Chapter 2 Objectives
3.1 State of the Art Biometry based on Electrophysio- Electrophysio-logical Signals
Crop prices have increased over time, and as crop prices increase farmers purchase more crop insurance to insure their higher expected revenues. In our model, the total amount of crop insurance purchased is Q = KΘD, where K is the number of farmers, Θ is the expected price of the crop, and D is the expected yield from farming that is insured. As crop prices increase, the total amount of crop insurance purchased increases; equilibrium agent compensation rates and agent effort levels change as a result.
For a symmetric equilibrium, a marginal increase in Θ has the following effect on the
effort level.
∂e∗
∂Θ = f w∗ ∂D∂e∗
(1 − α)
∂2CA
∂e∗2 −(1−α)f w∗Θ∂e∂2D∗2
≥ 0 (12)
Equation (12) is determined by implicitly differentiating equation (5) and is only the partial effect of a change in Θ on e∗. A marginal change in Θ also changes the equilibrium agent compensation rate, and the total effect of a change in Θ on e∗ is dedΘ∗ = ∂e∂Θ∗ + ∂w∂e∗∗
∂w∗
∂Θ. Ignoring for the moment any changes in the compensation rate due to a change in Θ, ∂e∂Θ∗ is non-negative. As Θ increases each farmer spends more on crop insurance increasing the effort level of agents.
The competitive agent compensation rate is defined in equation (7) and the collusive monopsony compensation rate is implicitly defined in equation (9). In a competitive equi-librium, a marginal change in Θ does not affect the equilibrium compensation rate: ∂w∂Θc = 0.
Insurance companies set the agent compensation rate as high as possible in a competitive equilibrium. Since the compensation rate is the agent’s share of total premiums, an increase in Θ does not permit insurance companies to increase the compensation rate; if they did in-surance company profits would be negative. An increase in Θ also does not allow competitive insurance companies to decrease the agent compensation rate. This is due to the perfectly elastic residual supply of insurance policies that each insurance company faces. While an increase in Θ does not change the competitive equilibrium compensation rate, agents expend more effort: dedΘc ≥ 0.
The change in the collusive monopsony compensation rate due to a marginal change in Θ is more complex. An increase in Θ increases insurance company profits and insurance com-panies may respond by lowering the compensation rate. The following proposition describes this behavior.
Proposition 12. If wm ≥ α2 + (1−α)s2f d and ∂w∂2em2m ≤ 0, then ∂w∂Θm ≤ 0.
Note that when wm ≤ α2 +(1−α)s2f d, the sign of ∂w∂Θm cannot be determined without additional
m α (1−α)s
increase in Θ induces insurance companies to decrease the collusive monopsony compensation rate. For insurance company profits to be positive, it is assumed that α + (1−α)sf d ≥ wm. Combining this condition with the condition from Proposition 12, if α + (1−α)sf d ≥ wm ≥
α
2+(1−α)s2f d, then a marginal increase in Θ results in a decrease in the monopsony compensation rate. In the collusive monopsony case, the sign of dedΘm cannot be determined.
8 Conclusion
This study develops a model to examine the effects of changes to the U.S. federal crop in-surance program on the structure of the agricultural inin-surance industry. This paper is one of the first to model the supply side of the industry and evaluate the interactions between policy variables and competition in these markets. The industry (and model) consists of insurance agents and crop insurance companies. Agents sell federally subsidized crop in-surance policies to farmers, and the inin-surance companies then buy those policies from the agents by offering them compensation rates per dollar of liability. Symmetric equilibria are solved under two alternative assumptions about competition among insurance companies.
In the first environment, insurance companies are competitive; in the second, they form a monopsonistic cartel to obtain insurance policies from independent insurance agents, who sell those policies to farmers. In both environments, agents are competitive suppliers of insurance policies to insurance companies, and agents are operating in a market where entry and exit are relatively easy.
In practice, the federal government subsidizes insurance contracts in two ways: by reduc-ing the premium paid by farmers through a premium subsidy rate and by payreduc-ing a further direct subsidy to insurance companies to cover A&O costs. The premium rate subsidy is the proportion of the total premium paid by the government, and the insurance companies’
direct subsidy is defined as a proportion of the total premium associated with the policies an insurance company acquires. However, the federal government also enters the market in
a third way by determining the premium rate for each policy by estimating the actuarially fair premium rate and then inflating that rate by adding a catastrophic loading factor.
The model evaluates the effects of changes in these three core elements of the federal program. Some of the results are straightforward, but others are somewhat unexpected and provide useful insights about the rent-seeking behavior of private insurance companies and insurance agents. We find that marginal increases in the A&O subsidy rate, sd, do not change the premium rates for farmers but increase agent compensation rates, agent effort levels, market quantity, insurance company profits, and subsidy payments by the government. Taking into consideration the competition among insurance companies, the direction of these changes is the same, but the magnitude of these effects depends on whether insurance companies collude or are perfectly competitive.
Increasing the premium subsidy rate, s, benefits farmers by reducing their out-of-pocket costs for crop insurance. The increase in the premium subsidy rate also has an added effect of reducing the returns to effort on the part of insurance agents, leading those agents to expend less effort per farmer (for sufficiently low compensation rates). On a per policy basis, increasing the premium subsidy rate has no effect on insurance company revenues, as the increase in the premium subsidy is exactly offset by the reduction in farmer paid premiums, leaving the overall premium rate (g) unchanged. Government expenditures on subsidies, however, are increased both on a per policy basis and because the amount of insurance demanded by farmers increases.
A major finding concerns the effects of the inflation of premium rates by adjusting the catastrophic loading factor reflected by increases in the value of α in the model. An increase in α generates higher premium rates for farmers and higher levels of subsidy payments on a per policy basis by the government and higher revenues on a per policy basis for the insurance companies. The overall effect on quantity due to a marginal increase in α is ambiguous. A marginal increase in α increases the premium rate and—holding compensation rates constant—insurance agents respond to the rate increase with an increase in effort. The
increase in effort increases the quantity of insurance purchased by farmers, but—depending on the parameterization of the model—the increase in effort may or may not offset the decrease in quantity due to the increase in the premium rate. For similar reasons, the overall effect of a marginal change in α on insurance company profits is ambiguous.
The costs and benefits of policy changes are evaluated using a convenient baseline of no prior government policy. Conditional on no prior government policy, farmers prefer a marginal increase in the premium subsidy rate. This change has the lowest associated net cost but is the insurance companies’ least preferred policy. The insurance companies’ most preferred policy is a marginal increase in the direct subsidy rate. A marginal increase in the direct subsidy rate has the highest associated net cost, the highest cost to the government, and does not benefit farmers.
Lastly, we investigate the impacts of changing crop prices on equilibrium outcomes. Hold-ing compensation rates constant, we find that a marginal increase in crop prices increases agent effort. If the market for insurance companies is competitive, then a change in crop prices does not change the agent compensation rate. If, instead, insurance companies collude, then the agent compensation rate will change and is likely to decrease as crop prices increase.
This result suggests an empirical test regarding insurance company market performance.
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Appendices
A Proofs
Proof of Lemma 1. Equation (3) is f Q −PK
k=1ckΘDk = 0. Assumption 4 states that ck = c for all k, and equation (3) becomes f Q = cPK
k=1ΘDk. ThePK
k=1ΘDk= Q, and equation (3) further reduces to f = c which is the intended result.
Proof of Proposition 1. Note that a solution to equation (1) exists because the objective function is concave. Assumption 2 states that the demand function is concave and Assumption 3 states that the joint variable cost function is convex.
There are k first order conditions of equation (1) and each first order condition is f wiΘ
(1 − α)
∂D(p, ek)
∂eak
−∂CA(ea)
∂eak
= 0.
If farmer k does not purchase insurance through agent a, then ∂D(p,e∂e k)
ak = 0 by Assumption 2. In this case the first order condition reduces to ∂C∂eA(ea)
ak = 0 and by Assumption 3 is only satisfied for effort levels of eak = 0. This establishes the result that eak = 0 if Ik∈a= 0.
If farmer k purchases insurance through agent a, then ∂D(p,e∂e k)
ak > 0 by Assumption 2 and
∂CA(ea)
∂eak ≥ 0 by Assumption 3.17 Now let e∗ > 0 be a solution to one of the first order conditions with Ik∈a = 1. The existence of e∗ has already been established. Since demand is homogeneous and CA is symmetric (Assumption 3), if e∗ is a solution to one of the first order conditions with Ik∈a = 1, then it is a solution for all the first order conditions with Ik∈a = 1. This establishes e∗ as the symmetric equilibrium effort level when Ik∈a = 1. Note that e∗ is implicitly determined from the equation in Proposition 1 as this is the first order condition of the insurance agent.
An alternative equilibrium one might consider is where each agent exerts the same amount of effort for all farmers regardless of whether or not a farmer buys a policy through them. This cannot be an equilibrium because effort is costly. An agent could reduce her effort level for farmers that do not purchase from her and be strictly better off.
Proof of Proposition 2. The four inequalities in Proposition 2 are shown by implicitly differentiating equation (5):
f wiΘ (1 − α)
∂D
∂e∗ = ∂CA
∂e∗ .
Note that D is a function of p and e∗ where p = (1−α)(1−s)f , and CA is a function of e∗. Implicitly differentiating equation (5) with respect to e∗ yields
∂e∗ ∂2CA
∂e∗2 − f wiΘ (1 − α)
∂2D
∂e∗2
.
17It is possible that ∂D(p,e∂e k)
ak = 0, but this condition results in a degenerate equilibrium where the effort level of all agents is zero. We ignore this trivial case.
∂2CA
∂e∗2 ≥ 0 by the convexity of CA (Assumption 3) and ∂∂e2∗2D ≤ 0 by Assumption 2. Let B ≡
∂2CA
∂e∗2 −(1−α)f wiΘ∂∂e2∗2D where B ≥ 0.
Now implicitly differentiate equation (5) with respect to e∗ and each of the exogenous variables to get the following.
∂e∗
The denominator for each equation is the same and is greater than or equal to zero. Regarding the numerators, we know that ∂D∂e∗ > 0 and ∂e∂p∂2D ≥ 0 from Assumption 2. Since p = (1−α)(1−s)f ,
∂p
∂s = −(1−α)f < 0 and ∂α∂p = (1−α)(1−s)2f > 0. Use the above information to sign each comparative static and what results are the inequalities listed in Proposition 2.
Proof of Proposition 3. The competitive symmetric equilibrium compensation rate is defined in equation (7). From this equation we know that
∂wc
The total change in the equilibrium effort level due to a change in either s, α, or sdis dec
where the comparative statics of ecare defined in equations (13) through (16) and signed in Propo-sition 2.
Proof of Proposition 4. The first order condition from the monopsonist’s problem (equation (8)) is
− f
Setting the first order condition equal to zero, replacing w with wm, e with em, and substituting
∂e∗
∂wi from equation (13) results in equation (9).
The second order condition of the monopsonist’s problem is
−2 f
and we want to show that the second order condition is less than or equal to zero so that wm is a maximum value of equation (8).
The signs of ∂D∂e and ∂∂e2D2 are non-negative and non-positive respectively, as assumed in Assump-tion 2. The sign of ∂e is non-negative as shown in Proposition 2. ∂2e is the remaining component
in the second order condition to be signed. If ∂w∂2e2 ≤ 0, then the second order condition is less than or equal to zero.
Note that it is possible to determine ∂w∂2e2 from equation (13). This would allow us to determine a second order condition that is both necessary and sufficient. This condition would depend on the third derivative of the demand function and the agent cost function. We opt instead for the simpler requirement of ∂w∂2e2 ≤ 0 which is just a sufficient condition.
Proof of Proposition 5. As before, let
B ≡ ∂2CA
where R ≥ 0 so that the profits of insurance companies are non-negative. Equation (9) can be represented as RΘB ∂e∂Dm
2
− D = 0. Implicit differentiation of the above equation yields the comparative statics of wm with respect to a change in the exogenous policy variables. These are as follows.
Now we specify the terms in the above comparative statics. R is defined in equation (24) and from equation (16). From equations (14) and (15) we have
∂em
dwm is defined in equation (22) and equation (20) defines ∂e∂Bm. The remaining terms are
∂B where ∂B∂p is defined in equation (21).
The denominators of ∂w∂sm, ∂w∂αm, and ∂w∂sm
d are all the same. Denote the denominator by T where T ≡ Θ
T ≤ 0 by Assumption 2 and equation (23).
The numerator of ∂w∂sm is
and cannot be signed without additional assumptions. The first term in U is non-negative by equation (23), the second term in U is non-positive by Assumption 2, the sign of the third term is determined by the sign of f w1−αm− 2R which may be positive or negative, and the sign of the fourth term is determined by the sign of ∂e∂m23D∂p which may be positive or negative. The comparative static
∂wm
∂s = −UT and since T ≤ 0, ∂w∂sm ≥ 0 if and only if U ≥ 0.
The numerator of ∂w∂αm is
and cannot be signed without additional assumptions. There are five terms in V . The first term in V is non-positive by equation (23), the second term is non-negative by Assumption 2, the sign of the third term is determined by the sign of f w1−αm − 2R which may be positive or negative, and the sign of the fourth term is determined by the sign of ∂e∂m23D∂p which may be positive or negative.
These first four terms correspond to the terms in U (the numerator of ∂w∂sm), but the last term does not. The sign of the fifth term depends on the sign of 1 − 2wm + wmBRΘ∂e∂2m2D. The comparative
The comparative static ∂w∂sm
d = −Θ(∂em∂D )2
BT and since T ≤ 0, ∂w∂sm
d ≥ 0. This proves the first result in the proposition.
Next we show that for a sufficiently low compensation rate ∂w∂sm ≤ 0 and ∂w∂αm ≥ 0. To prove our intended result, we show that U ≤ 0 and V ≥ 0 for a sufficiently small compensation rate.
Setting the compensation rate equal to zero, we have U |wm=0 =
where R ≥ 0 so that insurance company profits are non-negative and B ≥ 0. Assumption 2 provides the signs for the demand terms.
Proof of Proposition 6. dedsm
d = ∂e∂sm
d +∂w∂emm∂wm
∂sd is the total effect of an increase in sd on the effort level. The total effect contains a direct effect and an indirect effect through the change in the compensation rate. From Proposition 2, we know that ∂e∂sm
d = 0 and both ∂w∂emm and ∂w∂sm
d are defined
in Proposition 5.
The other results for effort levels are that for a sufficiently low compensation rate dem
The sign of these expressions is determined from results in Propositions 2 and 5.
Proof of Proposition 7. Conditional on no prior government policy, α = 0, s = 0 and sd= 0. When all these policy variables are zero, then wc= 0 (see equation 7) and e∗ = 0 (see equation 5). The comparative statics of wc are determined by making the relevant substitutions in equation (17).
Note that f ≤ 1 as f = c and c is the probability of a loss.
The total effect on ec is determined from equation (18). Simplifications from no prior govern-ment policy combined with equations (13) through (16) are useful.
Proof of Proposition 8. Conditional on no prior government policy, α = 0, s = 0 and sd= 0. When all these policy variables are zero, then wm= 0 and e∗ = 0 (see equation 5). Proposition 5 indicates that ∂w∂sm ≤ 0. Since conditional on no prior government policy, wm= 0 it must be that ∂w∂sm = 0.
Also from Proposition 5, ∂w∂sm
d = −Θ(∂em∂D )2
It is straightforward to show that ∂w∂sm
d
Equations (14) through (16) are all equal to zero since wm|N P = 0. The total effect for a change in em conditional on no prior government policy is dedxm
N P = ∂w∂emm∂wm total effect which includes a direct effect and an indirect effect through w. The comparative static effects on effort determined in the proof of Proposition 2 are used to simplify the above changes.
The marginal changes in market demand due to a change in either s, α, or sd, conditional on no prior government policy (N P ) are
∂Qc
where equation (19) in the Appendix defines B.
The agent compensation rate for a symmetric competitive equilibrium is wc(equation 7). Con-ditional on no prior government policy
∂wc
and the changes in market demand for the symmetric competitive equilibrium are as follows.
∂Qc
Note that from our symmetry assumption f = c which is the probability of a loss. Since f is a probability, it is bounded between 0 and 1. Lastly, from equation (19) and conditional on no prior government intervention, B|N P = ∂∂e2C∗2A.
Proof of Proposition 10. The proof here is similar to the proof of Proposition 9. The marginal changes in market demand due to a change in either s, α, or sd, conditional on no prior government policy (N P ) are
The agent compensation rate for a symmetric competitive equilibrium is wm (equation 7).
Conditional on no prior government policy
∂wm and the changes in market demand for the symmetric competitive equilibrium are as follows.
∂Qm
Note that from our symmetry assumption f = c which is the probability of a loss. Since f is a probability, it is bounded between 0 and 1. Lastly, from equation (19) and conditional on no prior government intervention, B|N P = ∂∂e2C∗2A.
Proof of Proposotion 11. The net costs of a competitive symmetric equilibrium are N Cc= T C − F B and the net costs of a collusive monopsony equilibrium are N Cm = T C − F B − πm. The results of the proposition are shown by using the benefit and cost measures defined in the paper.
Proof of Proposotion 12. Represent equation (9) as RΘB ∂e∂Dm
2
∂wm
Simplify both X and Y to get the following.
Y = wmRΘ
X ≤ 0 from Assumption 2 and the proof of Proposition 5. Equation (23) in the proof of Proposi-tion 5 is particularly useful, but the inequality is based on ∂w∂2em2m ≤ 0. If Y ≤ 0, then ∂w∂Θm = −YX ≤ 0 and is the intended result. If f w1−αm − R ≥ 0, then wm≥ α2 +(1−α)s2f d and Y ≤ 0.