Adverse Selection: A Graphical Framework
From “Selection in Insurance Markets: Theory and Empirics in
Pictures” Liran Einav and Amy Finkelstein 2011
The Textbook Environment for Insurance Markets
We start by considering the textbook case of insurance demand and
cost, in which perfectly competitive, risk-neutral firms offer a single
insurance contract that covers some probabilistic loss, risk- averse
individuals differ only in their (privately-known) probability of
incurring that loss, and there are no other frictions in providing
insurance such as administrative or claim-processing costs. Thus, more
in the spirit of Akerlof (1970) and unlike the well known environment
of Rothschild and Stiglitz (1976), firms compete in prices but do not
compete on the coverage features of the insurance contract. We return
to this important simplifying assumption in the end of this essay.
Figure 1 provides our graphical representation of this case, and
illustrates the resulting adverse selection as well as its consequences for
insurance coverage and welfare. The figure considers the market for a
specific insurance contract. Consumers in this market make a binary
choice of whether or not to purchase this contract, and firms in this
market compete only over what price to charge for the contract.
The vertical axis indicates the price (and expected cost) of that contract,
and the horizontal axis indicates the quantity of insurance demand.
Since individuals face a binary choice of whether or not to purchase the
contract, the “quantity” of insurance is simply the fraction of insured
individuals. With risk-neutral insurance providers and no additional
frictions, the social (and firms’) costs associated with providing
insurance are the expected insurance claims—that is, the expected
payouts on policies.
Figure 1 shows the market demand curve for the insurance contract.
Because individuals in this setting can only choose the contract or not,
the market demand curve simply reflects the cumulative distribution of
individuals’ willingness to pay for the contract. While this is a standard
unit demand model that could apply to many traditional product
markets, the textbook insurance context allows us to link willingness to
pay to cost. In particular, a risk averse individual’s willingness to pay
for insurance is the sum of her expected cost and her risk premium.
In the textbook environment, individuals are homogeneous in their risk
aversion (and all other features of their utility function). Therefore,
their willingness to pay for insurance is increasing in their risk type—
that is, their probability of loss, or expected cost—which is privately
known. This is illustrated in Figure 1 by plotting the marginal cost
(MC) curve as downward sloping: those individuals who are willing to
pay the most for coverage are those that have the highest expected cost.
This downward sloping MC curve represents the well-known adverse
selection property of insurance markets: the individuals who have the
highest willingness to pay for insurance are those who are expected to
be the most costly for the firm to cover.
The link between the demand and cost curve is arguably the most
important distinction of insurance markets (or selection markets more
generally) from traditional product markets. The shape of the cost curve
is driven by the demand-side customer selection. In most other
contexts, the demand curve and cost curve are independent objects;
demand is determined by preferences and costs by the production
technology. The distinguishing feature of selection markets is that the
demand and cost curves are tightly linked since the individual’s risk
type not only affects demand but also directly determines cost.
The risk premium is shown graphically in the figure as the vertical
distance between expected cost (the MC curve) and the willingness to
pay for insurance (the demand curve). In the textbook case, the risk
premium is always positive, since all individuals are risk-averse and
there are no other market frictions. As a result, the demand curve is
always above the MC curve and it is therefore efficient for all
individuals to be insured (Qeff = Qmax). Absent income effects, the
welfare loss from not insuring a given individual is simply the risk
premium of that individual, or the vertical difference between the
demand and MC curves.
When the individual-specific loss probability (or expected cost) is
private information to the individual, firms must offer a single price for
pools of observationally identical, but in fact heterogeneous,
individuals. Of course, in practice firms may vary the price based
on some observable individual characteristics (such as age or zip
code). Thus, Figure 1 can be thought of as depicting the market for
coverage among individuals who are treated identically by the
firm.
The competitive equilibrium price will be equal to firms’ average cost
at that price. This is a zero profit condition; offering a lower price will
result in negative profits, and offering higher prices than competitors
will not attract any buyers. The relevant cost curve the firm faces is
therefore the average cost (AC) curve, which is also shown in Figure 1.
The (competitive) equilibrium price and quantity is given by the
intersection of the demand curve and the AC curve (point C).
The fundamental inefficiency created by adverse selection arises
because the efficient allocation is determined by the relationship
between marginal cost and demand, but the equilibrium allocation is
determined by the relationship between average cost and demand.
Because of adverse selection (downward sloping MC curve), the
marginal buyer is always associated with a lower expected cost than
that of infra-marginal buyers. Therefore, as drawn in Figure 1, the AC
curve always lies above the MC curve and intersects the demand curve
at a quantity lower than Qmax. As a result, the equilibrium quantity of
insurance will be less than the efficient quantity (Qmax) and the
equilibrium price will be above the efficient price, illustrating the
classical result of under-insurance in the presence of adverse selection
(Akerlof, 1970; Rothschild and Stiglitz, 1976). That is, it is efficient to
insure every individual (MC is always below demand) but in
equilibrium the Qmax – Qeqm individuals with the lowest expected costs
remain uninsured because the AC curve is not always below the
demand curve. These individuals value the insurance at more than their
expected costs, but firms cannot insure these individuals and still break
even.
The welfare cost of this under-insurance depends on the lost surplus
(the risk premium) of those individuals who remain inefficiently
uninsured in the competitive equilibrium. In Figure 1, these are the
individuals whose willingness to pay is less than the equilibrium price,
Peqm. Integrating over all these individuals’ risk premia, the welfare loss
from adverse selection in this simple framework is given by the area of
the “dead-weight loss” trapezoid CDEF.
Even in the textbook environment, the amount of under-insurance
generated by adverse selection, and its associated welfare loss, can vary
greatly. Figure 2 illustrates this point by depicting two specific
examples of the textbook adverse selection environment, one that
produces the efficient insurance allocation and one that produces
complete unraveling of insurance coverage. The efficient outcome is
depicted in Panel (a). While the market is adversely selected (that is,
the MC curve is downward sloping), the AC curve always lies below
the demand curve. This leads to an equilibrium price Peqm, that,
although it is higher than marginal cost, still produces the efficient
allocation (Qeqm = Qeff = Qmax). This situation can arise, for example,
when individuals do not vary too much in their unobserved risk (that is,
the MC and consequently AC curve is relatively flat) and/or
individuals’ risk aversion is high (that is, the demand curve lies well
above the MC curve).
The case of complete unraveling is illustrated in Panel (b). Here, the
AC curve always lies above the demand curve even though the MC
curve is always below it. As a result, the competitive equilibrium is that
no individual in the market is insured, while the efficient outcome is for
everyone to have insurance.
FIGURE I: Adverse selection in the textbook setting
Price
B
Demand curve
A
AC curve
C
Peqm J
G
MC curve E
D
F
Qeqm Qmax Quantity
30
FIGURE 2: Specific Examples of extreme cases
Panel (a): Adverse selection with no efficiency cost
Price
Demand curve
AC curve
Peqm C
MC curve
Qmax Quantity
Panel (b): Adverse selection with complete unraveling
Price
AC curve
Demand curve
MC curve
Qmax Quantity
31
