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Market failure in light of non-expected utility

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Abstract

This paper merges the non-expected utility approach (Tversky and Kahneman, J Risk Uncertain 5:297–323, 1992 and Quiggin, J Econ Behav Organ 3:323–343, 1982) into Akerlof’s (Quart J Econ 84:488–500, 1970) model of Market for Lemons. We derive the results for different probability weighting functions and analyze the phenomenon of market failure in light of non-expected utility maximization. Our main finding suggests that when the proportion of traded lemons is high (low), the problem of market failure is mitigated (enhanced). In addition, for the case of Cumulative Prospect Theory, we show that (a) the higher the loss aversion is, the more pronounced is the market failure; (b) gain-domain elevation is negatively related to the extent of market failure; and (c) the value function is (i) negatively monotonic in the gain-domain diminishing sensitivity parameter when the market is characterized by a high proportion of “peaches,” and (ii) positively monotonic in the loss-domain diminishing sensitivity parameter when the market is characterized by a high proportion of “lemons.”

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Notes

  1. Lower and upper subadditivity are two characterizations of the PWF (Tversky and Wakker 1995, conditions 4.1 and 4.2). Let \(w\) denote the PWF. Lower subadditivity is formally defined as follows: \(w(q)\ge w(p+q)-w(p)\) whenever \(p+q\le 1-\varepsilon \); \(\varepsilon \ge 0\). The formal definition of upper subadditivity is: \(1-w(1-q)\ge w(p+q)-w(p)\) whenever \( p\le \varepsilon ^{\prime }\); \(\varepsilon ^{\prime }\ge 0\).

  2. Let \(x\) and \(y\) denote two alternatives such that \(x\succ y\), and let \(z\) denote an alternative that is the probability mixture of \(x\) and \(y\). According to the Betweenness Axiom, \(x\succ z\succ y\).

  3. Several alternative PWFs are presented and discussed in the sequel.

  4. Parameter estimates of various PWF specifications obtained by early and pioneering research in the area are detailed in Table 1. For further estimation results see, e.g., Bruhin et al. (2010) and Fehr-Duda and Epper (2012).

    Table 1 Survey of parameter estimates
    Full size table

  5. For further analysis of this PWF, see Prelec (1998, p. 504).

  6. Refer to Gonzalez and Wu (1999) for detailed psychological rationale for the shapes of PWFs.

  7. We present the analysis with the one-parameter PWF of Tversky and Kahneman (1992). Qualitatively similar results are obtained with all of the alternative PWF specifications.

  8. This result is formally presented in Therorem 1.

  9. The graph is plotted with \(\gamma _{L}=0.69\) and \(\gamma _{G}=0.61\), the estimated values in Tversky and Kahneman (1992). The diminishing sensitivity parameters, \(\alpha \) and \(\beta \), equal unity to comply with the risk neutrality assumption in Akerlof’s (1970).

  10. Recall that the graphs are plotted with the PWF specification of Tversky and Kahneman (1992), while the formal results hold for all the discussed PWFs.

  11. The Graph is plotted with \(\gamma _{L}=\gamma _{G}=0.44\) and \(\delta _{L}=0.77\), the gain-domain values from Gonzalez and Wu (1999), applied to both domains as a benchmark. The diminishing sensitivity parameters, \(\alpha \) and \(\beta \), equal unity to comply with the risk neutrality assumption in Akerlof’s (1970).

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Authors and Affiliations

  1. Department of Economics, Bar-Ilan University, 52900 , Ramat Gan, Israel

    Eyal Baharad

  2. Department of Economics, The University of Haifa, 31905 , Haifa, Israel

    Doron Kliger

Authors
  1. Eyal Baharad
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  2. Doron Kliger
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Correspondence to Eyal Baharad.

Appendix: Proofs

Appendix: Proofs

Theorem 1

For any APWF \(\Phi (\cdot )\), there exist \(\underline{Q_{L}}\) and \( \overline{Q_{L}}\) such that:

  1. (i)

    \( U^{RDEU}(q)<U^{EUT}(q)\) if \(0<q<\underline{Q_{L}}\); and

  2. (ii)

    \( U^{RDEU}(q)>U^{EUT}(q)\) if \(\overline{Q_{L}}<q<1\).

Proof

We first prove the Theorem for the special case of risk neutrality. Then we prove it for the general EUT setting allowing for risk aversion.

Risk Neutrality:

(i) Recall Eq. 4, asserting that the certainty equivalent according to RDEU, induced by participation in the lottery is:

$$\begin{aligned} \mathrm{CE}^\mathrm{RDEU}(q)=u^{-1}\left( U\left( \left[ W+X_{L}-P\left( q\right) ,q;W+X_{H}-P\left( q\right) \right] \right) \right) . \end{aligned}$$

In a market governed by risk neutrality pricing, the induced value is:

$$\begin{aligned} \mathrm{CE}^\mathrm{RDEU}\left( q\right) -W&= \left( 1-\omega \left( 1-q\right) \right) \cdot \left( X_{L}-\left. P\left( q\right) \right| _\mathrm{RN}\right) \\&+\omega \left( 1-q\right) \cdot \left( X_{H}-\left. P\left( q\right) \right| _\mathrm{RN}\right) . \end{aligned}$$

where \(\omega \left( \cdot \right) \) is the PWF. Due to the appropriateness of the PWF, there exists \(\underline{Q_{L}}\) such that for all \(0<q< \underline{Q_{L}}\), \( \omega \left( 1-q\right) <1-q\) and \(1-\omega \left( 1-q\right) >q\). Thus, for these \(q\) values,

$$\begin{aligned} \mathrm{CE}^\mathrm{RDEU}\left( q\right) -W&= \left( 1-\omega \left( 1-q\right) \right) \cdot \left( X_{L}-\left. P\left( q\right) \right| _\mathrm{RN}\right) \\&+\omega \left( 1-q\right) \cdot \left( X_{H}-\left. P\left( q\right) \right| _\mathrm{RN}\right) <q\cdot \left( X_{L}-\left. P\left( q\right) \right| _\mathrm{RN}\right) \\&+\left( 1-q\right) \cdot \left( X_{H}-\left. P\left( q\right) \right| _\mathrm{RN}\right) =0. \end{aligned}$$

Note that \(\mathrm{CE}^\mathrm{RDEU}(q)<W\) implies \(U^\mathrm{RDEU}(q)<U^\mathrm{EUT}(q)\), which completes the proof of (i).

(ii) Use again

$$\begin{aligned} \mathrm{CE}^\mathrm{RDEU}\left( q\right) -W&= \left( 1-\omega \left( 1-q\right) \right) \cdot \left( X_{L}-\left. P\left( q\right) \right| _\mathrm{RN}\right) \\&+\omega \left( 1-q\right) \cdot \left( X_{H}-\left. P\left( q\right) \right| _\mathrm{RN}\right) . \end{aligned}$$

where \(\omega \left( \cdot \right) \) is the PWF. Due to the appropriateness of the PWF, there exists \(\overline{Q_{L}}\) such that for all \(\overline{ Q_{L}}<q<1\),\( \omega \left( 1-q\right) >1-q\) and \(1-\omega \left( 1-q\right) <q\). Thus, for these \(q\) values,

$$\begin{aligned} \mathrm{CE}^\mathrm{RDEU}\left( q\right) -W&= \left( 1-\omega \left( 1-q\right) \right) \cdot \left( X_{L}-\left. P\left( q\right) \right| _\mathrm{RN}\right) \\&+\omega \left( 1-q\right) \cdot \left( X_{H}-\left. P\left( q\right) \right| _\mathrm{RN}\right) >q\cdot \left( X_{L}-\left. P\left( q\right) \right| _\mathrm{RN}\right) \\&+\left( 1-q\right) \cdot \left( X_{H}-\left. P\left( q\right) \right| _\mathrm{RN}\right) =0. \end{aligned}$$

Note that \(\mathrm{CE}^\mathrm{RDEU}(q)>W\) implies \(U^\mathrm{RDEU}(q)>U^\mathrm{EUT}(q)\), which completes the proof under risk neutrality.

Risk Aversion:

(i) Using Eq. 3, the utility an RDEU buyer derives from purchasing the product at a market governed by EUT pricing is:

$$\begin{aligned} u\left( W\right)&= \left( 1-\omega \left( 1-q\right) \right) \cdot u\left( W+X_{L}-P\left( q\right) \right) \\&+\omega \left( 1-q\right) \cdot u\left( W+X_{H}-P\left( q\right) \right) , \end{aligned}$$

where \(\omega \left( \cdot \right) \) is the PWF.

The RDEU buyer’s utility differential, with respect that of an EUT participant (using Eq. 2), in a market governed by EUT pricing, is, therefore:

$$\begin{aligned}&U^\mathrm{RDEU}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad -U^\mathrm{EUT}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad =\left( \begin{array}{c} \left( 1-\omega \left( 1-q\right) \right) \cdot u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) + \\ \omega \left( 1-q\right) \cdot u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \end{array} \right) \\&\quad \quad -\left( \begin{array}{c} q\cdot u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \\ +(1-q)\cdot u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \end{array} \right) \\&\quad =\left( \left( 1-\omega \left( 1-q\right) \right) -q\right) \cdot u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \\&\quad \quad +\left( \omega \left( 1-q\right) -(1-q)\right) \cdot u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) . \end{aligned}$$

Note that \(\left( \omega \left( 1-q\right) -\left( 1-q\right) \right) =-\left( 1-\omega \left( 1-q\right) -q\right) \), thus:

$$\begin{aligned}&U^\mathrm{RDEU}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad -U^\mathrm{EUT}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad =\left( \left( 1-\omega \left( 1-q\right) \right) -q\right) \cdot \left( \begin{array}{c} u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) - \\ u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \end{array} \right) . \end{aligned}$$

The monotonicity of the utility function implies that

\(u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) <u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \). Due to the appropriateness of the PWF, there exists \(\underline{Q_{L}}\) such that for all \(0<q<\underline{Q_{L}}\), \( \omega \left( 1-q\right) <1-q\) (i.e., \(\left( \omega \left( 1-q\right) -\left( 1-q\right) \right) <0\)). Thus, for these \(q\) values,

$$\begin{aligned}&U^\mathrm{RDEU}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad - \quad U^\mathrm{EUT}\left( \left[ W+X_{L} -\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) <0 \end{aligned}$$

which completes the proof of (i)

(ii) start from the RDEU buyer’s utility differential, with respect that of an EUT participant, derived in (i) above:

$$\begin{aligned}&U^\mathrm{RDEU}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad -U^\mathrm{EUT}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&\quad =\left( \left( 1-\omega \left( 1-q\right) \right) -q\right) \cdot \left( \begin{array}{c} u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) - \\ u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \end{array} \right) . \end{aligned}$$

Recall that the monotonicity of the utility function implies

\(u\left( W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) <u\left( W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right) \). Due to the appropriateness of the PWF, there exists \(\overline{Q_{L}}\) such that for all \(\overline{Q_{L}}<q<1\),\( \omega \left( 1-q\right) >1-q\) (i.e., \(\left( \omega \left( 1-q\right) -\left( 1-q\right) \right) >0\)). Thus, for these \(q\) values,

$$\begin{aligned}&U^\mathrm{RDEU}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) \\&- U^\mathrm{EUT}\left( \left[ W+X_{L}-\left. P\left( q\right) \right| _\mathrm{EUT},q;W+X_{H}-\left. P\left( q\right) \right| _\mathrm{EUT}\right] \right) >0, \end{aligned}$$

which completes the proof under risk aversion. \(\square \)

Theorem 2

For any pair APWFs, \(\Phi (\cdot )\), for the loss- and gain-domains, and risk neutral carrier of values, there exist \(\underline{Q_{L}}\) and \(\overline{Q_{L}}\) such that:

  1. (i)

    \({CE}^{CPT}(q)<0\) if \(0<q<\underline{Q_{L}}\); and

  2. (ii)

    \({CE}^{CPT}(q)>0\) if \(\overline{Q_{L}}<q<1\).

Proof

(i) Recall that according to CPT, the value of the prospect representing a purchase of the product is:

$$\begin{aligned} V\left( \left[ x_{L},q;x_{H}\right] \right) =\omega _{L}\left( q\right) \cdot v\left( x_{L}\right) +\omega _{G}\left( 1-q\right) \cdot v\left( x_{H}\right) , \end{aligned}$$

where \(\omega _{L}\left( \cdot \right) \) and \(\omega _{G}\left( \cdot \right) \) are the PWFs for losses and gains, respectively, and \(v\left( \cdot \right) \) is the value (utility) function. Under a linear value function, \(\mathrm{CE}^\mathrm{CPT}\left( q\right) \) is described by

$$\begin{aligned} \mathrm{CE}^\mathrm{CPT}\left( q\right) =\omega _{L}\left( q\right) \cdot x_{L}+\omega _{G}\left( 1-q\right) \cdot x_{H}. \end{aligned}$$

Due to the appropriateness of the PWF, there exists \(\underline{Q_{L}}\) such that for all \(0<q<\underline{Q_{L}},\;\omega _{L}\left( q\right) >q\) and \(\omega _{G}\left( 1-q\right) <1-q\). Thus, for these \(q\) values,

$$\begin{aligned} \mathrm{CE}^\mathrm{CPT}\left( q\right) <\omega _{L}\left( q\right) \cdot x_{L}+\omega _{G}\left( 1-q\right) \cdot x_{H}=0, \end{aligned}$$

which proves (i).

(ii) Use again

$$\begin{aligned} \mathrm{CE}^\mathrm{CPT}\left( q\right) =\omega _{L}\left( q\right) \cdot x_{L}+\omega _{G}\left( 1-q\right) \cdot x_{H}. \end{aligned}$$

Due to the appropriateness of the PWFs, there exists \(\overline{Q_{L}}\) such that for all \(\overline{Q_{L}}<q<1\) , \(\omega _{L}\left( \overline{Q_{L}}\right) <\overline{Q_{L}}\) and \(\omega _{G}\left( 1- \overline{Q_{L}}\right) >1-\overline{Q_{L}}\). Thus, for these \(q\) values,

$$\begin{aligned} \mathrm{CE}^\mathrm{CPT}\left( q\right) >\omega _{L}\left( q\right) \cdot x_{L}+\omega _{G}\left( 1-q\right) \cdot x_{H}=0, \end{aligned}$$

which completes the proof. \(\square \)

Theorem 3

For any APWF \(\Phi (\cdot ) ,\;V\left( \left[ x_{L},q;x_{H}\right] \right) \) is negatively monotonic in the loss aversion value \(\lambda .\)

Proof

The proof is established by showing that

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \lambda }<0 \end{aligned}$$

Recall that the value of the prospect representing a purchase of the product at the market price (Eq. 5):

$$\begin{aligned} V\left( \left[ x_{L},q;x_{H}\right] \right) =\omega _{L}\left( q\right) \cdot v\left( x_{L}\right) +\omega _{G}\left( 1-q\right) \cdot v\left( x_{H}\right) , \end{aligned}$$

where the value function (Eq. 6), is:

$$\begin{aligned} v\left[ x\right] =\left\{ \begin{array}{l@{\quad }ll} x^{\alpha } &{} \mathrm{if} &{} x\ge 0 \\ -\lambda \left( -x\right) ^{\beta } &{} \mathrm{if} &{} x<0 \end{array} \right. . \end{aligned}$$

Consider the derivative of the value function w.r.t. \(\lambda \):

$$\begin{aligned} \frac{\partial v\left[ x\right] }{\partial \lambda }=\left\{ \begin{array}{l@{\quad }ll} 0 &{} \mathrm{if} &{} x>0 \\ -\left( -x\right) ^{\beta } &{} \mathrm{if} &{} x<0 \end{array} \right. \end{aligned}$$

Thus, since \(x_{L}<0,\)

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \lambda }=\omega _{L}\left( q\right) \cdot \frac{\partial v\left[ x_{L} \right] }{\partial \lambda }=-\omega _{L}\left( q\right) \cdot \left( -x_{L}\right) ^{\beta }<0. \end{aligned}$$

which completes the proof. \(\square \)

Theorem 4

For any APWF \(\Phi (\cdot ), V\left( \left[ x_{L},q;x_{H}\right] \right) \) is positively monotonic in the elevation value \(\delta _{G}.\)

Proof

The proof is established by showing that

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \delta _{G}}>0 \end{aligned}$$

Noting that \(x_{L}\le 0\le x_{H}\):

$$\begin{aligned} V\left( \left[ x_{L},q;x_{H}\right] \right) =\omega _{L}\left( q\right) \cdot (-\lambda )(-x_{L})^{\beta }+\omega _{G}\left( 1-q\right) \cdot (x_{H})^{\alpha }, \end{aligned}$$

and:

$$\begin{aligned} \frac{\partial V\left( \cdot \right) }{\partial \delta _{G}}=\frac{\partial \omega _{G}\left( 1-q\right) }{\partial \delta _{G}}(x_{H})^{\alpha }. \end{aligned}$$

Since, by construction, \(\partial \omega _{G}\left( \cdot \right) /\partial \delta _{G}>0\), we obtain that \(\frac{\partial V\left( \cdot \right) }{ \partial \delta _{G}}>0,\)which completes the proof. \(\square \)

Theorem 5

For any APWF \( \Phi (\cdot \ )\), in a market characterized by a high proportion of peaches, \(V\left( \left[ x_{L},q;x_{H}\right] \right) \) is negatively monotonic in the gain-domain diminishing sensitivity parameter, \(\alpha \); and in a market characterized by a high proportion of lemons, \(V\left( \left[ x_{L},q;x_{H}\right] \right) \) is positively monotonic in the loss-domain diminishing sensitivity parameter, \(\beta \).

Proof

Recall the value of the prospect representing a purchase of the product at the market price, Eq. 5:

$$\begin{aligned} V\left( \left[ x_{L},q;x_{H}\right] \right) =\omega _{L}\left( q\right) \cdot v\left( x_{L}\right) +\omega _{G}\left( 1-q\right) \cdot v\left( x_{H}\right) , \end{aligned}$$

where the value function, Eq. 6, is:

$$\begin{aligned} v\left[ x\right] =\left\{ \begin{array}{l@{\quad }ll} x^{\alpha } &{} \mathrm{if} &{} x\ge 0 \\ -\lambda \left( -x\right) ^{\beta } &{} \mathrm{if} &{} x<0 \end{array} \right. . \end{aligned}$$

Recall that \(x_{L}=X_{L}-P<0\) with probability \(q\) and \(x_{H}=X_{H}-P>0\) with probability \((1-q)\). Evidently, ceteris paribus, the higher the proportion of peaches (lemons) is, the higher (lower) is \(P\). Consequently, \( x_{H}\) becomes lower, i.e., approaches zero (\(x_{L}\) becomes higher, i.e., approaches zero).

The proof is established by showing that

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \alpha }&= \omega _{G}\left( 1-q\right) \cdot \frac{\partial v\left[ x_{H}\right] }{ \partial \alpha }\\&= \omega _{G}\left( 1-q\right) \cdot \left( x_{H}\right) ^{\alpha }\ln \left( x_{H}\right) \left\{ \begin{array}{r@{\quad }rr} >0 &{} \mathrm{if} &{} x_{H}>1 \\ <0 &{} \mathrm{if} &{} 0<x_{H}<1 \end{array} \right. , \end{aligned}$$

and

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \beta }&= \omega _{L}\left( q\right) \cdot \frac{\partial v\left[ x_{L}\right] }{ \partial \beta }\\&= -\omega _{L}\left( q\right) \cdot \lambda \left( -x_{L}\right) ^{\beta }\ln \left( -x_{L}\right) \left\{ \begin{array}{r@{\quad }rr} <0 &{} \mathrm{if} &{} -x_{L}>1 \\ >0 &{} \mathrm{if} &{} 0<-x_{L}<1 \end{array} \right. . \end{aligned}$$

(i) Consider the derivative of the value function w.r.t. \(\alpha \):

$$\begin{aligned} \frac{\partial v\left[ x\right] }{\partial \alpha }=x^{\alpha }\ln \left( x\right) \left\{ \begin{array}{r@{\quad }rr} >0 &{} \mathrm{if} &{} x>1 \\ <0 &{} \mathrm{if} &{} 0<x<1 \end{array} \right. . \end{aligned}$$

Thus, noting that \(0<x_{H}<1\),

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \alpha }&= \omega _{G}\left( 1-q\right) \cdot \frac{\partial v\left[ x_{H}\right] }{ \partial \alpha }\\&= \omega _{G}\left( 1-q\right) \cdot \left( x_{H}\right) ^{\alpha }\ln \left( x_{H}\right) \left\{ \begin{array}{r@{\quad }rr} >0 &{} \mathrm{if} &{} x_{H}>1 \\ <0 &{} \mathrm{if} &{} 0<x_{H}<1 \end{array} \right. , \end{aligned}$$

(ii) Consider the derivative of the value function w.r.t. \(\beta \):

$$\begin{aligned} \frac{\partial v\left[ x\right] }{\partial \beta }=-\lambda \left( -x\right) ^{\beta }\ln \left( -x\right) \left\{ \begin{array}{r@{\quad }rr} <0 &{} \mathrm{if} &{} -x>1 \\ >0 &{} \mathrm{if} &{} 0<-x<1 \end{array} \right. . \end{aligned}$$

Thus, noting that \(-1<x_{L}<0\),

$$\begin{aligned} \frac{\partial V\left( \left[ x_{L},q;x_{H}\right] \right) }{\partial \beta }&= \omega _{L}\left( q\right) \cdot \frac{\partial v\left[ x_{L}\right] }{ \partial \beta }\\&= -\omega _{L}\left( q\right) \cdot \lambda \left( -x_{L}\right) ^{\beta }\ln \left( -x_{L}\right) \left\{ \begin{array}{r@{\quad }rr} <0 &{} \mathrm{if} &{} -x_{L}>1 \\ >0 &{} \mathrm{if} &{} 0<-x_{L}<1 \end{array} \right. . \end{aligned}$$

which completes the proof. \(\square \)

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Baharad, E., Kliger, D. Market failure in light of non-expected utility. Theory Decis 75, 599–619 (2013). https://doi.org/10.1007/s11238-013-9377-0

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