NIF B -3 ESTADO DE RESULTADOS

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

optimal insurance choice.

Lemma 1. Given any contribution rate set by PU and any feasible fee schedule of PR, it is optimal for customers to choose the insurance which offers the contract with the lowest fee.

Customers whose income is below the opt-out threshold can only choose PU’s con-tract. All other customers have the choice between PU and PR. As the utility function is strictly increasing, every customer chooses the contract which offers him the largest net benefit. Since the benefit level is fixed and equal for PU and PR, it is the contract’s fee 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 insurers charge the same fee.¹²

Having determined the population’s optimal insurance choice, we analyze PR’s opti-mal fee schedule. We call a customer profitable if, for a given contribution rate, the fee PU charges exceeds his health benefits; otherwise, we call the customer unprofitable.

Lemma 2. Given customers’ optimal choice and any contribution rate set by PU, it is op-timal for PR to set its fee equal to PU’s fee if a customer is profitable and to set the highest possible fee if a customer is unprofitable.

For a profitable customer, PR faces the trade-off between attracting the customer and charging a high fee. If PR’s fee exceeds PU’s fee, the customer rejects PR’s contract and chooses PU. If PR’s fee is strictly less than PU’s fee, PR can increase profits by increasing

12See also the remarks following Theorem1.

its fee slightly without losing the customer. Hence, it is optimal for PR to set its fee exactly equal to PU’s fee for all profitable customers.

If a customer is unprofitable and PR sets a fee below PU’s fee, PR incurs a loss.

Because PR may not reject customers, PR tries to deter unprofitable customers by setting its fee as high as possible. Note that PR may not deter all unprofitable customers because of the upper bound on its fee.

In contrast to PU, PR sets a flexible fee 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, if it would not be for the opt-out threshold, PR could cream skim all profitable customers in the population 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 this chapter.

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

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

Roughly, Assumption1says that the total income of all customer who must insure with PU covers the health costs of the whole population. It guarantees that the popu-lation structure is such that, at least potentially, PU can run a balanced budget. There are several reasons why Assumption1may be satisfied; some of which of may lie under the direct control of an exogenous regulator (benefit level, opt-out threshold, contribu-tion cap) some of which may not (health costs). In particular, Assumpcontribu-tion1holds if the benefit level is sufficiently low or the opt-out threshold is sufficiently high. With this assumption in place, we obtain the following theorem:

Theorem 1. Assume that the health insurance market is viable. Then the health insurance market has an equilibrium. Furthermore, the equilibrium contribution rateα^∗is unique if customers and PR behave as described in Lemma1and Lemma2.

The proof of Theorem1relies 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

in-¯e

¯h

K₁ K₂

0

min(c(h),L) α^∗

PR PR PU

PU

e h

(a)

¯e

¯h

K₂ K₁

0

min(c(h),L) α^∗

PR PR PU

PU

e h

(b)

Figure 1. Customers’ insurance choice by customer type for the cases (a)K₁< K2and (b) K₂< K1. The green area is composed of PU customers, the red area of PR customers. The cross-hatched area represents unprofitable customers, and the area with diagonal stripes

profitable customers.

come above the contribution cap and above the opt-out threshold choose PR. See Figure 1for a graphical illustration.

Interestingly, independent of their insurance choice, all customers pay the amount they would pay if they insured with PU. That is, PR’s fee is coupled to PU’s fee in equi-librium. Intuitively, its monopolistic power in the private sector allows PR to charge customers a fee that makes them indifferent to choosing their outside option which is insuring with PU.¹³ Attracting profitable customers entails two positive effects for PR:

First, there is an immediate gain in profits. Second, if PU loses profitable customers to PR, PU has to increase the contribution rate which leads to a higher fee for all customers.

This in turn allows PR to increase fees for all its customers as their outside option has

13We study the case of competing private insurers in Section4.5.

become less attractive. In fact, note that if profitable customers with income above the opt-out threshold would collectively choose to insure with PU instead of PR, PU could adjust the contribution rate downward which would lead to a lower fee for the entire population. Intuitively, PR prevents this by slightly undercutting PU’s fee.

What are the redistributional effects of the health insurance market? Profitable cus-tomers with income below the opt-out threshold subsidize all unprofitable cuscus-tomers of PU. Furthermore, the relative profitability of these two customer groups determines the fee 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 consequence 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 Theorem1for details.

A couple of technical remarks are in order. First, as can be seen from the proof of Theorem1, the assumption that health and income are affiliated is not required for the existence of equilibrium.

Second, note that PU’s contribution rate is only unique given the behavior of cus-tomers and PR. However, cuscus-tomers indifference behavior is not unique. Our specifica-tion that customers choose PR if they are indifferent resolves existence issues which stem from 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 fee ar-bitrarily close but not equal to PU’s fee. However, one could imagine other, plausible specifications for unprofitable customers with income above the contribution cap and above the opt-out threshold. For example, these customers could join PU. This could be justified as follows. Bauhoff (2012) shows that insurers do not not only cream skim ex-plicitly based on prices but also imex-plicitly, for example, by signaling poor service quality through delayed responses. A general theme of our model is that PR is more flexible than PU, thus, it is plausible to assume that PR is also more successful in deterring cus-tomers implicitly. Even if PR is only slightly better at deterring cuscus-tomers implicitly, all profitable customers with income above the opt-out threshold join PR, and all unprof-itable customers with income above the opt-out threshold join PU. For this and other specifications an analogous analysis applies.

Relatedly, PR’s optimal fee schedule 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 fee that exceeds PU’s fee. Note, however, that this does not change customers decisions and thus the equilibrium

contri-bution rate is the same as under Lemma2. Furthermore, our specification is particularly robust to tremble-like errors in the behavior of customers.