The effects of uncertainty on the WTA–WTP gap

Abstract

We analyze the effects of uncertainty on WTA, WTP and the WTA–WTP gap. Extending the approach of Weber (Econom Lett 80:311–315, 2003) to the case of lotteries, we develop an exact expression for the WTA–WTP gap that allows identification of its magnitude under different utility specifications. Reinterpreting and extending results by Gabillon(Econom Lett 116:157–160, 2012), we also identify generally the relationship between an agent’s utility of income and the gap’s algebraic sign, as well as the effects of risk increases on WTA and WTP. Finally, we derive the limit behavior of the gap as income or risk increase.

Appendices

Appendix 1

1.1 Proof of corollary 1

Proof of part (i)

(a) was established as Eq. (4) above. For (b), define $y_2 =y_0 -C(y_0 ,z)$. By definition of $C(y_0 ,z),\,v(y_0 )=\nu \left( {y_2 ,z}\right)$. Further, by definition of $E(y_2 ,z),\,v(y_2 ,z)=v\left[ {y_2 +E(y_2 ,z)}\right]$, Hence $v(y_0 )=v\left[ {y_0 -C(y_0 ,z)+E(y_2 ,z)}\right]$, and so $C(y_0 ,z)=E(y_2 ,z)$. $\square$

Proof of part (ii)

Theorem 2 in Pratt (1964) establishes the equivalence between strict DARA and a strictly decreasing risk premium, between CARA and a constant risk premium, and between strict IARA and a strictly increasing risk premium. Denoting the expected value of lottery $z$ by $\mu _z$ and recalling that the risk premium is defined as $\mu _z -E(y,z)$, Pratt’s theorem establishes (ii) for $E(y,z)$. $\square$

From part 1.i.b., $C(y_0 ,z)=E(y_2 ,z)$, so,

$$\begin{aligned} \frac{\partial C\left( {y_0 ,z}\right) }{\partial y_0}=\frac{\partial E\left[ {y_2 ,z}\right] }{\partial y_0}\!=\!\frac{\partial E\left[ {y_2 ,z}\right] }{\partial y_2}\frac{\partial y_2}{\partial y_0}\!=\!\frac{\partial E\left[ {y_2 ,z}\right] }{\partial y_2}\left( {1-\frac{\partial C\left( {y_0 ,z}\right) }{\partial y_0}}\right) .\qquad \end{aligned}$$

(9)

Since by definition $\nu \left( {y_0}\right) =\nu \left[ {y_0 -C\left( {y_0 ,z}\right) ,z}\right]$ holds for all $y_{0}$, we may differentiate it with respect to $y_{0}$ to obtain

$$\begin{aligned} \frac{\partial \nu \left( {y_0}\right) }{\partial y_0}=\frac{\partial \nu \left( {y_2 ,z}\right) }{\partial y_2}\left( {1-\frac{\partial C\left( {y_0 ,z}\right) }{\partial y_0}}\right) . \end{aligned}$$

Note that positive marginal utility of income (or alternatively, stochastic dominance) implies that both $\frac{\partial \nu \left( {y_0}\right) }{\partial y_0}>0$ and $\frac{\partial \nu \left( {y_2 ,z}\right) }{\partial y_2}>0$. Thus from the above equality, $\left( {1-\frac{\partial C\left( {y_0 ,z}\right) }{\partial y_0}}\right) > 0$.

This result, together with $\frac{\partial E\left[ {y_2 ,z}\right] }{\partial y_2}>(=,<)0$ under DARA (CARA,IARA) established for the $E(y,z)$ part of (ii), guarantees that the right hand side of Eq. (9) $>(=<)0$ under DARA (CARA,IARA). Hence $\frac{\partial C\left( {y_0 ,z}\right) }{\partial y_0} >(=<) 0$ under DARA (CARA,IARA), establishing the $C(y_0 ,z)$ part of (ii).

Proof of part (iii)

In the proof of Lemma 2 it was established that for any initial income $y_0 ,\,1+\frac{\partial E\left( {y_0 ,z}\right) }{\partial y_0}>0$. Thus the denominator of the integrand on the right hand side of (7) is positive for all $\delta$ in the interval of integration.

Theorem 2 in Pratt (1964) establishes the equivalence between strict DARA and a strictly decreasing risk premium, and between strict IARA and a strictly increasing risk premium. In the development here, the risk premium of the lottery $z$ at initial income $y_{0}$ is the expected outcome of the lottery minus its certainty equivalent. For a risk averter, the certainty equivalent of lottery $z$ is necessarily less than its expected outcome. As noted earlier, the certainty equivalent of the lottery at initial income $y_{0}$ is the willingness to accept, $E\left( {y_0 ,z}\right)$.

Consider first the case of DARA. A decreasing risk premium for $z$, together with a constant expected value, implies a rising certainty equivalent, i.e., that $\frac{\partial E\left( {y_0 ,z}\right) }{\partial y_0}>0$. Thus, the numerator of the integrand on the right hand side of (7) is positive for any initial income $y_{0}$. Hence, the integrand is positive across the entire interval of integration. This implies that the integral itself is necessarily positive. Thus, under DARA, the right hand side of (7) is positive.

Matters are reversed in the case of IARA. An increasing risk premium combined with a positive expected value implies a falling certainty equivalent, that is, $\frac{\partial E\left( {y_0 ,z}\right) }{\partial y_0}<0$ and thus the numerator of the integrand on the right hand side of (7) is negative. Using the same logic as in the case of DARA the integrand across the entire interval of integration is negative, meaning that the right hand side of (7) is negative.

Under CARA, the risk premium is constant and thus $\frac{\partial E\left( {y_0 ,z}\right) }{\partial y_0}=0$. Hence, the numerator of the integrand on the right hand side of (7) is identically zero, and thus the integral is zero as well, establishing that the equivalent and compensating variations are equal in that case. $\square$

Appendix 2

1.1 Proof of corollary 2

Proof

Consider first the inequality on the left hand side. In their Theorem 2, Rothschild and Stiglitz (1970) establish, that given an income level $y_0 , \nu \left( {y_0 ,z_2}\right) <\nu \left( {y_0 ,z_1}\right)$. Hence, $\nu \left( {y_0 +E\left[ {y_0 ,z_2}\right] }\right) <\nu \left( {y_0 +E\left[ {y_0 ,z_1}\right] }\right)$, and so by $\nu _y >0,\,E\left( {y_0 ,z_1}\right) >E\left( {y_0 ,z_2}\right)$.

Now consider the right hand inequality. Suppose that $C\left( {y_0 ,z_1}\right) <C\left( {y_0 ,z_2}\right)$. Then $\nu \left( {y_0}\right) \!=\!\nu \left( {y_0 \!-\!C\left[ {y_0 ,z_1}\right] ,z_1}\right) \ge \nu \left( {y_0 \!-\!C\left[ {y_0 ,z_2}\right] ,z_1}\right) >\nu \left( {y_0 -C\left[ {y_0 ,z_2}\right] ,z_2}\right) = \nu \left( {y_0}\right)$ where the strict inequality is again implied by Theorem 2 of Rothschild and Stiglitz. Thus $\nu \left( {y_0}\right) >\nu \left( {y_0}\right)$, a contradiction. Hence $C\left( {y_0 ,z_1}\right) >C\left( {y_0 ,z_2}\right)$. $\square$