite result, namely, as the innovation efficiency grows, the regulator raises his subsidy instead. The intuition behind is that a higher innovation efficiency leads to a lower unit innovation cost, inducing the new drug producer to invest in more efforts and thus raising the total innovation costs, which in turn lowers her incentives to conduct pharmaceutical innovation. To motivate innovation, the regulator has to increase his subsidy rate.

4.3 Inclusion of New Drugs in the Health Insurance Plan

When the government incorporates new drugs in the health insurance (i.e.,

I_{n} = 1

), the new drug producer first determines her innovation level

θ

. Then, the new drug producer and the general drug producer decide on individual prices simultaneously. The profit functions are then

\begin{array}{rcl} Π_{n} (p_{n}, θ) = p_{n} (\frac{1}{2} + \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}) - κ θ^{2}, \end{array}

(12)

\begin{array}{rcl} Π_{g} (p_{g}) = p_{g} (\frac{1}{2} - \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}) . \end{array}

(13)

By backward induction, we obtain the equilibrium solutions, which summarizes in Lemma 4.

Lemma 4 When new drugs are included in the health insurance plan, there exists a unique competitive equilibrium, if $κ > \frac{1}{9 (1 - α) t}$ holds. The equilibrium innovation level, price and market share of the new drug are $θ^{M *} = \frac{3 t}{18 (1 - α) κ t - 1}$ , $p_{n}^{M *} = \frac{18 κ t^{2}}{18 (1 - α) κ t - 1}$ , and $d_{n}^{M *} = \frac{9 (1 - α) κ t}{18 (1 - α) κ t - 1}$ , respectively; the equilibrium price and market share of the general drug are $p_{g}^{M *} = \frac{2 t (9 (1 - α) κ t - 1)}{18 (1 - α) κ t - 1}$ , and $d_{g}^{M *} = \frac{9 (1 - α) κ t - 1}{18 (1 - α) κ t - 1}$ . Accordingly, the equilibrium profits of the new drug producer and the general drug producer are $Π_{n}^{M *} = \frac{9 κ t^{2}}{18 (1 - α) κ t - 1}$ , $Π_{g}^{M *} = \frac{2 t (9 (1 - α) κ t - 1)^{2}}{(18 (1 - α) κ t - 1)^{2}}$ .

From Lemma 4, we can clearly see that an increase in copayment level (larger

α

) benefits to increase the innovation level of the new drug and expand the accessibility of the new drug (characterized by the market demand of new drug), but pushes up the new drug's price and reduces the general drug's price. This implies that the inclusion of new drugs in the health insurance plan not only expands the accessibility of the new drug but helps bring down the general drug.

When the new drugs are included in the health insurance plan, consumer surplus function and social welfare function are respectively given by

\begin{array}{rcl} {C S}^{M} = \int_{0}^{\frac{1}{2} + \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}} U_{n} d x + \int_{\frac{1}{2} + \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}}^{1} U_{g} d x, \end{array}

(14)

\begin{array}{rcl} {S W}^{M} = Π_{n}^{M} + Π_{g}^{M} + {C S}^{M} - α p_{n}^{M} d_{n}^{M} . \end{array}

(15)

Substituting the results shown in Lemma 3 into (14) and (15), we obtain equilibrium consumer surplus and social welfare in a competitive market as

{C S}^{M *} = v - \frac{(810 (1 - α)^{2} κ^{2} t^{2} - 144 (1 - α) κ t + 5) t}{2 (18 (1 - α) κ t - 1)^{2}}

{S W}^{M *} = v - \frac{(162 (1 - α)^{2} κ^{2} t^{2} - 18 (3 - 4 α) κ t + 1) t}{2 (18 (1 - α) κ t - 1)^{2}}

4.4 Evaluation of Incentive Effects of the Two Policies

In this subsection, we evaluate the incentive effects of innovation subsidies and the inclusion of new drugs in the health insurance plan. To guarantee the compatibility of conditions under different policies, we require

κ > max (\frac{2}{9 t}, \frac{1}{9 (1 - α) t})

We start with the comparison of innovation level and market share of the new drug under different scenarios, i.e., without regulator intervention, innovation subsidies, and the inclusion of new drugs in the health insurance plan. The results are summarized in Proposition 1.

Proposition 1 $(i)$ $θ^{M *} > θ^{N *}$ , $θ^{S *} > θ^{N *}$ , $d_{n}^{M *} > d_{n}^{N *}$ ; $(i i)$ $θ^{M *} \geq θ^{S *}$ , $d_{n}^{M *} \geq d_{n}^{S *}$ , if $α \geq α_{M}$ , and $θ^{M *} < θ^{S *}$ , $d_{n}^{M *} < d_{n}^{S *}$ , if $α < α_{M}$ , where $α_{M} = \frac{9 κ t + 1}{27 κ t}$ . Moreover, $α_{M}$ is decreasing in $t$ .

Proposition 1 indicates that in a competitive market, innovation subsidies and the inclusion of new drugs in the health insurance plan can enhance both the new drug producer's incentives to conduct innovation and the accessibility of the new drug. Intuitively, the inclusion of new drugs in the health insurance plan can lower the actual expenditure for purchasing the new drug, which will enhance consumers' willingness to purchase the new drug and then help expand the market demand for the new drug. On the other hand, innovation subsidies can reduce the innovation cost for the new drug producer, and then raise the new drug producer's innovation incentives as well as the new drug's innovation level.

However, Part (ⅱ) in Proposition 1 shows that this is not always the case. When the copayment level is sufficiently high (i.e.,

α \geq α_{M}

), the inclusion of new drugs in the health insurance plan is superior to innovation subsidies in improving the innovation level and expanding the accessibility of the new drug. Conversely, when the copayment level is relatively low (i.e.,

α < α_{M}

), innovation subsidies perform better than the inclusion of new drugs in the health insurance plan in improving the innovation level and expanding the accessibility of the new drug. The intuition behind stems from the two opposite effects of pharmaceutical innovation. First, pharmaceutical innovation can increase patients' willingness to choose new drugs, and thereby expanding the market demand for the new drug, which we call the demand-expansion effect. Second, pharmaceutical innovation will bring about additional cost, which we call the cost-incurring effect. When

α \geq α_{M}

, the inclusion of the new drugs in the health insurance plan can lower patients' actual expenditure for purchasing the new drug. In such a situation, the demand-expansion effect dominates the cost-incurring effect, which leads to higher market share and innovation level of the new drug. On the opposite, when

α < α_{M}

, the cost-incurring effect dominants. In such a situation, the innovation subsidy policy is better than the inclusion of new drugs in the health insurance plan in improving innovation level and accessibility of the new drug, because the former can directly reduce the innovation cost. Note that

α_{M}

is decreasing in

t

, which indicates that as the therapeutic competition between the new drug and the general drug become less intense (larger

t

), the inclusion of new drug in the health insurance plan is more likely to outperform innovation subsidies in improving the innovation level and accessibility of the new drug.

Next, we discuss the preference of the new drug producer for different incentive policies by comparing her profits among without regulator intervention, innovation subsidies and the inclusion of new drugs in the health insurance plan. The results are summarized in Proposition 2.

Proposition 2 $(i)$ $Π_{n}^{S *} > Π_{n}^{N *}$ , $Π_{n}^{M *} > Π_{n}^{N *}$ ; $(i i)$ $Π_{n}^{M *} \geq Π_{n}^{S *}$ , if $α \geq \tilde{α}$ , and $Π_{n}^{M *} < Π_{n}^{S *}$ , if $α < \tilde{α}$ , where $\tilde{α} = \frac{9 κ t + 1}{18 κ t (18 κ t - 1)}$ . Moreover, $\tilde{α}$ is decreasing in $t$ .

Proposition 2 demonstrates that in a competitive market, innovation subsidies and the inclusion of new drugs in the health insurance plan can both increases the profitability of the new drug's producer, but the new drug producer's preferences for the above two policy tools depend on the copayment level. More specifically, when the copayment level is sufficiently high (i.e.,

α \geq \tilde{α}

), the new drug producer prefers the inclusion of new drugs in the health insurance plan to reduce patients' actual expenditure for new drugs and thus expanding the new drug's accessibility. On the opposite, when the copayment level is low, the new drug producer prefers innovation subsidies. This is largely because, with a low copayment level, the demand-expansion effect of the inclusion of new drugs in the health insurance plan is so limit that cost-incurring effect dominants. In this regard, innovation subsidies help lower the innovation cost and increase the new drug producer's profits.

We further explore the impacts of

t

and

κ

on the preference of the new drug producer for innovation subsidies and the inclusion of new drugs in the health insurance plan. To make it observable, we conduct numerical analysis in Figure 2. As illustrated in Figure 2, as the therapeutic competition between the new drug and the general drug increases (smaller

t

), the new drug producer performs better under innovation subsidies, while as the innovation efficiency decreases (larger

κ

), the new drug producer prefers the inclusion of new drugs in the health insurance plan. The intuition behind is that a smaller

t

leads to a decreased demand-expansion effect of pharmaceutical innovation and thus is replaced by the cost-incurring effect. That is, the cost-incurring effect becomes the dominant effect. In such a situation, innovation subsidies superior the inclusion of new drugs in the health insurance plan in terms of boosting the new drug producer's profitability. However, a lager

κ

allows the cost-incurring effect to be weaker, and thus the demand-expansion effect become the dominant effect, which makes the inclusion of new drugs in the health insurance plan performs better in improving the new drug's producer's profit.

Figure 2 Preferences of the new drug producer over two incentive policies with $t$ under different $κ$

Full size|PPT slide

We also compare the social welfare and consumer surplus under different incentive schemes and obtain the following propositions.

Proposition 3 $(i)$ ${S W}^{S *} > {S W}^{M *}$ , ${S W}^{S *} > {S W}^{N *}$ ; $(i i)$ ${S W}^{M *} > {S W}^{N *}$ , if $κ \geq \frac{11}{36 t}$ ; Otherwise, there exists a threshold $\bar{α} = \frac{2 (9 κ t + 1) (18 κ t - 1)}{9 κ t (72 κ t - 1)}$ such that, ${S W}^{M *} \geq {S W}^{N *}$ , if $α \geq \bar{α}$ , and ${S W}^{M *} < {S W}^{N *}$ , if $α < \bar{α}$ .

Proposition 3 suggests that from the perspective of social welfare improvement, innovation subsidies perform better than does the inclusion of new drugs in the health insurance plan and without regulator's intervention. This is largely because innovation subsidies are more flexible than the inclusion of new drugs in the health insurance plan. Recalling that the copayment level is exogenous and does not change over time, while under innovation subsidies, the regulator acting as Stackelberg leader can decide the optimal subsidy level to achieve social welfare maximization.

It is worth noting that from the perspective of social welfare maximization, the inclusion new drugs into the health plan does not always work. As shown in Proposition 3(ii), when the innovation efficiency is sufficient high, and the copayment level is relatively low, the inclusion new drugs into the health plan is not a preferred choice for the regulator in terms of social welfare maximization. The possible reasons are as follows. On the one hand, when the innovation efficiency is high enough, the new drug producer has more incentives to invest in pharmaceutical innovation, which leads to increased total innovation cost. On the other hand, when the copayment level is relatively low, the demand-expansion from the inclusion of new drugs in the health insurance plan does not offset the increase in total innovation cost.

Proposition 4 $(i)$ ${C S}^{S *} > {C S}^{N *}$ , ${C S}^{M *} > {C S}^{N *}$ ; $(i i)$ ${C S}^{M *} \geq {C S}^{S *}$ , if $α \geq α_{M}$ , and ${C S}^{M *} < {C S}^{S *}$ , if $α < α_{M}$ .

Proposition 4(ⅰ) indicates that innovation subsidies and the inclusion of new drugs in the health insurance plan can both improve patient welfare. This is understandable, because the above two incentive policies can increase the innovation level of the new drug and patients' willingness to choose the new drug. Intuitively, the inclusion of new drugs in the health insurance plan performs better than does innovation subsidies in improving patient welfare, because such scheme helps reduce patients' actual expenditure for new drugs. However, Proposition 4(ⅱ) shows that this intuition holds only when the copayment level is sufficiently high (i.e.,

α \geq α_{M}

). Conversely, when the copayment level is relatively low (i.e.,

α < α_{M}

), this intuition will break, that is, patients are better off under innovation subsidies. The underlying reason is as follows. The inclusion of new drugs in the health insurance plan can stimulate the new drug's demand, but when the copayment level is relatively low, its incentive effect is limit, and the new drug's innovation level is lower than that with innovation subsidies. Recall that

α_{M}

is decreasing in

t

. Thus, we can conclude that with a larger

t

, the inclusion of new drug in the health insurance plan is more likely to perform better than innovation subsidies from the perspective of patient welfare improvement.

Combining results in Propositions 1

\sim

4, we show that when the copayment level is relatively low (i.e.,

α < \tilde{α}

), innovation subsidies strictly superior to the inclusion of new drugs in the health insurance plan, because the innovation level and accessibility of the new drug, the new drug's profit, patient welfare and social welfare under innovation subsidies are all higher than those under the inclusion of new drugs in the health insurance plan. On the other extreme, when the copayment level is high enough (i.e.,

α \geq α_{M}

), the inclusion of new drugs in the health insurance plan superior to innovation subsidies in terms of improving innovation level and accessibility of the new drug, the new drug producer's profit as well as patient welfare. This finding provides valuable managerial insights for the regulator who should implement policies to promote pharmaceutical innovation in a competitive therapeutic market. More specifically, if the regulator aims to improve social welfare, he should choose innovation subsidies to reduce the new drug producer's innovation cost and raise her incentives to innovate. Nevertheless, if the regulator devotes to improve patient welfare, he should choose the inclusion of new drugs in the health insurance plan and enhance the copayment level properly to incentivize pharmaceutical innovation by expanding market demand for the new drug.

5 Hybrid Policy

In previous section, we focus on the regulator's optimal selection between innovation subsidies and the inclusion of new drugs in the health insurance plan and show that from the perspective of social welfare maximization, innovation subsidies are preferred, while from the perspective of patient welfare maximization, the inclusion of new drugs in the health insurance plan is optimal, when the copayment level is sufficiently high. Therefore, an immediate question arises: Can innovation subsidies and the inclusion of new drugs in the health insurance plan be adopted together to coordinate the above two policy objectives and become the optimal policy? For ease of exposition, we call this the hybrid policy. We first characterize the equilibrium outcomes under hybrid policy and then conduct comparison among scenarios

S

M

and

B

to evaluate the incentive effect of the hybrid policy and provide the feasible scope of the hybrid policy.

5.1 Equilibrium Outcomes Under Hybrid Policy

Under the hybrid policy, the profit functions of the new drug producer and the general drug producer are given by

\begin{array}{rcl} Π_{n} (p_{n}, θ) = p_{n} (\frac{1}{2} + \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}) - (1 - s) κ θ^{2}, \end{array}

(16)

\begin{array}{rcl} Π_{g} (p_{g}) = p_{g} (\frac{1}{2} - \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}) . \end{array}

(17)

Accordingly, the social welfare function is

{S W}^{B} (s) = Π_{n}^{B} + Π_{g}^{B} + {C S}^{B} - α p_{n}^{B} d_{n}^{B} - s κ θ^{2},

(18)

where

{C S}^{B} = \int_{0}^{\frac{1}{2} + \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}} U_{n} d x + \int_{\frac{1}{2} + \frac{θ + p_{g} - (1 - α) p_{n}}{2 t}}^{1} U_{g} d x

Similar to the analytical logic in Section 4, we solve the above game by backward induction, and obtain the following lemma.

Lemma 5 Under hybrid policy, there exists a threshold $α_{M} (= \frac{9 κ t + 1}{27 κ t})$ such that, $(i)$ when $α \geq α_{M}$ , the equilibrium innovation subsidy level is $s^{B *} = 0$ ; the equilibrium innovation level of the new drug is $θ^{B *} = \frac{3 t}{18 (1 - α) κ t - 1}$ ; the prices of the new drug and the general drug are $p_{n}^{B *} = \frac{18 κ t^{2}}{18 (1 - α) κ t - 1}$ , $p_{g}^{B *} = \frac{2 t (9 (1 - α) κ t - 1)}{18 (1 - α) κ t - 1}$ ; the market shares of the two types of drugs are $d_{n}^{B *} = \frac{9 (1 - α) κ t}{18 (1 - α) κ t - 1}$ , $d_{g}^{B *} = \frac{9 (1 - α) κ t - 1}{18 (1 - α) κ t - 1}$ . Accordingly, the equilibrium profits of the new drug producer and the general drug producer are $Π_{n}^{B *} = \frac{9 κ t^{2}}{18 (1 - α) κ t - 1}$ , and $Π_{g}^{B *} = \frac{2 t (9 (1 - α) κ t - 1)^{2}}{(18 (1 - α) κ t - 1)^{2}}$ , the consumer welfare is ${C S}^{B *} = v - \frac{(810 (1 - α)^{2} κ^{2} t^{2} - 144 (1 - α) κ t + 5) t}{2 (18 (1 - α) κ t - 1)^{2}}$ , and the social welfare is ${S W}^{B *} = v - \frac{(162 (1 - α)^{2} κ^{2} t^{2} - 18 (3 - 4 α) κ t + 1) t}{2 (18 (1 - α) κ t - 1)^{2}}$ . $(i i)$ When $α < α_{M}$ , the equilibrium innovation subsidy level is $s^{B *} = \frac{1 + 9 κ t - 27 α κ t}{27 (1 - α) κ t}$ ; the equilibrium innovation level is $θ^{B *} = \frac{9 t}{36 κ t - 5}$ ; the prices of the new drug and the general drug are $p_{n}^{B *} = \frac{2 t (18 κ t - 1)}{(36 κ t - 5) (1 - α)}$ , $p_{g}^{S *} = \frac{4 t (9 κ t - 2)}{36 κ t - 5}$ ; the equilibrium market demands of two types of drugs are $d_{n}^{B *} = \frac{18 κ t - 1}{36 κ t - 5}$ , $d_{g}^{B *} = \frac{2 (9 κ t - 2)}{36 κ t - 5}$ . Accordingly, the equilibrium profits of the new drug producer and the general drug producer are $Π_{n}^{B *} = \frac{(18 κ t - 1) t}{36 κ t - 5}$ , $Π_{g}^{B *} = \frac{8 t (9 κ t - 2)^{2}}{(36 κ t - 5)^{2}}$ , the consumer welfare is ${C S}^{B *} = v - \frac{t (3240 κ^{2} t^{2} - 1224 κ t + 103)}{2 (36 κ t - 5)^{2}}$ , and the social welfare is ${S W}^{B *} = v - \frac{(18 κ t - 7) t}{2 (36 κ t - 5)}$ .

The implication of Lemma 5 is that the hybrid policy is not always the best choice for the regulator. Only when the copayment level is relatively low (i.e.,

α < α_{M}

), can the regulator adopt innovation subsidies and include new drugs in the health insurance plan simultaneously. Nevertheless, when the copayment is sufficiently high (i.e.,

α \geq α_{M}

), the regulator has no incentives to implement the hybrid policy.

5.2 Comparison of Scenarios B, S and M

Next, we further compare the price, market share, and innovation level of the new drug, the profit of the new drug producer, the regulator's subsidy level, as well as consumer surplus and social welfare under three incentive policies. The results are summarized in Proposition 5.

Proposition 5 When $α < α_{M}$ , we have, $(i)$ $p_{n}^{B *} > p_{n}^{S *}$ , $d_{n}^{B *} = d_{n}^{S *} > d_{n}^{M *}$ , $θ^{B *} = θ^{S *} > θ^{M *}$ , $Π_{n}^{B *} > Π_{n}^{S *} > Π_{n}^{M *}$ ; $(i i)$ $s^{B *} < s^{S *}$ , ${S W}^{B *} = {S W}^{S *} > {S W}^{M *}$ , ${C S}^{B *} = {C S}^{S *} > {C S}^{M *}$ .

Intuitively, the hybrid policy is the best choice for the regulator in increasing innovation level and accessibility of the new drug, raising the new drug producer's profit as well as improving patient welfare and social welfare. However, Proposition 5 shows the hybrid policy adoption helps lower the regulator's subsidy level and increase the new drug producer's profit, but it will push up the price the new drug. Since the increase in price of the new drug completely absorbs the demand-expansion effect resulted from the inclusion of new drugs in the health insurance plan, the hybrid policy is not better than innovation subsidies in enhancing the innovation level and the accessibility of the new drug as well as patient welfare and social welfare. This result suggests that although the hybrid policy is not necessarily superior than innovation subsidies from the perspective of social welfare maximization, it benefits to save the financial expenditure of the regulator.

We further discuss preferences of the regulator, the new drug producer and consumers on three different incentive schemes for pharmaceutical innovation in a competitive therapeutic market. The results are summarized in Table 3.

Table 3 Preferences of the regulator, the new drug producer and patients on different incentives schemes for pharmaceutical innovation in a competitive therapeutic market

Conditions	Regulator prefers	Producer prefers	Patients prefers
$α \geq α_{M}$	S	M	M
$α < α_{M}$	S/B	B	S/B

Note: S, M and B represent the scenarios of innovation subsidies, the inclusion of new drugs in the health insurance plan and hybrid policy respectively.

From Table 3, we can clearly see that the copayment level is an important factor affecting policy preference of the regulator, the new drug producer, and patients. More specifically, when the copayment level is high enough (i.e.,

α \geq α_{M}

), the innovation subsidy is optimal for pharmaceutical innovation from the perspective of social welfare maximization, but both the producer and consumers prefer the inclusion of new drugs in the health insurance plan. This implies that the regulator, the new drug producer and patients have inconsistent policy preference. On the contrary, when the copayment level is relatively low (i.e.,

α < α_{M}

), the hybrid policy can align the preferences of the regulator, the producer and patients. In other words, all players can achieve a "win-win-win" situation under hybrid policy. Recalling from Proposition 5, the hybrid policy can also reduce the regulator's subsidy expenditure. Thus, hybrid policy becomes the optimal incentive scheme, when

α < α_{M}

We also examine the impacts of therapeutic competition and innovation efficiency on all players' policy preferences, and we numerically show the results in Figure 3. From Figure 3, we can clearly see that as the therapeutic competition between two types of drugs decreases (larger

t

) and the innovation efficiency decreases (larger

κ

), the "win-win-win" zone shrinks, implying that it is more likely for the regulator, the producer and consumers to disagree on preference for the hybrid policy, which results in the risk of policy failure. This finding provides valuable insights for the regulator who is devoted to implementing policies to promote pharmaceutical innovation. On the one hand, the regulator should induce the producers committing to pharmaceutical innovation to enhance their innovation efficiency and increase therapeutic competition intensity appropriately. On the other hand, the regulator should lower the copayment level properly. Thus, the regulator can eliminate the inconsistency of all players' policy preference by simultaneously adopting innovation subsidies and the inclusion of new drugs in the health insurance plan (i.e., the hybrid policy), and achieve a "win-win-win" situation for the regulator, the drug producer, and patients in a competitive environment.

Figure 3 Changes of "win-win-win" zone with $κ$ and $t$

Full size|PPT slide

6 Extension: Endogenous Copayment Level

In the base model, we investigate a scenario of exogenous copayment level employed very widely in practice. Here, we extend the base case by considering the scenario where the regulator determines the copayment level endogenously to maximize social welfare. The sequence of the events is similar to our base model except that the regulator determines the copayment level

α

in the first stage to maximize social welfare. For ease of exposition, we use the superscript "ME" to denote this scenario. By backward induction, we obtain the equilibrium outcomes summarized in the following lemma.

Lemma 6 In the case where the copayment level $α$ is endogenous, there exists a unique competitive equilibrium, if $κ > \frac{5}{18 t}$ holds. The optimal copayment level is $α^{M E *} = \frac{162 κ^{2} t^{2} - 18 κ t - 1}{36 κ t (18 κ t - 5)}$ ; the innovation level of the new drug is $θ^{M E *} = \frac{6 (8 κ t - 5) t}{486 κ^{2} t^{2} - 198 κ t + 11}$ ; the prices of new drug and general drug are $p_{n}^{M E *} = \frac{36 κ (8 κ t - 5) t^{2}}{486 κ^{2} t^{2} - 198 κ t + 11}$ ; $p_{g}^{M E *} = \frac{3 (162 κ^{2} t^{2} - 78 κ t + 7) t}{486 κ^{2} t^{2} - 198 κ t + 11}$ ; the market share of new drug and general drug are $d_{n}^{M E *} = \frac{486 κ^{2} t^{2} - 162 κ t + 1}{2 (486 κ^{2} t^{2} - 198 κ t + 11)}$ ; $d_{g}^{M E *} = \frac{3 (162 κ^{2} t^{2} - 78 κ t + 7)}{2 (486 κ^{2} t^{2} - 198 κ t + 11)}$ . Accordingly, the equilibrium profits of the new drug producer and the general drug producers, patient welfare and social welfare are $Π_{n}^{M E *} = \frac{18 κ (18 κ t - 5) t^{2}}{486 κ^{2} t^{2} - 198 κ t + 11}$ , $Π_{g}^{M E *} = \frac{9 (162 κ^{2} t^{2} - 78 κ t + 7)^{2} t}{2 (486 κ^{2} t^{2} - 198 κ t + 11)^{2}}$ , ${C S}^{M E *} = v - \frac{(211 - 3852 κ t + 19278 κ^{2} t^{2} - 17496 κ^{3} t^{3}}{2 (72 κ t - 7)^{2}}$ , ${S W}^{M E *} = v - \frac{((5832 κ^{3} t^{3} + 2430 κ^{2} t^{2} - 1062 κ t + 67) t}{2 (72 κ t - 7)^{2}}$ .

According to Lemma 6, we can clearly see that when the new drug producer's innovation is not efficient (i.e.,

κ > \frac{5}{18 t}

), there exists an optimal copayment level, which depends on both the new drug producer's innovation efficiency and the therapeutic competition intensity. Given that the endogenous determination of copayment level is quite complicated, the exogenous copayment level is widely adopted in reality even though it may lead to suboptimal results.

We further evaluate incentive effects between innovation subsidies and the inclusion of new drugs in the health insurance plan with endogenous copayment level. By comparing the innovation level, market share of the new drug, the new drug producer's profit, patient welfare and social welfare, we obtain the following proposition.

Proposition 6 In the case where $α$ is endogenous, we have, $(i)$ $θ^{M E *} > θ^{S *}$ and $d_{n}^{M E *} > d_{n}^{S *}$ , if $t > \frac{3 + \sqrt{43}}{18 κ}$ ; $(i i)$ there exists a threshold $t_{1}$ such that, $Π_{n}^{M E *} > Π_{n}^{S *}$ , if $t > t_{1}$ and $Π_{n}^{M E *} < Π_{n}^{S *}$ , otherwise; $(i i i)$ ${S W}^{S *} > {S W}^{M E *}$ ; $(i v)$ there exists a threshold $t_{2}$ such that, ${C S}^{S *} > {C S}^{M E *}$ , if $t < t_{2}$ and ${C S}^{S *} < {C S}^{M E *}$ , if $t > t_{2}$ .

As shown in Proposition 6, consistent with our main model, when

α

is endogenous, from the perspective of social welfare maximization, the innovation subsidy scheme superiors to the inclusion of new drugs in the health insurance plan (see Figure 4). However, the inclusion of new drugs in the health insurance plan may either be better or be worse than the innovation subsidy scheme in improving the accessibility and innovation level of the new drug, the new drug producer's profitability and patient welfare, which depends on the therapeutic competition between the new drug and the general drug (i.e.,

t

). Specifically, if the therapeutic competition between the two drugs is less fierce (i.e., larger

t

), the inclusion of new drugs in the health insurance plan outweighs the innovation subsidy policy in improving the accessibility and innovation level of the new drug, the new drug producer's profitability and patient welfare, and verse versa.

Figure 4 The comparison of social welfare

Full size|PPT slide

Recalling from Propositions 1

\sim

4, when

α

is exogenous, with a larger

t

, the inclusion of new drugs in the health insurance plan is more likely to outperform the innovation subsidy scheme in terms of innovation level and accessibility of the new drug, the profitability of the new drug producer, and patient welfare. As such, the endogenous copayment level does not change our main results qualitatively, which verifies the robustness of our main findings.

7 Conclusion

To address the market failure in new drug creation, regulators of various countries constantly put forward incentive policies from national strategic level to promote pharmaceutical innovation and improve the accessibility of new drugs. In this paper, we evaluate incentive effects of three incentive policies, namely, innovation subsidies, the inclusion of new drugs in the health insurance plan, and a hybrid policy. More specifically, we focus on impacts of the above three policies on drug's innovation level, the accessibility of the new drug, drug producers' profits as well as social welfare and patient welfare by examining the strategic interaction among the regulation, drug producers and consumers in a competitive therapeutic market.

The analytical results show that although innovation subsidies and the inclusion of new drugs in the health insurance plan can both enhance the innovation incentives and profits of the new drug producer, expand the accessibility of the new drug, and improve social welfare. Comparatively, innovation subsidies always perform better than the inclusion of new drugs in the health insurance plan in terms of social welfare improvement, but worsen in increasing innovation level and accessibility of the new drug, and the new drug producer's profitability as well as patient welfare when the copayment level is relatively high. When it comes to a hybrid policy combined with innovation subsidies and the inclusion of new drugs in the health insurance plan, we identify that the hybrid policy benefits to reduce the regulator's subsidy expenditure and increase the innovative producer's profits, but push up the new drug's price, which allows the hybrid policy does not perform better than the single incentive policy. We also show that when the copayment level is relatively high, the hybrid policy may result in over-innovation and bring about a deadweight loss for social welfare. In such a situation, the regulator will abandon the hybrid policy. However, when the copayment level is relatively low, the regulator, the producer, and consumers can achieve a "win-win-win" situation under the hybrid policy, implying that in such a situation, the hybrid policy is the optimal choice from the perspectives of all players. Our results provide valuable insights for the regulator who should implement incentive policies to promote pharmaceutical innovation. The details are as follows.

First, the copayment level plays significant role in policy choice of the regulator for pharmaceutical innovation. Specifically, if the regulator aims to improve patient welfare, he should incorporate new drugs to health insurance plan and increase the copayment level appropriately, by which promotes pharmaceutical innovation by expanding the market demand of the new drug. On the other hand, if the regulator aims to improve social welfare, he should choose innovation subsidies, if the copayment level is relatively high, and the hybrid policy, if the copayment is low. Second, the regulator's incentive policies should match the innovation efficiency of new drug producer. Our analysis shows that the producer with high innovation efficiency performs better under innovation subsidies, while the producer with low innovation efficiency is better off under the inclusion of new drugs in the health insurance plan. Finally, we reveal that the copayment level can adjust the producer's innovation output and the incentive effects of different policies, which partially confirms the feasibility of health insurance plan guiding the innovation and upgrade of pharmaceutical industry. Moreover, the regulator should determine appropriate copayment level, which can not only enhance pharmaceutical producers' innovation incentives but also benefits to solve practical problem of "expensive medical treatment".

There are several limitations in this paper, which also offer some opportunities for future research. First, we evaluate the incentive effects of different policies for pharmaceutical innovation using a static model. In the future, we can take pharmaceutical industry evolution into account, and investigate how the regulator should adjust the innovation incentive policies over time. Second, we assume the regulator does not face subsidy budget constraints. However, the innovation subsidy may be difficult to reach the optimal level because of the limited financial budget in reality. Therefore, it would be worth considering subsidy budget constraint when determining the optimal incentive scheme.

Appendix

Proof of Lemma 1 By backward induction, we first consider the pricing decisions. From

\frac{\partial^{2} Π_{n} (p_{n}, θ)}{\partial {p_{n}}^{2}} = - 1 / t

\frac{\partial^{2} Π_{g} (p_{g})}{\partial {p_{g}}^{2}} = - 1 / t

, we derive that

Π_{i}

(i = n, g)

, is concave in

p_{i}

. Letting

\frac{\partial Π_{n} (p_{n}, θ)}{\partial p_{n}} = 0

\frac{\partial Π_{g} (p_{g})}{\partial p_{g}} = 0

, we get

p_{n}^{R} = t + \frac{θ}{3}

p_{g}^{R} = t - \frac{θ}{3}

. Next, we consider the new drug producer's innovation decision. Substituting these into the profit function of the new drug producer yields

Π_{n} (θ) = \frac{{(3 t + θ)}^{2}}{18 t} - κ θ^{2}

. From

\frac{\partial^{2} Π_{n} (θ)}{\partial θ^{2}} = - 2 κ + \frac{1}{9 t}

, we know that

Π_{n} (θ)

is concave in

θ

\frac{1}{18 t} < κ

holds. Let

\frac{\partial Π_{n} (θ)}{\partial θ} = 0

, we have

θ^{N *} = \frac{3 t}{18 κ t - 1}

. Substituting this into the expression of prices, we get

p_{n}^{N *} = \frac{18 κ t^{2}}{18 κ t - 1}

p_{g}^{N *} = \frac{2 t (9 κ t - 1)}{18 κ t - 1}

. Substituting

p_{n}^{N *}

p_{g}^{N *}

and

θ^{N *}

into the demand functions of the new drug producer and the general producer yields

d_{n}^{N *} = \frac{9 κ t}{18 κ t - 1}

d_{g}^{N *} = \frac{9 κ t - 1}{18 κ t - 1}

. Clearly, when

κ > \frac{1}{9 t}

holds, we have

d_{n}^{N *}, d_{g}^{N *} > 0

. Accordingly, the equilibrium profits of the new drug and the general drug producers are

Π_{n}^{N *} = \frac{9 κ t^{2}}{18 κ t - 1}

Π_{g}^{N *} = \frac{2 t (9 κ t - 1)^{2}}{(18 κ t - 1)^{2}}

. On the other hand, when

κ \leq \frac{1}{9 t}

d_{g}^{N *} \leq 0

, implying that the general drug producer is squeezed out of the market, and the new drug producer monopolies the whole market, that is,

d_{n} (θ) = 1

d_{g} (θ) = 0

. Next, we consider the monopolist's pricing decisions. To guarantee the patient located at 1 to choose the new drug, we have

U_{n} = v + θ - p_{n} - t \geq 0

, which leads to

p_{n} = v + θ - t

. Substituting

p_{n}

into the

Π_{n} (θ)

yields

Π_{n} (θ) = v + θ - t - κ θ^{2}

. From

\frac{\partial^{2} Π_{n} (Π (θ))}{\partial θ^{2}} = - 2 κ < 0

, we derive that

Π_{n} (θ)

is concave in

θ

. Let the first order condition be equal to 0, that is,

\frac{\partial Π_{n} (θ)}{\partial θ} = 1 - 2 κ θ = 0

, we have

θ^{N *} = \frac{1}{2 κ}

. Substituting

θ^{N *}

into

p_{n}

, we have

p_{n}^{N *} = v - t + \frac{1}{2 κ}

, Substituting

θ^{N *}

into

Π_{n} (θ)

yields

Π_{n}^{N *} = v - t + \frac{1}{4 κ}

Proof of Lemma 2 Similar to the scenario without regulator intervention, we first consider pricing decisions. From

\frac{\partial^{2} Π_{n} (p_{n}, θ)}{\partial p_{n}^{2}} = - \frac{1}{t}

\frac{\partial^{2} Π_{g} (p_{g})}{\partial p_{g}^{2}} = - \frac{1}{t}

, we show that

Π_{i}

(i = n, g)

, is concave in

p_{i}

. Let

\frac{\partial Π_{n} (p_{n}, θ)}{\partial p_{n}} = 0

\frac{\partial Π_{g} (p_{g})}{\partial p_{g}} = 0

, we get

p_{n}^{R} = t + \frac{θ}{3}

p_{g}^{R} = t - \frac{θ}{3}

. Next, we turn to the new drug producer's innovation decisions. Substituting these into the expression

Π_{n}

, we get

Π_{n} (θ | s) = \frac{(3 t + θ)^{2}}{18 t} - (1 - s) κ θ^{2}

. From

\frac{\partial^{2} Π_{n} (θ | s)}{\partial θ^{2}} = \frac{1}{9 t} - 2 (1 - s) κ

Π_{n} (θ | s)

is concave in

θ

, if

κ > \frac{1}{18 (1 - s) t}

. From the first order condition

\frac{3 t + θ}{9 t} - 2 (1 - s) κ θ = 0

. we get

θ^{R} (s) = \frac{3 t}{18 (1 - s) κ t - 1}

. Substituting

θ^{R} (s)

into the expressions of prices, market share and profits of the two drug producers yields the results of Lemma 2.

Proof of Lemma 3 Based on results in Lemma 2, we have,

S W (s) = - \frac{\begin{array}{l} 162 κ^{2} s^{2} t^{3} - 648 κ^{2} s^{2} t^{2} v - 324 κ^{2} s t^{3} + 1296 κ^{2} s t^{2} v + 162 κ^{2} t^{3} - 648 κ^{2} t^{2} v + 72 κ s t^{2} \\ - 72 κ s t v - 54 κ t^{2} + 72 κ t v + t - 2 v \end{array}}{2 (18 (1 - s) κ t - 1)^{2}} .

From

\frac{\partial^{2} S W (s)}{\partial s^{2}} = - \frac{1458 κ^{2} t^{3} (12 κ t s - 1)}{(18 (1 - s) κ t - 1)^{4}}

, we show that

S W (s)

is concave in

s

, if

s > \frac{1}{12 κ t}

. Letting

\frac{\partial S W (s)}{\partial s} = 0

, we get

s^{S *} = \frac{9 κ t + 1}{27 κ t}

. Substituting

s^{S *}

into

θ^{R *}

, we get

θ^{S *} = \frac{9 t}{36 κ t - 5}

. Moreover, we substitute

θ^{S *}

into

p_{n}^{R}

and

p_{g}^{R}

, and derive

p_{n}^{S *} = \frac{2 t (18 κ t - 1)}{36 κ t - 5}

p_{g}^{S *} = \frac{4 t (9 κ t - 2)}{36 κ t - 5}

. Substituting

s^{S *}

θ^{S *}

p_{n}^{S *}

and

p_{g}^{S *}

into the demand functions, profit functions, consumer surplus and social welfare, we get

d_{n}^{S *} = \frac{18 κ t - 1}{36 κ t - 5}

d_{g}^{S *} = \frac{4 (9 κ t - 2)}{36 κ t - 5}

Π_{n}^{S *} = \frac{(18 κ t - 1) t}{36 κ t - 5}

Π_{g}^{S *} = \frac{8 t (9 κ t - 2)^{2}}{(36 κ t - 5)^{2}}

{C S}^{S *} = v - \frac{t (3240 κ^{2} t^{2} - 1224 κ t + 103)}{2 (36 κ t - 5)^{2}}

{S W}^{S *} = v - \frac{(18 κ t - 7) t}{2 (36 κ t - 5)}

. To guarantee the existence of competitive equilibrium, we require

κ > \frac{2}{9 t}

. This condition also ensures decision variables, profits, consumer surplus, and social welfare are all positive.

Proof of Lemma 4 By backward induction, we first consider pricing decisions. From

\frac{\partial^{2} Π_{n} (p_{n}, θ)}{\partial p_{n}^{2}} = - \frac{1 - α}{t}

\frac{\partial^{2} Π_{g} (p_{g})}{\partial p_{g}^{2}} = - \frac{1}{t}

, we know that

Π_{i}

(i = n, g)

, is concave in

p_{i}

. Letting

\frac{\partial Π_{n} (p_{n}, θ)}{\partial p_{n}} = 0

\frac{\partial Π_{g} (p_{g})}{\partial p_{g}} = 0

, we get

p_{n}^{R} (θ) = \frac{θ + 3 t}{3 (1 - α)}

p_{g}^{R} (θ) = t - \frac{θ}{3}

. Next, we consider the new drug producer's innovation decision. Substituting the response functions of prices into the expression of

Π_{n} (p_{n}, θ)

, and solving the second derivative of

Π_{n} (p_{n}, θ)

with respect to

θ

, we obtain

\frac{\partial^{2} Π_{n} (θ)}{\partial θ^{2}} = - 2 κ + \frac{1}{9 (1 - α) t}

. Clearly,

Π_{n} (p_{n}, θ)

is concave in

θ

, if

κ > \frac{1}{18 (1 - α) t}

. From the first order condition

- 2 κ θ + \frac{3 t + θ}{9 (1 - α) t} = 0

, we get

θ^{M *} = \frac{3 t}{18 (1 - α) κ t - 1}

. Substituting

θ^{M *}

into to

p_{n}^{R} (θ)

and

p_{g}^{R} (θ)

, we have

p_{n}^{M *} = \frac{18 κ t^{2}}{18 (1 - α) κ t - 1}

p_{g}^{M *} = \frac{2 t (9 (1 - α) κ t - 1)}{18 (1 - α) κ t - 1}

. Substituting

θ^{M *}

p_{n}^{M}

and

p_{g}^{M *}

into the demand functions, the profit functions, consumer surplus and social welfare yields

d_{n}^{M *} = \frac{9 (1 - α) κ t}{18 (1 - α) κ t - 1}

d_{g}^{M *} = \frac{9 (1 - α) κ t - 1}{18 (1 - α) κ t - 1}

Π_{n}^{M *} = \frac{9 κ t^{2}}{18 (1 - α) κ t - 1}

Π_{g}^{M *} = \frac{2 t (9 (1 - α) κ t - 1)^{2}}{(18 (1 - α) κ t - 1)^{2}}

{C S}^{M *} = v - \frac{(810 (1 - α)^{2} κ^{2} t^{2} - 144 (1 - α) κ t + 5) t}{2 (18 (1 - α) κ t - 1)^{2}}

{S W}^{M *} = v - \frac{(162 κ^{2} t^{2} - 54 κ t + 1) t}{2 (18 κ t - 1)^{2}}

. To guarantee the existence of competitive equilibrium, we require

κ > \frac{1}{9 (1 - α) t}

. This condition also ensures positive decision variables and profits.

Proof of Proposition 1 Comparing

θ^{M *}

θ^{S *}

, and

θ^{N *}

yields

\begin{array}{rcl} θ^{M *} - θ^{N *} = \frac{54 κ α t^{2}}{(18 (1 - α) κ t - 1) (18 κ t - 1)}, θ^{S *} - θ^{N *} = \frac{6 (9 κ t + 1)}{(36 κ t - 5) (18 κ t - 1)}, \\ θ^{M *} - θ^{S *} = \frac{6 (9 κ t (3 α - 1) - 1) t}{(18 (1 - α) κ t - 1) (36 κ t - 5)} . \end{array}

Clearly, when

κ > max (\frac{2}{9 t}, \frac{1}{9 (1 - α) t})

holds,

θ^{M *}, θ^{S *} > θ^{N *}

. When

α \geq \frac{9 κ t + 1}{27 κ t} = α_{M}

holds, we have

θ^{M *} \geq θ^{S *}

; otherwise,

θ^{M *} < θ^{S *}

. Similarly, by comparing

d_{n}^{M *}

d_{n}^{S *}

, and

d_{n}^{N *}

, we have

d_{n}^{M *}, d_{n}^{S *} > d_{n}^{N *}

d_{n}^{M *} - d_{n}^{S *} = \frac{9 κ t (3 α - 1) - 1}{(18 (1 - α) κ t - 1) (36 κ t - 5)}

. We can easily prove that

d_{n}^{M *} \geq d_{n}^{S *}

, if

α \geq α_{M}

; otherwise,

d_{n}^{M *} < d_{n}^{S *}

. Moreover,

\frac{\partial α_{M}}{\partial t} = - \frac{1}{27 κ t^{2}} < 0

Proof of Propositions 2

\sim

4. The proof for Proposition 2, Proposition 3, and Proposition 4 are similar to that for Proposition 1 and thus omitted for brevity. Comparing the two thresholds

α_{M}

and

\tilde{α}

yields

Δ α = α_{M} - \tilde{α} = \frac{(9 κ t + 1) (36 κ t - 5)}{54 κ t (18 κ t - 1)} > 0

when

κ > max (\frac{2}{9 t}, \frac{1}{9 (1 - α) t})

Proof of Lemma 5 The proof of Lemma 5 is analogous with the proof of Lemma 3 and Lemma 4, and so omitted here.

Proof of Proposition 5 First, we compare the innovation level, market demand and price of new drug as well as the profit of the new drug producer across Scenarios B, S and M. To ensure the compatibility of conditions under different policies, we require

κ > max (\frac{2}{9 t}, \frac{1}{9 (1 - α) t})

. From the results of Lemmas 2

\sim

5, we can clearly see that

θ^{B *} = θ^{S *} > θ^{M *}

d_{n}^{B *} = d_{n}^{S *} > d_{n}^{M *}

Π_{n}^{S *} > Π_{n}^{M *}

, if

α < α_{M} = \frac{9 κ t + 1}{27 κ t}

\begin{array}{rcl} p_{n}^{B *} - p_{n}^{S *} = \frac{2 t (18 κ t - 1) α}{(36 κ t - 5) (1 - α)} > 0, Π_{n}^{B *} - Π_{n}^{S *} = \frac{(18 κ t - 1) α}{(36 κ t - 5) (1 - α)} > 0. \end{array}

Similarly, comparing the innovation subsidy, social welfare and consumer surplus under Scenarios

B

S

, and

M

yields

s^{B *} - s^{S *} = \frac{(- 18 α κ t)}{27 (1 - α) κ t} < 0

{S W}^{B *} = {S W}^{S *} > {S W}^{M *}

, and

{C S}^{B *} = {C S}^{S *} > {C S}^{M *}

, if

α < α_{M} = \frac{9 κ t + 1}{27 κ t}

Proof of Lemma 6 Let the first order condition

\frac{\partial S W (α)}{\partial α} = 0

, we have

α^{M E *} = \frac{162 κ^{2} t^{2} - 18 κ t - 1}{36 κ t (18 κ t - 5)}

. It is easy to prove that

0 < α^{M E *} < 1

, when

κ > \frac{5}{18 t}

, Substituting

α^{M E *}

into

θ (α)

p_{n} (α)

p_{g} (α)

d_{n} (α)

and

d_{g} (α)

yields

θ^{M E *} = \frac{6 (8 κ t - 5) t}{486 κ^{2} t^{2} - 198 κ t + 11}

p_{n}^{M E *} = \frac{36 κ (8 κ t - 5) t^{2}}{486 κ^{2} t^{2} - 198 κ t + 11}

p_{g}^{M E *} = \frac{3 (162 κ^{2} t^{2} - 78 κ t + 7) t}{486 κ^{2} t^{2} - 198 κ t + 11}

d_{n}^{M E *} = \frac{486 κ^{2} t^{2} - 162 κ t + 1}{2 (486 κ^{2} t^{2} - 198 κ t + 11)}

and

d_{g}^{M E *} = \frac{3 (162 κ^{2} t^{2} - 78 κ t + 7)}{2 (486 κ^{2} t^{2} - 198 κ t + 11)}

. Substituting

θ^{M E *}

p_{n}^{M E *}

p_{g}^{M E *}

into the expressions of profit functions, consumer surplus, and social welfare yields

Π_{n}^{M E *} = \frac{18 κ (18 κ t - 5) t^{2}}{486 κ^{2} t^{2} - 198 κ t + 11}

Π_{g}^{M E *} = \frac{9 (162 κ^{2} t^{2} - 78 κ t + 7)^{2} t}{2 (486 κ^{2} t^{2} - 198 κ t + 11)^{2}}

{C S}^{M E *} = v - \frac{(211 - 3852 κ t + 19278 κ^{2} t^{2} - 17496 κ^{3} t^{3}}{2 (72 κ t - 7)^{2}}

{S W}^{M E *} = v - \frac{((5832 κ^{3} t^{3} + 2430 κ^{2} t^{2} - 1062 κ t + 67) t}{2 (72 κ t - 7)^{2}}

Proof of Proposition 6 Comparing the innovation level and market share of the new drug and the new drug producer's profit under scenario ME and those under scenario S, we have

\begin{array}{rcl} θ^{M E *} - θ^{S *} = \frac{3 t (162 κ^{2} t^{2} - 54 κ t - 17)}{(486 κ^{2} t^{2} - 198 κ t + 11) (36 κ t - 5)}, \\ d_{n}^{M E *} - d_{n}^{S *} = \frac{(162 κ^{2} t^{2} - 54 κ t - 17)}{2 (486 κ^{2} t^{2} - 198 κ t + 11) (36 κ t - 5)}, \\ Π_{n}^{M E *} - Π_{n}^{S *} = \frac{11664 κ^{3} t^{2} - 8748 κ^{3} t^{3} - 4860 κ^{2} t^{3} + 4050 κ^{2} t^{2} + 450 κ t^{2} - 396 κ t + 11}{2 (486 κ^{2} t^{2} - 198 κ t + 11) (36 κ t - 5)} . \end{array}

With the help of graphic analysis, we can prove that

θ^{M E *} > θ^{S *}

and

d_{n}^{M E *} > d_{n}^{S *}

, if

t > \frac{3 + \sqrt{43}}{18 κ}

, and

Π_{n}^{M E *} > Π_{n}^{S *}

, if

t > t_{1}

, where

t_{1}

is one real root of

11664 κ^{3} t^{2} - 8748 κ^{3} t^{3} - 4860 κ^{2} t^{3} + 4050 κ^{2} t^{2} + 450 κ t^{2} - 396 κ t + 11 = 0

Furthermore, comparing social welfare under Scenario S and Scenario ME, we have

\begin{array}{rcl} {S W}^{S *} - {S W}^{M E *} = \frac{(324 κ^{2} t^{2} - 27 κ t + 5)^{2}}{(72 κ t - 7)^{2} (36 κ t - 5)} > 0 . \end{array}

Finally, comparing consumer surplus under Scenario S and Scenario ME, we have

\begin{array}{rcl} {C S}^{S *} - {C S}^{M E *} = - \frac{3 t (3779136 κ^{5} t^{5} - 2414448 κ^{4} t^{4} + 459756 κ^{3} t^{3} - 35937 κ^{2} t^{2} + 1410 κ t - 38)}{(72 κ t - 7)^{2} (36 κ t - 5)^{2}} . \end{array}

We can prove that there exists a

t_{2}

such that,

{C S}^{S *} \geq {C S}^{M E *}

, if

t \leq t_{2}

and

{C S}^{S *} < {C S}^{M E *}

, if

t > t_{2}

, where

t_{2}

is one real root of

3779136 κ^{5} t^{5} - 2414448 κ^{4} t^{4} + 459756 κ^{3} t^{3} - 35937 κ^{2} t^{2} + 1410 κ t - 38 = 0

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Abstract
Key words
Cite this article
1 Introduction
2 Literature Review
Table 1 Summary of literature
3 Model Setup
Figure 1 The relationships between decisions makers
Table 2 Notations and definitions
4 Equilibrium Analysis
4.1 The Benchmark: Without Regulator Intervention
4.2 Innovation Subsidies
4.3 Inclusion of New Drugs in the Health Insurance Plan
4.4 Evaluation of Incentive Effects of the Two Policies
Figure 2 Preferences of the new drug producer over two incentive policies with $t$ under different $κ$
5 Hybrid Policy
5.1 Equilibrium Outcomes Under Hybrid Policy
5.2 Comparison of Scenarios B, S and M
Table 3 Preferences of the regulator, the new drug producer and patients on different incentives schemes for pharmaceutical innovation in a competitive therapeutic market
Figure 3 Changes of "win-win-win" zone with $κ$ and $t$
6 Extension: Endogenous Copayment Level
Figure 4 The comparison of social welfare
7 Conclusion
Appendix
References

Received	Accepted	Published
2020-01-22	2020-07-24	2021-10-25
Issue Date
2021-10-25

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Abstract

Key words

Cite this article

1 Introduction

2 Literature Review

Table 1 Summary of literature

3 Model Setup

Figure 1 The relationships between decisions makers

Table 2 Notations and definitions

4 Equilibrium Analysis

4.1 The Benchmark: Without Regulator Intervention

4.2 Innovation Subsidies

4.3 Inclusion of New Drugs in the Health Insurance Plan

4.4 Evaluation of Incentive Effects of the Two Policies

Figure 2 Preferences of the new drug producer over two incentive policies with $t$ under different $κ$

5 Hybrid Policy

5.1 Equilibrium Outcomes Under Hybrid Policy

5.2 Comparison of Scenarios B, S and M

Table 3 Preferences of the regulator, the new drug producer and patients on different incentives schemes for pharmaceutical innovation in a competitive therapeutic market

Figure 3 Changes of "win-win-win" zone with $κ$ and $t$

6 Extension: Endogenous Copayment Level

Figure 4 The comparison of social welfare

7 Conclusion

Appendix

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References

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Footnotes

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1 Introduction

2 Literature Review

Table 1 Summary of literature

3 Model Setup

Figure 1 The relationships between decisions makers

Table 2 Notations and definitions

4 Equilibrium Analysis

4.1 The Benchmark: Without Regulator Intervention

4.2 Innovation Subsidies

4.3 Inclusion of New Drugs in the Health Insurance Plan

4.4 Evaluation of Incentive Effects of the Two Policies

Figure 2 Preferences of the new drug producer over two incentive policies with t under different κ

5 Hybrid Policy

5.1 Equilibrium Outcomes Under Hybrid Policy

5.2 Comparison of Scenarios B, S and M

Table 3 Preferences of the regulator, the new drug producer and patients on different incentives schemes for pharmaceutical innovation in a competitive therapeutic market

Figure 3 Changes of "win-win-win" zone with κ and t

6 Extension: Endogenous Copayment Level

Figure 4 The comparison of social welfare

7 Conclusion

Appendix

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{{custom_sec.title}}

References

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Footnotes

Figure 2 Preferences of the new drug producer over two incentive policies with $t$ under different $κ$

Figure 3 Changes of "win-win-win" zone with $κ$ and $t$