Equilibrium Existence

Retaining obligatory health insurance, we prove existence of equilibrium in the health insur-ance market. We proceed backward from the second period, first studying customers’ optimal insurance choice.

Lemma 3.3.1. Given any contribution rate set by PU and any feasible payment scheme of PR, it is optimal for customers to choose the insurance which offers the contract with the lowest payment.

This result is immediate since both insurances offer contracts with equivalent benefit level L.

Customers whose income is below the opt-out threshold can only choose PU’s contract. All other customers have the choice between PU and PR. As the utility function is strictly increasing, every

customer chooses the contract which offers her the largest net benefit. Since the benefit level is fixed and equal for PU and PR, it is the contract’s payment that determines the net benefit and, thereby, its attractiveness for customers.

In the following, we assume that customers choose PR when they are indifferent, i.e., if both insurances charge the same payment.¹⁰

Having determined the population’s optimal insurance choice, we analyze PR’s optimal payment scheme. We call a customer profitable at a given PU contribution rate, if the payment charged by PU exceeds health benefits payable to the customer; otherwise, we call the customer unprof-itable.¹¹

Lemma 3.3.2. Given customer’s optimal contract choice and an arbitrary contribution rate set by PU, it is optimal for PR to set its payment equal to PU’s payment if a customer is profitable and to set the highest possible payment if a customer is unprofitable.

For a profitable customer, PR faces the trade-off between attracting the customer and charging a high payment. If PR’s payment exceeds PU’s payment, the customer turns down PR’s contract and chooses PU. If PR’s payment is strictly less than PU’s payment, PR can increase profits by increasing its payment slightly without losing the customer. Hence, it is optimal for PR to set its payment exactly equal to PU’s payment for all profitable customers.

If a customer is unprofitable and PR sets a payment below PU’s payment, PR incurs a loss since she attracts the customer. Since PR may not reject customers, PR tries to deter unprofitable customers by setting its payment as high as possible. Note that PR may not deter all unprofitable customers because of the upper bound on its payment (feasibility constraint).

In contrast to PU, PR sets a flexible payment and discriminates based on both health and income.

The above argument shows that PR exploits its greater flexibility to cream skim all profitable customers with sufficiently high income, i.e., with income exceeding the opt-out threshold. In fact, without an opt-out threshold, PR would cream skim all profitable customers in the pop-ulation which would make it impossible for PU to run a balanced budget. Hence, the opt-out threshold is essential for the existence of equilibrium in the health insurance market.

This observation motivates the following assumption which we maintain throughout the paper.

Assumption 1. (Viable health insurance market.) The aggregated income of customers with income below the opt-out threshold and below the contribution cap exceeds the entire population’s health benefits:

E[min(L, c(h))]<E[e1{e<min(K1,K2)}].

10See also the remarks following Theorem 3.3.1.

11Note that such a situation may only arise since insurance is obligatory.

Roughly, Assumption 1 says that the total income of all customer who must insure with PU covers the health costs of the whole population. It guarantees that the population structure is such that at least potentially PU can run a balanced budget. There are several reasons why Assumption 1 may be satisfied; some of those may be under direct control of an exogenous regulator (benefit level, opt-out threshold, contribution cap) but some of those may not (health costs). In particular, for a given health income distributionF Assumption 1 holds if the benefit level is sufficiently low or the opt-out threshold and contribution cap are sufficiently high. With this assumption in place, we obtain the following theorem:

Theorem 3.3.1. Assume that the health insurance market is viable. Then the health insurance market has an equilibrium and the equilibrium contribution rate α^∗ is unique.

The proof of Theorem 3.3.1 relies on the intermediate value theorem. The key step is to establish continuity of PU’s objective in the contribution rate. See the Appendix for details.

In equilibrium, customers with income below the opt-out threshold choose PU. Above the opt-out threshold customers who are profitable insure with PR; unprofitable customers insure with PU.

However, all customers, profitable or unprofitable, with income above the contribution cap and above the opt-out threshold choose PR. See Figure 3.1 for a graphical illustration.

Figure 3.1: Customers’ insurance choice by customer type. The function of renormalized health benefits, min(c(h),L)

α , takes values on the e-axis.

Interestingly, independent of their choice of insurance, all PR customers pay the same amount they would pay if they insured with PU. That is, PR’s payment is coupled to PU’s payment in equilibrium. Intuitively, its monopolistic power allows PR to charge customers a payment that makes them indifferent to choosing their outside option which is insuring with PU.¹² Attract-ing profitable customers entails two positive effects for PR: Firstly, there is an immediate gain in profits. Secondly, if PU loses profitable customers to PR, PU has to increase the contribution rate leading to a higher payment for all customers. This in turn allows PR to increase payments for all its customers as their outside option has become less attractive. In fact, note that if profitable customers with income above the opt-out threshold would collectively choose to insure with PU

12We study the case of competing private insurances in subection 5.3.

instead of PR, PU could adjust the contribution rate downward leading to a lower payment for the entire population. Intuitively, PR prevents this by slightly undercutting PU’s payment.

What are the redistributional effects of the health insurance market? Profitable customers with income below the opt-out threshold subsidize all unprofitable customers with income below the contribution cap. Furthermore, the relative profitability of these two customer groups determines the payment for the entire population through their effect on the contribution rate. The surplus of profitable customers with high income above the opt-out threshold is transformed one-to-one into a profit for PR and is lost for the population. PR may incur a loss on unprofitable customers with income above the contribution cap and above the opt-out threshold. However, as a con-sequence of the organizational structure of the health insurance market, PR obtains an overall profit: PU runs a balanced budget; relative to PU, PR attracts customers with higher income.

As health and income are positively correlated, a higher income entails also a better health type.

Thus, PR draws upon a more lucrative part of the population and earns positive profits. See the proof of Theorem 3.3.1 for details.

A couple of technical remarks are in order: Firstly, as can be seen from the proof of Theorem 3.3.1, the assumption that health and income are affiliated is not required for the existence of equilibrium.

Secondly, note that PU’s contribution rate is only unique given the behavior of PR and customers.

However, customers indifference behavior is not unique. Our specification that customers choose PR if they are indifferent resolves existence issues for profitable customers with income exceeding the opt-out threshold: If these customers would choose PU when they are indifferent, PR would like to set a payment arbitrarily close but not equal to PU’s payment. However, one could imagine different specifications for unprofitable customers with income above the contribution cap and above the opt-out threshold. For these specifications an analogous analysis applies.

Relatedly, PR’s optimal payment scheme may not be unique (even on a set with positive measure):

In order to deter unprofitable customers with income between the opt-out threshold and the contribution cap, PR can set any payment that exceeds PU’s payment. Note, however, that this does not change customers decisions and thus the equilibrium contribution rate is the same as under Lemma 3.3.2. Furthermore, our specification is particularly robust against tremble-like errors in the behavior of customers.