parameters of the MDCEV model by

The multiple discrete-continuous extreme value (MDCEV) model: Using a varying budget to jointly identify whole

parameters of the MDCEV model by

Dongchen Wu

University of Groningen

Faculty of Economics and Business MSc Marketing

19-04-2017

Tutor: Dr. Keyvan Dehmamy

Abstract: In previous literatures, researchers used a fixed budget situation to estimate parameters of the MDCEV model and to measure consumers’ choice behavior in multiple-choice buying activity. The problem of fixing the budget is that it does not provide the ability to jointly identify whole parameters in MDCEV model. Especially the parameter 𝛾 cannot be estimated. However, this study focuses on the MDCEV model using an unfixed budget. So, it improves the identification of the parameter 𝛾 and allows for estimation of the shadow quantity. This study varies the budget and contributes to the gap of previous literatures through using a data set to prove that parameter 𝛾 can be identified.

Keywords: Multiple discrete-continuous extreme values, conjoint, parameters estimation, corner solution, shadow quantity

University of Groningen Faculty of Economics and Business

P.O. Box 72, 9700 AB, Groningen (06) 11056955

d.wu.5@student.rug.nl

Student Number: S2747227

Table of Contents

1. Introduction……….………..………….3

2. Literature review of models……….….4

2.1. Classical approach of conjoint analysis……….………….………4

2.2. Multiple-discreteness model…….………….………...…………5

2.3. MDCEV models……….………...………5

2.4. Hierarchical Bayesian Model. ………..………...……….7

3. Methodology……….………..……….…8

3.1.Research design…….………….………...……….…….8

3.2.Data analysis…….………….………...……….……….9

3.3.Model estimation…….………….………...………...….16

3.4.The insights between each taste and consumer………...………...….17

4. Discussion and conclusion………..…….19

5. Limitation…………...………...……..22

6. Appendixes………..24

Appendix 1………..24

Appendix 2………..25

Appendix 3………..28

Appendix 4………..30

Appendix 5……….………32

Appendix 6……….………36

7. References………..43

1. Introduction

Products are innovated and produced in a fast way, the distinct features of a product persuade consumers to make multiple choices (Bruno et al., 2016). Based on the model of Kuhn-Tucker and the KAR model (Kim et al., 2002), multiple discrete-continuous extreme value (MDCEV) models are developed (Bhat, 2005).

Bhat (2005) suggested that it plays a crucial role in observing consumers’ buying behavior in a situation where they face multiple discrete-continuous quantity choices in a choice set. The model is expressed as u(𝑞)=𝑒𝑥𝑝(𝑥𝛽)(𝑞 + 𝛾)

. Where 𝑞 stands for the quantities a consumer selected. Parameter 𝛽 determines how adding one additional unit will increase 𝛽 to the utility. And 𝛼 represents the marginal decrease of utility if one additional unit of product is added. The term 𝛾 accounts for a corner solution, such as when a consumer chooses zero quantity for a product. In previous studies, many authors applied MDCEV models, but they set the budget as fixed in order to estimate 𝛽 and 𝛼 (such as Kim, 2002; Dubé, 2004; Bhat, 2005; Bhat, 2015;

Wang, 2015). So the parameter 𝛾 is defined as a fixed value that is either “1” or “0”

(with zero indicating the outside option) in existing literature. The problem of using a fixed budget is that it can only estimate two parameters 𝛽 and 𝛼. Full parameters of the MDCEV model cannot be identified. However, even though researchers can answer which is the most and least preferred product in previous studies, the identification of the MDCEV model’s parameters are still weak (Dehmamy，2015).

Dehmamy (2015) suggested an assumption that parameter 𝛾 might not always be

given as “0” or “1” and proposed the idea that parameter 𝛾 can be estimated by

MDCEV models if the budget is set as unfixed. In addition, Dehmamy (2015) stated

that parameter 𝛾 might reflect some stock quantities consumers already owned

before, which can be defined as shadow quantity. But Dehmamy (2015) only

proposed an assumption without using a data set to estimate it. In order to contribute

to existing research and prove the assumption posited by Dehmamy (2015) is correct,

this study comes up with empirical research to create a data set and uses MDCEV

models with unfixed budget to do estimation. The hypothesis in this paper is “if the

budget is set as unfixed, MDCEV models can be used to find out and estimate the

shadow quantities”. The goal of this paper is to estimate the parameter 𝛾 in a

situation where the budget varies, rather than being fixed by using MDCEV models

and to prove the shadow quantities exist. This paper is constructed as follows. First

the theory, limitations of existing research, research design, and data analysis will be

discussed. Then the contributions of this study and the managerial implications.

2. Literature review of models

In order to give a clear description of the MDCEV model and discuss the benefit of varying the budget, this part includes four sections. Firstly, a classical conjoint will be introduced. This will provide insight in how to select the most preferred product.

After that, the multiple-discreteness model will be discussed. The problem of fixing the budget will be described. And the way to identify all parameters will be proposed, which is to vary budget.

2.1. Classical approach of conjoint analysis

Consumers are facing various products in the market with all kinds of choice. For instance, consumers can make a purchase decision for more than hundred possible candy combinations among five tastes, four shapes, four types of quantity and five price levels. The full factorial alternatives are 5×4×4×5=400 possibilities. When using three stimuli per choice set, there are (400

3 ) =

(!""!!)!×!!

= 10,586,800 possible sets. Choice-based conjoint is a traditional discrete choice model. It is a useful approach based on products’ attribute bundles to estimate the buying possibility and find out which alternative product choice consumers prefer most and are willing to buy. As the choice-based conjoint presents a selection of stimuli and identifies the most preferred option, the utility of a product is an important factor to know, as it equals the sum of each attribute levels’ utilities.

According to the utility model and function, choices are based on overall utilities of

alternatives, so the utility of one consumer n for the product j can be written

as:  𝑈

= 𝑉

+ 𝜀

, with 𝑉

=

𝛽

𝑋

. Where V is systematic utility

component (rational utility) and 𝜀 describes the stochastic utility component (error

term). The k stands for the number of attributes (1,…,k), X means the dummy that

indicates the specific attribute level of product j, and 𝛽 is the Part-worth utility

(preferences) of consumer n for attribute k. Consumers will select the alternative with

the highest utility. In a traditional discrete choice model, consumers may decide to

select one choice from a variety of exclusive alternatives. Hence, consumers only

choose one alternative from mutually exclusive alternatives in a choice set. However,

in the highly developed and competitive market, consumers can select multiple

alternatives simultaneously (Bhat, 2005). Traditional discrete choice models can helps

researchers to find out the most and least preferred options, but the problem is that it

cannot measure multiple alternatives simultaneously.

2.2. Multiple-discreteness model

The multiple-discrete model allows consumers to make a selection among multiple alternatives such as units, prices, types and brands simultaneously (Hanemann, 1984;

McFadden, 1989; Hendel, 1999). Compared with classical discrete choice models, using a multiple-discrete model can measures consumers’ decision-making choosing behavior for multiple-units as well as multiple-brands or types of a product category.

Hendel (1999) suggested that the advantage of using a multiple-discrete model is that it helps us to investigate the relationship between products’ characteristics or features and a large number of variety types of alternatives in a choice task. In addition, it also allows estimating links between different prices and purchased quantities. Hendel (1999) stated that the multiple-discrete model can be applied in widespread area in the market. It can be used to capture the features and estimate demands for any products, such as in household appliances, consumption goods or airline flights and so on.

Dubé (2004) posited that observation of consumers’ multiple decisions is crucial for studying consumers’ decision making of choices and preferences in daily purchasing activities. Consumers are looking for a variety of products within multiple categories and switching among many alternatives all the time. In addition, consumers’ buying decisions might be based on their experiences of previous variety seeking. If consumers do not know which one to buy, they may just select one from the alternatives to try and give the feedback as whether they made a good choice. That is why consumers are always making multiple decisions during shopping. According to that, Dub é (2004) suggested that a multiple discreteness model can observe consumers’ purchasing of multiple units from a set of alternatives, not only selecting the unit of either 0 or 1. Therefore, compared to the single-units model, the multiple-discreteness model can observe and investigate consumers’ buying activity by allowing them to select different kinds of product alternatives and multiple units for each product.

However, in existing literature, researchers used fixed budgets in their studies (e.g.

Bhat, 2005, 2008; Wang et al., 2015). Consumers cannot spend their budget in a flexible manner. The limitation of fixing the budget is that the estimated model cannot be applied for investigation and observation of purchasing activity in the same consumer when he or she uses different budgets. In addition, fixing the budget does not facilitate identification of all the parameters from the MDCEV model (Dehmamy，

2015).

2.3. MDCEV models

According to Bhat et al. (2005), the MDCEV model is a theory based utility function

and developed based on the Kuhn-Tucker model (Kim et al., 2002; Bhat, 2005;

Sobhani, 2013; Wang et al., 2015). It considers how a consumer receives utility from buying a product in quantities from 𝑞

, … , 𝑞

. The utility function is denoted through the expression u( 𝑞

, … , 𝑞

) =

𝑢( 𝑞

), and u(𝑞

)=  𝑒𝑥𝑝(𝑥

𝛽)(𝑞

+ 𝛾

)

. This formulation reflects the intuition that the marginal utility will be less compared to a previous quantity when a consumer receives one additional unit in quantity. In existing literature (e.g. Kim, 2002; Bhat, 2005; Bhat, 2008; Wang et al., 2015), researchers always use a fixed budget to estimate MDCEV models. For instance, Bhat (2005, 2008) mentioned that parameter 𝛾

determines if a corner solution exists (e.g.

a product j received “0” quantity), but they still set a fixed budget and treat 𝛾

as a binary value of either “1” or “0” (zero means outside option) in order to identify the model. So in a fixed budget situation, the MDCEV model is usually denoted by u(𝑞

)=  𝑒𝑥𝑝(𝑥

𝛽)(𝑞

+ 1)

. In this case, only parameter 𝛽 and 𝑎

can be estimated.

Parameter 𝛽 accounts for adding each additional number of quantity, which increases 𝛽 to the utility in product j, and parameter 𝑎

accounts for the law of diminishing marginal utility. Similar with Bhat (2005, 2008), Kim et al. (2002) also use a fixed budget and set 𝛾

as a fixed value to identify 𝛽

and 𝑎

.

In previous literature, researchers did not estimate parameters 𝛾

because of the fixed budget setting. Otherwise the model will be over parameterized and difficult to use for identification (Dehmamy，2015). However, Dehmamy (2015) suggested that the parameter 𝛾

can be estimated through varying the budget. According to Dehmamy (2015), parameter 𝛾

cannot only be shown as value “1” or “0”. It can be estimated as different values in multiple alternatives. Therefore, in the unfixed budget case, the MDCEV model formulation is written as u(𝑞

)=  𝑒𝑥𝑝(𝑥

𝛽)(𝑞

+ 𝛾

)

and parameter 𝛾

is not given as value “1” anymore. The advantage of varying the budget is that all three parameters 𝛽, 𝑎

and 𝛾

can be estimated. The shadow quantity can also be discovered and the identification of the model is strengthened (Dehmamy，2015).

Shadow quantity can also be called corner solution. It is defined as the quantity that consumers have already owned (Dehmamy，2015). In several articles, authors defined the corner solution as just one of two offerings is chosen. When parameter 𝛾 is larger than zero, this indicates that a product receives none quantity (Satomura et al., 2011).

According to Dehmamy (2015), parameter 𝛾 accounts for shadow quantity and it can

be discovered in an unfixed budget situation. Shadow quantity explains the reason

why an individual selects lower amounts of a product. So if an individual chooses a

lower quantity, this may be indicative of this individual already having some stock of

the product. The estimation of shadow quantity is beneficial for understanding

consumers’ purchasing behavior correctly and identifies the reason why a product has

a lower chosen quantity.

2.4. Hierarchical Bayesian Model

According to Rossi et al. (2005), the hierarchical Bayes model helps in estimating the parameters like beta, alpha and gamma from the MDCEV model. It is interesting to learn that how these parameters are embodied in the Bayesian estimation. Dehmamy and Otter (2015) suggested that a standard multivariate normal hierarchical prior could be applied for beta and gamma parameters.

Therefore, in order to get the population mean and population variance, the formulation (𝛽, 𝛾)~  𝑀𝑉𝑁   [𝛽, 𝛾], 𝑉

0

0 𝑉

can be used for estimating 𝛽 and 𝛾, where parameter 𝛾 is estimated by the formulation of 𝛾 = 𝑒𝑥𝑝(𝛾). MVN is described as a multi-variate normal distribution. Therefore, the population mean of beta can be measured by 𝛽 and population variance can be measured by 𝑉

(which is also defined as square root diagonal 𝑉

). And by this analogy, parameter 𝛾 are measured by 𝑒𝑥𝑝(𝛾)  and  𝑉

for the population mean and square root diagonal respectively.

On the other hand, Rossi et al. (2005) suggested that the natural conjugate prior for a

multinomial distribution is defined as a Dirichlet distribution. And Dehmamy and

Otter (2015) also explained that usually an exercised prior in logit-transformation of

normally distributed variables is sensitive to the choice of the subjective prior

parameter. Thus, it is suggested to use the non-standard hierarchical prior for the

alpha parameter. Unite interval will be divided with equal range of intervals. A

subjective Dirichlet prior for the hierarchical prior probabilities will be specified as a

parameter from a specific interval. Therefore, the alpha parameter can be estimated by

the model 𝛼  ~  𝐷𝑖𝑟𝑖𝑐ℎ𝑙𝑒𝑡  (𝑃

, 𝑃

, … , 𝑃

). As Dehmamy and Otter (2015) described,

the subjective Dirichlet prior is updated for each product type respectively. The

specific hierarchical prior probabilities of a product among those selected intervals

will be predicted as posteriori.

3. Methodology

This part contains the research design, sample and data description, data analysis and results. The research design part discusses the method and progress of the experiment. A brief description of the sample will be mentioned as well. Data analysis will be indicated here.

3.1. Research Design

In order to investigate the model of MDCEV, an experiment is designed and built up on a selection of Haribo candies in a supermarket with different consumption budgets.

There are six tastes, which are lemon, apple, pineapple, strawberry, raspberry and orange. Respondents are asked to fill in a questionnaire that contains 12 questions (see questionnaire example in appendix 1). The questionnaire scale is called constant sum, which allows respondents to fill in the numbers of quantity in numeric style.

Entered quantities will be summed and displayed to respondents. Three tastes are mentioned in each choice task randomly with a differing consumption budget. The price is one euro per each 100g candies. There are four options in each question, which are three random tastes and one outside option (no-choice). Respondents are required to answer their choices among three tastes by filling in the amount they want to buy based on the different budgets in each question. In this way, it is convenient to observe the change of respondents’ choices when they own a different budget. After that, respondents’ answers will be collected for further analysis.

Sample Size

The questionnaire is designed by Sawtooth software and sent to 82 respondents through a link to an online survey. The final sample size after data cleaning is 68, because the answers from 14 other respondents contain mistakes and missing values.

For example, they did not fill in the numbers correctly and just filled in “0” for all.

There are 32 males and 36 females. Respondents are students from the University of Groningen, aged around 23 to 28 years old.

Manipulation

Haribo candy is a very popular brand of sweets on the market. Whatever male or female, young or old, they all have experiences of buying Haribo candies in the supermarket by weighing up the quantities freely according to their favorite tastes.

Therefore, Haribo candy in this experiment case is suitable to use as a research

product for respondents. On the other hand, when weighing a batch of Haribo candies,

respondents need to balance their budget and the amount they want to buy. Also, they

need to consider what their preferences are among variety tastes. Thus, using Haribo candy is valid to conduct research on multiple-discreteness since respondents need to make choices among multiple tastes and quantities based on the different available budgets.

3.2. Data Analysis

The collected data contains variety dimensions as it is based on multiple tastes, quantities and different budgets (see table 1 listed below).

Attributes Levels

Tastes Apple Lemon Orange Pineapple Raspberry Strawberry Budget 1 euro 2 euro 3 euro 4 euro 5 euro 6 euro Quantities 100g 200g 300g 400g 500g 600g

Table 1: three attributes and six levels of each from collected data

In order to analyze the data and present parameters for alpha, beta and gamma under the unfixed budget, the full parameters of the MDCEV model are distributed as:

u =  

𝑢

𝑢

= 𝑒𝑥𝑝(𝑥

𝛽)(𝛾

+ 𝑞

)

Based on the model, beta (𝜷) is a parameter showing the taste that people prefer. It measures the preference of the candy flavor, and higher beta values present the most preferred taste. Gamma (𝜸) is the parameter that indicates whether people own the shadow quantity for a certain taste of candy. Which means gamma measures high-low quantities a respondent selected for a flavor at the beginning. The few selected amounts mean high shadow quantity could exist. The magnitude of q shows the quantity of candy that consumers selected in each choice task. Another parameter is alpha (a), the marginal utility effect is measured by this parameter. It is assumed that an increase of buying one unit of candy will lead to a decrease of its marginal utility to the consumer. As is known, the baseline utility exp(𝑥

𝛽) is presented as linear, the incremental one unit of product will lead to a higher utility for the consumer.

The estimation of the model was done using the procedure and the code provided by

Dehmamy (2015). By following the coding process in Dehmamy (2015), the

formulation of the MDCEV model is coded as well to estimate parameters 𝛼, 𝛽 and

𝛾 . Table 2 provides the results about parameter 𝛽 for each candy flavor, 𝛽

(population mean) and 𝑉

(population variance). As the table showed in below (see table 2), consumers have a higher preference to buy strawberry, as it has the highest population mean ( 𝛽)  at 4.14 and population  variance (𝑉

) at 1.35. This result indicates that strawberry flavor is the most preferred.

𝜷 Apple Lemon Orange Pineapple Raspberry Strawberry

𝜷 ^{2.77 1.86 1.47 2.39 0.74 4.14}

Square root

diagonal ( 𝑽

) ^{1.26 1.09 0.81 1.21 0.90 1.35}

Table 2: the population mean and square root diagonal of 𝛽

The table 3 shows the overview values of about parameter gamma. This investigates that whether consumers have shadow quantities for a taste of candy. It also includes one outside option (no-choice). According to the formulation 𝛾 = 𝑒𝑥𝑝(𝛾), table 3 emerges the results that taste of apple has the highest value at 4.42 of and 0.76 for 𝑒𝑥𝑝(𝛾) and square root diagonal 𝑉

. For the apple flavor in this case, even the outside option (no-choice) is always showed in each time, but apple flavor still receives positive quantities with just a little amounts. The previous stocks of this flavor are treated as shadow quantity for parameter gamma and explain the reasons why respondents chose smaller amounts.

𝜸

(no-choice) Apple Lemon Orange Pineapple Raspberry Strawberry

𝐞𝐱𝐩(𝛄) .0049

1.69 0.90 0.87 1.10 0.99

Square root

diagonal

( 𝑽

) ^2.09

^{0.40 0.15 0.13 0.20 0.15}

Table 3: the 𝑒𝑥𝑝(𝛾) and square root diagonal 𝑉

of parameter 𝛾

The table 4 provides the general information about parameter alpha’s standard

deviation and population mean. Alpha is the parameter to estimate the marginal utility

of taste. It is supposed to have lower marginal utility of a taste when consumers buy a

higher quantity of it. Which means that the incremental of buying one unit more, the

marginal utility of this taste to him or her will be lower than before. As the table 4

indicates, the taste of raspberry has the highest value at 0.65 mean and its standard

deviation is at .011. Therefore, it accounts for diminishing marginal utility of every

additional quantity of raspberry. The marginal utility of raspberry taste is easier diminished than other flavors.

𝜶

(no-choice) Apple Lemon Orange Pineapple Raspberry Strawberry

Mean (𝛂) 0.18 0.35 0.46 0.56 0.41 0.65 0.21

Std (𝜶) .0071 .0066 .0025 .013 .0057 .011 .0034

Table 4: the mean and standard deviation of parameter

On the other hand, the MCMC (Markov Chain Monte Carlo) traces are for the draws made for the hyperparameters of parameter 𝛽 (𝛽 and 𝑉

), parameter 𝛾 (𝑒𝑥𝑝(𝛾) and 𝑉

) and parameter 𝛼 (mean and standard deviation) from which the population mean and variance are drawn from the preferences of each individual comes from multivariate normal (MVN distribution). So the traces for 𝛽 and 𝑉

can have a different mean and standard deviation than those of the population distribution showed in table 2, 3 and 4. Therefore, three graphs of MCMC draws are displayed for parameter 𝛼, 𝛽 and 𝛾 to give the visual comparison about the population mean (see appendix 2.1 to 2.3). Since result of MCMC draws show the similar results as table 2, 3 and 4 presented, this indicates that the estimated results are reliable and correctly.

In appendix 3, the table shows an overview of beta values for each respondent’s preference among 6 Haribo candy tastes. In order to learn that in a more perceptual intuition way, six histogram charts (see appendix 4, chart 4.1 to 4.6) supply the overview description of every Haribo candy taste based on the parameter of beta value. The number of histogram 1 to 6 represents the taste of Apple, Lemon, Orange, Pineapple, Raspberry and Strawberry respectively. To give an example, by looking at the histogram about the beta value of strawberry flavor (see appendix 4, chart 4.6), there is increasing tendency in the positive side and thus more respondents prefer to choose strawberry flavor than other tastes.

The selected quantities of each candy flavor by respondents are also distributed from 5% to 95% respondents (see table 5, 6 and 7, start in next page 14). Compared to the results showed in table 2, 3 and 5, the results from table 5, 6 and 7 are the population distribution based on the actual choices and quantity selection from 68 respondents.

While the values from table 2, 3 and 4 are estimated results. Therefore, studying at the

values showed in table 5, 6 and 7 can help to understand the actual preference and

choice behavior of respondents.

Parameter 𝛽 accounts for the preference of candy flavor. The higher value of 𝛽 stands for a flavor receives higher quantities and is the most preferred. According to the results in table 5 (see page 14), the values of orange and pineapple flavors are displayed quite average and they are not much preferred by consumers. In addition, consumers’ preferences of lemon and raspberry are not high. There are 5% of respondents choosing the taste of lemon below the number of -.052 and 5% of respondents having the raspberry flavor below the number of -0.64. However, there is obviously incremental tendency of lemon’s beta value towards the positive side. From 25% to 95% of respondents, their preferences are increasing from below the number of 1.38 to 3.40.The taste of apple is more preferred than orange and pineapple flavors.

As the data showed, 55% of respondents prefer to have the taste of apple below the number of 3.07. When looking at 95% of respondents, they prefer to have apple flavor below the number at 4.31. Compare to the other four tastes, raspberry and strawberry tastes are more popular. Strawberry flavor has the highest beta values and they are all emerged as positive. There are 95% of respondents choose the strawberry flavor below the number of 6.09. The higher value of beta means consumers do like the taste more. Based on that, consumers have a higher preference to buy strawberry flavor.

Raspberry taste has the lowest beta value and is therefore the least preferred taste for consumers.

Table 6 (see page 15) provides the information about parameter gamma for each taste.

The gamma value of apple is the highest among six tastes, means that consumers do not select this taste a lot during most of time. 5% of respondents have apple flavor below the number around 1.45 and 95% of respondents choose apple flavor below the number around 14.86. It can be seen that apple taste is not very high in demand.

However, even consumers that do not choose apple a lot still select lower amounts of this flavor from the whole category and this is always a positive number. One explanation is that consumers have some stocks at home, so that they do not need to choose higher quantities, but just few apple flavors. This is assumed as the shadow quantity that is hidden in the gamma parameter. On the contrary, consumers select the taste of strawberry and pineapple very frequently, since they have lower gamma values. Which means consumers do like to buy these two flavors and they do not have a lot of stocks at home. Since 95% of respondents are willing to have strawberry and pineapple flavor below the number around 1.23 and 1.07 respectively.

In the table 7 (see page 16), the information of parameter alpha for each taste is

presented. Alpha is defined as diminishing of marginal utility when adding every

additional quantity of a taste. From the table 7, it is clear to find out that the taste of raspberry has the highest values. There are 95% of the respondents have raspberry flavor below the number of 0.85 and 5% of people choose raspberry below the number of around 0.49. Therefore, the marginal utility will diminished if raspberry taste receives every additional quantity. If raspberry taste appears in a choice set together with the other two random tastes, consumers might want to buy raspberry taste only instead of choosing the other two. The taste of strawberry has the lowest value in the alpha parameter and the tendency of it increases very slowly. To look at the numbers, 5% of respondents are willing to have strawberry taste below the number around 0.11, 45% respondents do like to choose it below figure 0.21 and 95%

respondents select strawberry below the number around 0.28. Therefore, when people

see strawberry, they will tend to choose some of it but they will not restrict their

choice to strawberry alone.

5% 15% 25% 35% 45% 55% 65% 75% 85% 95%

Apple

0. 7 5 1. 3 3 2. 1 4 2. 5 9 2. 8 2 3. 0 7 3. 3 1 3. 7 1 3. 9 7 4. 3 1

Lemon

- .0 5 2 0. 6 9 1. 38 1. 6 9 1. 84 2. 09 2. 24 2. 5 5 2. 98 3. 40

Orange

0. 1 3 0. 6 0 0. 9 5 1. 1 4 1. 5 1 1. 6 3 1. 8 6 2. 0 1 2. 2 7 2. 7 7

Pineapple

.0 5 4 1. 1 1 1. 4 8 1. 9 7 2. 4 2 2. 7 9 3. 0 5 3. 3 4 3. 6 2 4. 0 9

Raspberry

- 0. 6 4 - .06 9 0. 2 8 0. 4 6 0. 6 2 0. 7 8 0. 9 9 1. 2 6 1. 7 7 2. 10

Strawberry

1. 8 7 2. 8 8 3. 2 9 3. 5 8 4. 14 4. 5 4 4. 76 5. 13 5. 5 8 6. 09 T ab le 5: T he qua nt iti es of popul ati on di str ibut ions for pa ra m ete r be ta of e ac h ca ndy fla vor a m ong re sponde nt s f rom 5% to 95%

5% 15% 25% 35% 45% 55% 65% 75% 85% 95%

Outside .00057 .00089 .0013 .00207 .0025 .0030 .0046 .0073 .049 0.86

Apple 1.45 2.0012 2.84 3.46 4.01 4.70 5.11 7.04 11.09 14.86

Lemon 0.84 1.18 1.36 1.45 1.52 1.62 1.96 2.21 2.59 3.43

Orange 0.72 0.77 0.82 0.88 0.90 0.92 0.94 0.97 1.04 1.13

Pineapple 0.70 0.76 0.79 0.83 0.86 0.87 0.91 0.96 1.01 1.07

Raspberry 0.80 0.90 0.95 1.01 1.08 1.14 1.19 1.24 1.32 1.51

Strawberry 0.79 0.86 0.90 0.95 0.98 1.02 1.04 1.10 1.16 1.23

T ab le 6: T he qua nt iti es of popul ati on di str ibut ions for pa ra m ete r ga m m a of e ac h ca ndy fla vor a m ong re sponde nt s f rom the vi ew of 5% to 95%

5% 15% 25% 35% 45% 55% 65% 75% 85% 95%

Outside .0035 .0045 .0063 .0079 .012 .0201 .034 0.28 0.69 0.76

Apple 0.24 0.30 0.31 0.32 0.33 0.34 0.35 0.37 0.38 0.43

Lemon 0.41 0.42 0.43 0.44 0.45 0.453 0.46 0.468 0.47 0.51

Orange 0.40 0.50 0.52 0.54 0.55 0.57 0.58 0.59 0.63 0.66

Pineapple 0.24 0.33 0.35 0.36 0.37 0.39 0.40 0.43 0.47 0.60

Raspberry 0.49 0.58 0.60 0.61 0.63 0.65 0.67 0.69 0.74 0.85

Strawberry 0.11 0.14 0.17 0.19 0.21 0.22 0.23 0.24 0.26 0.28

T ab le 7 : T he qua nt iti es of popul ati on di str ibut ions for pa ra m ete r a lpha of e ac h ca ndy fla vor a m ong re sponde nt s f rom the vi ew of 5% to 95%

3.3. Model Prediction

This part is to examine that whether the results of the MDCEV model are close to the true value. It is based on 68 respondents’ selection of the quantity to each taste in 12 questions. For example, table 8 shows the result of estimated values for one respondent’s (respondent ID 1) selection of quantity for each taste in each choice task and table 9 shows the true value.

Choice

set Outside Apple Lemon Orange Pineapple Raspberry Strawberry

1 2.53 28.87 43.12 0.00 25.49 0.00 0.00

2 1.71 0.00 0.00 62.21 0.00 54.39 81.69

3 3.38 63.25 0.00 115.70 0.00 117.67 0.00

4 1.80 0.00 53.26 73.90 0.00 71.05 0.00

5 7.99 0.00 84.31 0.00 0.00 162.18 145.53

6 4.32 47.03 0.00 124.18 0.00 0.00 124.47

7 14.26 0.00 132.49 0.00 93.76 259.49 0.00

8 63.38 147.77 0.00 0.00 106.96 0.00 281.89

9 2.95 0.00 0.00 94.76 0.00 90.40 111.89

10 8.61 108.10 147.90 235.41 0.00 0.00 0.00

11 9.34 58.16 0.00 0.00 0.00 180.52 151.99

12 8.91 93.76 0.00 215.24 82.10 0.00 0.00

Table 8: the predicted quantity of choosing each taste among 12 times for one respondent (ID No.1)

Choice set Outside Apple Lemon Orange Pineapple Raspberry Strawberry

1 0 60 20 0 20 0 0

2 40 0 0 30 0 80 50

3 50 100 0 50 0 100 0

4 0 0 50 30 0 120 0

5 0 0 150 0 0 50 200

6 40 100 0 60 0 0 100

7 0 0 300 0 150 50 0

8 0 100 0 0 150 0 250

9 0 0 0 100 0 50 150

10 0 200 100 200 0 0 0

11 0 50 0 0 0 100 150

12 0 150 0 100 150 0 0

Table 9: the true value of quantity that a respondent (ID No.1) selected for each taste among 12 times

To give more perceptual intuition of whether the model estimates in good fit, six plots of scatter diagrams are listed in below (see graph 1) to give the expression between true values and estimated values based on six respondents (ID No. 1, 20, 30, 35, 40, 64). From scatter diagrams, it becomes clear that splashes are quite close to the straight line and emerge as linear alignment. The results manifest that the estimated values are close to the true values. Therefore, the model shows good fit in estimating values.

Graph 1: plots of six scatter diagrams to give the comparing for estimated quantities with the true value for respondents (ID 1, 20, 30, 35, 40, 64), by reading in row from left to right side.

3.4. The insights between each taste and consumer

Following the positive result of model estimation, it is also interesting to make a comparison for each person across six candy flavors. This will mainly look at the relationship between the utility of each taste (including one outside option) and the quantity of a consumer selected. Here four persons (ID 1, 20, 40, 64) are selected from all 68 respondents to make the comparison. The four curve graphs (see appendix 5) give the expression of the utility tendency under the change of budget and quantity.

It is obvious that the linear function does not exist between the utility of each flavor to consumers and incremental purchasing. The purchasing of one more candy flavor leads to a decrease in its utility. Comparing among these four persons, the utility results are still not similar between each other. For example, when taking a look at the taste of strawberry for each person (compare appendix 5.1 to 5.4), the curve in respondent No.20 (see appendix 5.2) increases more dramatically than others at the initial stage, and then the increase slows down and gently smoothens.

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In addition, it is also interesting to see the part of parameter gamma (see all four graphs appendix 5.1 to 5.4). This is the part in the negative district and it explains why consumers do not choose this. One possibility is that consumers have a so-called pre-stock of a candy flavor, so that they do not really need it at the beginning.

However, each person’s gamma parameter for each taste is different. For example, the curve of the outside option contains a relatively large area of gamma parameter.

Generally, the hidden part of gamma can only be explained as assumption, since it is still hard to give the precise explanation for the hidden part.

In the other way around, the relationship between person and taste can also be observed by comparing each taste across persons. 6 respondents (ID 1, 20, 30, 35, 40, 64) are selected to observe the changes between utility and quantity according to each taste and includes one outside. Seven curve graphs are displayed (see appendix 6) to show the results. From the graphs of strawberry flavor across each person (see appendix 6.6), it is interesting to see that all six selected respondents are choosing strawberry quite frequently at the beginning. There is an obviously dramatic increase in the function between utility and quantity in strawberry graph. Then its curve gradually tends to flatten afterwards. By looking at the others, the utility of raspberry to respondent No.30 nearly emerges as linear (see appendix 6.5). It is possible that the utility of raspberry flavor to this person does still keep adding, and that for this respondent it is hard to reach saturation of satisfy the purchase of raspberry candy.

Relatively, the utility tendency of orange and raspberry flavor for each person is quite

stable and without rapid increase (see appendix 6.3 and 6.5).

4. Discussions and Conclusions

In this section, the present research findings will be discussed. According to the current findings and the other articles’ findings that were discussed in the literature review, the whole study and MDCEV models will be discussed. After that, some managerial application will be mentioned as well

In this paper, MDCEV models are used for the estimation under the product choice condition of multiple types, quantities and with unfixed budgets. The investigation is based on six Haribo candies with six different flavors, variety quantities and different prices. According to the results of data analysis, this study proves that parameter 𝛾 can be estimated if the budget varies. Such as in each candy flavor, it is able to show parameter 𝛾 in different values. It also allows making jointly identifying all parameters of the MDCEV model when the budget is unfixed. In addition, the predicted value is close to the true value. Therefore, the significant positive results suggest that MDCEV models can be applied in the study of products with multiple dimensions and to give a prediction of consumer preference.

The measurement in a classical model such as a choice-based conjoint is limited. It can only use fixed and unitary product attributes to estimate the purchase probability.

In addition, when adding more dimensions such as variety quantity or budget, the model does not work out. Moreover, the traditional discrete choice model allows consumers to only choose one from multiple alternatives, but it still cannot work with more dimensions of products selection. MDCEV models solve the problem of choice singleness and make the models more flexible to do measurement. It allows consumers to choose a product with more possibilities. However, in existing literatures, researchers fixed the budget and set it to a given value of parameter 𝛾 (e.g. “1” or “0”). Since in most studies, the consumers’ budget is set as fixed and there is no additional assumption. For instance, in most purchasing activities, consumers are provided with a fixed budget and the attribute such as price is the only variation. However, in real life, consumers’ budget cannot always be fixed or the same. When it changes, it is difficult to calibrate the model and identify the function of all parameters. This causes the problem that full parameters of a MDCEV model cannot be jointly identified and it is not possible to estimate the shadow quantity.

In order to evaluate this problem and contribute to the existing literature, this study

adds to the idea that the full three parameters (i.e. alpha, beta and gamma) of the

MDCEV model can be investigated, through varying the budget only. By doing it in

this way, it adds a third dimension that respondents can have multiple budgets to

purchase candies with the quantity they prefer. Thus, based on these three dimensions, the changes of parameter alpha, beta and gamma are observed. Based on the data analysis results, parameter 𝛾 is estimated. It has different values, not only a fixed value such as “1” or “0”. The values of parameter 𝛾 reflect the existence of shadow quantity. This is the evidence that shadow quantity can be estimated when the budget is unfixed. It also shows MDCEV models are able to estimate and identify the parameters under the condition of budget variation.

This study is also able to show the goodness of model fit since the estimated results of full parameters are close to the true values. In addition, this study includes more dimensions like budget variations which improves the observation of model and consumers’ buying activities. Also, it helps to define the parameters in the model more precisely and gives interpretation to the relationship among those three parameters alpha, beta and gamma. Meanwhile, the law of diminishing marginal utility can also be embodied in this case.

This study suggests that MDCEV models with budget variation can give the expression to each respondent’s utility or preference for each kind of product. The models can also portray the changes of parameters and show the connections among each variable. For instance, the parameter beta from MDCEV models will show which product or taste is more preferred by people from all kinds of multiple options.

It is convenient and quick to highlight the key product or favorite preference that managers should focus on. Under the unfixed budget situation, MDCEV models are also able to find out the shadow quantity. This is measured by the parameter gamma in the models. Some respondents only choose a low quantity for a candy flavor at the beginning, but the reasons why they do not choose a larger amount are not explained.

Parameter 𝛾 could not be estimated in previous literature, as the budget was fixed.

This paper varies the budget and it proves the assumption that consumers might have some stocks of product at home or may have been fed up with this product. By studying the gamma parameter, this helps in finding out which taste has potential shadow quantity and has the tendency to reach saturation. On the other hand, MDCEV models can supply a flexible comparison between each taste and each respondent. It increases the flexibility of comparison for each parameter and also increases the comparability for each attribute.

In managerial implications, this study shows that it is feasible to use these MDCEV

models to estimate a full set of parameters when the budget is unfixed. Since

consumers are facing a variety of types of products and multiple options in the choice

set, and use their budget in a more flexible way for each payment in real life. Hence,

researchers can do the multiple discrete continuous data analysis together with multiple dimensions based on this MDCEV model and provide a prediction for product preference and utility. Since parameter beta can tell managers which product is more preferred, managers can enhance the production plan and build up a marketing strategy to increase sales. According to the gamma parameter, managers can figure out for which product a so-called shadow quantity exists and is not selected by consumers quite often. Therefore, management teams can coordinate production plans or rearrange the product assortment to display a lower quantity of that product.

On the other hand, the alpha parameter can tell managers the diminishing of marginal utility of product.

In conclusion, this study’s contribution is that it is able to jointly identify all three parameters 𝛼, 𝛽 and 𝛾 in an MDCEV model when the budget is varied.

Furthermore, the shadow quantity can be estimated by the MDCEV model under the

unfixed budget case. Dehmamy (2015) proposed the problem of weakness of

parameters identification in the MDCEV model and proposed that a solution is to vary

the budget when applying an MDCEV model. However, this assumption lacked the

support of empirical data to prove it. This paper contributes to the gap of it. In this

study, a survey was built up to collect a data set. The results provide the evidence that

the hypothesis of “if the budget is set as unfixed, MDCEV models can be used to find

out and estimate the shadow quantities” is true. In addition, it can also be concluded

that MDCEV models can be applied to identify all three parameters of beta, gamma

and alpha in unfixed budget case. And managers can build up the marketing strategy

and sales plan according to the result of parameters prediction.

5. Limitations

This section contains the discussion about the limitations in present research. There are still several limitations that need to be improved in the future research. Relative and potential limitations in this paper will be described with reasoning

MDCEV models are investigated with full parameters, but using shadow quantity in the gamma parameter explain why consumers choose to buy less of one flavor of candy is difficult. The reason can be only predicted. Models can only reveal which product is not selected a lot. For future studies, it is suggested to improve the model to estimate and explain all parameters more precisely. Thus, it could be much better for managerial implications to learn the reasons why consumers are purchasing less of certain flavors.

In this research, the sample frame might limit the generalizability of the current findings, because the target group contains university students only and the sample size is limited to 68 recipients only. Respondents are from a young age group. They may have quite a different way of thinking and attitude from middle aged people toward buying sweets. People from different nationalities might have different flavor preferences as well. Some people from an older generation might pay more attention to the health-care aspect, so they could choose lower quantities of sweets. It might also influence the choice and taste preference. Therefore, it is also necessary to conduct the research across different groups of people, such as older generations, different occupations or nationalities. Besides, a larger sample size will make the research results more reliable, because a larger sample size can expand the range of data and be more representative of the whole population. In the current research, gender is not equally distributed. There are more female than male respondents, so it may not reflect the result in average. Future research should try to reach an equal distribution of gender.

On the other hand, this research applies MDCEV models for the estimation on Haribo candy product and it has not been used for investigation on other types of products.

MDCEV models currently are only suitable for the study of product categories like

Haribo candy that are sold in bulk and allow consumers to choose with multiple

dimensions. So product categories that allow consumers to freely weigh the quantity,

choose the taste they prefer and arrange the different budget for buying. Perhaps the

result cannot really reflect the overall view about choice behavior in all products. For

example, if the study is about household appliance, MDCEV models maybe not very

suitable in that case. In future research, it is recommended to develop the model in

wider extension and make it able to do the investigation on more products.

On the other hand, conjoint analysis contains the issue of incentive incompatibility.

Which means respondents might have selected a certain product in such large

amounts, but perhaps they maybe will not buy it in actual life. Sometimes,

respondents just deal with or evade the questionnaire task, so they could fill in the

amount without carefully thinking. In this study, there is no incentive mechanism to

avoid the problem of conjoint analysis incompatibility. Therefore, in future research,

it is strongly recommended to add the part of incentive alignment (IA) to motivate

respondents to provide true answers. The idea is to reward a respondent with the

alternative he or she has selected in one randomly selected choice set. Therefore,

when respondents know that they could be rewarded with the product they have

chosen, they will pay more attention to answering the question and provide genuine

choices. And incentive alignment can have a large effect on the predictive validity of

the research.

6. Appendices

Appendix 1. Examples of questionnaire

1. Below  are  3  tastes  of  Haribo  candies  and  the   price  of  each  100  grams  is  1   euro.   Assume   you   can   have   1   euro   in   your   bag.   Please   allocate   100   grams   among   the   3   tastes   such   that   the   allocation   represents   the   favorite   of   each   taste  to  you.  When  you  finished,  please  double  check  to  make  sure  that  your   total  adds  to  100  grams.    

Lemon                         Apple                   Pineapple                 Non                   Total    

(         )                       (           )                     (             )                 (       )                 (       )    

2. Below  are  3  tastes  of  Haribo  candies  and  the  price  of  each  100  grams  is  1   euro.   Assume   you   can   have   2   euro   in   your   bag.   Please   allocate   200   grams   among   the   3   tastes   such   that   the   allocation   represents   the   favorite   of   each   taste  to  you.  When  you  finished,  please  double  check  to  make  sure  that  your   total  adds  to  200  grams.  

      Strawberry               Raspberry                   Orange                 Non                   Total    

      (               )                   (               )                 (             )             (         )             (           )

Appendix 2.1.

MCMC draw of the 𝛽(𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛_𝑚𝑒𝑎𝑛) for each candy taste kept in every 20

draw from 20000 draws. X axis stands for draws and Y axis stands for hyper-parameter for the population mean of 𝛽 per taste

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Beta

be tM[ , 1 :6 ]

apple lemon orange

pineapple

raspberry

strawberry

Appendix 2.2.

MCMC draw of the 𝑥𝑝(𝛾) for each candy taste kept in every 20

draw from 20000 draws. X axis stands for draws and Y axis stands for hyper-parameter for the population mean of 𝛾 per taste

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Gamma

exp (b et M[ , 7 :1 3] )

no choice apple lemon orange

pineapple raspberry strawberry

Appendix 2.3.

MCMC draw of the 𝛼 population mean for each candy taste kept in every 20

draw from 20000 draws. X axis stands for draws and Y axis stands for the population mean of alpha per taste

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Alpha

t(a lp hM)

no choice apple lemon orange

pineapple

raspberry

strawberry

Appendix 3:

Apple Lemon Orange Pineapple Raspberry Strawberry

[1] 1.14 1.77 0.45 -1.29 -0.41 1.50

[2] -1.10 -1.57 -3.28 -1.27 -1.91 1.23

[3] 0.69 0.90 0.49 1.39 -0.82 -0.46

[4] -0.46 0.23 -0.98 -1.78 -0.06 0.27

[5] -0.12 -0.80 -1.12 1.09 0.34 -3.23

[6] -0.89 0.63 0.27 0.65 -1.54 0.02

[7] 2.59 -0.19 -0.28 1.13 -1.25 0.66

[8] -0.58 0.11 0.73 -1.11 -0.02 -0.10

[9] 0.84 1.39 -0.03 0.21 -0.28 0.57

[10] 0.42 0.11 -0.53 0.40 0.07 1.13

[11] -1.32 -1.09 -0.92 0.42 0.29 1.62

[12] -1.22 -0.30 -0.01 -0.36 0.21 1.37

[13] 2.15 -0.46 -1.00 -1.78 1.69 -0.90

[14] -1.44 -0.03 0.86 0.15 -1.29 -0.61

[15] -0.54 -0.10 -0.31 0.00 0.28 -2.23

[16] -0.53 0.49 -0.53 -0.49 0.04 0.45

[17] 0.25 0.03 0.17 1.60 1.28 -0.27

[18] -0.74 0.82 1.44 0.06 -0.13 -0.48

[19] -0.11 -0.06 0.42 1.54 -0.33 0.21

[20] 0.31 0.69 1.32 0.24 0.68 0.87

[21] -0.75 0.23 0.12 -0.29 0.11 0.16

[22] -0.53 1.43 -0.26 -0.81 0.78 1.15

[23] 0.66 -0.96 1.53 -0.30 -0.52 1.30

[24] 0.86 0.70 0.18 -0.41 0.79 -0.29

[25] -0.27 1.30 0.76 -0.52 -1.26 -0.23

[26] 0.60 1.16 -0.57 0.85 1.02 -0.50

[27] -0.43 -0.85 -0.91 -0.51 0.12 -0.68

[28] -0.98 -1.44 1.54 3.62 0.38 0.11

[29] -2.18 -1.28 1.17 0.64 0.30 0.31

[30] -0.50 -1.28 -0.34 0.53 -1.47 -0.39

[31] -0.90 0.71 -0.67 1.96 0.24 -1.63

[32] 0.32 -0.89 -1.03 -0.80 1.79 0.49

[33] 1.70 -1.22 -0.37 -0.04 -0.15 0.40

[34] 1.09 0.75 -0.75 2.34 1.19 -1.15

[35] -0.16 0.79 0.84 -0.18 0.06 -0.65

[36] 0.36 1.50 0.20 -0.34 -0.37 -0.45

[37] -1.62 1.46 -0.25 -1.20 -0.42 0.68

[38] -1.39 -0.05 -0.49 1.62 1.12 0.26

[39] -1.99 0.40 0.74 1.18 0.66 -1.37

[40] -2.18 0.95 -0.77 -0.56 -1.93 -0.24

[41] -1.82 0.31 -0.44 -1.22 -0.93 -0.61

[42] 0.72 -0.18 -1.14 0.06 1.56 1.56

[43] -1.57 -1.94 1.59 -1.31 -0.04 0.40

[44] 0.14 -0.98 0.80 0.70 0.01 0.52

[45] -0.88 0.19 0.63 -1.06 -0.50 -0.56

[46] -0.24 0.05 -0.91 0.38 1.18 0.03

[47] -0.84 0.66 0.53 1.63 1.15 0.67

[48] -0.63 -2.01 -0.63 -1.30 -0.85 -0.02

[49] 1.23 -0.38 -0.55 -0.63 -0.15 -1.11

[50] 0.39 -0.25 -0.83 -1.07 0.94 0.99

[51] 1.73 -0.93 1.17 -0.25 -2.16 0.78

[52] -1.58 1.33 -0.69 1.14 -1.16 0.63

[53] 1.20 0.45 1.11 1.01 0.48 0.99

[54] 1.69 -1.15 0.47 0.98 2.05 1.03

[55] -0.02 -1.42 0.25 -0.51 -0.58 -1.12

[56] 0.35 0.66 -0.18 -0.46 0.46 -2.45

[57] 0.07 0.20 -0.01 -1.91 0.55 0.85

[58] 1.64 -1.89 -0.25 0.98 0.97 1.25

[59] 0.99 -1.05 0.64 0.50 -2.18 0.25

[60] 0.48 -0.11 -1.66 -1.55 -1.22 1.53

[61] -1.22 0.94 -0.54 0.34 -0.53 0.22

[62] -0.02 -1.54 -1.28 1.39 0.89 -0.32

[63] -0.89 0.75 -0.57 -1.02 -0.81 -1.06

[64] -1.42 -1.07 -0.38 -0.10 0.32 1.74

[65] -1.11 -0.37 0.64 -0.36 -0.74 0.68

[66] 1.78 -0.49 0.23 1.63 -0.81 0.18

[67] -1.41 2.11 -0.56 -0.56 -0.81 0.26

[68] -1.26 -0.36 0.64 0.18 -0.35 0.74

Appendix 3: Beta values of each taste among each person from total 68 respondents

Appendix 4

chart 4.1: beta value of apple flavor chart 4.2: beta value of lemon flavor among each person among each person

chart 4.3: beta value of orange flavor chart 4.4: beta value of pineapple flavor among each person among each person

Histogram of betaN[, 1]

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chart 4.5: beta value of raspberry flavor chart 4.6: beta value of strawberry flavor among each person among each person

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Appendix 5 Appendix 5.1

Graphs set of app 5.1: relationship between utility and quantity for respondent No. 1 across each taste and includes one no-choice option

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