The Value of Demand Information in an Insurance Market Under Demand and Cost Uncertainty

Abstract

The purpose of this research is to investigate the value of demand information in an insurance market under both demand and cost uncertainty, something not previously considered in the literature. Thus, the contribution of this research is to shed light on the characteristics of the value of demand information in an insurance market under demand and cost uncertainty. The main results of this research are as follows. First, the value of demand information is always positive in an insurance market under both demand and cost uncertainty. This result shows that the value of demand information is also positive even if both demand and cost uncertainty coexist. Second, increases in demand uncertainty and the investment parameter and a decrease in cost uncertainty raise the value of demand information. From this result, we find that demand and cost uncertainty have opposite effects on the value of demand information. Third, whether an increase in the number of insurance firms raises the value of demand information depends on the magnitude of cost uncertainty. When cost uncertainty is very low (high), an increase in the number of insurance firms lowers (raises) the value of demand information.

Appendices

Appendix A

Consider the case in which all insurance firms have demand information. In the second stage, each insurance firm chooses its own quantity to satisfy the following first-order condition⁹:

$$ \frac{\partial C{E}_i}{\partial {q}_i}=a-2{q}_i-{\displaystyle \sum_{j\ne i}{q}_j}-{\mu}_x-\frac{1}{2}\left(\overline{r}-{r}_i\right){\sigma}_x^2=0. $$

(A.1)

Because all insurance firms are identical, q ^∗₁  = q ^∗₂  = ⋯ = q ^∗ _n is realized. From Eq. A.1, the equilibrium quantity can be derived as

$$ {q}_i^{\ast }=\frac{2\left(a-{\mu}_x\right)-\left(\overline{r}-2{r}_i+{\displaystyle \sum_{j\ne i}{r}_j}\right){\sigma}_x^2}{2n+1}, $$

(A.2)

where an asterisk denotes an equilibrium value. Substituting Eq. A.2 into Eq. 9 yields

$$ C{E}_i=\frac{\left\{2\left(a-{\mu}_x\right)-\left(\overline{r}-2{r}_i+{\displaystyle \sum_{j\ne i}{r}_j}\right){\sigma}_x^2\right\}\left\{2\left(a-{\mu}_x\right)+\left(\overline{r}-n{r}_i+{\displaystyle \sum_{j\ne i}{r}_j}\right){\sigma}_x^2\right\}}{4{\left(n+1\right)}^2}-\frac{1}{2}c{r}_i^2. $$

(A.3)

In the first stage, each insurance firm chooses its investment level to satisfy the following first-order condition:

$$ \frac{\partial C{E}_i}{\partial {r}_i}=\frac{\left[2\left(n+2\right)\left(a-{\mu}_x\right)+\left\{\left(5n+2\right){r}_i-\left(n+2\right)\left(\overline{r}+{\displaystyle \sum_{j=1}^n{r}_j}\right)\right\}{\sigma}_x^2\right]{\sigma}_x^2}{4{\left(n+1\right)}^2}-c{r}_i=0. $$

(A.4)

In order to satisfy the second-order condition, the following condition is assumed to be satisfied¹⁰:

$$ \frac{\partial^2C{E}_i}{\partial {r_i}^2}=\frac{n{\sigma}_x^4}{{\left(n+1\right)}^2}-c<0\Rightarrow {\sigma}_x^2<\sqrt{\frac{c{\left(n+1\right)}^2}{n}}. $$

(A.5)

Because all insurance firms are identical, r ^∗₁  = r ^∗₂  = ⋯ = r ^∗ _n is realized. From Eq. A.4, the equilibrium amount of investment is derived as follows:

$$ {r}_i^{\ast }=\frac{\left(n+2\right)\left\{2\left(a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}{\sigma}_x^2}{4c{\left(n+1\right)}^2+\left\{\left(n-3\right)n-2\right\}{\sigma}_x^4}. $$

(A.6)

In order to confirm that r ^∗ _i is always positive, the following lemma must be checked.

Lemma 1: 4c(n + 1)² + {(n − 3)n − 2}σ ⁴ _x  > 0 is always satisfied.

Proof:

Let f(n) ≡ 4c(n + 1)² + {(n − 3)n − 2}σ ⁴ _x . f(n) is a monotone increasing function of n. Thus, we confirm that f(n) becomes strictly positive in the case of n = 2. From Eq. A.5, we find that ${\sigma}_x^2<\sqrt{9c/2}$, and thus, f(2) = 4(9c − σ ² _x ) > 0 is proved. Q. E. D.

In order to guarantee that $\overline{r}\ge {r}_i$, it is assumed that $\overline{r}$ satisfies the following condition: ¹¹

$$ \overline{r}-{r}_i^{\ast }=\frac{4c{\left(1+n\right)}^2\overline{r}-\left\{2\left(n+2\right)\left(a-{\mu}_x\right)-n\left(n-2\right)\overline{r}{\sigma}_x^2\right\}{\sigma}_x^2}{4c{\left(n+1\right)}^2+\left\{\left(n-3\right)n-2\right\}{\sigma}_x^4}\ge 0 $$

$$ \Rightarrow \overline{r}\ge \frac{2\left(n+2\right)\left(a-{\mu}_x\right){\sigma}_x^2}{4c{\left(n+1\right)}^2+n\left(n-2\right){\sigma}_x^4}. $$

(A.7)

Combining Eqs. 10 and A.7 yields

$$ \frac{2\left(n+2\right)\left(a-{\mu}_x\right){\sigma}_x^2}{4c{\left(n+1\right)}^2+n\left(n-2\right){\sigma}_x^4}<\overline{r}<\frac{2\left(a-{\mu}_x\right)}{\sigma_x^2}. $$

(A.8)

By substituting Eq. A.6 into Eqs. 3, A.2, and A.3, the following equilibrium values of the quantity, insurance premium, and certainty equivalent can be derived respectively as follows¹²:

$$ {q}_i^{\ast }=\frac{\left\{2\left(a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}\left\{2c{\left(n+1\right)}^2-\left(n-2\right){\sigma}_x^4\right\}}{\left(n+1\right)\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}, $$

(A.9)

$$ {p}^{\ast }=a-\frac{n\left\{2\left(a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}\left\{2c{\left(n+1\right)}^2-\left(n-2\right){\sigma}_x^4\right\}}{\left(n+1\right)\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}, $$

(A.10)

$$ C{E}_i^{\ast }=\frac{{\left\{2\left(a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}}^2\left[8{c}^2{\left(n+1\right)}^4+c{\left(n+1\right)}^2\left\{4+\left(n-12\right)n\right\}{\sigma}_x^4-n{\left(n-2\right)}^2{\sigma}_x^8\right]}{2{\left(n+1\right)}^2{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^2}. $$

(A.11)

Appendix B

To determine the value of demand information in an insurance market, we compare the case in which insurance firms have demand information with the case in which they do not. Having discussed the former case in Appendix A, we now consider the other case.

When no insurance firms have demand information, the demand function is given by Eq. 4 rather than Eq. 3. Although these two demand functions differ, the same procedure is used to derive the equilibrium. Thus, under the case in which insurance firms do not have demand information, the equilibrium certainty equivalent, which is denoted by CE ^0 ∗ _i , is

$$ C{E}_i^{0\ast }=\frac{{\left\{2\left({\mu}_a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}}^2\left[8{c}^2{\left(n+1\right)}^4+c{\left(n+1\right)}^2\left\{4+\left(n-12\right)n\right\}{\sigma}_x^4-n{\left(n-2\right)}^2{\sigma}_x^8\right]}{2{\left(n+1\right)}^2{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^2}. $$

(B.1)

To show how the certainty equivalent differs depending on whether there is demand information or not, the following equation is derived by Eqs. A.11 and B.1.

$$ C{E}_i^{\ast }=C{E}_i^{0\ast }+\varDelta, $$

(B.2)

where

$$ \varDelta \equiv \left[{\left\{2\left(a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}}^2-{\left\{2\left({\mu}_a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}}^2\right] $$

$$ \times \frac{\left[8{c}^2{\left(n+1\right)}^4+c{\left(n+1\right)}^2\left\{4+\left(n-12\right)n\right\}{\sigma}_x^4-n{\left(n-2\right)}^2{\sigma}_x^8\right]}{2{\left(n+1\right)}^2{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^2}. $$

(B.3)

Equation B.2 can be computed as

$$ E\left[C{E}_i^{\ast}\right]=E\left[C{E}_i^{0\ast}\right]+\frac{2{\sigma}_a^2\left[8{c}^2{\left(n+1\right)}^4+c{\left(n+1\right)}^2\left\{4+\left(n-12\right)n\right\}{\sigma}_x^4-n{\left(n-2\right)}^2{\sigma}_x^8\right]}{{\left(n+1\right)}^2{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^2}. $$

(B.4)

The second term on the right-hand side of Eq. B.4 can be interpreted as the value of demand information because it represents the difference between certainty equivalents in the presence and absence of demand information. To simplify the expression of that differential, the second term on the right-hand side of Eq. B.4 is defined as follows:

$$ \varOmega \equiv \frac{2{\sigma}_a^2\left[8{c}^2{\left(n+1\right)}^4+c{\left(n+1\right)}^2\left\{4+\left(n-12\right)n\right\}{\sigma}_x^4-n{\left(n-2\right)}^2{\sigma}_x^8\right]}{{\left(n+1\right)}^2{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^2}. $$

(B.5)

In order to confirm that the value of demand information is always positive, Ω > 0 must be confirmed. From Lemma 1, the denominator in Eq. B.5 is always positive. The following lemma is the result in the sign of the numerator in Eq. B.5.

Lemma 2: 8c ²(n + 1)⁴ + c(n + 1)²{4 + (n − 12)n}σ ⁴ _x  − n(n − 2)² σ ⁸ _x  > 0.

Proof: Let g(σ ² _x ) ≡ c(n + 1)²{4 + (n − 12)n}σ ⁴ _x  − n(n − 2)² σ ⁸ _x . In order to prove the above lemma, we check g(σ ² _x ) > − 8c ²(n + 1)⁴ for any σ ² _x . Also suppose that λ ≡ σ ⁴ _x ; then, g(λ) = c(n + 1)²{4 + (n − 12)n}λ − n(n − 2)² λ ². g(λ) is the quadratic concave function of λ and has one maximum value.

In order to check g(λ) > − 8c ²(n + 1)⁴ for any λ, two corner values, λ = 0 (σ ² _x  = 0) and λ = c(n + 1)²/n (${\sigma}_x^2=\sqrt{c{\left(n+1\right)}^2/n}$), must be investigated. We find that g(0) = 0 and g(c(n + 1)²/n) = − 8c ²(n + 1)⁴. Thus, we know that − 8c ²(n + 1)⁴ is the minimum value. Because λ < c(n + 1)²/n (${\sigma}_x^2<\sqrt{c{\left(n+1\right)}^2/n}$) must be satisfied by Eq. A.5, g(λ) > − 8c ²(n + 1)⁴ is always satisfied. Q. E. D. From Lemma 2, we find that the numerator in Eq. B.5 is always positive and that Ω > 0 is confirmed.

Appendix C

In order to determine how the value of demand information changes when the exogenous variables change, the following comparative statics must be calculated:

$$ \frac{\partial \varOmega }{\partial {\sigma}_a^2}=\frac{2\left[8{c}^2{\left(n+1\right)}^4+c{\left(n+1\right)}^2\left\{4+\left(n-12\right)n\right\}{\sigma}_x^4-n{\left(n-2\right)}^2{\sigma}_x^8\right]}{{\left(n+1\right)}^2{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^2}, $$

(C.1)

$$ \frac{\partial \varOmega }{\partial c}=\frac{2\left(n+2\right){\sigma}_a^2{\sigma}_x^4\left[12c\left(n-2\right){\left(n+1\right)}^2+\left\{n\left(24+n\left(n-9\right)\right)-4\right\}{\sigma}_x^4\right]}{{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^3}, $$

(C.2)

$$ \frac{\partial \varOmega }{\partial {\sigma}_x^2}=-\frac{4c\left(n+2\right){\sigma}_a^2{\sigma}_x^2\left[12c\left(n-2\right){\left(n+1\right)}^2+\left\{n\left(24+n\left(n-9\right)\right)-4\right\}{\sigma}_x^4\right]}{{\left[4c{\left(n+1\right)}^2+\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^3}, $$

(C.3)

From Lemma 2 in Appendix B and n(24 + n(n − 9)) − 4 > 0, we find that ∂Ω/∂σ ² _a  > 0, ∂Ω/∂c > 0, and ∂Ω/∂σ ² _x  < 0.¹³

Appendix D

In order to determine how the value of demand information changes when the number of insurance firms changes, comparative statics are calculated as follows:

$$ \frac{\partial \varOmega }{\partial n}=-\frac{2{\sigma}_a^2\varXi }{{\left[4c{\left(n+1\right)}^3+\left(n+1\right)\left\{n\left(n-3\right)-2\right\}{\sigma}_x^4\right]}^3}\begin{array}{c}\hfill >\hfill \\ {}\hfill =\hfill \\ {}\hfill <\hfill \end{array}0, $$

(D.1)

where

$$ \varXi \equiv 64{c}^3{\left(n+1\right)}^6+24{c}^2\left(n-4\right)\left(n-1\right){\left(n+1\right)}^4{\sigma}_x^4+2c{\left(n+1\right)}^2\left[n\left\{n\left\{n\left(n-23\right)+48\right\}-52\right\}-16\right]{\sigma}_x^8 $$

$$ -\left(n-2\right)\left[n\left\{16+n\left\{17+3n\left(n-4\right)\right\}-4\right\}{\sigma}_x^{12}\right]. $$

(D.2)

Consider the two extreme cases σ ² _x  = 0 (σ ² _x is very low) and ${\sigma}_x^2\approx \sqrt{c{\left(n+1\right)}^2/n}$ (σ ² _x is very high).¹⁴ In the first case, ∂Ω/∂n < 0 is realized because Ξ = 64c ³(n + 1)⁶ > 0. In the second case, ∂Ω/∂n > 0 is realized because Ξ ≈ − c ³(n − 1)²(n + 1)⁶(n + 2)³/n ³ < 0.¹⁵

Appendix E

By differentiating Eq. A.9 with respect to n and evaluating σ ² _x  = 0 and ${\sigma}_x^2=\sqrt{c{\left(n+1\right)}^2/n}$,

$$ {\left.\frac{\partial {q}_i^{\ast }}{\partial n}\right|}_{\sigma_x^2=0}=-\frac{a-{\mu}_x}{{\left(n+1\right)}^2}<0, $$

(E.1)

$$ {\left.\frac{\partial {q}_i^{\ast }}{\partial n}\right|}_{\sigma_x^2=\sqrt{c{\left(n+1\right)}^2/n}}=-\frac{6\left(2n-1\right)\left\{2\left(a-{\mu}_x\right)-\overline{r}{\sigma}_x^2\right\}}{\left(n+2\right){\left({n}^2-1\right)}^2}<0. $$

(E.2)

In contrast, by differentiating Eq. A.10 with respect to n and evaluating at σ ² _x  = 0, we show that

$$ {\left.\frac{\partial {p}^{\ast }}{\partial n}\right|}_{\sigma_x^2=0}=-\frac{a-{\mu}_x}{{\left(n+1\right)}^2}<0. $$

(E.3)

In the case of ${\sigma}_x^2=\sqrt{c{\left(n+1\right)}^2/n}$, we derive the following:

$$ {\left.\frac{\partial {p}^{\ast }}{\partial n}\right|}_{\sigma_x^2=\sqrt{c{\left(n+1\right)}^2/n}}=\frac{4\left({n}^2+1\right)\left(a-{\mu}_x\right)-{\left(n+1\right)}^2\overline{r}{\sigma}_x^2}{2{\left({n}^2-1\right)}^2}. $$

(E.4)

In order to confirm that the sign of Eq. E.4 is positive, it is sufficient to confirm the case of the maximum $\overline{r}$. Substituting $\overline{r}=2\left(a-{\mu}_x\right)/{\sigma}_x^2$ into Eq. E.4, we have¹⁶

$$ {\left.\frac{\partial {p}^{\ast }}{\partial n}\right|}_{\sigma_x^2=\sqrt{c{\left(n+1\right)}^2/n}}=\frac{a-{\mu}_x}{{\left(n+1\right)}^2}>0. $$

(E.5)