Exclusive vs. Independent Agents:
A Separating Equilibrium Approach
by
Itzhak Venezia*
Dan Galai*
Zur Shapira**
_________________
* The Hebrew University of Jerusalem, **Stern School of Business, New York University. The authors
wish to thank Yaacov Bergman, Sari Carp, Claude Fluet, Eugene Kandel, Yoram Peles, Sridhar
Seshadri, and two anonymous referees for their constructive comments and suggestions. They also
acknowledge the financial support of The Gallanter Center, The Picker Center for Insurance, The Stern
School of Business, the Krueger Center of Finance and the hospitality of the Center for Rationality.
Address: Itzhak Venezia, School of Business, Hebrew University, Jerusalem, Israel. Fax: 972-2-5881341, e-mail: msvenez@pluto.huji.ac.il.
Abstract
We provide a separating equilibrium explanation for the existence of
the independent insurance agent system despite the potentially higher costs of this
system compared to those of the exclusive agents system (or direct underwriting). A
model is developed assuming asymmetric information between insurers and insureds;
the formers do not know the riskiness of the latter. We also assume that the claims
service provided by the independent agent system to its clients is superior to that
offered by direct underwriting system, that is, insureds using the independent agent
system are more likely to receive reimbursement of their claims. Competition compels
the insurers to provide within their own system the best contract to the insured. It is
shown that in equilibrium the safer insureds choose direct underwriting, whereas the
riskier ones choose independent agents. The predictions of the model agree with
previous research demonstrating that the independent agent system is costlier than
direct underwriting. The present model suggests that this does not result from
inefficiency but rather from self-selection. The empirical implication of this analysis
is that, ceteris paribus, the incidence of claims made by clients of the independent
agents system is higher than that of clients of direct underwriting. Implications for the
co-existence of different distribution systems due to unbundling of services in other
industries such as brokerage houses and the health care industry are discussed.
I. Introduction
Two distribution channels are dominant in the sale of insurance in the US and
other countries: the independent agent system and the exclusive agent system (or direct
underwriting). Exclusive agents are autonomous entities, which are contractually bound
to represent just one insurer. Exclusive dealing arrangements include insurers who sell
through employees (direct underwriters), companies who use exclusive agents, or
companies who use mass merchandizing without employing salespersons. 1
The independent agent is also an autonomous contractor, but he/she represents the
products of several competing insurers. One important aspect of the independent agents
system is the right to ownership of the customer list. The independent agent has the legal
rights over the customer list, which implies that the insurer may neither solicit the agent's
clients directly, nor may he/she reassign a client to a different insurer. The independent
agent, however, has the right to move his/her clients to a competing insurer.
Many researchers find that exclusive agents enjoy cost advantages compared to
independent agents.2 It has been debated whether one system dominates the other (see
Callahan, 1990, Hammes, 1990 and Nicosia, 1990): and some speculate that eventually
the costlier independent system will be eliminated (Kenney, 1974).
Previous research attempted to resolve the seeming paradox of the existence of
the two systems where one dominates the other. Most of this research focuses on the
relationship between the insurer and the agent, as well as on the advantages and
disadvantages for the insurer in using exclusive vs. independent agents (see, e.g., Barrese
1
We use the terms “exclusive agents” and “direct underwriters” interchangeably.
1
and Nelson, 1992, D’arcy and Doherty, 1990, Mayers and Smith, 1981, Cummins and
Weiss, 1992, Marvel, 1982, Sass and Gisser, 1989, Kim, Mayers and Smith, 1996, Regan
and Tzeng, 1998, Regan, 1997, and Regan and Tennyson, 1996).
Recently, Posey and Yavas (1995) and Posey and Tennyson (1997) suggested that
the key to explaining the co-existence of the two distributions systems lie in the different
client-agent, rather than agent-insurer, relationships. The two systems may be able to
coexist because they differ in the services they offer and the clienteles they attract. Posey
and Yavas suggest that an important advantage with which independent agents provide
their clients is in the search for an “appropriate” insurer. They show that in equilibrium,
the two systems coexist: clients with higher search costs choose independent agents,
while those with lower search costs opt for direct underwriters.
This paper follows the line of reasoning of Posey and Yavas in justifying the coexistence of the two systems by the differences in services and clienteles. Our major
assumption is that independent agents are expected to act on behalf of the insured, as
opposed to direct underwriters who represent the insurers. In case of conflict over claims,
the independent agent is more likely to assist the insured in the claims settlement process
than the direct underwriter is.
The above presumption is based on Cummins and Weiss (1992), Mayers and
Smith (1981), and Kim, Mayers and Smith (1996), who argue that independent agents are
more effective in representing the interests of their clients since they own the
policyholder list (in the exclusive agents system the insurance company owns the list).
Independent agents can therefore, threaten more credibly, to switch their clients to an
2
See, e.g., Joskow, 1973, Cummins and VanDerhei, 1979, Barrese and Nelson, 1992, and Pritchett and
Brewster, 1979.
2
alternative insurer if their claims are not treated fairly and promptly. As Peretz (1998)
noted, attempts by insurance companies to lower the commissions they pay to
independent agents fail due to the agents' threat to move clients to other insurers. Indeed,
Etgar (1976) shows that independent agents typically provide their clients with more
generous claims settlements. In addition, a study by Barrese, Doerpinghaus, and Nelson
(1993), finds that independent agents provide higher quality of services than exclusive
agents do.
We show that equilibrium may exist where riskier clients choose the more
expensive independent agents with lower deductibles and the safer clients prefer the
cheaper direct underwriting. This occurs because riskier clients believe that there are
higher chances they may need an agent’s help in case of damage. They are willing to pay
for the superior services offered by the higher-priced independent agents. Our
model
provides, therefore, an alternative explanation for the co-existence of the two systems. It
also provides a testable hypothesis different from that of Posey and Yavas; namely, that
ceteris paribus, clients of the independent agents system experience either a higher
incidence of claims, or higher levels of claims.
The paper is structured as follows: In section II we lay out the basic assumptions,
in Section III we present the model, in Section IV we provide a numerical example and
discuss the viability of the equilibrium. Conclusions and possible extensions of the model
to other industries are presented in the last section. Technical material is presented in the
Appendix.
3
II. Assumptions
There are two types of insurance companies in the market, differing in the
distribution systems they use. One employs direct underwriting, the other, independent
agents. It is assumed for simplicity of exposition that if the client is insured via an
independent agent, it is guaranteed that in case of damage, he or she will be reimbursed.
If the client is insured via direct underwriting there is a probability  (<1), but no
certainty of reimbursement.3
It is commonplace that insurance contracts sometimes fail to provide appropriate
coverage from the point of view of the insured. That is, the insured sometimes does not
get paid even when he or she is certain they should be paid. This may occur because of
insurer insolvency, or more often, from disputes between the insurer and the insured. The
vast number of contested claims resolved by courts demonstrates that there exists a
significant probability that a claim will not be covered. Also, delays in payments which
often occur are tantamount to partial default (see Doherty and Schlesinger, 1990).
An important task of the agent is to help the client in settling claims. This is
necessary because sometimes damages are not clear-cut, and contracts do not always
encompass all eventualities. We assume that every agent makes an effort to settle claims.
The amount of effort is determined so as to optimize the agent’s value function. The more
effort expended, the better the claims settlements for the clients are, and hence the higher
3
This assumption is made to simplify the analysis and can be relaxed. A necessary condition for our claim
is that a > d, where a and d are the probabilities of reimbursement for the independent agent and
direct underwriter, respectively. We show in Appendix 2 that the separating equilibrium results can also be
obtained in this case. Analysis of the case of certain reimbursement has an additional advantage as it
allows comparison of the deductible choice in our case with the more traditional models of deductible
choice (see, e.g., Mossin, 1968) which assume a guaranteed repayment.
4
is the reputation of the agent, as well as the future demand for his/her services. On the
other hand, the effort invested by the agent for settling claims is costly to the agent. We
assume that the benefit and cost functions of effort are about the same for both types of
agents. However, costs of reimbursement of claims may differ: in contrast with
independent agents, direct underwriters experience higher costs since payments to their
clients are made by their own firm. It is, therefore, reasonable to assume that the direct
underwriters make less effort to help their clients settle claims compared with
independent agents. Consequently, we assume that clients using the independent agent
system enjoy higher probability of reimbursement.
We denote by A the excess (effort) costs that the independent agent chooses to
incur in claims settlement. For the direct underwriter these costs are assumed to be zero.
It is also assumed that if a damage occurs, its amount is known and fixed, denoted here
by K.
We assume there are two types of potential insurance buyers in the market: Highrisk (H) clients and Low risk (L) clients, differing in the probability of incurring a
damage. The probabilities of damages for insureds of types H and L are denoted P H and
PL, respectively. All potential insureds are assumed to be risk averse with exponential
utility function with risk aversion measure . (This assumption is made only to facilitate
the numerical examples, and does not affect the qualitative results). All clients are small
(and hence price takers), and have full information about contracts available from all
insurers. Each firm offers a contract, or a menu of contracts, from which the insured may
choose just one. The objective of the firm is to maximize its expected profits (implying
all firms are risk neutral). Perfect competition drives insurers to provide the optimal form
5
of insurance contract, which under the assumptions made above, is full coverage above a
deductible (see, Arrow, 1963). Thus, each firm provides contracts of the form [D, (D)]
which specify the deductible D and the associated price (D). Insurers can provide any
deductible in the range [0,K].4 Full information concerning prices and contracts available
forces all contracts with the same deductible D to have the same price. Under
competition, the profits under all contracts available in the market are driven to zero.
III. The Model
If a separating equilibrium exists where the only clients buying from independent
agents are the riskier clients and those buying from direct underwriting are the safer ones,
and if expected profits are zero, the pricing formulas become:
a(D) = PH[KD + A],
(1)
for the independent agent system, and:
d(D) = PL[(KD)],
(2)
for direct underwriting.
Next, we derive the optimal contract for each system, based on which we infer the
contracts to be offered in equilibrium. The optimal deductible that can be offered in a
direct underwriting system for an insured of type i, i = H, L, can be evaluated by
The function (D) must satisfy -1 <' <0, where ' denotes differentiation with respect to D. ' <0 since
a higher deductibles implies lower payments to the insured. ' >-1 since otherwise the marginal pay for a
$1 of a deductible equals or exceeds $1. This will not be worthwhile unless the probability of damage is 1,
which rules out insurance.
4
6
maximizing his/her expected utility subject to the pricing function d(D). The deductible
is calculated by choosing D so as to maximize:
Ei{U[W|direct]}
= Pi{U[W0-d(D)-D] + (1-)U[W0-d(D)-K]} +
(1-Pi)U[W0-d(D)],
for i=H,L,
(3)
where Ei denotes expectations of a type i client, d(D) denotes the pricing function under
the direct underwriting system, W denotes the insured (random) final wealth, W0 denotes
initial wealth, and U(.) the client’s utility function. The assumption of an exponential
utility function implies (see Appendix 1) that the optimal deductible for an insured i (i =
H, L) with a probability of damage, Pi, buying from direct underwriting is given by:
D*id = (1/)ln{-d/((1+d)) [((1-Pi)/Pi)+(1-)exp(K)]}
(4)
where ’d denotes the derivative of d(D) with respect to D.
The expected utility of a client i, (i = H,L) buying a policy from the independent
agent system is given by:
Ei{U[W|Agent]}
= PiU[W0-a (D)-D]+ (1-Pi)U[W0-a (D)],
(5)
where a(D) denotes the pricing function in the independent agent system. The optimal
deductible for this client is obtained by choosing D to maximize the above expression
subject to the pricing function a(D) and is given (see Appendix 1) by:
D*ia = (1/)ln{- [ a/(1+a)] [(1-Pi)/Pi)]},
7
(6)
The pricing functions (1) and (2) imply that a and d, equal -PH, and -PL,
respectively. 5 Inserting these values into (4) and (6) we obtain:
D*Ha = (1/)Ln[ (1-PH) / (1-PH) ] = 0
(7)
D*Ld = (1/)ln{ [(1-PL) + PL((1-)/)exp(K)] /(1-PL)}
(8)
and
Note that in the above scenario, as in Mossin (1968), the optimal deductible for
the insured in the case of the independent agency system is zero when the insurer has
zero profits. In the case of direct underwriting however, the optimal deductible is strictly
positive. The reason is that buying insurance from direct underwriters does not eliminate
all risks even if the deductible is zero because of the non-guaranteed reimbursement.6
Moreover because of uncertain reimbursement, the lowest wealth (“worst eventuality”)
that could be obtained is (W0-K-d(D)), which, in the case of fair pricing equals [W0-KPL(K-D)]. The value of this wealth is minimized when D equals zero and the risk
averse insured who wants to avoid this alternative chooses a positive deductible.7
It is shown below that the assumption of perfect competition which leads to zero
expected profits, and the provision of optimal deductibles for the insured, results in a
reactive separating equilibrium given a wide range of parameters values (cf., Riley,
5
If the monopolistic competition drives the profits of all firms to the same proportion k above costs, then
the pricing formulas become:a (D) = PH(1+k)(K-D +A), and,d (D) = (1+k)PL(K-D). The
corresponding optimal deductibles are then given by: D*Ha =(1/)ln{ (1+k)(1-PH) / [1-(1+k)PH] }, and
D*Ld=(1/)ln{ (1+k)[ (1-PL) + PL(1-)exp(K) ] /[(1-(1+k)PL] }. All the results carry through to this
case.
6
Doherty and Schlesinger (1990) also analyzed optimal contracts when reimbursement is uncertain.
7
Another way to see why D*Ld> 0 must be positive, is to consider the effect of a marginal increase,  in the
deductible from zero for type L clients. This decreases the premium by PL, and hence increases the
wealth of the client by PL in all states of nature, but also decreases his or her wealth by  in the case of
damage and reimbursement (with a probability of P L). However, the positive effect on utility of the lower
premium is higher than the negative effect on utility of the lower reimbursement. This occurs because the
benefits of a lower price are obtained also when the client has a damage and is not reimbursed, a case
where the final wealth is extremely low and hence the marginal utility loss is very high.
8
1979). The concept of a reactive equilibrium has been shown to be an appropriate
equilibrium concept in situations such as the one discussed here. It applies where a Nash
equilibrium may not exist, that is, for situations where for any potential equilibrium
contract, deviations may be profitable.8 The Nash equilibrium makes the strong
assumption that other firms remain passive to such deviations. Reactive equilibrium has
been introduced to “correct” for that passivity by postulating that a firm which considers
diverging from equilibrium, can anticipate that other firms will react to its action.
Consequently, the firm will not deviate if it anticipates that competitors’ reactions may
annul any benefit from its initial deviation.
In the situation discussed in this paper, a (reactive) separating equilibrium is
obtained by direct underwriting firms offering lower prices and higher deductibles. The
price/deductible combination should be set so as to deter the high risk clients from buying
policies from direct underwriting firms. To that end, the difference in prices between the
agent system and direct underwriting should be set so that it is worthwhile for the low
risk clients (but not for high risk clients) to face a higher probability of no reimbursement
and a higher deductible.
The formal conditions for a reactive separating equilibrium are proved in the
following proposition.
Proposition 1: If inequalities (9) and (10) below hold, then the contract [D*Ld, (D*Ld)]
defined by (2) and (8) and offered by direct underwriters, and the contract [D*Ha, (D*Ha)]
defined by (1) and (7) and offered by the independent agent system, provide a reactive
8
This concept has been used widely in finance and insurance (see, e.g., Miller and Rock, 1985, Cresta and
Laffont, 1987, and Venezia, 1991).
9
separating equilibrium. All high-risk clients choose the independent agent system while
low risk clients opt for direct underwriting.
EH{U[W0-X-a(D*Ha)]} > EH{U[W0-X-d(D*Hd)]},
(9)
EL{U[W0-X-a(D*La)]} < EL{U[W0-X-d(D*Ld)]},
(10)
where X is the random variable denoting damages.
Proof: To show that the set of contracts is an equilibrium set, one needs to show that if
these contracts are offered, neither buyers nor sellers have an incentive to purchase a
different contract. That is, buyers do not have an incentive to buy other contracts and
sellers do not have an incentive to offer other contracts.
Suppose the parameters of the utility functions and loss distributions are such that
inequalities (9) and (10) hold. In this case, high-risk clients prefer the optimal contract
under the independent agent system [D*Ha,  (D*Ha)] whereas the safer clients prefer the
contract offered by the direct underwriters,
[D*Ld, d (D*Ld)].
Next, we show that the independent agents and the direct underwriters offer the
above-specified contracts without deviating from their terms. Suppose a firm in the agent
system9 considers deviating from equilibrium by offering the new contract [D’,(D’)]
that will attract low risk clients only. Such a contract can usually be constructed.
Competition however will drive the price of this contract to equal its fair value, and hence
(D’) must equal PL[K-D’+A]. This contract is not sustainable since another firm in the
independent agent system can come up with a better contract for the low risk clients. It
can offer [D*La, (D*La)] which was derived in (6) as the optimal among all fairly priced
10
contracts. Therefore, the firm that initially deviates from equilibrium gains nothing from
this deviation as its contract is merely eliminated by other contracts. The contract [D*La,
(D*La)] will not prevail either, since by (10), it is dominated by the contract [D*Ld,
(D*Ld)] offered by direct underwriters. It thus follows that firms do not deviate from the
equilibrium contract [D*Ha, (D*Ha)] offered by independent agents, and [D*Ld, (D*Ld)]
offered by direct underwriters. QED
IV.
Sensitivity to the Parameters and a Numerical Example
We now examine the conditions (parameters) under which a reactive equilibrium
is reached. The key to reaching this equilibrium is that the clients separate, namely, that
inequalities (9) and (10) hold. To find when this occurs, we examine the clients' monetary
tradeoffs when choosing between the systems.
A risky client, H, who chooses the independent agents system over the direct
underwriters, actually prefers paying the additional price a - d in order to avoid the
chance of losing (KD) with probability PH(1). This probability denotes the chance
that a direct underwriter’s client will suffer a damage and will not get paid. In a similar
situation, the client of the independent agent will receive (KD).
From (1) and (2) it follows that for a given deductible, D, the extra price to be
paid, a  d, is:
a  d = PH (KD+A) PL  (KD),
(11)
which, after some rearrangement of terms can be written as:
a  d = (KD) (PH  PL) + PL (1) (K  D) + APH
9
(12)
The argument for firms in the direct underwriting system is very similar, and hence will not be repeated
11
We note from (12) that the higher the difference (PH  PL) between the
probabilities, the higher the differences in prices. It then becomes more difficult to
convince the high-risk clients not to mimic the low risk clients by buying policies from
direct underwriters. If, on the other hand, the difference between PH and PL is small (in
the extreme PH = PL), both type of clients would again want to buy from the same system.
If A is low, both types patronize independent agents; if A is large, both types buy from
direct underwriters.
Another way to view the choice between the systems is as follows: the direct
system describes a “no frills” system without guaranteed reimbursement. The agent
system describes a “deluxe” system, which offers extra insurance against no repayment.
The riskier clients want to buy this extra insurance, for an additional price. The larger (P H
 PL), the higher the additional price paid for the extra insurance, and the lower the
chance that the high risk clients want to buy it. To examine what happens when PH and PL
are very close it is convenient to set PH = PL . In this case the two types of clients exhibit
similar behavior, either both buy the extra insurance or both decline it (again depending
on A). This is also the case, by continuity argumentation, when PH is close to PL but not
identical to it.
For the purpose of illustration we present below the equilibrium contracts for
some sets of parameters. The following parameters are considered: PH = .05, K = 500, PL
= .01,  = .01, A = 10,  = .9.
In this case the optimal contracts are given by: D*Ha = 0, D*La = 0,
D*Hd = 57.2, D*Ld = 13.9, *a = 25.5, *d = 4, and the corresponding expected utilities are:
here.
12
EH{U[W0-X -a(D*Ha)]} = -1.29, EH{U[W0-X-d(D*Hd)]} = -1.83
EL{U[W0-X -d(D*Ld)]} = -1.19 , EL{U[W0-X-a(D*La)]} = -1.29
It can be seen from the above data that the proposed contracts
provide a
separating equilibrium. The high-risk clients enjoy a higher expected utility when buying
through the independent agent system rather than from direct underwriting, and the
reverse is true for the safer clients. It can also be verified that the expected profits of the
insurers in the agent system, as well as those in the direct distribution system, are zero.
As suggested in the above Proposition, a separating reactive equilibrium is achieved. In
Appendix 2 we show that a separating equilibrium exists even when reimbursement is not
certain in the independent agent system, but is more likely than in the direct underwriting
system.
To evaluate how robust is the existence of a separating equilibrium to the set of
parameters chosen for the illustration, we looked at the range of parameters that allows
separation. Table 1 shows the range of values that each variable can assume (holding the
other variables constant at their initial values described above), while a separating
equilibrium is still maintained.
Table 1 shows that the parameters could change considerably and the equilibrium
is still maintained. In particular, the existence of a separating equilibrium is only weakly
sensitive to the fixed collection costs. This can be attributed in part to the low chances of
damages. The difference between high risk and low risk clients’ respective chances of
damage must be reasonably large to prevent the riskier clients from choosing the less
expensive direct underwriting. Finally, one relatively surprising result is that the highest
13
value of damage, K, permitting a separating equilibrium is bounded from above. The
reason for this is that when the possible value of damage is too high, even lower risk
clients prefer the independent agent system.
V. Conclusion
We have shown that the coexistence of the independent agent system
alongside the lower cost exclusive agent system may be plausibly explained by the
different clienteles being catered by the two systems. Our model predicts that insureds in
the independent agent system have on average higher claims. The empirical implications
of the model agree with the observation of Berger, Cummins and Weiss (1995) that the
independent agent system is more costly than the direct underwriting, but that the
differences in cost efficiency are less than price efficiency differentials. This observation
reinforces the argument that independent agents provide a different and/or a higher
quality product. Future research should attempt to test the predictions of the theory
directly, namely that ceteris paribus, clients of direct underwriting have a lower
incidence or lower levels of claims than policy holders of independent agents. 10
Independent agents often "sell" themselves as protectors of the insured in case of
a claim (this of course is not a proof they are right). In one commercial on Israeli radio,
the commentator asks: "In case you are sued, who are you going to hire for your lawyer,
the prosecutor? Of course not, then why would you have a direct underwriter handle your
10
As a matter of casual empiricism, we have the following piece of data. In Israel, there are only two firms
(established in 1996) that sell through direct agents: AIG, and "Direct Insurance". In 1997, the
percentages of these firms' premiums, out of all firms' premiums in property and liability insurance were
0.3% and 0.7%, respectively (Israel Insurance Association, Financial Reports Analysis, 1998). Their
corresponding claims' payments were 0.1% and 0.6%, respectively. This may indicate that relative to the
14
claim?" In another commercial they describe a medical patient in need, trying to call
some anonymous hospital to help him in his predicament. He either gets evasive answers,
or an impossible answering machine. The announcer then says: "In medical need you will
not call the 'direct hospital services,' you will call your personal physician, why would
you act differently in case you need help in settling claims? Call your independent
insurance agent!"
Note that our framework explains the existence of the independent agent system
by these agents’ superior claims service, whereas Posey and Yavas explain it by the
search services offered by independent agents. As search technology continues to
advance and consumers are increasingly able to buy insurance via the Internet, the
relative merit of the differential claims service based framework may become apparent.
The implications of this study can be extended to other industries that employ
different distribution channels, such as security brokerage firms and health care
providers. Security brokerage firms are often classified into two types: discount brokers
and full service brokers. The latter are more expensive, and unlike the former, provide
also investment recommendations. In general, as is the case with insurance agents,
brokerage firms provide many services to their customers. In addition to the execution of
trades, they also provide services such as margin lending, advice, bookkeeping, and
holding of the stocks for the client (see, e.g., Brown, 1996).
Brokerage firms choose the bundle they offer. They can use the bundle to separate
clients with differing unknown traits (risk tolerance, informativeness, income, etc.)11 In
analogy with our study, one could test what traits are separated by the different types of
premiums they charged direct insurers had either fewer and/or lower claims than firms using independent
agents.
15
distributions systems by comparing the portfolios of clients of discount brokers to those
of full service ones, and examine whether they differ in their risks, performance (which
may indicate the degree to which they are informed), or other attributes.12 The less
informed investors (analogous to the riskier clients in the insurance industry) need the
investment counseling of full service brokers more than the better-informed investors do.
Consequently, the less-informed turn to full service brokers more frequently, but their
performance lags behind that of the more informed investors who use discount brokers.
In the health care industry there are also two main distribution systems: HMOs
which provide few frills and are usually less expensive, and private insurers, who are
more expensive but provide better services. As in the case of the insurance industry and
brokerage firms, it can be conjectured that the two systems separate between different
types of clients, according to risk, risk aversion, social status, education, income, etc.
(Browne and Doerpinghaus, 1993, find adverse selection in HMO’s although they did not
compare clienteles of HMO’s to those of private insurers, an endeavor worth pursuing).
In all these cases (the insurance industry, health care providers, and brokerage
houses) there is a common trend of unbundling of services. The more expensive system
that provided more services, has been established first and enjoyed exclusivity for a long
period. The less expensive systems came later, offered fewer services and threatened to
erode the market share of the veteran systems. However, the more expensive systems
were not eliminated. For example, in 1994 independent agency insurers controlled 52
percent of the total market (see Bests’s Aggregates and Averages 1995), and full service
11
See Brennan and Chordia, 1994, Irvine, Kandel and Wiener, 1997.
Shapira and Venezia (1998) found significant differences between the patterns of behavior of investors in
Israel, between those who had their portfolios managed by their brokers and those who managed their
portfolios on their own.
12
16
brokers still accounted for 75% of all individual investor trading in 1992 (Wall Street
Journal, September 18, 1992). Different distribution systems provide different bundles of
services according to the different preferences and attributes of the clients. One may
expect to observe different systems coexisting, while the costs of operating these systems
differ considerably. These cost differentials, however, may not necessarily imply
differences in efficiency.
17
Table 1
Ranges of Values of the Parameters
Allowing Equilibrium to Exist
Variable
Original value
Minimal value







Maximal value



PH
. 
.036
.579
PL
.

.0145
K
500
350
560
.0069
.011
0
722

A

 
10


*Any number  > 0.
18
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21
Appendix 1
The Optimal Deductible
Differentiating (3) with respect to D and assuming U(W) = -exp(-W), one obtains:
Ei[U (W|Direct)]/D
= -exp(-Wo) {’d exp (d) [Piexp(D) + Pi(1-)exp(K)
+ (1-Pi)] + exp(d)PiexpD)}
(A1)
= -exp (-Wo)exp(d)
{Piexp(D)(1+’d) + (1-Pi)’d + Pi (1-)exp(K)}
(A2)
Equating the derivative to 0 and rearranging terms one obtains:13
D*id=(1/)ln{-’d’d))[((1-Pi)/Pi)+(1-)eexp(K)]}
(A3)
We next provide conditions under which the second derivative becomes negative.
2Ei[U(W|Direct)]/D2 = -exp(-W0){’dexp(d) G + exp(dG’}
(A4)
where G is the term in curly brackets in (A2). From (A1) it follows that at the optimum
G equals 0, and hence the sign of the second derivative is determined by the sign of
- G’. G’, however, is given by:
G’ = Pexp(D)(1 + ’d) + Piexp(D)’’d + (1-Pi)’’d
(A5)
Assuming that d is linear (e.g.d = (1+k)Pi(K-D)), and since we have already assumed
that 'd >- 1, it follows that G’ > 0 and that iU(W|DirectD2 < 0.
13
The optimal deductible in case the client buys from the agent system is obtained as a special case of the
above analysis with  = 1, and a replacing d., and to save space we do not repeat the derivations.
22
Appendix 2
Uncertain reimbursement in the agent system
If there is uncertainty in the reimbursement of the client in the agent system, then
this system becomes a special case of direct underwriting. In this case we denote the
respective probabilities of reimbursement by a and d, where 1 > a > d The
expected utility of a client of type i, i = H, L, buying from system s, s=a, d (where a and
d denote the agent system and direct underwriting, respectively), becomes in this case:
Ei{-exp(-W)|s}
=-{Pi[sexp(-(W0-s-D)) + (1-s)exp(-(W0-s-K))]+
(1-Pi)-exp(-(W0-s))},
for i=H,L, s= a,d
The optimal deductibles are given in this case by:
D*is =(1/)ln{-’s’s))[((1-Pi)/Pi)+Pi(1-)exp(K)]}
for i=H,L, s= a,d
If the prices are given by:

a(D) = PHa(K-D +A),and d(D) = PLd(K-D),

then the equilibrium optimal deductibles are given by
D*Ld = (1/) ln{ [(1-PL) + PL(1-d)exp(K)] / dPL) }
D*Ha = (1/) ln{ [(1-PH) + PH(1-a)exp(K)] / aPH) }
The analysis thereafter is quite similar to the case of sure reimbursement for the agent
system. One difference which is somewhat tangential to the analysis is that with certain
reimbursement the deductibles in the agent system are lower than in direct underwriting
(in the agent system they must be zero). This, however, is not necessarily true with
uncertain reimbursement. The reason being that the optimal deductible is an increasing
function of the chances of damage. This occurs because as the probability of damage
increases, so does the price.
Existence of a separating equilibrium can be shown with the parameters of Table
1, even if the probability of reimbursement in the agent system is less than 1. Actually as
long as this probability is higher than .939, a separating equilibrium exists.
23