香港代写assignment|Buyer Uncertainty WTP

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Buyer Uncertainty WTP

Due to the importance of willingness-to-pay (WTP) in the area of pricing, many different methods have been proposed on measuring WTP. However, significant discrepancies are obvious among these elicitation methods when the estimates are compared. Attempts to reconcile these inconsistencies suggest that certain biases may be inherent of many measurement methods, e.g. hypothetical bias and strategic bias.

In this paper, we propose that besides the two known biases, inconsistencies of WTP estimates may be a result of overlooking buyer uncertainties (e.g. information uncertainty and preference uncertainty.) When the level of buyer uncertainty is high, not only will it generate large discrepancy among estimates of extant methods, using these estimates of WTP for pricing purpose can also be highly misleading. Therefore, a practical guidance on dealing with buyer uncertainty when measuring WTP is warranted.

In this paper, we first review the extant definitions and provide a synthesis when buyer uncertainty is presented. Then we review the extant methods of measuring WTP on their pros and cons in practice. We then provide empirical evidences with two experiments that WTP estimates can be significantly biased when buyer uncertainty is overlooked.

Review of Extant Definitions and Synthesis with Buyer Uncertainty

The economics literature interprets WTP as the combined economic value to a consumer of all the benefits that would accrue from the product (Ryan and Miguel 2000). It is noteworthy that extant definitions either assume buyer uncertainty does not exist or it can be solved by consumers when a price is presented to him/her.

In other words, a single price point exists across which buy/not buy decision is clearly defined. The researchers may not be able to observe this price threshold accurately. If the estimates of WTP are not in line with the actual behavior, the usual suspect is measurement errors instead of buyer’s uncertainty.

According to Wang, Venkatesh and Chatterjee (2007), the inconsistencies in the definitions emerge when they clarify how a consumer would behave if the seller’s price is set at the threshold representing the WTP. Since buyer uncertainty is best represented as purchase probability, all the extant definitions differ remarkably in the implied level of purchase probability.

The lack of consensus goes beyond the economics studies. We classify in Table 1 a number of studies on WTP and highlight how they differ in their definitions and their implicit purchase probability when buyer uncertainty is presented.

As proposed by Wang, Venkatesh and Chatterjee (2007), the meaning of convergence would be that the transition in purchase probability from 100% to 0% occurs at a particular point, as shown in Figure 1A.

Figure 1B schematically represents the alternative argument of Wang, Venkatesh and Chatterjee (2007) – that of the WTP as a range of thresholds. (The specific shape of the curve is obviously individual specific.) Their range conceptualization seems able to unify the various definitions of WTP and has the single point representation as a special case. When buyer uncertainty does not exist, all these definitions converge to a single point.

An Assessment of Extant Approaches for WTP Measurement under Buyer Uncertainty

Given our primary interest in WTP at the individual level, we will review related methods only. The six prominent methods that we consider are direct self-explication, Vickrey auction, BDM’s point-of-purchase procedure, ICERANGE, conjoint analysis and choice experiment. The first four are methods for direct elicitation of WTP whereas the last two involve indirect estimation.

Remarkably, when buyer uncertainty is overlooked, the estimates from the alternative methods diverge in systematic ways. For example, the estimates of WTP from the BDM are found to be lower than those from direct self-explication and choice experiment (Wertenbroch and Skiera 2002). The estimates from conjoint analysis based methods are all found to be higher than those from direct self-explication (Chung and Rao 2003, Jedidi and Zhang 2002).

The biases that the alternative methods ignore or overcome are arguably a key source of the observed differences. Two important types of biases examined in the WTP literature are incentive compatibility bias and strategic response behavior bias (Wang, Venkatesh and Chatterjee 2007). We proposed another bias from overlooking buyer uncertainty.

  • Incentive Compatibility Bias (including hypothetical bias):An incentive-compatible mechanism gives the respondent the incentive to respond truthfully and a disincentive for not revealing the truth. Incentive incompatibility arises, for example, when respondents are not responsible to pay their stated (or inferred) WTP. The willingness to pay is higher in such instances than when respondents are responsible to pay (Posavac 2001). Wertenbroch and Skiera (2002) show that this upward bias extends to such other non-incentive-compatible methods as open-ended and double-ended contingent valuation.
  • Strategic Response Bias: This arises when respondents anticipate a link between their response and the ultimate outcome and act so “as to maximize their utility taking into account how their responses may affect the decision” (Carson et al. 1997). The direction of strategic bias depends on the context of elicitation. When consumers perceive that stating a higher WTP will increase the likelihood of subsequently receiving a good, they are more likely to state a higher value than their true value (Posavac 2001). On the other hand, when anticipating a higher final price for a product, buyers might indicate a lower willingness to pay hoping to induce the seller to set prices lower (Monroe 1990, p.107).
  • Bias from Overlooking Buyer Uncertainty: This is an inherent bias of most methods (with the exception of Wang, Venkatesh, and Chatterjee 2007) due to the overlooking of buyer uncertainty while significant one exists.

In Table 3 we briefly point out the salient features of each method, their pros and cons, and representative studies that invoke these approaches. Under pros and cons, we note the biases that the methods control or ignore and their (in) ability to capture buyer uncertainty of an individual consumer. A key conclusion that we draw from the table is that ICERANGE approach appears to be a “best method” among available options based on its avoidance of the strategic, incentive compatibility biases, and buyer uncertainty.

The direction of strategic bias depends on the context of elicitation:

(a) When consumers perceive that stating a higher WTP will increase the likelihood of subsequently receiving a good, they are more likely to state a higher value than their true value (Posavac 2001)

(b) On the other hand, “when facing a direct question as to which a set of prices would be preferred or acceptable, buyers often may indicate the lowest price option, since this would be a ‘rational’ answer.” (Monroe 1990, p.107), the reason for providing lower value than their true value is that consumers are trying to use strategic thinking to keep price low in the real market. This downward bias has been used to explain higher estimates of WTP from indirect mechanisms than those from simple self-explication (Chung and Rao 2000; Jedidi and Zhang 2002).

However, after taking all the above biases into consideration, the extant literature still cannot agree with each other on which method provides the “true” (or closest to “true”) value of WTP: As shown experimentally that simple self explication is biased upwards due to incentive incompatibility (Wertenbroch and Skiera 2002) or hypothetical bias (Posavac 2001), one may conclude that the “true” WTP should be lower than the estimate value from self explication.

This is, however, in contrary to the findings of indirect elicitation methods: the supposedly unbiased estimates of indirect elicitation (which is incentive indifferent as consumers has no incentive to either tell the truth or lie) turned out to have even higher value than those from self explication format (Chung and Rao 2000, Jedidi and Zhang 2002).

This raises further concern on whether it is sufficient to use these biases to explain the difference on estimates from alternative methods. Could it be the case that despite all these biases, different methods are simply measuring different points of a range of “reservation price” under buyer’s uncertainty?

The lack of convergence among extant methods can only partly be attributed to the two known biases. If, as shown empirically, simple self explication is biased upward (overestimation) due to incentive incompatibility or hypothetical bias, one may conclude that the “true” WTP should be lower than the estimated value from self explication. This is, however, not supported by the findings of indirect elicitation methods: the supposedly unbiased estimates of indirect elicitation turn out to be even higher than self-explicated measures (Chung and Rao 2003, Jedidi and Zhang 2002).

There is no easy way to reconcile these differences if buyer uncertainty is overlooked. On the other hand, it can only be explained when these estimates are tied to different purchase probabilities. The underlying uncertainty of consumer’s valuation is supposed to result in a range estimate of WTP that contains different point estimates of conventional methods within the range.

Table 3, under salient features, contains the probability of purchase that the WTP measure (or estimate) implicitly maps on to. We surmise that without any elicitation bias, a WTP estimate associated with 50% of purchase probability would likely be higher than an alternative estimate associated with 100% of purchase probability.

As BDM and Vickrey auction procedures are tied to real choice (i.e., one has to buy the good if certain conditions are met), respondents may provide a value corresponding to their floor WTP with 100% of purchase probability (we will test this hypothesis in the next section). On the other hand, the nature of trade-off analysis in conjoint analysis and choice experiment—tied to indifference between buying and not buying—is likely to yield a higher estimate of WTP close to the “indifference” WTP with 50% of purchase probability.

Pricing implication when WTP estimated with buyer uncertainty overlooked

The purpose of this section is to prove analytically that ignoring buyer uncertainty in WTP estimation when it exists leads to sub-optimal pricing. To do so, we will set up a simple model of WTP, derive the closed form solution for optimal price and investigate the nature and extent of biases by ignoring buyer uncertainty.

Let the seller’s profit-maximizing price to an individual consumer be P. Let the consumer’s floor WTP (in Figure 1B) be a and the ceiling WTP be a+b, implying a range of b. Let the shape of the curve be S-shaped with the indifference WTP, at , halfway between the floor and ceiling WTP. In other words, the curve we assume is an S-shaped cumulative distribution function (cdf), a more widely accepted form in marketing than the ramp shaped (i.e., linear) cdf arising from a uniform density. The specific cdf we assume is:

for (1a)

and for (1b)

This functional form is better than, say, that normal cdf in that it is analytically tractable. Moreover, better than the normal and logistic cdfs, the one above is bounded between the floor and the ceiling WTP. The marginal cost is set to zero.

The seller’s profit would be:

Õ = aif P = a(2a)

Õ = if (2b)

Õ = if (2c)

Of the above possible levels for price, the optimal price occurs in the interval . Specifically, the seller’s optimal price to the consumer has the closed form solution:


Notice that the optimal price converges to a when b = 0. Notice that when the range is zero, the floor-, indifference-, and ceiling-WTP all converge to a. Obviously, the seller maximizes profit by setting the price at this point.

When the range is positive (b>0), we have the following proposition:

P1a:When a consumer is uncertain of her WTP, prices corresponding to the floor-, indifference- and ceiling WTP are all sub-optimal.

P1b:A price level corresponding to the floor WTP is biased downward, and those corresponding to the indifference or ceiling levels are biased upward. The price corresponding to the ceiling WTP has the most bias.

Empirical Evidence of WTP Elicitation Bias when Buyer Uncertainty is Overlooked

To demonstrate that conventional elicitation of WTP is likely to be biased due to overlooking of buyer uncertainty, we conducted two paper-and-pencil tasks. Our study 1 was in the context of a digital handheld. Our sample consisted of 87 MBA students, a relevant segment for this product category.

To assess whether the bias indeed exists when buyer uncertainty is overlooked, we asked each respondent three questions that map on to WTP associated with 100%, 50% and 0% purchase probability. In other words, we used a within-subjects design. The actual measures are provided in Appendix I. Specifically, we tested the following hypotheses:

H1a: Consumers’ mean indifference WTP is greater than their mean floor WTP.

H1b: Consumers’ mean ceiling WTP is greater than their mean indifference WTP.

There is some concern among marketing scholars that self-explicated measures of WTP, like the ones we used, could be downward biased as respondents might act strategically to keep prices low. The mitigating circumstance in our context is our interest in the differences between pairs of measurements, so the biases would largely cancel out.

We summarize the results in Table 2A. As seen here, and in support of Hypothesis 1(a), the mean indifference WTP exceeds the mean floor WTP by $44.67 (significantly greater than zero with p < 0.01). Further, in support of Hypothesis 1(b), the mean ceiling WTP is greater than the mean indifference WTP by $49.87 (p < 0.01). Together they support the Hypothesis 1 that WTP of individual consumers are better represented as a range of valuations, at least in the context of digital handhelds.

In Study 2, we address two potential problems with the earlier study: (a) As Study 1 is a within-subjects design, asking the three WTP related questions in succession might give rise to testing effects; (b) The digital handheld in Study 1 is just one category. Therefore, in Study 2, we used a between-subjects design with random assignment and considered three products: a digital camera, a ticket for a regular season NFL game and a bottle of French wine. We tested the following, considerably more conservative hypotheses.

H2a: (Mean Indifference WTP)Group j > (Mean Floor WTP)Group i

H2b: (Mean Ceiling WTP)Group k > (Mean Indifference WTP)Group j

The measures for a sample individual were: (1) What is the maximum price at or below which you would definitely buy the above shown <digital camera>? (At or below this price there is a 100% chance that you would buy.) (2) What is the price at which you would be indifferentbetween buying and not buying the above shown <NFL home ticket>? (At this price there is only a 50% chance that you would buy.) (3) What is the minimum price at or above which you would definitely not buy the above shown <French wine>? (At or above this price there is nochance that you would buy.)

The sample for Study 2 consisted of 82 MBA students, without any overlap with the earlier sample. To avoid potential sequence bias, we rotated the products and the measures in the sense that respondents in each of the three groups provided the floor WTP in one category, the indifference WTP in another and the ceiling WTP in the third category. The page layout and filler questions ensured that the floor-, indifference- and ceiling-measures were “far apart.” The relevant results are in Table 2C.

Despite the conservative nature of this test, all six observed differences are in the hypothesized direction, with four of these differences statistically significant. The mean ceiling WTP is significantly greater than the mean indifference WTP for the football ticket ($49.38, p<0.01) and digital camera ($111.13, p<0.01). The mean indifference WTP is significantly more than the mean floor WTP for the football ticket ($16, p<0.01) and wine ($9.63, p<0.01).

We believe that the above theoretical and empirical evidence lend strong support for the bias that exists when buyer uncertainty (purchase probability) is overlooked during WTP elicitation.


WTP is widely used concept in marketing and economics. The measurement of WTP is key to the pricing practice of marketing managers, especially value pricing and one-to-one pricing. However, when WTP is elicited using conventional methods (with the exception of Wang, Venkatesh and Chatterjee 2007), significant bias exist for all point estimates when buyers are uncertain about their valuation.

This paper reviewed the extant literature on the definitions and measurement methods of WTP. Our analytical model shows that when buyer uncertainty is present, point estimates can lead to serious bias and sub-optimal pricing. We further demonstrate that buyer uncertainty is evident for several popular consumer products.

In summary, researchers need to be careful when WTP is elicited under buyer uncertainty. The higher the level of uncertainty, the more bias the point estimate will be. The most severe bias happens when ceiling WTP is elicited.


由于面积计价的重要性,愿意支付( WTP ) ,已经提出了许多不同的方法测量WTP 。然而,显着的差异是明显的在这些启发式方法估计比较。试图调和这些矛盾表明,某些偏见可能是固有的许多测量方法,例如假设的偏差和战略偏差。
经济学文献中解释WTP的综合经济价值,给消费者的所有的好处, ( Ryan和米格尔2000 )从产品累计。值得注意的是,现存的定义假设买方不确定性不存在或可以解决消费者的价格呈现给他/她。
另据介绍,文卡塔斯和查特吉(2007) , ,在定义中出现的矛盾时,他们澄清消费者将如何表现,如果卖家的价格定在阈值代表WTP 。由于买家的不确定性购买概率是最好的代表,所有现存的定义明显不同的购买概率的引伸水平。
所提出的王,卡塔斯和Chatterjee (2007) ,收敛的含义。将购买的过渡在从100 %到0%的概率发生在一个特定的点,在图1A中所示。
图1B示意王,文卡塔斯和查特吉(2007) - 代表的另一种说法,作为一个阈值范围内的污水处理厂。 (具体的曲线形状显然是个别的具体的)的范围的概念化似乎能够统一的WTP的各种定义,具有单点表示,作为一种特殊情况。当买方不确定性不存在,所有这些定义收敛到一个单一的点。
鉴于我们的主要兴趣是在WTP在个人层面上,我们会检讨相关的方法。六个突出的方法,我们认为是直接的自我解释,维克瑞拍卖, BDM点购买过程, ICERANGE ,联合分析和选择实验。前四个是直接引出的WTP的方法,而最后两个涉及间接估计。
值得注意的是,当买方不确定性被忽视,从替代方法估计偏离系统的方式。例如,估计WTP从BDM发现是低于直接自我解释和选择的实验( 2002年Wertenbroch Skiera ) 。基于联合分析方法的估计都高于那些从直接的自我解释(涌, 2003年饶, Jedidi和2002年张) 。
替代方法忽略或克服的偏见,可以说是观察到的差异的一个重要来源。 WTP文学研究的两类重要偏见是激励相容的偏见和战略响应行为偏差的(王, 2007年卡塔斯和Chatterjee ) 。我们提出了另一个偏见从俯瞰买家不确定性。
激励相容偏置(包括假想偏置) :一个激励相容的机制,给出了受访者的激励如实答复,不显露真相有抑制作用。出现激励不相容,例如,当受访者不负责支付他们的说明(或推断) WTP 。在这样的情况下,支付意愿较高时相比,受访者负责支付( 2001年Posavac ) 。 Wertenbroch Skiera (2002)显示,这个向上的偏差,延伸到其他非激励相容的开放端和双端然估值的方法。
战略反应偏差:出现这种情况,当受访者预期他们的反应和最终结果之间的联系,并采取行动“最大化效用的考虑,他们的反应可能会影响决定” (卡森等人, 1997) 。战略偏差的方向依赖于上下文的启发。当消费者感知,说明较高的污水处理厂,随后接收到良好的可能性将增加,他们更容易说出一个值高于其真实价值( 2001年Posavac ) 。另一方面,当预期最终价格较高的产品,买家可能会表示希望能诱使卖家设置价格较低(第107页) 1990年,梦露支付意愿较低。
偏置从俯瞰买家不确定性:这是一个大多数方法(除王,文卡塔斯, 2007年和Chatterjee )由于买方不确定性,而忽视明显存在一个固有的偏见。
(a)当消费者感知,说明较高的污水处理厂的可能性将增加,随后接收到良好的,他们更容易说出一个值高于其真实价值( Posavac 2001 )
(二)另一方面, “当面对一个直接的问题,其中一组的价格将首选还是可以接受的,购房者往往可能表明价格最低的选项,因为这将是一个”理性“的答案。 ” ( 1990年梦露,第107页) ,提供较低的值比其真实价值的原因是,消费者试图用战略思维,以保持在真正的市场价格低。这种向下的偏差已被用来解释更高的WTP估计比那些简单的自我解释(东涌及2000年饶; 2002年Jedidi张)间接机制。
然而,上述所有的偏见考虑后,现存文献仍然不能同意对方上的方法提供“真正的” (或最接近为“true” )的WTP值:正如所示实验,简单的自我解释是偏颇向上由于激励的不兼容性( 2002年Wertenbroch Skiera )或假设性偏误( 2001年Posavac ) ,可以推断,“真实”的污水处理厂应低于自解释的估计值。
然而,这是间接启发式方法的结果:间接引出了所谓的无偏估计(这是激励无动于衷,因为消费者有没有动力去告诉真相或谎言)横空出世,具有更高的价值比那些从相反自我解释格式(仲和2000年饶, Jedidi和2002年张) 。
之间的现存方法的收敛性的缺乏只能部分归因于两个已知的偏差。经验显示,如果简单的自我解释是偏于上行由于激励不相容或假设性偏误(高估) ,人们可能会得出结论,“真实”的污水处理厂应低于从自我解释的估计值。然而,这是不支持的结果间接引出方法:所谓的无偏估计间接启发甚至高于的自我阐释措施(仲和2003年饶, Jedidi 2002年,张) 。
由于BDM和维克瑞拍卖程序与真正的选择(即,买的好,如果某些条件得到满足) ,受访者可能会提供一个值100 %的购买概率(我们将检验这一假设对应楼面WTP与下一节) 。另一方面,权衡分析,联合分析和选择的性质实验绑在购买之间的冷漠与不买很可能产生一个更高的“冷漠”的污水处理厂有50 %的购买概率估计的WTP 。
本节的目的是为了证明当它存在,忽略买家在WTP估算的不确定性会导致次优定价分析。要做到这一点,我们将建立一个简单的模型, WTP ,获得全封闭形式的解决方案,价格最优和调查的性质和程度的偏见忽略买家的不确定性。
让卖家的利润最大化的价格,个别消费者P.让消费者的地板WTP (图1B)和天花板WTP是A + B ,这意味着B的范围。让曲线的形状是S形WTP的冷漠,在地板和天花板WTP之间的一半。换句话说,我们假设的曲线是一个S形的累积分布函数(CDF) ,更广泛接受的形式的斜坡状的(即,线性的)累积分布函数所产生的均匀的密度比在市场营销。具体CDF我们假设是:
(1A )
此功能的形式,它是易于分析说,这比正常的CDF更好。此外,优于正常和后勤CDFS ,上面一个是有界的地板和天花板污水处理厂。的边际成本被设置为零。
O = AIF P = A( 2A )
O = (2B )
O = (2C )
请注意,价格最优收敛到当b = 0 。请注意,范围为0时,在地板上,冷漠,和天花板WTP都收敛到一个。显然,卖家设定价格,在这一点上实现利润最大化。
范围为阳性( B> 0 )时,我们得到如下命题:
P1A :当消费者不确定她的WTP ,相应地板,冷漠和天花板WTP的价格都是次优的。
P1B :其价格水平对应的地板WTP偏于下行,与相应的冷漠或天花板水平偏于上行。对应到天花板WTP价格最偏见。
为了证明WTP可能有偏差由于买家不确定性俯瞰,传统的启发,我们进行了两个纸和铅笔任务。我们的研究[ 1 ]中的数字手持式的上下文中。我们的样本包括87名MBA学生,该产品类别相关的分部。
为了评估是否偏置买家的不确定性时,确实存在被忽视,我们要求每位受访者100%,50 %和0%的购买概率映射WTP相关的三个问题。换句话说,我们使用了一个内设计。实际措施附录一具体来说,我们测试了以下假设:
H1A :消费者平均冷漠WTP是大于他们的均值楼WTP 。
H1B :消费者平均天花板WTP是大于他们的冷漠平均WTP 。
我们总结表2A中的结果。如这里所见,而在图1(a )假说的支持,平均冷漠弟超过:平均楼面弟44.67美元(显着大于零,具有p <0.01) 。此外,在天花板的平均WTP是支持假设1 (B ) ,大于平均冷漠WTP 49.87美元( P < 0.01 ) 。他们一起支持假设1 ,污水处理厂中,个体消费者更好地表示,至少在数字​​手持设备的背景下,作为一个估值范围。
在研究2中,我们要解决两个潜在的问题与早期的研究: ( a)作为研究1是一个内设计,要求三个污水处理厂在继承的相关问题可能会引起测试效果; (b)本数字手持研究图1是一个类别。因此,在研究2中,我们使用了一个主体间的设计与随机分配,并考虑了三个产品:一台数码相机,一票为常规赛季NFL比赛和一瓶法国葡萄酒。我们测试了以下,相当多的保守假设。
H2A : (平均冷漠WTP ) I组J组(平均楼层WTP )
H2B : (平均天花板WTP ) K组(平均冷漠WTP ) J组
样本个体的措施是:(1)什么是最大的售价或低于它,你肯定会买<数字显示camera>的? (或低于这个价格,有100%的机会,你会买。 )(2)是什么价格,你会买与不买如上图所示<NFL家庭ticket>之间漠不关心? (在这个价位上只有50%的机会,你会买。 )(3)或以上,你绝对不会购买上述所示<French的wine>什么是最低价格吗? (等于或高于这个价格,就没有机会了,你会买。 )
2研究的样本包括82名MBA学生,与早期的样品没有任何重叠。为了避免潜在的序列偏见,我们的产品和措施,在这个意义上,受访者中每三个组第三类地板WTP在一个类中, WTP在另一个冷漠和天花板WTP旋转。页面布局和填料的问题地板,冷漠和天花板措施确保“相距甚远。 ”相关的结果是在表2C 。
尽管在此测试中的保守性,所有六个观察到的差异是在虚拟的方向,有四个这些统计学上显着的差异。天花板的平均WTP是明显大于足球票( 49.38美元, P <0.01)和数码相机( 111.13美元, P < 0.01 )的的冷漠平均支付意愿。平均冷漠WTP明显超过平均楼面WTP足球票($ 16 , P <0.01)和葡萄酒( $ 9.63, P < 0.01 ) 。
WTP被广泛使用的概念,在市场营销和经济学。 WTP的测量是关键的营销经理,特别是价值定价和一对定价的定价方式。然而,当WTP是引起使用传统的方法(除王,文卡塔斯和Chatterjee 2007 ) ,存在重大偏见时,所有的点估计买家不确定其估值。
综上所述,研究人员需要时要小心WTP是根据买方的不确定性引起的。水平较高的不确定性,将偏置点估计。时,发生最严重的偏见引起天花板WTP 。