Precautionary Effort Investment under Cross Risk Aversion

Proof of Theorem 2

Using second-order two-variable Taylor expansion around the point (x̄, ȳ), where x̄ = Ex̃ and ȳ = Eỹ, the bivariate function u(x, y) is approximated as

By the above Taylor approximation (14), we get

where Var(⋅) and Var(∵) are variance and covariance operators, respectively. It is easily seen from (15) that, for small risks x̃ and ỹ with Cov(x̃, ỹ) ≥ 0, Eu(x̃, ỹ) ≤ Eu(x̃, Eỹ) if u^(0,2) ≤ 0 and u^(1,1) ≤ 0.

Proof of Theorem 3

By the Taylor approximation (14) in the proof of Theorem 2, we obtain

This implies that, for small risks x̃ and ỹ with Cov(x̃, ỹ) ≥ 0, Eu(x̃, ỹ) ≤ u(Ex̃, Eỹ) if u^(0,2) ≤ 0, u^(2,0) ≤ 0 and u^(1,1) ≤ 0.

Proof of Proposition 1

For the effect of initial background variable uncertainty, by (4) and the concavity of objective problem, we have

So, by (3), we obtain e^∗ ≥ e1∗ ⇔ Eu(x̃₁, ỹ₁) ≤ Eu(x̃₁, Eỹ₁). By Theorem 1, for independent x̃₁ and ỹ₁, the inequality Eu(x̃₁, ỹ₁) ≤ Eu(x̃₁, Eỹ₁) holds if, and only if, u^(0,2) ≤ 0.

For the effect of final background variable uncertainty, by (5) and the concavity, we get

So, by (3), we have e^∗ ≤ e2∗ ⇔ Eu(x̃₂, ỹ₂) ≤ Eu(x̃₂, Eỹ₂). By Theorem 1, for independent x̃₂ and ỹ₂, the inequality Eu(x̃₂, ỹ₂) ≤ Eu(x̃₂, Eỹ₂) holds if, and only if, u^(0,2) ≤ 0.

Proof of Proposition 2

From the proof of Proposition 1, it is obvious that e^∗ ≥ e1∗ ⇔ Eu(x̃₁, ỹ₁) ≤ Eu(x̃₁, Eỹ₁) (e^∗ ≤ e2∗ ⇔ Eu(x̃₂, ỹ₂) ≤ Eu(x̃₂, Eỹ₂)). By Theorem 2, we get that, for small risks with Cov(x̃₁, ỹ₁) ≥ 0 (Cov(x̃₂, ỹ₂) ≥ 0), Eu(x̃₁, ỹ₁) ≤ Eu(x̃₁, Eỹ₁) (Eu(x̃₂, ỹ₂) ≤ Eu(x̃₂, Eỹ₂)) holds if u^(0,2) ≤ 0 and u^(1,1) ≤ 0.

Proof of Proposition 3

For the joint impact of initial two-source uncertainties, by (6) and the concavity, we obtain

Therefore, by (3), we have e^∗ ≥ e3∗ ⇔ Eu(x̃₁, ỹ₁) ≤ u(Ex̃₁, Eỹ₁). By Theorem 3, for small risks with Cov(x̃₁, ỹ₁) ≥ 0 the inequality Eu(x̃₁, ỹ₁) ≤ u(Ex̃₁, Eỹ₁) holds if u^(0,2) ≤ 0, u^(2,0) ≤ 0 and u^(1,1) ≤ 0.

For the joint impact of final two-source uncertainties, by (7) and the concavity, we obtain

Thereby, by (3), we have e4∗ ≥ e^∗ ⇔ Eu(x̃₂, ỹ₂) ≤ u(Ex̃₂, Eỹ₂). By Theorem 3, for small risks with Cov(x̃₂, ỹ₂) ≥ 0, the inequality Eu(x̃₂, ỹ₂) ≤ u(Ex̃₂, Eỹ₂) holds if u^(0,2) ≤ 0, u^(2,0) ≤ 0 and u^(1,1) ≤ 0.

Proof of Proposition 4

From the proof of Proposition 3, it is evident that e^∗ ≥ e3∗ ⇔ Eu(x̃₁, ỹ₁) ≤ u(Ex̃₁, Eỹ₁) (e4∗ ≥ e^∗ ⇔ Eu(x̃₂, ỹ₂) ≤ u(Ex̃₂, Eỹ₂)). From Theorem 4, we capture that, for PQD(x̃₁, ỹ₁) (PQD(x̃₂, ỹ₂)), the inequality Eu(x̃₁, ỹ₁) ≤ u(Ex̃₁, Eỹ₁) (Eu(x̃₂, ỹ₂) ≤ u(Ex̃₂, Eỹ₂)) holds if, and only if, u^(0,2) ≤ 0, u^(2,0) ≤ 0 and u^(1,1) ≤ 0.

Proof of Proposition 5

For the case of initial risks, by (10) and the assumption of SOC, we get ê^∗ ≥ e^1∗ if, and only if,

Hence, by (9), we obtain ê^∗ ≥ e^1∗ if, and only if,

For convenience, it is useful to define a bivariate function:

Thereby, (22) can be rewritten as Ef(x͠₁, y͠₁) ≤ Ef(x͠₁, Ey͠₁). From Theorem 1, we know that, for independent (x͠₁, y͠₁), Ef(x͠₁, y͠₁) ≤ Ef(x͠₁, Ey͠₁) ⇔ f^(0,2) ≤ 0.

Since

when u^(0,2) ≤ 0 and u^(1,2) ≥ 0, we have

That is to say, ê^∗ ≥ e^1∗.

For the case of target risks, from (11) and the SOC, we have ê^∗ ≤ e^2∗ if, and only if,

Therefore, by (9), we capture ê^∗ ≤ e^2∗ if, and only if,

Define the bivariate function:

Accordingly, (27) can be rewritten as Eg(x͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂). From Theorem 1, we see that, for independent (x͠₂, y͠₂), Eg(x͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂) ⇔ g^(0,2) ≤ 0

Because

we obtain

In other words, ê^∗ ≤ e^2∗.

Proof of Proposition 6

From the proof of Proposition 5, it is evident that ê^∗ ≥ e^1∗ ⇔ Ef(x͠₁, y͠₁) ≤ Ef(x͠₁, Ey͠₁). By Theorem 2, for small risks with Cov(x͠₁, y͠₁) ≥ 0, the inequality Ef(x͠₁, y͠₁) ≤ Ef(x͠₁, Ey͠₁) holds if f^(0,2) ≤ 0 and f^(1,1) ≤ 0.

In addition,

when u^(1,1) ≤ 0 and u^(2,1) ≤ 0, we get

Thus, if u^(0,2) ≤ 0, u^(1,1) ≤ 0, u^(1,2) ≥ 0 and u^(2,1) ≥ 0, then (25) and (32) imply ê^∗ ≥ e^1∗.

On the other hand, ê^∗ ≤ e^2∗ ⇔ Eg(x͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂). By Theorem 2, for small risks with Cov(x͠₂, y͠₂) ≥ 0, we have Eg(x͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂) if g^(0,2) ≤ 0 and g^(1,1) ≤ 0.

Moreover,

Then,

From (30) and (34), we know that u^(0,2) ≤ 0, u^(1,1) ≤ 0, u^(1,2) ≥ 0 and u^(2,1) ≥ 0, imply ê^∗ ≤ e^2∗.

Proof of Proposition 7

For the joint impact of initial risks, from (12) and the SOC, we obtain ê^∗ ≥ e^3∗ if, and only if,

Thus, from (9), we obtain ê^∗ ≥ e^3∗ if, and only if,

By (23), we can write (36) as Ef(x͠₁, y͠₁) ≤ f(Ex͠₁, Ey͠₁). By Theorem 3, for small risks with Cov(x͠₁, y͠₁) ≥ 0, the inequality Ef(x͠₁, y͠₁) ≤ f(Ex͠₁, Ey͠₁) holds if f^(0,2) ≤ 0, f^(2,0) ≤ 0 and f^(1,1) ≤ 0.

Due to

when u^(2,0) ≤ 0 and u^(3,0) ≥ 0, we have

Thus, if u^(0,2) ≤ 0, u^(2,0) ≤ 0, u^(1,1) ≤ 0, u^(1,2) ≥ 0, u^(2,1) ≥ 0 and u^(3,0) ≥ 0, then (25) and (32) imply ê^∗ ≥ e^3∗.

For the joint impact of target risks, from (13) and the SOC, we have ê^∗ ≤ e^4∗ if, and only if,

By (9), we capture ê^∗ ≤ e^4∗ if, and only if,

By (28), write (40) as g(Ex͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂). By Theorem 3, we observe that, for small risks with Cov(x͠₂, y͠₂) ≥ 0, the inequality g(Ex͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂) holds if g^(0,2) ≤ 0, g^(2,0) ≤ 0 and g^(1,1) ≤ 0.

Owing to

we have

From (30), (34) and (42), it is easily seen that u^(0,2) ≤ 0, u^(2,0) ≤ 0, u^(1,1) ≤ 0, u^(1,2) ≥ 0, u^(2,1) ≥ 0 and u^(3,0) ≥ 0 imply ê^∗ ≤ e^4∗.

Proof of Proposition 8

From the proof of Proposition 7, it is obvious that ê^∗ ≥ e^3∗ ⇔ Ef(x͠₁, y͠₁) ≤ f(Ex͠₁, Ey͠₁). By Theorem 4, we know that, for PQD(x͠₁, y͠₁), the inequality Ef(x͠₁, y͠₁) ≤ f(Ex͠₁, Ey͠₁) holds if, and only if, f^(0,2) ≤ 0, f^(2,0) ≤ 0 and f^(1,1) ≤ 0, which are guaranteed by the conditions that u^(0,2) ≤ 0, u^(2,0) ≤ 0, u^(1,1) ≤ 0, u^(1,2) ≥ 0, u^(2,1) ≥ 0 and u^(3,0) ≥ 0, (25), (32) and (38).

On the other hand, it is evident that ê^∗ ≤ e^4∗ ⇔ g(Ex͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₂). By Theorem 4, for PQD(x͠₂, y͠₂), the inequality g(Ex͠₂, Ey͠₂) ≥ Eg(x͠₂, y͠₁) holds if, and only if, g^(0,2) ≤ 0, g^(2,0) ≤ 0and g^(1,1) ≤ 0, which are insured by the conditions that u^(0,2) ≤ 0, u^(2,0) ≤ 0, u^(1,1) ≤ 0, u^(1,2) ≥ 0, u^(2,1) ≥ 0 and u^(3,0) ≥ 0. This ends the proof.