Optimizing Moral Hazard Management in Health Insurance Through Mathematical Modeling of Quasi-Arbitrage

Abstract

Moral hazard in health insurance arises when insured individuals are incentivized to over-utilize healthcare services, especially when they face low out-of-pocket costs. While existing literature primarily addresses moral hazard through qualitative studies, this paper introduces a quantitative approach by developing a mathematical model based on quasi-arbitrage conditions. The model optimizes health insurance design, focusing on the transition from Low-Deductible Health Plans (LDHPs) to High-Deductible Health Plans (HDHPs), and seeks to mitigate moral hazard by aligning the interests of both insurers and insured. Our analysis demonstrates how setting appropriate deductible levels and offering targeted premium reductions can encourage insured to adopt HDHPs while maintaining insurer profitability. The findings contribute to the theoretical framework of moral hazard mitigation in health insurance and offer actionable insights for policy design.

1. Introduction

As Barberis (2013) discusses in his comprehensive review of prospect theory, the concept of moral hazard in health insurance, where low out-of-pocket costs incentivize excessive use of healthcare services, can be understood through the lens of risk aversion and loss aversion in decision-making. Early studies have examined the relationship between copayment levels and healthcare utilization, while recent research has explored various approaches to mitigate moral hazard. Despite these efforts, most studies are qualitative or empirical, leaving a gap in structured, quantitative models for effectively addressing moral hazard. This paper seeks to address this gap by developing a novel, “quasi”-arbitrage-based model to optimize insurance plan designs in a way that benefits both insurers and the insured.

The focus on health insurance, specifically the transition from Low-Deductible Health Plans (LDHPs) to High-Deductible Health Plans (HDHPs), stems from both practical and theoretical considerations. Our interest in this issue was sparked by an insurance company’s recommendation for us to switch from an LDHP to an HDHP. This personal experience highlighted the potential for reducing moral hazard through plan design, especially in terms of deductible and copay structures. Moreover, the health insurance company presents a valuable context for applying quasi-arbitrage models due to the unique challenges it poses for moral hazard management.

Notably, although much of the existing literature on moral hazard in health insurance has been centered around qualitative analysis or empirical studies, few works have attempted to establish a comprehensive quantitative framework to address this issue. The novelty of this paper lies in the development of a quasi-arbitrage-based mathematical model that provides a structured, quantitative approach to mitigate moral hazard. By focusing on optimizing deductible levels and incentivizing plan selection, this study offers a novel solution to align the interests of both insurers and the insured. Building on qualitative insights, we develop theoretical models for the quantitative analysis of moral hazard. By focusing on the two-stage optimal insurance design problem, this study considers both the insured’s response to varying levels of deductibles and copay rates, and the insurer’s need to maximize profitability or minimize capital requirements.

One of the fundamental trade-offs between LDHPs and HDHPs lies in monthly premiums versus out-of-pocket costs. HDHPs are often characterized by lower monthly premiums but significantly higher deductibles, creating an incentive structure that discourages unnecessary medical utilization.

Table 1. Hypothetical Comparison between a Low-Deductible Health Plan (LDHP) and a High Deductible Health Plan (HDHP) (for illustrative purposes only).

	LDHP	HDHP
Monthly Premium	USD 90	USD 35
Deductible	USD 250	USD 1500
Coinsurance	10%	10%
OOPL (out-of-pocket limit)	USD 1250	USD 2500

presents a hypothetical comparison to illustrate structural differences in premiums, deductibles, and out-of-pocket limits between LDHP and HDHP. These values are conceptual and do not represent actual market data. In this example, the incentive is USD 600. Zou and Petrick (2019) explore how price framing impacts individuals’ perceptions and decisions regarding dual pricing, which parallels how the framing of premiums and deductibles in HDHPs can influence employee choices.

One of the key strategies to mitigate moral hazard is the transition from Low-Deductible Health Plans (LDHPs) to High-Deductible Health Plans (HDHPs). This shift is often encouraged by employers, who offer financial incentives to employees switching to HDHPs. At first glance, this transition appears beneficial to both employees and employers:

• Employees receive incentives that offset the initial higher deductible costs;
• Employers reduce their insurance contribution costs.

It turns out that both the employer and the employee benefit from the switch from the LDHP to the HDHP; hence, the following question arises: could it be that the insurance company is losing money? Specifically, when switching from the LDHP to the HDHP, the decrease in premium is steeper than the decrease in the insurance coverage. In other words, the HDHP is underpriced or the LDHP is overpriced, and thus leaves an “arbitrage” opportunity. After careful examination, we find that the arbitrage opportunity is based on the assumption that the underlying risk does not change when the policyholder switches from the LDHP to the HDHP. However, this outcome does not occur in reality, because the underlying risks being priced are actually different, due to the alternation in the insured’s behavior under different plans. Under the HDHP, the insured individuals are incentivized to reduce risks. After switching from the LDHP to the HDHP, the number of the doctor visits will be reduced. With the HDHP, the policyholder has to pay for doctor visits out of the pocket before meeting the high deductible. This requirement prevents nonessential visits. At this point, we realized that the seemingly inconsistent pricing between the LDHP and the HDHP did not create an arbitrage. Rather, it is a reasonable design for mitigating moral hazards by incentivizing the insured to reduce excessive utilization of medical services for their own benefits.

Therefore, we call the quasi-arbitrage condition the “quasi”-arbitrage condition. We want to determine the conditions that motivate the insured to select the HDHP and then make efforts to reduce unnecessary medical expenses and thus mitigate the moral hazard. Specifically, we seek to answer the following:

• How do different deductible levels shape healthcare consumption patterns?
• What level of premium reduction or financial incentive optimally encourages HDHP adoption?
• How can insurers design health plans that mitigate moral hazard while maintaining financial sustainability?

2. Literature Review and Contribution

Moral hazards in health insurance have been a significant focus in economic and insurance literature. Foundational studies, such as those by Pauly (1968) and Arrow (1978) laid the groundwork by examining how low out-of-pocket costs encourage the overuse of healthcare services, thereby creating inefficiencies in insurance systems. Building on this, Zeckhauser (1970) explored the balance between risk-spreading and incentivization, highlighting the trade-offs in designing efficient insurance coverage and coining the term “indemnity insurance” for systems that address these trade-offs. Pauly (1971) further developed this notion, emphasizing the need for payment models that vary by the severity of illness to manage over-utilization.

Later studies incorporated more nuanced analyses within the general theory of agency. Harris and Raviv (1978) applied these principles to health insurance, discussing under what conditions moral hazards lead to inefficiencies and how appropriate contract structures can help mitigate such risks. With the evolution of empirical methodologies, Cutler and Zeckhauser (2000); Einav et al. (2013) provided empirical evidence quantifying moral hazard, establishing it as an economically significant issue. Five years later, Einav and Finkelstein (2018) used the moral hazard to illustrate the value of and important complementarities between different empirical approaches.

Despite these valuable insights, much of the current literature is either focused on qualitative analysis or empirical quantification, with limited emphasis on developing a structured mathematical framework for addressing moral hazard. Specifically, the question of how to mitigate moral hazard by optimizing deductible and copayment levels remains under-explored, particularly from a perspective that balances the interests of both insurers and the insured. Kairies-Schwarz et al. (2017) highlight the influence of risk preferences on health insurance choices, suggesting that individuals’ decisions to adopt specific plans are not solely driven by cost but also by their personal risk attitudes. Similarly, Ebrahimigharehbaghi et al. (2022) demonstrate how the cumulative prospect theory can be applied to decision-making in contexts such as energy retrofit projects, showing that individuals’ risk perceptions significantly shape their decisions in resource allocation, much like in health insurance plan selection.

This paper addresses this gap by introducing a quantitative model that optimizes deductible levels and incentivizes high-deductible plan selection, ultimately aiming to reduce unnecessary medical utilization. Unlike prior empirical studies, our approach leverages probabilistic modeling and optimization techniques to provide theoretical insights into moral hazard mitigation.

We begin by modeling insured behavior, defining $N\left(d\right)$ as the number of doctor visits, which depends on the deductible level d. We incorporate Poisson and Binomial distributions to model the frequency of doctor visits, capturing the probabilistic nature of healthcare utilization under different deductible structures. To formalize the insurer’s decision-making process, we introduce the quasi-arbitrage condition, which models the trade-offs in insurance pricing and insured behavior. This condition allows us to derive optimal incentive structures that encourage the insured to switch to high-deductible plans while ensuring financial sustainability.

By integrating behavioral modeling with an optimization-based framework, this study provides a structured, theoretical approach to balancing risk-sharing and cost control in health insurance. The quasi-arbitrage condition serves as a key innovation, offering a novel perspective on the design of deductible and incentive structures. Our findings contribute to the theoretical modeling of moral hazard by providing a structured mathematical framework. Although we do not validate our model with empirical data, our approach offers conceptual insights that could inform future research and policy discussions on sustainable health insurance design.

3. “Quasi”-Arbitrage Condition

It has been recognized in the insurance literature that medical insurance may increase usage by lowing the marginal cost of care to the individual, which is a characteristic that has been termed “moral hazard” by Pauly (1968). A review of the relevant academic literature shows that a great number of scholars realize the negative impact of moral hazard. For example, the moral hazard is defined by Boyd et al. (1998) as “the intangible loss-producing propensities of the individual assured” or as “comprehends all of the nonphysical hazards of risk”. Moral hazard represents a deviation from correct human behavior that may pose a problem for an insurer. If a health insurance plan imposes little copayment on the insured, the insured may be motivated to seek more medical services than necessary, which would raise the insurer’s share of cost. The extreme case is full insurance, and policyholders will not worry about the out-of-pocket monetary expenses during illness. Furthermore, the involvement of third-party healthcare providers adds more complications to the moral hazard. Healthcare providers and patients might choose to collaborate to benefit more from insurance reimbursement, which consequently results in unnecessary loss for the insurer. This paper aims to motivate the insured to switch from a Low-Deductible Health Plan to a High-Deductible Health Plan and then make efforts to reduce unnecessary medical expenses and thus mitigate the moral hazard.

The variables used in our model were selected based on both theoretical relevance and empirical grounding in insurance economics. Specifically, we use the Poisson and Binomial distributions to model the frequency of illness and care-seeking behavior because they are widely accepted in actuarial science for modeling discrete health events. Let N be the random counts of natural illness of an insured person during a one-year period, not all of which result in doctor visits. The decision of whether the insured person visits a doctor depends on their specific health insurance plan. Generally, individuals on a Low-Deductible Health Plan are more willing to visit a doctor compared to those on a High-Deductible Health Plan. The deductible level d measures the direct cost impact on the insured, while $p\left(d\right)$ represents the probability of doctor visits under different deductible levels. This function, which decreases as d increases, aligns with the moral hazard theory in health insurance studies, as it reflects the insured’s reduced likelihood of visiting a doctor as out-of-pocket costs rise. Specifically, when $d=0$, the insured is fully covered and, therefore, has little incentive to limit risk exposure, leading to a higher likelihood of doctor visits. Conversely, as $d=\infty$, the insured bears the full cost of care but may still visit a doctor when necessary.

To quantitatively describe these behavioral patterns, we model $N\left(d\right)$ as the number of doctor visits, which depends on d. The function $p\left(d\right)$, which governs the probability of seeking care, is assumed to be strictly decreasing and convex:

$$p\left(d\right)=p_{\infty }+(1-p_{\infty })e^{-d}$$

(1)

The function $p\left(d\right)$ is assumed to satisfy the following properties:

• $p\left(0\right)=1$ and $p(\infty )=p_{\infty }$ for some $p_{\infty }$> 0.
• $p\left(d\right)$ is strictly decreasing and convex.

The convexity of $p\left(d\right)$ is assumed to reflect the marginal diminishing effect. That is, $p\left(d\right)$ is decreasing in d, but the rate of this decrease slows down as d increases. These two assumptions immediately imply that $p^{\prime }\left(d\right)<0$ for any d.

To model the randomness in medical conditions and healthcare utilization, we assume that N follows either a Poisson or Binomial distribution, both commonly used in insurance mathematics.

By structuring the model in this way, we provide a framework that not only characterizes insured behavior under different deductible levels but also offers practical insights into the design of health plans. In particular, it helps identify how deductible adjustments influence insured decisions, allowing insurers to optimize premium structures and incentive strategies while mitigating moral hazard. This structured approach strengthens the theoretical foundation for health insurance modeling and offers insights into designing more effective and financially sustainable insurance plans.

To achieve the “quasi”-arbitrage condition, we start with the following model to characterize the relationship between the variables:

$$N\left(d\right)=\sum _{i=1}^N{I}_i$$

(2)

where {$I_i$, i = 1,2,...} are i.i.d. Bernoulli random variables with P{$I_i$ = 1} = $p\left(d\right)$. Intuitively, $N\left(d\right)$ is a thinning version of N.

Medical expenses are assumed to be random. Specifically, let $\{X_1,X_2,...\}$ denote the medical expenses for different visits. When a deductible is imposed, the insured may start to screen their doctor visits. To model this screening, we introduce a screening indicator that is independent of medical expenses. Specifically, the loss random variable is defined as follows:

$$L\left(d\right)=\sum _{i=1}^N{X}_i\times I_i,$$

(3)

where $\{I_1,I_2,...\}$ are independent of $\{X_1,X_2,...\}$ and $\{I_1,I_2,...\}$$\stackrel{i.i.d.}{\sim }$I and $P\{I=1\}$ = $p\left(d\right)$ and $P\{I=0\}$ = $1-p\left(d\right)$. Note that this screening process is equivalent to the following expression:

$$L\left(d\right)=\sum _{i=1}^{N\left(d\right)}{X}_i$$

(4)

In the following, we learn that the conditions ensuring the “quasi”-arbitrage condition holds if N follows a Poisson distribution or Binomial distribution.

Proposition 1.

We assume that the natural illness count N follows a Poisson distribution with mean λ. During each visit, the medical expenses $X_i$ follows an independent and identical distribution. For simplicity, we assume the copay rate of the health insurance plan is $\alpha =1$; then, the quasi-arbitrage condition holds if this inequality is satisfied for all d,

$$-{\displaystyle \frac{1}{P\left({\sum }_{i=1}^{N\left(\lambda \right)}{X}_i>d\right)}}\ge E\left[X\right]\lambda p^{\prime }\left(d\right)-1$$

(5)

Proof. 

The cost (or pure premium) of this insurance coverage is defined as $\pi \left(d\right)=E\left[{(L\left(d\right)-d)}_{+}\right]$, where $L\left(d\right)={\sum }_{i=1}^{N\left(d\right)}{X}_i$, and $N\left(d\right)$ represents the random count of doctor visits when the deductible level is d. Assume that $N\sim Poi\left(\lambda \right)$, which means that $N\left(d\right)\sim Poi\left(\lambda p\right(d\left)\right)$, where $\lambda p\left(d\right)$ reflects the adjusted rate under deductible d. For simplicity, let us define a function $\phi (\lambda ,y)=E\left[{(L\left(\lambda \right)-y)}_{+}\right]$, where $L\left(\lambda \right)={\sum }_{i=1}^{N\left(\lambda \right)}{X}_i$, and $N\left(\lambda \right)\sim Poi\left(\lambda \right)$ denotes the baseline number of visits. Define $u\left(n\right)=E\left[{({\sum }_{i=1}^n{X}_i-y)}_{+}\right]$.

Then, we have

$${\pi }^{\prime }\left(d\right)={\displaystyle \frac{\partial }{\partial \lambda }}\phi (\lambda ,y){{|}_{(\lambda ,y)=\left(\lambda \right(d),d)}\times {\displaystyle \frac{\partial \lambda \left(d\right)}{\partial d}}+{\displaystyle \frac{\partial }{\partial y}}\phi (\lambda ,y)|}_{(\lambda ,y)=\left(\lambda \right(d),d)}$$

The second term is direct as follows:

$${\displaystyle \frac{\partial }{\partial y}}{\phi (\lambda ,y)|}_{(\lambda ,y)=\left(\lambda \right(d),d)}=-P(L\left(\lambda \right)>d)=-P(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d)$$

In the first term,

$${\lambda }^{\prime }\left(d\right)=\lambda p^{\prime }\left(d\right)$$

We focus on the other part:

$$\begin{array}{cc}\hfill \phi (\lambda ,y)& =E\left[{(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-y)}_{+}\right]\hfill \\ & =E\left[E\left[{(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-y)}_{+}|N\left(\lambda \right)\right]\right]\hfill \\ & \triangleq E\left[u\right(N\left(\lambda \right)\left)\right]\hfill \end{array}$$

where $u\left(N\right(\lambda \left)\right)$ represents the inner expectation based on the Poisson-distributed count $N\left(\lambda \right)$.

To compute ${\displaystyle \frac{\partial }{\partial \lambda }}\phi (\lambda ,y)$, we expand it using the probability mass function of a Poisson distribution, $p_n\left(\lambda \right)={\displaystyle \frac{{\lambda }^n}{n!}}{e}^{-\lambda }$; thus,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial \lambda }}\phi (\lambda ,y)& ={\displaystyle \frac{\partial }{\partial \lambda }}\sum _{n=0}^{\infty }{p}_n\left(\lambda \right)u\left(n\right)\hfill \\ & =\sum _{n=0}^{\infty }{\displaystyle \frac{\partial }{\partial \lambda }}{\displaystyle \frac{{\lambda }^n}{n!}}{e}^{-\lambda }u\left(n\right)\hfill \\ & =\sum _{n=0}^{\infty }{\displaystyle \frac{n{\lambda }^{n-1}}{n!}}{e}^{-\lambda }u\left(n\right)+\sum _{n=0}^{\infty }{\displaystyle \frac{{\lambda }^n}{n!}}{e}^{-\lambda }(-1)u\left(n\right)\hfill \\ & =\sum _{n=1}^{\infty }{\displaystyle \frac{{\lambda }^{n-1}}{(n-1)!}}{e}^{-\lambda }u\left(n\right)-\sum _{n=0}^{\infty }{\displaystyle \frac{{\lambda }^n}{n!}}{e}^{-\lambda }u\left(n\right)\hfill \\ & =E\left[u\left(N\right(\lambda )+1)\right]-E\left[u\left(N\right(\lambda \left)\right)\right]\hfill \end{array}$$

Now, we compute $u\left(N\right(\lambda \left)\right)$ and $u\left(N\right(\lambda )+1)$ as follows:

$$u\left(N\left(\lambda \right)\right)=E\left[{(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-y)}_{+}|N\left(\lambda \right)\right]=E\left[(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-y)·\mathbb{I}\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y\}\right].$$

Similarly, for $u\left(N\right(\lambda )+1)$,

$$\begin{array}{cc}\hfill u\left(N\right(\lambda )+1)& =E\left[{(\sum _{i=1}^{N\left(\lambda \right)+1}{X}_i-y)}_{+}|N\left(\lambda \right)\right]\hfill \\ & =E\left[{(\sum _{i=1}^{N\left(\lambda \right)}{X}_i+X_{N\left(\lambda \right)+1}-y)}_{+}\right]\hfill \\ & =E\left[(\sum _{i=1}^{N\left(\lambda \right)}{X}_i+X_{N\left(\lambda \right)+1}-y)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y-X_{N\left(\lambda \right)+1}\right\}\right]\hfill \\ & =E\left[(\sum _{i=1}^{N\left(\lambda \right)}{X}_i+X_{N\left(\lambda \right)+1}-y)·\left(\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y\right\}+\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (y-X_{N\left(\lambda \right)+1},y)\right\}\right)\right]\hfill \\ & =E\left[(\sum _{i=1}^{N\left(\lambda \right)}{X}_i+X_{N\left(\lambda \right)+1}-y)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y\right\}\right]+\hfill \\ & E\left[(\sum _{i=1}^{N\left(\lambda \right)}{X}_i+X_{N\left(\lambda \right)+1}-y)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (y-X_{N\left(\lambda \right)+1},y)\right\}\right]\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill E\left[u\right(N\left(\lambda \right)+1\left)\right]-E\left[u\right(N\left(\lambda \right)\left)\right]& =E\left[E\left[\left(X_{N\left(\lambda \right)+1}\right)\mathbb{I}\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y\}\right|N\left(\lambda \right)]\right]+\hfill \\ & E\left[E\left[\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-(y-X_{N\left(\lambda \right)+1})\right)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (y-X_{N\left(\lambda \right)+1},y)\right\}|N\left(\lambda \right)\right]\right]\hfill \\ & =E\left[X_{N\left(\lambda \right)+1}\mathbb{I}\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y\}\right]+\hfill \\ & E\left[\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-(y-X_{N\left(\lambda \right)+1})\right)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (y-X_{N\left(\lambda \right)+1},y)\right\}\right]\hfill \\ & =E\left[X\right]·P(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>y)+\hfill \\ & E\left[\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-(y-X_{N\left(\lambda \right)+1})\right)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (y-X_{N\left(\lambda \right)+1},y)\right\}\right]\hfill \end{array}$$

By setting $y=d$, we obtain

$$\begin{array}{cc}\hfill {\pi }^{\prime }\left(d\right)& =\left(E\left[X\right]·P(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d)+E\left[\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-(d-X_{N\left(\lambda \right)+1})\right)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (d-X_{N\left(\lambda \right)+1},d]\right\}\right]\right)·\hfill \\ & \lambda p^{\prime }\left(d\right)-P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d\right)\hfill \\ & =\left(E\left[X\right]\lambda p^{\prime }\left(d\right)-1\right)·P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d\right)+\hfill \\ & E\left[\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i-(d-X_{N\left(\lambda \right)+1})\right)·\mathbb{I}\left\{\sum _{i=1}^{N\left(\lambda \right)}{X}_i\in (d-X_{N\left(\lambda \right)+1},d]\right\}\right]\lambda p^{\prime }\left(d\right)\hfill \\ & ⩽\left(E\left[X\right]\lambda p^{\prime }\left(d\right)-1\right)·P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d\right)\hfill \end{array}$$

To demonstrate that ${\pi }^{\prime }\left(d\right)⩽-1$, it is sufficient to satisfy the inequality $-{\displaystyle \frac{1}{P\left({\sum }_{i=1}^{N\left(\lambda \right)}{X}_i>d\right)}}\ge E\left[X\right]\lambda p^{\prime }\left(d\right)-1$ for the quasi-arbitrage condition to hold. □

Corollary 1.

There exists a constant ${\lambda }_0$ such that Inequality (5) holds for all $\lambda >{\lambda }_0$.

Proof. 

We assume that $N\left(\lambda \right)\to \infty$ is almost certain as $\lambda \to \infty$ on the given probability space. This mild technical assumption allows the application of the strong law of large numbers in the following argument. The total medical cost ${\sum }_{i=1}^{N\left(\lambda \right)}{X}_i$ is the sum of $N\left(\lambda \right)$ i.i.d. random variables $X_i$, we have

$$E\left[\sum _{i=1}^{N\left(\lambda \right)}{X}_i\right]=E\left[N\left(\lambda \right)\right]E\left[X\right]=\lambda E\left[X\right]$$

Furthermore, by the strong law of large numbers, we obtain

$${\displaystyle \frac{{\sum }_{i=1}^{N\left(\lambda \right)}{X}_i}{N\left(\lambda \right)}}\to E\left[X\right]$$

almost surely, as $N\left(\lambda \right)\to \infty$.

This implies that for any $ϵ>0$, there exists a ${\lambda }_0$ such that for all $\lambda >{\lambda }_0$,

$$P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>(E\left[X\right]-ϵ)\lambda \right)>1-ϵ.$$

Since d is a finite constant, we can always choose $\lambda$ large enough such that

$$\left(E\right[X]-ϵ)\lambda >d.$$

Consequently, for sufficiently large $\lambda$,

$$P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d\right)\ge P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>(E\left[X\right]-ϵ)\lambda \right)>1-ϵ$$

Thus, we have

$$P\left(\sum _{i=1}^{N\left(\lambda \right)}{X}_i>d\right)\to 1$$

As $\lambda \to \infty$, and $E\left[X\right]\lambda p^{\prime }\left(d\right)-1\to -\infty$. Consequently, Inequality (5) holds for all $\lambda >{\lambda }_0$. □

Corollary 2.

When $d=0$, the quasi-arbitrage condition is satisfied if the following inequality holds:

$$-{\displaystyle \frac{1}{1-e^{-\lambda }}}\ge -E\left[X\right]\lambda (1-p_{\infty })-1$$

Proof. 

When $d=0$, we have $p^{\prime }\left(0\right)=-(1-p_{\infty })$ and $P\left({\sum }_{i=0}^{N\left(\lambda \right)}{X}_i>0\right)=1-e^{-\lambda }$. Consequently, ${\pi }^{\prime }\left(0\right)⩽-1$ holds if $-{\displaystyle \frac{1}{1-e^{-\lambda }}}\ge -E\left[X\right]\lambda (1-p_{\infty })-1$. □

Proposition 2.

Assuming that the natural illness count N follows a Binomial distribution with mean $nq$, the probability that the insured decides to visit the doctor is given by $p\left(d\right)=p_{\infty }+(1-p_{\infty })e^{-d}$. During each visit, the medical expenses $X_i$ are assumed to be independently and identically distributed. For simplicity, we assume the health insurance plan has a copay rate $\alpha =1$. The quasi-arbitrage condition holds if the following inequality is satisfied for all d:

$$-{\displaystyle \frac{1}{P({\sum }_{i=1}^{N^{\prime }}{X}_i>d)}}\ge E\left[X\right]nq·p^{\prime }\left(d\right)-1$$

(6)

where $N^{\prime }\sim Bin(n-1,q·p\left(d\right))$

Proof. 

Since $N\sim Bin(n,q)$, it follows that $N\left(d\right)\sim Bin(n,q^{\prime })$, where $q^{\prime }=p\left(d\right)q$. Here, $\pi \left(d\right)=E\left[{({\sum }_{i=1}^{N\left(d\right)}{X}_i-d)}_{+}\right]$. Let $\phi (q^{\prime },y)=E\left[{({\sum }_{i=1}^{N\left(d\right)}{X}_i-y)}_{+}\right]$, which can be rewritten as $E\left[E\left[{({\sum }_{i=1}^{N\left(d\right)}{X}_i-y)}_{+}\right|N\left(d\right)]\right]$ = $E\left[u\right(n\left)\right]$, where $u\left(n\right)=E\left[{({\sum }_{i=1}^n{X}_i-y)}_{+}\right]$. Therefore,

$${\pi }^{\prime }\left(d\right)={\displaystyle \frac{\partial }{\partial q^{\prime }}}\phi (q^{\prime },y){{|}_{(q^{\prime },y)=(p\left(d\right)q,d)}\times {\displaystyle \frac{\partial p\left(d\right)}{\partial d}}+{\displaystyle \frac{\partial }{\partial y}}\phi (q^{\prime },y)|}_{(q^{\prime },y)=(p\left(d\right)q,d)}$$

In the first term,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial q^{\prime }}}E\left[u\left(n\right)\right]& ={\displaystyle \frac{\partial }{\partial q^{\prime }}}{P}_n\left(q^{\prime }\right)·u\left(n\right)\hfill \\ & =n·(E\left[u(N^{\prime }+1)\right]-E\left[u\left(N^{\prime }\right)\right])\hfill \end{array}$$

Since $N^{\prime }\sim Bin(n-1,p\left(d\right)q)$, therefore,

$$u\left(N^{\prime }\right)=E\left[{(\sum _{i=1}^{N^{\prime }}{X}_i-y)}_{+}|N^{\prime }\right]=E\left[(\sum _{i=1}^{N^{\prime }}{X}_i-y)·\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i>y\}\right]$$

$$\begin{array}{cc}\hfill u(N^{\prime }+1)& =E\left[{(\sum _{i=1}^{N^{\prime }+1}{X}_i-y)}_{+}|N^{\prime }\right]\hfill \\ & =E\left[(\sum _{i=1}^{N^{\prime }}{X}_i+X_{N^{\prime }+1}-y)·\left(\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i>y\}+\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i\in (y-X_{N^{\prime }+1},y)\}\right)\right]\hfill \\ & =E\left[(\sum _{i=1}^{N^{\prime }}{X}_i+X_{N^{\prime }+1}-y)·\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i>y\}\right]+\hfill \\ & E\left[(\sum _{i=1}^{N^{\prime }}{X}_i+X_{N^{\prime }+1}-y)·\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i\in (y-X_{N^{\prime }+1},y)\}\right]\hfill \end{array}$$

By the law of total expectation,

$$E\left[u(N^{\prime }+1)\right]-E\left[u\left(N^{\prime }\right)\right]=E\left[X_{N^{\prime }+1}·\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i>y\}\right]+$$

$$E\left[(\sum _{i=1}^{N^{\prime }}{X}_i+X_{N^{\prime }+1}-y)·\mathbb{I}\{\sum _{i=1}^{N^{\prime }}{X}_i\in (y-X_{N^{\prime }+1},y)\}\right]$$

Therefore,

$${\pi }^{\prime }\left(d\right)⩽n{p}^{\prime }\left(d\right)q·E\left[X\right]·P(\sum _{i=1}^{N^{\prime }}{X}_i>d)-P(\sum _{i=1}^N{X}_i>d)$$

To show that ${\pi }^{\prime }\left(d\right)⩽-1$, it is equivalent to requiring that $n{p}^{\prime }\left(d\right)q·E\left[X\right]·P({\sum }_{i=1}^{N^{\prime }}{X}_i>d)-P({\sum }_{i=1}^N{X}_i>d)⩽-1$. Since $N\left(q^{\prime }\right)$∼$Bin(n,q^{\prime })$, and $N^{\prime }\left(q^{\prime }\right)$∼$Bin(n-1,q^{\prime })$, we have $P(N^{\prime }\left(q^{\prime }\right)>d)⩽P(N\left(q^{\prime }\right)>d)$. Therefore, $(n{p}^{\prime }\left(d\right)q·E\left[X\right]-1)P({\sum }_{i=1}^{N^{\prime }}{X}_i>d)⩽-1$ is a sufficient condition for the quasi-arbitrage condition to hold. □

Corollary 3.

There exists a constant $n_0$ such that Inequality (6) holds for all $n>n_0$.

Proof. 

If $n\to \infty$, we have $n{p}^{\prime }\left(d\right)q·E\left[X\right]-1\to -\infty$. Consequently, Inequality (6) holds for all $n>n_0$. □

Corollary 4.

When $d=0$, the quasi-arbitrage condition is satisfied if the following inequality holds:

$$-{\displaystyle \frac{1}{1-{(1-q)}^{n-1}}}\ge -n(1-p_{\infty })q·E\left[X\right]-1$$

Proof. 

Since $P^{\prime }\left(0\right)=-(1-p_{\infty })$, and $N^{\prime }\left(q^{\prime }\right)$∼$Bin(n-1,p(d\left)q\right)$, then $P(N^{\prime }\left(q^{\prime }\right)>0)=1-P(N^{\prime }\left(q^{\prime }\right)=0)$ = $1-{(1-p\left(d\right)q)}^{n-1}$. Therefore, $-{\displaystyle \frac{1}{1-{(1-q)}^{n-1}}}\ge -n(1-p_{\infty })q·E\left[X\right]-1$ is the condition for the quasi-arbitrage condition to hold. □

4. Optimal Design of Health Insurance Policy

In this section, we examine how the quasi-arbitrage condition helps to mitigate the moral hazard from the insured’s perspective and establish conditions under which this condition holds when N, and the random counts of natural illnesses of an insured person during a one-year period follows specific distributions. Shifting our focus to the insurer’s perspective, we analyze the expected profit under varying deductible levels and incentives designed to encourage policyholders to switch to a High-Deductible Health Plan (HDHP).

Suppose the insurer charges $\pi \left(d_1\right)$ for a Low-Deductible Health Plan (LDHP) and $G\left(d_2\right)$ for an HDHP. Here, $G\left(d_2\right)$ is not necessarily equal to $\pi \left(d_2\right)=E\left[\left(d_2\right)\right]$, the pure premium for risk. When $G\left(d_2\right)=\pi \left(d_2\right)$, the insurer makes no profit on the HDHP. However, if $G\left(d_2\right)$ is set too high, fewer policyholders may choose to switch to the HDHP. Thus, $G\left(d_2\right)$ must be determined with the constraint $G\left(d_2\right)\ge \pi \left(d_2\right)$ to ensure profitability. For simplicity, we assume the insurer charges a pure premium for risk on the LDHP. Our goal is to identify an optimal incentive structure to maximize insurer profit following policyholder transitions from LDHP to HDHP.

When a policyholder switches from LDHP to HDHP, the maximum reduction in coverage value is $d_2-d_1$, and the savings in premium are $\pi \left(d_1\right)-G\left(d_2\right)$. Therefore, we define the incentive amount I as:

$$I=\pi \left(d_1\right)-G\left(d_2\right)-(d_2-d_1)$$

The incentive I must satisfy the inequality $0\le I\le \pi \left(d_1\right)-\pi \left(d_2\right)-(d_2-d_1)\triangleq I_m$. On one hand, I must be positive to encourage policyholders to switch; otherwise, they will prefer to remain with the LDHP. On the other hand, the insurer provides maximum incentive when they earn minimum profit, that is, when $G\left(d_2\right)=\pi \left(d_2\right)$. We denote this maximum incentive as $I_m$, with a positive value that is guaranteed by the quasi-arbitrage condition.

To determine the optimal incentive level I that maximizes profit, we define the probability of policyholders switching from LDHP to HDHP as $\lambda \left(I\right)$. As the incentive amount increases, the probability of switching also increases, but the rate of increasing slows due to diminishing marginal returns. To establish some basic properties of $\lambda \left(I\right)$, consider two boundary cases:

• If $I=0$, then $\lambda \left(0\right)=0$, as no policyholder has an incentive to switch;
• If $I=I_m$, then $\lambda \left(I_m\right)\triangleq {\lambda }_{max}<1$, since some policyholders may still choose to stay with LDHP despite the maximum incentive.

We assume a specific functional form for $\lambda \left(I\right)$:

$$\lambda \left(I\right)=1-{(1-{\displaystyle \frac{I}{A}})}^{\alpha }$$

where $\alpha >1$ and A is a constant such that $I<A$ for all I. This form is chosen to model diminishing marginal returns in switching behavior as I approaches $I_m$. Given the probability of switching $\lambda \left(I\right)$, we maximize the insurer’s expected profit across two groups: one group chooses to remain with LDHP with probability $1-\lambda \left(I\right)$, while the other group switches to HDHP with probability $\lambda \left(I\right)$.

Assume there are n policyholders initially under LDHP, each making independent decisions. Let M be the number of policyholders who decide to switch to HDHP so that, after HDHP’s introduction, there are $n-M$ remaining on LDHP. Here, $M\sim Bin(n,\lambda (I\left)\right)$ and $n-M\sim Bin(n,1-\lambda (I\left)\right)$. The total profit B of this insurance portfolio is composed of two components: the first part represents the profit from policyholders remaining on LDHP, and the second part represents the profit from those switching to HDHP.

$$B=\sum _{i=1}^{n-M}(\pi \left(d_1\right)-L_i\left(d_1\right))+\sum _{j=1}^M(\pi \left(d_1\right)+d_1-d_2-I-L_j\left(d_2\right))$$

(7)

where $\left\{L_i\left(d_1\right)\right\}$ are individual losses from LDHP policyholders, and $\left\{L_j\left(d_2\right)\right\}$ are losses from HDHP policyholders.

In our model, we assume that individual losses for LDHP and HDHP policyholders follow the same distribution. This simplification abstracts the direct impact of deductibles on the distribution of individual losses, allowing us to focus on the primary effects of switching behaviors and premium adjustments. Specifically, we model the transition from LDHP to HDHP based on the expected cost differences rather than variations in individual loss distributions. While, in reality, insured individuals may adjust their healthcare consumption due to behavioral or financial considerations, our model isolates the impact of deductible structures and incentives on switching decisions. This assumption maintains analytical tractability while capturing the core trade-offs in insurance plan design.

The expected profit $E\left[B\right]$ is then

$$E\left[B\right]=n(1-\lambda \left(I\right))(\pi \left(d_1\right)-E\left[L\left(d_1\right)\right])+n\lambda \left(I\right)(\pi \left(d_1\right)+d_1-d_2-I-E\left[L\left(d_2\right)\right])$$

The insurer’s expected profit for each LDHP policyholder is

$$\pi \left(d_1\right)-E\left[L\left(d_1\right)\right]=0$$

The expected gain from each HDHP is

$$\lambda \left(I\right)(G\left(d_2\right)-E\left[L\left(d_2\right)\right])=\lambda \left(I\right)(I_m-I)\triangleq g_1\left(I\right)$$

where $G\left(d_2\right)=\pi \left(d_1\right)+d_1-d_2-I$. Our goal is to maximize $g_1\left(I\right)$ over $[0,I_m]$.

Lemma 1.

The function $g_1\left(I\right)=\lambda \left(I\right)(I_m-I)$ has the following properties:

$\left(i\right)$  $g_1\left(0\right)=0,g_1\left(I_m\right)=0$

$\left(ii\right)$ $g_1\left(I\right)$ is concave in I and obtains a maximum at $I^{*}\left(d_2\right)$; in other words, $g_1^{\prime }\left(I^{*}\left(d_2\right)\right)=0$.

Proof. 

$\left(i\right)$ $g_1\left(0\right)=\lambda \left(0\right)\left(I_m\right)=0$, $g_1\left(I_m\right)=\lambda \left(I_m\right)(I_m-I_m)=0$

$\left(ii\right)$ Since $\lambda \left(I\right)$ is an increasing concave function, ${\lambda }^{\prime }\left(I\right)>0$ and ${\lambda }^{\prime \prime }\left(I\right)<0$. Differentiating $g_1\left(I\right)$ with respect to I, we obtain $g_1^{\prime }\left(I\right)={\lambda }^{\prime }\left(I\right)(I_m-I)-\lambda \left(I\right)$. Further differentiation gives $g_1^{\prime \prime }\left(I\right)={\lambda }^{\prime \prime }\left(I\right)(I_m-I)-2{\lambda }^{\prime }\left(I\right)<0$, confirming $g_1\left(I\right)$ is concave in I and that there exists $I^{*}\left(d_2\right)$, such that $g_1^{\prime }\left(I^{*}\left(d_2\right)\right)=0$. □

Proposition 3.

If $I_m<I^{*}\left(d_2\right)$, the maximum expected profit is obtained at $I=I_m$, thus $max$$E\left[B\right]=g_1\left(I_m\right)$. If $I_m\ge I^{*}\left(d_2\right)$; then, the maximum expected profit is achieved at $I=I^{*}\left(d_2\right)$, hence $max$$E\left[B\right]=g_1\left(I^{*}\left(d_2\right)\right)$.

In the insurance model, different policyholders have their own utility functions. Based on the profit formula (6), the total profit B can be expressed as

$$\begin{array}{cc}\hfill B& =\sum _{i=1}^{n-M}(\pi \left(d_1\right)-L_i\left(d_1\right))+\sum _{j=1}^M(\pi \left(d_1\right)+d_1-d_2-I-L_j\left(d_2\right))\hfill \\ & =n\pi \left(d_1\right)-[(\sum _{i=1}^{n-M}{L}_i\left(d_1\right)+\sum _{j=i}^M(L_j\left(d_2\right)-(d_1-d_2-I))]\hfill \end{array}$$

Assuming that the insurer has utility function u, the optimization problem can be formulated as maximizing the expected utility of the profit B, as follows:

$$\underset{(I,d_2)\in D}{max}E\left[u(n\pi \left(d_1\right)-\left[\left(\sum _{i=1}^{n-M}{L}_i\left(d_1\right)+\sum _{j=i}^M(L_j\left(d_2\right)-(d_1-d_2-I))\right]\right)\right]\triangleq g_1(I,d_2)$$

(8)

where

$$D=\left\{(I,d_2)\right|I\le E\left[L\left(d_1\right)\right]+d_1-E\left[L\left(d_2\right)\right]-d_2=I_m\}$$

Assume the natural illness count N follows a geometric distribution. However, the medical expenses $X_i$ are different for each doctor’s visit, which is much more realistic. We assume that each medical expense $X_i$ follows an exponential distribution with mean $\theta$, $\{X_i,i=1,2,...\}$$\stackrel{i.i.d.}{\sim }$ Exp($\theta$). Thus, the total profit B can be written as

$$B=\sum _{i=1}^{n-M}\left(\pi \left(d_1\right)-L_i\left(d_1\right)\right)+\sum _{j=1}^M\left(\pi \left(d_1\right)+d_1-d_2-I-L_j\left(d_2\right)\right)$$

where $L_i\left(d_1\right)={\sum }_{k=1}^{N\left(d_1\right)}{X}_k$, $L_j\left(d_2\right)={\sum }_{k=1}^{N\left(d_2\right)}{X}_k$, $X_i\sim Exp\left(\theta \right)$, and $N\left(d\right)$ is the geometric over the exponential model. Therefore, the loss function $L\left(d\right)$ follows an exponential distribution given by

$$L\left(d\right)\sim Exp\left(\theta \right(1+\beta p\left(d\right)\left)\right)$$

Assuming that the insurer has a utility function u, the optimization problem can be reformulated to maximize the expected utility of the profit B under the aggregate loss model, as follows:

$$\underset{(I,d_2)\in D}{max}E\left[u\left(\sum _{i=1}^{n-M}\left(\pi \left(d_1\right)-L_i\left(d_1\right)\right)+\sum _{j=1}^M\left(\pi \left(d_1\right)+d_1-d_2-I-L_j\left(d_2\right)\right)\right)\right]$$

(9)

5. Conclusions

In this paper, we analyzed optimal health insurance designs from the perspectives of both the insured and the insurer. In health insurance, addressing moral hazard—where lowering the marginal cost of care may increase healthcare utilization—is essential. Wu (2022) underscores the importance of effective insurance designs in balancing cost control with incentivizing responsible care utilization, a principle that is central to our model’s emphasis on optimizing deductible levels. While much of the existing literature focuses on qualitative or empirical studies, our research contributes a quantitative approach to model and mitigate moral hazard. We developed theoretical models to quantify moral hazard and derived optimal health insurance designs that balance the interests of both insurers and the insured.

Our findings highlight the significant role that deductible levels play in shaping insured behavior. Higher deductibles discourage unnecessary medical utilization by increasing out-of-pocket costs, effectively reducing moral hazard. By modeling the probability of doctor visits as a function of the deductible, we quantitatively demonstrate how healthcare consumption patterns change under different plan structures.

We also examine how financial incentives impact the adoption of High-Deductible Health Plans (HDHPs). Our results indicate that premium reductions and employer-provided incentives play an important role in encouraging policyholders to switch from Low-Deductible Health Plans (LDHPs) to HDHPs. Through optimization modeling, we determine the optimal level of financial incentives that maximizes plan adoption while maintaining insurer profitability.

From the insurer’s perspective, designing sustainable health plans requires carefully structuring deductibles, premiums, and incentives to maintain financial stability. Our model demonstrates how insurers can strategically adjust these factors to influence insured behavior and control costs. The concept of quasi-arbitrage, introduced in this study, offers a new perspective on designing pricing structures that mitigate moral hazard without compromising financial sustainability.

By developing a structured, quantitative approach, this study contributes primarily theoretically, offering a mathematical framework for analyzing moral hazard in health insurance. While the model provides insights into how deductible levels and incentives influence insured behavior, its direct applicability requires empirical validation. Future research could enhance this framework by incorporating real-world data and diverse insured risk profiles to refine health insurance design strategies.