One-sided commitment in dynamic insurance contracts: Evidence from private health insurance in Germany

Abstract

This paper studies long-term private health insurance (PHI) in Germany. It describes the main actuarial principles of premium calculation and relates these to existing theory. In the German PHI policyholders do not commit to renewing their insurance contracts, but insurers commit to offering renewal at a premium rate that does not reflect revealed future information about the insured risk. We show that empirical results are consistent with theoretical predictions from one-sided commitment models: front-loading in premiums generates a lock-in of consumers, and more front-loading is generally associated with lower rates of lapse. Due to a lack of consumer commitment, dynamic information revelation about risk type implies that high-risk policyholders are more likely to retain their PHI contracts than are low-risk types.

Appendix

Consider a competitive two-period health insurance market with a high number of buyers and sellers. Buyers wish to insure potentially worse health states in the future. A health status can be described as a certain probability of suffering medical loss and a certain severity of the loss in the event the individual experiences medical loss. In the first period, individuals are identical with regard to loss probability and severity, i.e., they face identical medical expenses m in case of illness, and they have identical probability p of suffering an illness.⁴¹ In period 2, multiple health statuses are possible so that policyholders differ in both their probability $p_{i}$ of experiencing some medical loss and the severity or size of the loss. If an individual is in health status i in period 2, he or she faces potential medical loss $m_{2}^{i}$, which may represent physician visits, time and money invested in obtaining treatment, etc. We order health states so that $p_{1}<p_{2}<...<p_N$ and $m_{2}^{1}<m_{2}^{2}<....<m_{2}^N$ and assume $p\leq p_{1}$ and $m\leq m_{2}^{1}$, i.e., health worsens over time in both frequency and severity. The probability of being in health state i is given by $\pi _{i}$. Individuals’ utility function is $u(c)$ when they consume $c \geq 0$ in each period. Individuals are risk-averse: the utility function is strictly concave and twice continuously differentiable.

Period 1 involves three stages: first, insurers offer contracts; second, buyers choose a contract; third, uncertainty about medical loss is revealed and consumption takes place. Period 2 involves four stages: first, uncertainty about a policyholder’s health status is revealed. The realized health status of a policyholder can be observed by the insurer. The following stages are equivalent tostages 1 to 3 in the first period.

A first-period premium consists of a first-period premium $P_{1}$ and coverage amount $C_{1}$, and a vector of premiums and coverage amounts $(P_{2}^{1},C_{2}^{1})...(P_{2}^N,C_{2}^N)$ indexed by the second-period health states. Thus a first-period contract is a long-term contract to which the insurer unilaterally commits. In contrast, the second-period contract is ashort-term contract. Note that a short-term contract may depend on information revealed at the beginning of the second period. It consists of a premium and coverage amount $(P_{2}^{i},C_{2}^{i})$ indexed by second-period health status. There is one-sided commitment: insurers can commit tofuture premiums while consumers freely choose between staying with their period 1 contract and switching to a competitor offering a short-term spot contract in period 2.

Consumer heterogeneity is given by differences in the income process. We assume that consumers differ in income growth. While some minimum income $\tilde {y}$ is needed in order to enter into a PHI contract, consumers with minimum income $\tilde {y}$ may differ in income growth. To capture this, we assume that a consumer receives an income of $y-g \geq \tilde {y}$ in the first period and $y+g$ in the second period, and the variation in consumers’ income growth is represented by $g>0$. In our model, $\tilde {y}$ represents the minimum income needed in order to enter the PHI market.⁴²

In competitive equilibrium, allocations maximize individuals’ expected utility subject to a zero expected profit constraint and a set of no-lapse constraints that capture individuals’ in ability to commit to the PHI contract. Solving this constrained maximization problem gives the set of prices and coverage amounts that must be available to individuals in a competitive PHI market.⁴³

In equilibrium, premiums and coverage amounts that are fully contingent on health states, $(P_{1},C_{1})$, and $(P_{2}^{1},C_{2}^{1})...(P_{2}^N,C_{2}^N)$ must maximize individuals’ expected utility:

$$\begin{array}{rll} EU &=& (1-p)u(y-g-P_{1})+pu(y-g-P_{1}-m+C_{1}) \\ && + \sum\limits_{i=1}^N \pi_{i} \left [(1-p_{i}) u\big(y+g-P_{2}^{i}\big)+ p_{i} u\big(y+g-P_{2}^{i}-m_{2}^{i}+C_{2}^{i}\big) \right ]\end{array}$$

(8)

subject to a zero profit constraint (i.e., contracts break even on average)

$$ P_{1}-pC_{1}+\sum\limits_{i=1}^N \pi_{i} \left [P_{2}^{i}-p_{i} C_{2}^{i} \right ]=0 $$

(9)

and no-lapse constraints imposed by lack of consumer commitment: for all possible future health states, $i=1,...N$, and for all $\tilde {P}_{2}^{i},\tilde {C}_{2}^{i}$ such that $\tilde {P}_{2}^{i}-p_{i} \tilde {C}_{2}^{i}>0$, we must have $EU(P_{2}^{i},C_{2}^{i})\geq EU(\tilde {P}_{2}^{i},\tilde {C}_{2}^{i})$ or equivalently

$$\begin{array}{lll} &&(1-p_{i}) u\big(y+g-P_{2}^{i}\big)+p_{i} u\big(y+g-P_{2}^{i}-m_{2}^{i}+C_{2}^{i}\big) \\&& \quad \geq\, (1-p_{i}) u\big(y+g-\tilde{P}_{2}^{i}\big)+p_{i} u\big(y+g-\tilde{P}_{2}^{i}-\tilde{m}_{2}^{i}+\tilde{C}_{2}^{i}\big).\end{array}$$

(10)

The no-lapse constraints imply that an equilibrium contract is such that there isno other contract that is profitable and offers potential buyers higher expectedutility in any state of period 2. We refer to the actuarially fair premium that guarantees full insurance in health state i and guarantees zero expected profits as $P_{2}^{i}(FI)$.

Proposition 1

In the equilibrium set of contracts:

• (i) All consumers obtain full insurance in period 1 and for all health states of period 2.

• (ii) For health states s and worse, premiums are capped at a price that is below the fair price for each of these states, i.e. for every g there is an s such that $P_{2}^{i}=P_{2}^{i}(FI)$ for $i=1,...,s-1$ and $P_{2}^{i}<P_{2}^{i}(FI)$ for $i=s+1,...,N$.

• (iii) Insurance contracts are front-loaded as long as g is not “too high”. That is, there is a $\hat {g}$ such that, if $g<\hat {g}$ , then PHI contracts involve front-loading.

• (iv) More front-loaded contracts appeal to buyers with higher first-period income (i.e. consumers with lower g who find front-loading less costly). Contracts with higher $P_{1}$ involve a lower cutoff s. Policyholders with higher income growth g choose PHI contracts with less front-loading.

• (v) Consumers are eligible for PHI as long as g is not “too high”. That is, there is a $\tilde {g}$ so that $y-\tilde {g}<\tilde {y}$ and the consumer drops out of the PHI market.

Proof of Proposition 1

We can replace the set of constraints (10) with the following simpler set

$$ P_{2}^{i}-p_{i} C_{2}^{i} \leq 0 \forall {i} $$

(11)

This is because if $P_{1}$, $C_{1}$, $(P_{2}^{1},C_{2}^{1})...(P_{2}^N,C_{2}^N)$ maximize Eq. 8 subject to Eqs. 9 and 11 then there is no state i and no $(\tilde {P}_{2}^{i},\tilde {C}_{2}^{i})$ that results in positive expected profits and offers consumers a higher expected utility in state i. Hence, Eq. 10 is satisfied. Conversely, if $(P_{2}^{i},C_{2}^{i})$ are such that Eq. 11 is violated, then Eq. 10 is violated as well since a competing company may offer terms slightly better for consumers than $(P_{2}^{i},C_{2}^{i})$ while still making positive expected profits.

Let $\mu$ be the Lagrange multiplier for the constraint in Eq. 9 and $\lambda _{i}$ be the multiplier for the ith constraint in Eq. 11. Then the Lagrangian is

$$\begin{array}{rll} \mathcal{L} &=& (1-p)u(y-g-P_{1})+pu(y-g-P_{1}-m+C_{1}) \\&&+\, \sum\limits_{i=1}^N \pi_{i} \left [(1-p_{i}) u\big(y+g-P_{2}^{i}\big)+ p_{i} u\big(y+g-P_{2}^{i}-m_{2}+C_{2}^{i}\big) \right ] \\&&+\, \mu \cdot \left [P_{1}-pC_{1}+\sum\limits_{i=1}^N \pi_{i} \big(P_{2}^{i}-p_{i} C_{2}^{i}\big) \right ] \\&&+\, \lambda_{i} \cdot \left [P_{2}^{i}-p_{i} C_{2}^{i} \right ]\end{array}$$

and the first-order conditions for an optimum are given by

$$ \mu=(1-p)u'(y-g-P_{1})+pu'(y-g-P_{1}-m+C_{1})$$

(12)

$$ \mu=u'(y-g-P_{1}-m+C_{1}) $$

(13)

$$\begin{array}{lll} && \pi_{i} \left [ -p_{i} u'\big(y+g-P_{2}^{i}-m_{2} + C_{2}^{i}\big)-(1-p_{i}) u'\big(y+g-P_{2}^{i}\big) \right ] \\ &&\qquad\; +\, \pi_{i} \mu + \lambda_{i}=0 \forall {i} \end{array}$$

(14)

$$ \pi_{i} u'\big(y+g-P_{2}^{i}-m_{2} +C_{2}^{i}\big)- \pi_{i} \mu- \lambda_{i} =0 $$

(15)

$$ \big(P_{2}^{i}-p_{i} C_{2}^{i}\big)\lambda_{i}=0 \forall {i}; \lambda_{i}\leq 0. $$

(16)

Combining first-order conditions (12) and (13) gives $C_{1}=m$ which implies full insurance in period 1. Combining conditions (14) and (15) results in $m_{2}^{i}=C_{2}^{i} \forall i$ which implies full insurance in all states of period 2. This proves part (i) of Proposition 1.

To prove part (ii), note that if constraint i in Eq. 11 is binding, then $P_{2}^{i}=p_{i} C_{2}^{i}$ which implies actuarially fair insurance. It follows that if i and j are two binding constraints and $i>j$ (and thus $p_{i}>p_{j}$ and $m_{2}^{i}>m_{2}^{j}$) then it must be that $P_{2}^{i}>P_{2}^{j}$. In contrast, if k in Eq. 11 is non-binding, then $P_{2}^{k}<p_{k} C_{2}^{k}$ and thus $\lambda _{k}=0$. Then Eq. 14 reduces to

$$ u'\big(y+g-P_{2}^{k}\big)= \mu $$

(17)

As a result, if constraints k and l are non-binding, then $P_{2}^{k}=P_{2}^{l}$. If in contrast constraint i in Eq. 11 is binding, then Eq. 14 becomes

$$ u'\big(y+g-P_{2}^{i}\big)= \mu + \frac{\lambda_{i}}{\pi_{i}}<\mu $$

(18)

where the inequality on the right-hand side holds because $\lambda _{i}<0$ if constraint i in Eq. 11 is binding. As a consequence, if constraint i is binding and k is not, it follows that $P_{2}^{i}<P_{2}^{k}$. Finally, we show that if i is binding and k is not, then $i<k$. To see this, note that since k is non-binding, we have $P_{2}^{k}<p_{k} C_{2}^{k}$, which, together with $P_{2}^{i}<P_{2}^{k}$, leads to

$$ p_{i} C_{2}^{i}<p_{k} C_{2}^{k}. $$

(19)

Assume $i>k$. Then $C_{2}^{i}>C_{2}^{k}$ (due to $m_{2}^{i}>m_{2}^{k}$) and $p_{i}>p_{k}$. This would render $P_{2}^{i}<P_{2}^{k}$ impossible and sowe must have $i<k$. This proves part (ii).

To show that part (iii) holds, we need to show that the first-period premium $P_{1}$ is larger than the actuarially fair premium $P_{1}(FI)$. If any of the no-lapse constraints (11) is nonbinding, then$P_{1}>P_{1}(FI)$ is immediate from the zero expected profit condition (9) above. Assume instead that all no-lapse constraints in Eq. 11 are binding. Keeping in mind full insurance in optimum and then substituting Eq. 12 into Eq. 14, we obtain

$$ u'\big(y+g-P_{2}^{i}\big)=u'(y-g-P_{1})+\frac{\lambda_{i}}{\pi_{i}} $$

(20)

which, since $\lambda _{i}<0$ if all no-lapse constraints are binding, gives $P_{1}>P_{2}^N(FI)-2g=p_N \cdot m_{2}^N-2g$. This inequality requires that $P_{1}>P_{1}(FI)$ if g is small enough since $p_N>p$ and $m_{2}^N>m$.

To prove part (iv), note that as g increases, more and more of the no-lapse constraints become binding and thus as g grows so does the cutoff s. When s becomes larger, $P_{1}$ decreases.

Finally, part (v) is obvious from our assumptions. □

Following Hendel and Lizzeri (2003), Proposition 1 can be extended to non-contingent contracts.⁴⁴ A non-contingent contract is also an equilibrium contract if a consumer for whom the contingent contract is optimal can obtain the same utility from the noncontingent contract (in each state), and insurance companies earn the same profits. In states $i=1,...,s-1$, premiums and coverage amounts of the contingent contract equal those offered on the spot market. Fixing the terms of the non-contingent contract to be the same as those of the contingent contract in the first period and in the bad states of the second period (i.e. states $s,...,N$), both contracts are equivalent except for states $1,...,s-1$, in which the contingent contract offers better terms. Note, however, that a consumer can drop out of the alternative non-contingent contract and purchase spot contracts. As a result, the alternative contracts are equivalent.⁴⁵ Taking into account equivalence of contingent and non-contingent contracts, we state:

Proposition 2

Consider two PHI contracts that are offered in period 1 and not contingent on the health state in period 2. In competitive equilibrium, the contract with the higher first-period premium is chosen by those consumers with lower income growth (i.e., higher first-period income $y-g$), and has lower lapse rates.

Proof of Proposition 2

The proof follows Hendel and Lizzeri (2003). The contract with the higher first-period premium must involve a lower second-period premium, otherwise no consumer would chooseit. Thus, in the second period this contract retains a healthier pool of consumers (since a higher $P_{1}$ involves a lower cutoff s). This implies that the average cost of this contract is lower. Under competition (and due to the equivalence principle used in German PHI premium calculations), the present value of the premiums must be lower. Proposition 1 implies that the contract with the higher first-period premium is chosen by consumers with lower income growth. □