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POLICY DESIGN UNDER RISK (Continued - 1)
** SOME APPLICATIONS:
1- Risk Sharing:
Consider a group of n individuals making a public choice x that provides an uncertain return π(x,e) to be distributed among the n members of the group, where e is a discrete random variable that can take m possible values (e1, ..., em).
The maximization of aggregate sure net benefit gives:
W(u) = Maxx,T {Si ti1 - (-p(x, e1)): ui = EUi(-tij), i = 1, …, n;
Sj ti1 + p(x, e1) = Si tij + p(x, ej), j = 2, …, m}.
where u = (u1, ..., un) is the vector of reference utility levels, T = {tij, i = 1, ..., n; j = 1, ..., m}, EUi(×) denotes the objective function of the i-th individual, and (-π(x,e)) is interpreted as the "cost of the project".
We make the following specialized assumptions:
. π(x,e) = x e, where E(e) = m > 0 and Var(e) = E(e2) = s > 0,
. The n individuals have CARA risk preferences represented by the mean-variance utility function: EUi(-ti) = E(-ti) - ½ ri Var(ti), where ri is the risk aversion coefficient of the i-th individual, Ri = ½ ri Var(ti) denoting the Arrow-Pratt risk premium, i = 1, ..., n.
. All individuals are assumed to be risk averse, ri > 0, i = 1, ..., n.
. The amount paid (or received if negative) tij is restricted to the linear rule
tij = -[ai + bi p(x,ej)] = -[ai + bi x ej],
where [ai + bix ej] is the amount received by the i-th individual, i = 1, ..., n.
Then, the above problem becomes:
W(u) = Maxx,a,b {Si -[ai + bi x e1] + x e1: ui = ai + bi x m - .5 ri bi2 x2 s, i = 1, …, n;
Si -[ai + bi x e1] + x e1 = Si -[ai + bi x ej] + x ej, j = 2, …, m},
which, given λ1 = 1 - Sj³2λj, can be written in terms of the Lagrangean L:
W(u) = Maxx,a,b Ming,l {Sj³1 lj [x ej (1 - Si bi) - Si ai] + Si gi [ui - ai - bi x m + .5 ri bi2 x2 s]}
where the λ's and γ's are Lagrange multipliers. The first-order conditions for this optimization problem are:
¶L/¶x = (Sj³1 lj ej)(1 - Si bi) - Si gi [bi m - ri bi2 x s] = 0, (1)
¶L/¶ai = - Sj³1 lj - gi = 0, i = 1, …, n, (2)
¶L/¶bi = -(Sj³1 lj ej) x - gi [x m - ri bi x2 s] = 0, i = 1, …, n, (3)
¶L/¶gi = ui - ai - bi x m + .5 ri bi2 x2 s = 0, i = 1, …, n, (4)
¶L/¶li = x ej (1 - Si bi) - Si ai = 0, j = 1, …, m. (5)
Note that equation (2) implies that
gi = - Sj³1 lj = -1, i = 1, …, n. (6)
Also, from (5), we have
Si ai = 0 and Si bi = 1, (7)
which corresponds to a Pareto optimal policy (since it implies that W(u) = 0). Using expressions (6) and (7), equations (1) and (3) become respectively:
m = Si ri bi2 x s, (8)
and
m = Sj³1 lj ej + ri bi x s. (9)
Multiplying (9) by bi, summing across all i, using (7), and combining the result with (8) implies:
Sj³1 lj ej = 0.
Combining this result with (9) gives:
m = ri bi x s. (10)
Again, combining result (10) with (8) yields:
ri bi = Si ri bi2.
Using (7), this implies that the optimal proportion of returns π(x,e) paid to the i-th individual is:
bi = (1/ri)/[Si (1/ri)].
Combining this result with (10) yields the optimal provision of the public good x:
x = (m/s)[Si (1/ri)]. (11)
Note that there exists an infinite number of Pareto optimal solutions for the ai's : any ai's that satisfy Siai = 0 in (7) would do. If we choose the Pareto optimal point ai = 0, i = 1, ..., n, then the optimal transfers are:
-tij = bi x ej = (m ej)/(ri s), i = 1, …, n. (12)
The Pareto optimal policy (11) and (12) provides useful information about the nature of collective choices under risk. First, consider the properties of the optimal provision of the public good x in (11):
.¶x/¶μ > 0,
.¶x/¶σ < 0,
.¶x/¶ri < 0, i = 1, ..., n.
This indicates that higher expected returns (μ) have a positive effect on the provision of the public good x. It also shows that both higher risk (σ) and higher degree of risk aversion (ri) by any group member has a negative influence of the choice of x.
Second, consider the optimal transfers (-tij) in (12):
.¶(-tij)/¶μ > 0,
.¶(-tij)/¶ej > 0, j = 1, ..., m,
.¶(-tij)/¶σ < 0,
.¶(-tij)/¶rk < 0, for k = i,
= 0 for k ¹ i, i = 1, ..., n.
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This shows that a higher expected return (μ) as well as a higher realized value of the random variable e tend to increase the transfers toward any member of the group. It also indicates that a higher risk (σ) decreases the payments toward all individual. Finally, it shows that a higher degree of risk aversion by the i-th individual (ri) tends to decrease the transfer payment to this individual, but leaving the payments to other members of the group unaffected.
2- Insurance:
Consider a group of n agents, i = 1, ..., n, where i = 1 represents an insurance firm, and i = 2, ..., n, denote a set of (n-1) individuals interested in insurance coverage. Each agent is facing a stochastic return πi(e), i = 1, ..., n,. Under the EUH, the objective function of the i-th agent is given by
EiUi[-tij + πi(ej)],
which can be represented equivalently by its certainty equivalent
Ei(-tij) + Ei[πi(ej)] - Ri, i = 1, ..., n,
where Ri is the Arrow-Pratt risk premium for the i-th individual: Ri = 0 under risk neutrality; and Ri > 0 under risk aversion. The problem is to design an efficient insurance contract between the insurance firm and the (n-1) individuals.
The associated welfare problem is:
W(u) = MaxT {Si ti1: ui = EiUi[-tij + pi(ej)], i = 1, …, n; Si ti1 = Si tij, j = 2, …, m}.
Here, we make the following assumptions:
. there is no asymmetric information within the group.
. the insurance firm is risk neutral (R1 = 0).
. the insured individuals are risk averse (Ri > 0, i = 2, ..., n).
Then, the optimal transfer (or the optimal insurance contract) is of the form:
tij = πi(ej) - Ki for some (non random) constant Ki, i = 2, ..., n,
t1j = -Si³2 tij = Si³2[Ki - πi(ej)], j = 1, ..., m.
To see that, note that the above rule implies that [-tij + πi(ej)] = Ki, i.e. that the Arrow-Pratt risk premium Ri = 0 for i = 2, ..., n. And R1 = 0 because the insurance firm is risk neutral. Then it is is sufficient to note that any other transfer rule would imply Ri > 0 for some i ³ 2, making at least one individual worse-off. Also note that the above rule is Pareto optimal since it satisfies the Pareto optimality criterion: W(u) = Si³1 tij = 0, for all j = 1, ..., m.
Thus, in the absence of asymmetric information, the optimal transfer is as follows:
. the risk neutral insurance firm bears all the risks;
. the optimal insurance contract eliminates all private risk bearing by the risk averse insured individuals.
3- The Principal-Agent problem: (n = 2) (Shavell; Holmstrom)
Consider a group of two agents: the principal (i = 1) and the agent (i = 2). The principal designs a contract for the agent to perform a specific economic task generating a stochastic return π to be shared between the two parties. Under the EUH, the objective function of the principal is:
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E1U1(-t1j),
which has for certainty equivalent
E1(-tij) - R1,
where R1 is the Arrow-Pratt risk premium.
And the objective function of the agent is
Maxy E2U2[-t2j + π(y,e)],
which has for certainty equivalent
Maxy [E2(-t2j) + E2π(y,e) - R2(y)],
where R2(y) is the corresponding Arrow-Pratt risk premium.
We make the following assumptions:
. the principal decides the form of the contract.
. the agent makes the decisions about the activities y which generate a uncertain profit π(y,e).
. the agent has more information than the principal: the states (e1, ..., ek-1) are observable by both the principal and the agent, but the states (ek, ..., em) are observable only by the agent (and not by the principal). We will assume that part of the asymmetric information between the two parties relates to the observations of the "effort level" y, which is known to the agent but at least partly unknown to the principal.
The efficient contract between the two parties corresponds to the welfare problem:
W(u) = MaxT {Si ti1: u1 = E1U1(-t1j), u2 = E2U2(-t2j + p(y, ej)); Si ti1 = Si tij, j = 2, …, m;
tik = tij, j = k, …, m}.
Note that Pareto optimality requires that W(u) = 0 = t1j + t2j, for all j = 1, ..., m.
a/ The case of a Risk Neutral Agent:
If the agent is risk neutral (R2 = 0) and the principal is either risk averse (R1 > 0) or risk neutral (R1 = 0), then the optimal transfer takes the form:
t2j = - t1j = K, j = 1, ..., m,
where K is a non stochastic constant. Thus, under an optimal contract:
. the principal is paid the constant K,
. the agent receives (π - K).
In this case, R1 = R2 = 0 at the optimum, as all the observable risk is efficiently transferred to the risk neutral agent.
b/ The Case of a Risk Averse Agent:
If the agent is risk averse (R2 > 0) and the principal is either risk averse (R1 > 0) or risk neutral (R1 = 0), then at the optimum:
. at least some risk will be faced by the principal (otherwise, the benefits of risk sharing among risk averse individuals would not be obtained)
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. at least some risk will be faced by the agent. To see that, assume the contrary where the agent would face no risk and receive K = -t2j + π(y,ej) = a constant; then, the agent would receive U(K) no matter what the choice of y is, implying no incentive to choose the "socially desirable effort level" y. If the effort level y is not observable by the principal, this lack of incentive has been called "moral hazard". The optimal shift of some of the risk to the agent is an incentive effect that attempts to correct for the existence of moral hazard in the contract due to asymmetric information.
Example 1: Liability Rules between a firm (the agent) and society typically represented by a government agency (the principal). There are two broad categories of liability rules:
. strict liability, where the agent fees are:
t2j = 0 in the absence of an accident
t2j > 0 in the event of an accident.
. negligence, where the agent fees depend on whether the firm has been negligent or not.
The above results suggest that strict liability is appropriate if the firm is risk neutral. Alternatively, negligence standards are appropriate if the firm is risk averse.
Example 2: Moral hazard and insurance where the agent is the insured individual and the principal is the insurance firm. In the context of asymmetric information, our results show that, if the agent is risk averse, then he/she must bear some of the risk because of moral hazard. Then, optimally, the insurance coverage should not be complete and should include a deductible.
Example 3: Sharecropping (Stiglitz).
Let the principal be the landlord, and the agent be the tenant. Then, if the tenant is risk neutral, a cash rent contract would be optimal. Alternatively, if the principal and the agent are both risk averse, then some form of sharecropping would be optimal. In this case, a sharecropping contract could be motivated by both risk sharing and incentive issues.
4- Adverse Selection: (Rothschild and Stiglitz)
Consider a competitive insurance industry composed of risk neutral insurance firms. There are two types of potentially insurable individuals:
. type a: low risk individuals facing a prospect of loss πa(e) >0,
. type b: high risk individuals facing a prospect of loss πb(e) >0,
with E(πa) < E(πb).
Assume that all individuals have the same risk averse preferences U(-π), implying
EU(-πa) = U(E(-πa) - Ra),
and
EU(-πb) = U(E(-πb) - Rb),
where Ra > 0 and Rb > 0 are the Arrow-Pratt risk premium.
The insurance firms know that there are α percent individuals of type a, and (1-α) percent individuals of type b, but they do not know the type of each individual. This is a situation of asymmetric information not about the actions of individuals but about their "type".
Under competition, the insurance firms may want to offer an insurance contract for the loss π, with premium set equal to the expected value of the loss among all individuals:
αE(πa) + (1-α)E(πb).
Type b individuals would always accept this contract since
U[-E(πb) - Rb] = EU(-πb) < U[-αE(πa) - (1-α)E(πb)]
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or
0 > α[E(πa) - E(πb)] < Rb > 0.
However, type a individual would not accept this contract if:
U[-E(πa) - Ra] = EU(-πa) > U[-αE(πa) - (1-α)E(πb)]
or
(1- α)[E(πb) - E(πa)] > Ra.
In this case, low risk individuals would self-select and would not purchase a contract. The insurance firms would face higher losses than anticipated (because only high risk individuals would purchase the contract). As result, the proposed contract cannot be an equilibrium contract. Thus, under asymmetric information, low risk individuals may not be able to obtain an equilibrium insurance contract, resulting in a market failure. This has been called a problem of adverse selection.
Note: Other examples of adverse selection can be in product quality, labor market, etc. To reduce the problem of adverse selection, the quality of information can be improved by the generation and use of proper "signaling" (e.g. the case of education used as a signal for the underlying unknown ability of individuals).
Note: Self selection is not necessarily always "adverse". In some cases, it may be beneficial. For example, under imperfect information, some policy rules may help self-select out "undesirable" individuals and thus save on possibly costly information requirements (e.g. the case of transfer-in-king of "merit goods" to the poor).