Optimal policy design for the sugar tax
Kelly Geyskens, Alexander Grigoriev, Niels Holtrop, Anastasia Nedelko
OOptimal policy design for the sugar tax
K. Geyskens A. GrigorievN. Holtrop
Maastricht University School of Business and EconomicsP.O.Box 616, 6200 MD Maastricht, The Netherlands { K.Geyskens;A.Grigoriev;N.Holtrop } @maastrichtuniversity.nl A. Nedelko
Higher School of EconomicsP.O.Box 33, Kirpichnaya street, Moscow, [email protected]
Abstract
Healthy nutrition promotions and regulations have long been re-garded as a tool for increasing social welfare. One of the avenues takenin the past decade is sugar consumption regulation by introducing asugar tax. Such a tax increases the price of extensive sugar contain-ment in products such as soft drinks. In this article we consider atypical problem of optimal regulatory policy design, where the taskis to determine the sugar tax rate maximizing the social welfare. Wemodel the problem as a sequential game represented by the three-level mathematical program. On the upper level, the governmentdecides upon the tax rate. On the middle level, producers decideon the product pricing. On the lower level, consumers decide upontheir preferences towards the products. While the general problem iscomputationally intractable, the problem with a few product types ispolynomially solvable, even for an arbitrary number of heterogeneousconsumers. This paper presents a simple, intuitive and easily imple-mentable framework for computing optimal sugar tax in a market witha few products types. This resembles the reality as the soft drinks, forinstance, are typically categorized in either regular or no-sugar drinks,e.g. Coca-Cola and Coca-Cola Zero. We illustrate the algorithm usingan example based on the real data and draw conclusions for a specificlocal market. a r X i v : . [ ec on . T H ] O c t eywords: Three-level mathematical program, nutrition promotion, socialwelfare optimization, governmental regulations, enumeration algorithms
Since the 2000s, interest in the proper nutrition promotion has dramaticallyincreased in many countries. According to the World Health Organization(WHO), worldwide obesity has nearly tripled since 1975. In 2016, more than1.9 billion adults were overweight, with over 650 million of these adults beingobese. Moreover, 41 million children under the age of 5 were overweight orobese in 2016 [FAO and WHO, 2017]. In addition to obesity, adults, childrenand adolescents often face risks such as depletion, stunted growth, lack ofvitamins and minerals, nutritional non-communicable diseases (heart disease,stroke, diabetes, certain cancers) that affect their health in the short and longterm periods [FAO and WHO, 2017]. Therefore, the formation of a propernutrition culture is very important, especially in youth, since all the humanhabits and life values are formed at a young age [Ferreira et al., 2007]. Duringadolescence, young people adjust their lifestyle easier than adults. Thereby,nutrition habit is one of the basic elements for their health in future.Solving obesity problems is also a challenging task for the governments,as inadequate nutrition increases health care costs, reduces productivity andslows economic growth. These consequences, in turn, are the basis for per-manent poverty and poor population health [FAO and WHO, 2017].In contrast, the main concern of the companies is the average market shareand/or profit [Smith et al., 2013]. This incentives are not always directed athealthy products, and it is always a challenge for a government to introduceand to maintain mechanisms stimulating the companies to promote propernutrition. Clearly, there might exist opposing interests, when the companiesfocus on purely financial indicators, while the government seeks to improvethe social welfare. In this interaction, the end-consumers play crucial role.On the one hand, consumers choose products guided by various market-ing stimuli such as advertising, pricing, and branding [Solomon et al., 2012].This way consumers strongly support the companies. On the other hand, thegovernmental and societal information provision programs create awarenessof proper nutrition importance, shift the demand towards healthy productsand, as a result, customer valuations for healthy products become higher thanfor unhealthy ones. This way consumers strongly support the government.Therefore, a combination of information provision and direct regulations,e.g. extra taxation for unhealthy products, is a powerful instrument of thegovernment to improve the social welfare.2 .1 Nutrition promotion instruments
In this section we describe possible nutrition promotion mechanisms for a gov-ernment and for a company/firm. Later, some of the mechanisms, namely,taxes and prices, will be explicitly introduced as variables in the utility func-tions of the companies and consumers, respectively.The government is typically the first mover which sets monetary and/ornon-monetary product/market regulations. Among monetary regulations,the most popular ones are taxes (for unhealthy products), subsidies (forhealthy products), and caps (maximal price for a product). These regula-tions directly influence the utility function of a company and rarely affect theconsumer utility functions. Furthermore, government can use non-monetaryinstruments to stimulate nutrition consumption, e.g., certification, label-ing, obligatory description of ingredients and nutritional value on the pack-ages [Stevenson and Ingwersen, 2012, Minkov et al., 2015]. More specifically,the government can impose an obligation on firms to use particular size, colorand shape for nutritious food packages or for price tags. For instance, recentresearch shows that the consumers perceive products in vivid packaging asless healthy than food in muted color packages [Mead and Richerson, 2018].
Minimal font size of ingredients’ inscription and nutritional value also canbe used by government as non-monetary regulation. This tool may attractconsumers’ attention on containment of harmful to health ingredients suchas sugar, preservatives and dyes. The last but not the least tool is point-of-sales merchandising . One study showed that joint presentation of healthyand unhealthy products forces consumers to choose the first one because suchway increases guilt and the difficulty of social justification [Okada, 2005].Consequently, merchandising can be an effective instrument which may forcepeople to buy healthy products. These non-monetary regulatory mechanismsserve for information provision and do have influence on the consumer utilityfunctions, though the effect of the information provision is sometimes is notimmediate. The company utility functions are rarely directly affected by thenon-monetary governmental regulations.In turn, a company/firm (producer or retailer) is the follower. Know-ing the governmental regulations, companies maximize the profit applyingtheir toolkit to influence the consumer behavior. It should be noted thatinterests of producers and retailers can be different, but within the frame-work of this research we do not distinguish these two players. It is widelyaccepted that the most powerful tool of a firm is the price set for a prod-uct. Next to the price, discounts is the most popular instrument. To illus-trate the possible efficiency of this tool, prior research has provided resultsof experiments where price reduction can be a reason of increased consump-3ion [Geliebter et al., 2013, Ball et al., 2015]. For example, [Ball et al., 2015]have found that a 20% discount for fruit and vegetables categories causedincreased purchasing per household of 35% for fruit and 15% for vegeta-bles. Furthermore, it is proved that even temporary discounts can stim-ulate proper nutrition consumption. In one study [Geliebter et al., 2013],there were three periods (baseline - no discount, intervention - 50% dis-count, follow-up - no discount) during which obese respondents were buyingfruits and vegetables during this periods. As a result, purchasing of fruitsand vegetables during follow-up period became higher than during baseline.This research demonstrates effectiveness of applying discounts to increaseproper nutrition consumption. At the same time, firms may use such non-monetary instruments as availability in stores , merchandising , and packagedesign [Glanz et al., 2012]. Availability in stores means that wide accessto proper nutrition food increase nutrition consumption. It happens be-cause people prefer to buy what is in every supermarket instead of tryingto find something special for everyday meals [Desai and Ratneshwar, 2003,Morales, 2005]. Merchandising was already discussed in the context of thegovernmental tools and it can also be effective on the company’s level. More-over, unusual packages can be used by marketers to increase demand forproper nutrition [Rettie and Brewer, 2000]. In this study we assume that (1) effective information provision programstake place; (2) consumers are aware of proper nutrition importance; and (3)consumers already formed their utility functions (product valuations) and inthe near future they are not going to change their preferences significantly.Notice, without these assumptions the efficiency of direct regulations is ques-tionable. We take the consumer utility functions as granted. This is also avalid assumption, given a number and variety of consumer behavior modelsavailable in the literature. In the end of the paper, we provide an insight-ful example and analysis based on a specific consumer behavior model andactual purchase data. Such data can be routinely obtained from the marketresearch agencies such as Nielsen, GfK and Kantar. We base our exampleand experiments on the data provided by the latter party.Specifically targeting the optimal regulations of the soft drinks markets,this research focuses on the most popular, powerful and long-run instrumentsof the government and of the firm, namely on the sugar tax and prices,respectively. In the past decade, many countries initiated sugar consumptionregulation by introducing a sugar tax. Such a tax increases the price ofextensive sugar containment in products such as soft drinks. In this article4e consider a typical problem of optimal regulatory policy design, wherethe task is to determine the sugar tax rate maximizing the social welfare.We model the problem as a sequential game represented by the three-levelmathematical program. On the upper level, given the utility functions of thesoft drink producer/retailer and consumers, the government decides upon thesugar tax rate with the goal to optimize the social welfare. On the middlelevel, given the sugar tax rate and the utility functions of the consumers,a company decides on the product prices maximizing the company utility.On the lower level, given the product prices, consumers decide upon theirpreferences towards either sugar containing or sugar-free products. All utilityfunctions and the social welfare are taken from the classic economic literature.This ensures the generality of the approach and applicability to not only thesoft drinks markets and sugar taxes, but rather to a broad variety of themarkets needing governmental regulations.
The sugar tax means each liter of sugary drink will have an extra tax chargeup to 50%, depending on how much sugar is in the drink . Tax rates dependon government policy in a country and can be expressed in percentage or inmonetary units. Moreover, taxation schemes vary from country to country,see Table 1. Nowadays, there are two common schemes: a one-level tax rateand a multi-level tax rate. In the case of a one-level tax rate, governmentestablishes a single tax rate for all drinks containing sugar, or for drinks witha sugar containment above a specific threshold. Such approaches are usedin France, Chile, Mexico, Belgium, Colombia, India, Portugal, Saudi Arabia,UAE, USA, South Africa. Alternatively, several countries (e.g. Thailand,Ireland, UK) apply a multi-level taxation scheme, which assumes differenttax rates according to sugar content , see Table 1.In this paper we assume a one-level tax rate for any positive sugar contain-ment in the drink. This is the current sugar tax practice in many countries,e.g., France, Mexico, Belgium, Colombia, India, Saudi Arabia, UAE andUSA.The approach is straightforwardly extendable to a multi-level tax ratewith a constant number of (a few) levels. ountry Tax rate Effectivesince France 7.53 euro per 100 liters 2013Chile > ,
25 grams of sugar per 0.1 liter →
16% 2014Mexico 1 peso per liter 2014Belgium 3.7284 euro per 100 liters 2013Colombia 20% per liter 2016India 40% 2017Portugal >
80 grams of sugar per liter →
16% 2017Thailand 14% + 5-stage sugar tax according to 2017sugar contentSaudi Arabia 50% 2017UAE 50% 2017USA (several cities) 12 cents per ounce 2017Ireland 58 grams of sugar per 0.1 liter →
21 cents; > →
31 cents 2018South Africa > → → > →
24% 2018
Table 1: Sugar tax rates across countries
Since the tax per cent varies widely from country to country, it makes sense todevelop a general mathematical model determining the taxation mechanismmaximizing the social welfare. Such a model should coordinate the interestsacross the three players: government, firms and consumers. As a startingpoint for the model the utility functions of the players are defined. The gov-ernment utility is usually understood as social welfare , see [Bernoulli, 2011].Let the social welfare be referred as W . It is expressed as the total utility ofconsumers and firms plus the tax: W = U c + U f + T, where U c is the total utility of all consumers, U f is the total utility of allfirms, and T is the total tax collected.The total utility of the consumers U c is defined as follows. Let the setof consumers be denoted by N . For simplicity of presentation, consider amarket with only two products: one is containing sugar and another oneis sugar-free. Let the product containing sugar be indexed with 1, and thesugar-free product be indexed with 0, and let M = { , } be the productindex set. Later we explain how to generalize the model and how to adjustthe algorithms in the case of several (a constant number of) products. Let6 i,j be an individual utility of consumer i ∈ N for product j ∈ M , and let x i,j be a binary decision variable taking value 1 if consumer i prefers product j to any other products and 0 otherwise. Assuming consumer’s rationality, wehave x i,k = 1 , i ∈ N, k ∈ M , if only if u i,k = max j ∈ M u i,j and u i,k ≥
0. Here,for all consumers i ∈ N we assume that (cid:80) j ∈ M x i,j ≤
1. For this assumptionto be true, the ties on the maximal utilities are broken in favor of the companyrevenue — this is also a folklore economic assumption. Clearly, if a consumerhas negative utilities for all products, she does not purchase anything, i.e., (cid:80) j ∈ M x i,j = 0.In the literature on marketing, there are various models describing be-havior and individual utilities of the consumers for a product. Vast majorityof the models are linear in the product price. For instance, consumer utilityfunction suggested in [Holtrop et al., 2017] consists of (1) a constant whichincludes different psychological, economical, and sociological factors such asbrand loyalty, consumer’s budget, readiness to pay etc; (2) a term dependingon the number of claims and nutritional values which are written on productpackages, e.g., labels as ”low in fat”, ”high in fiber” and other which mayinfluence consumer’s choice; and (3) deduction of the price related factor: u i,j = (cid:0) β i,j + β · N rClaims + β · N utrV al − β · p j (cid:1) + . Here, β i,j , β , β , β are the coefficients determined by a multinominal choicemodel , N rClaims is the average number of claims on product packages incategory,
N utrV al is the average nutritional value in category, and p k is theprice of product j . Notice, the intercept β i,j is a constant including factorscausing heterogeneity across the consumers. Finally, U c = (cid:88) i ∈ N (cid:88) j ∈ M ln(1 + u i,j ) · x i,j , under a classic assumption of diminishing marginal utilities of consumers,see, e.g., [Bernoulli, 2011].The consumer behavior model in [Holtrop et al., 2017] does not assumeprice elasticity of the demand. This is a reasonable assumption for the presentresearch as for the soft drinks, the volume of purchase depends rather onthe preferences and demographic characteristics of the consumer’s householdthan on the product price. Therefore, we may assume that the demand of aconsumer i ∈ N is given in two quantities: if the consumer prefers sugar-freedrink, the realized demand is D i, , and if the sugary drink is preferred, the https://eml.berkeley.edu/books/choice2.html D i, . Then, the utility of a company U f is defined by its revenue U f = (cid:32)(cid:88) i ∈ N (cid:88) j ∈ M D i,j · p j · x i,j (cid:33) − T, while the tax T is defined by T = α · (cid:88) i ∈ N D i, · p · x i, , where 0 ≤ α ≤ ≤ α ≤ (cid:88) i ∈ N (cid:88) j ∈ M (cid:0) ln(1 + u i,j ) + D i,j · p ∗ j (cid:1) · x ∗ i,j (1)subject to p ∗ = arg max p ≥ (cid:32)(cid:88) i ∈ N D i, · p · x ∗ i, (cid:33) + (1 − α ) · (cid:32)(cid:88) i ∈ N D i, · p · x ∗ i, (cid:33) (2)subject to x ∗ = arg max x (cid:88) i ∈ N (cid:88) j ∈ M u i,j · x i,j (3)subject to u i,j = (cid:0) β i,j + β · N rClaims + β · N utrV al − β · p j (cid:1) + , (4) (cid:88) j ∈ M x i,j ≤ ∀ i ∈ N, (5) x i,j ∈ { , } ∀ i ∈ N, j ∈ M. (6)On the first level, the government decides upon the sugar tax rate 0 ≤ α ≤ α , thecompany decides on the prices p ≥ p ≥ p , the consumers select preferredproducts maximizing their utility. 8 Solution of the three-level program
In this section we provide intuition and sketch the algorithms solving thethree-level mathematical problem (1)-(6). Then, we present the formal pseudo-codes of the algorithms. As the algorithms are rather straightforward andvery intuitive, we leave it for a reader to prove their correctness.To find the optimal tax rate, we first suggest to find all potentially optimalpricing strategies of the company. Notice, the consumer utility functions arelinear in prices. Therefore, for every consumer i ∈ N , the preference half-spaces are determined by inequalities β i, − β p ≥ β i, − β p , where theconsumer prefers the sugar-free drink, and β i, − β p ≤ β i, − β p , wherethe consumer might prefer the sugary drink. The indifference hyperplane β i, − β p = β i, − β p does, actually, represent the prices where the consumeris indifferent which product to purchase. Next to indifference hyperplanes,let us introduce the budget hyperplanes : β i, = β p and β i, = β p , asthe consumer purchases a product only if her utility for the product is non-negative, i.e., β i, ≥ β p or β i, ≥ β p . The union of all indifference andbudget hyperplanes splits the price space in many polyhedras. Let us referto a polyhedra as a choice region if the interior of the polyhedra is notintersected by any of the indifference or budget hyperplanes.By construction, for all consumers, their preferences for a product remainthe same for any prices taken from the same choice region. Then, by linearityof the revenue function in prices, only vertices of the choice regions mightbecome optimal pricing strategies of the company. Therefore, having O ( n )indifference and budget hyperplanes for only two products, where n is thenumber of consumers, we may have at most O ( n ) potentially optimal pricingstrategies. When the number of products increases to m , the number ofpotentially optimal strategies increases to O ( n m ). Thus, keeping m a smallconstant, brute-force enumeration of potentially optimal pricing strategiesremains polynomial and can be efficiently implemented. Moreover, since theconsumer utilities are independent on the sugar tax, for any choice of the taxrate, one of the computed pricing strategies will be optimal. Hence, the firststep towards the optimal policy design is computing all potentially optimalprices, which is done by Algorithm 1 below.Given a set of potentially optimal pricing strategies of the company, it ispossible to compute the optimal sugar tax rate. Since the number of pricingstrategies and, consequently, the number of consumer responses is finite, thesocial welfare function is a staircase function in α with break points possible9 lgorithm 1 Computing potentially optimal prices Input : N rClaims , N utrV al , β , β , β , β i,j for all i ∈ N, j ∈ { , } P := ∅ (cid:46) Set of potentially optimal prices3: Create a list B of budget hyperplanes: β · p j = β i,j + β · N rClaims + β · N utrV al for all i ∈ N and j ∈ { , }
4: Create a list I of indifference hyperplanes: β i, − β · p = β i, − β · p for all i ∈ N for h ∈ B ∪ I do for h (cid:48) ∈ B ∪ I such that h (cid:48) (cid:54) = h do
7: Compute prices p = h ∩ h (cid:48) P := P ∪ p Output : P only in the break-evens of the company’s revenue: (cid:32)(cid:88) i ∈ N D i, · p (cid:48) · x (cid:48) i, (cid:33) + (1 − α ) · (cid:32)(cid:88) i ∈ N D i, · p (cid:48) · x (cid:48) i, (cid:33) = (cid:32)(cid:88) i ∈ N D i, · p (cid:48)(cid:48) · x (cid:48)(cid:48) i, (cid:33) + (1 − α ) · (cid:32)(cid:88) i ∈ N D i, · p (cid:48)(cid:48) · x (cid:48)(cid:48) i, (cid:33) , where p (cid:48) and p (cid:48)(cid:48) are two potentially optimal pricing strategies, and x (cid:48) and x (cid:48)(cid:48) are the respective consumer choices. Evaluating the social welfare inevery break point, and choosing for α the break point that maximizes thesocial welfare does solve the three-level mathematical program. We call thisprocedure Algorithm 2. Overall complexity of Algorithm 2 is O ( n m ) for n consumers and m products, which is still polynomial if the number ofproducts m is fixed.Notice, the approach remains polynomial even for the multi-level sugartaxes if the number of levels L is also fixed. In this case, one has to determinenot a single optimal α , but an optimal set of 2 L − L tax levelsand L − regulatory space R L − , where the revenuesof 2 L potentially optimal pricing strategies meet each other. The overall timecomplexity of the algorithm in this case is O ( n mL ), which is still polynomialin the input size. The following example describes the process of obtaining an optimal sugartax rate from the purchase data set. For simplicity of presentation, consider10 lgorithm 2
Computing optimal sugar tax rate Input : P from Algorithm 1, D i,j for all i ∈ N, j ∈ { , } α ∗ := 0 (cid:46) Optimal sugar tax rate for p ∈ P do for i ∈ N do if u i, ( p ) ≥ u i, ( p ) then x i, ( p ) = 1 and x i, ( p ) = 0 else x i, ( p ) = 1 and x i, ( p ) = 0 if max j ∈{ , } u i,j < then x i, ( p ) = 0 and x i, ( p ) = 0 W ∗ := max p ∈P U c ( p, x ( p )) + U f ( p, x ( p )) (cid:46) Optimal social welfare for p (cid:48) ∈ P do for p (cid:48)(cid:48) ∈ P do Compute 0 ≤ α ≤ (cid:88) i ∈ N D i, · p (cid:48) · x i, ( p (cid:48) ) + (1 − α ) · (cid:88) i ∈ N D i, · p (cid:48) · x i, ( p (cid:48) ) = (cid:88) i ∈ N D i, · p (cid:48)(cid:48) · x i, ( p (cid:48)(cid:48) ) + (1 − α ) · (cid:88) i ∈ N D i, · p (cid:48)(cid:48) · x i, ( p (cid:48)(cid:48) ) p ( α ) := arg max p ∈P (cid:88) i ∈ N D i, · p · x i, ( p ) + (1 − α ) · (cid:88) i ∈ N D i, · p · x i, ( p ) W ( α ) := (cid:88) i ∈ N (cid:88) j ∈ M (ln(1 + u i,j ( p ( α ))) + D i,j · p j ( α )) · x i,j ( p ( α )) if W ( α ) ≥ W ∗ then α ∗ = α and W ∗ := W ( α ) Output : α ∗ and W ∗ a market with two basic products, say Coca-Cola ( j = 1) and Coca-ColaZero ( j = 0), where the first one contains sugar and the second one is sugar-free. For a generalization of this case one may consider two generic typesof products and deal with the averages per product type. Given a purchasedata set and applying multinomial choice model [Holtrop et al., 2017], weobtain utility functions for three types of consumers, see Table 2.When all budget and indifference lines/hyperplanes are drawn in the pricespace ( R in our case), we obtain the diagram as depicted in Figure 1. Thecrossing points of the budget and indifference lines are the potential opti-mal pricing strategies of the company (Coca-Cola). Having three lines perconsumer, we have nine lines in total, which might lead to at most 36 cross-ings, which will be the output of Algorithm 1. However, not all pairs ofthe lines cross each other, e.g., the budget lines for a product are parallel.This shrinks the number of potentially optimal pricing strategies to 27 pointslisted in Table 3 together with consumer preferences (“CC” stands for Coca-Cola and “Z” stands for Coca-Cola Zero), realized utilities of consumers11 oca-Cola Coca-Cola Zero(H)igh sugar u H, = (0 . − . p ) + u H, = (0 . − . p ) + consumers(like sugar)(M)edium sugar u M, = (0 . − . p ) + u M, = (0 . − . p ) + consumers(indifferent to sugar)(L)ow sugar u L, = (0 . − . p ) + u L, = (0 . − . p ) + consumers(do not like sugar) Table 2: Consumer preferences in soft drinks with different sugar contentand respective company revenues. In the case at hands, “Low” consumershave demand 11441 units, “Medium” consumers have demand 9433 units,and “High” consumers have demand 9942 units independent on the product(Regular or Zero). This allows us to compute the revenue of the company forevery potentially optimal pricing strategy. As an intermediate result, we ob-tain that if the sugar tax rate α = 0, then the maximal revenue is 1093019.67currency units (CU) in the following point: Coca-Cola is priced at 4.7 CUand Coca-Cola Zero is priced at 5.47 CU.12igure 1: Price spaceNow, we are ready to calculate the optimal sugar tax rate 0 ≤ α ≤ p (cid:48) = 0 .
94 and p (cid:48) = 1 .
96, and pricing strategy 13is defined by p (cid:48)(cid:48) = 2 .
13 and p (cid:48)(cid:48) = 1 .
58. In point 9, “High” and “Medium”consumers purchase Coca-Cola while “Low” consumer buys Zero. In point13, “High” consumer purchases Coca-Cola, and “Medium” joins “Low” inher preference to Zero. The break-even in revenue under these two pricingstrategies is achieved with α being a solution to the equation: · .
96 + (1 − α ) · (9433 + 9942) · .
94 = (11441 + 9433) · .
58 + (1 − α ) · · . , r Coordinates High sugar Medium sugar Low sugar Revenue1 (0; 1.25) CC : u = 0 . CC : u = 0 . Z : u = 0 .
72 14301 .
252 (0; 1.58) CC : u = 0 . CC : u = 0 . Z : u = 0 .
66 18076 .
783 (0; 1.96) CC : u = 0 . CC : u = 0 . Z : u = 0 . .
364 (0; 2.35) CC : u = 0 . CC : u = 0 . CC : u = 0 .
53 05 (0; 5.47) CC : u = 0 . CC : u = 0 . CC : u = 0 .
53 06 (0.44; 1.58) CC : u = 0 . CC : u = 0 . Z : u = 0 .
66 26601 .
787 (0.94; 0) CC : u = 0 . Z : u = 0 . Z : u = 0 .
93 9345 .
488 (0.94; 1.58) CC : u = 0 . Z : u = 0 . . Z : 0 .
66 42326 .
49 (0.94; 1.96) CC : u = 0 . CC : u = 0 Z : 0 . . CC : u = 0 . CC : u = 0 CC : 0 .
31 28967 . CC : u = 0 . CC : u = 0 CC : u = 0 .
31 28967 . CC : u = 0 . Z : u = 0 . Z : u = 0 .
93 21176 . CC : u = 0 . Z : u = 0 . Z : u = 0 .
66 54157 . CC : u = 0 . Z : u = 0 Z : u = 0 . .
515 (2.13; 2.85) CC : u = 0 . Z : u = 0 .
45 53783 . CC : u = 0 . CC : u = 0 .
04 45545 . CC : u = 0 . Z : u = 0 . Z : u = 0 .
93 26346 .
318 (4.7; 0) Z : u = 0 . Z : u = 0 . Z : u = 0 .
93 019 (4.7; 1.58) CC : u = 0 Z : u = 0 . Z : u = 0 .
66 79708 . CC : u = 0 Z : u = 0 Z : u = 0 . . CC : u = 0 Z : u = 0 .
12 101415 . CC : u = 0 Z : u = 0
23 (4.7; 8.7) CC : u = 0 46727 .
424 (5.19; 1.96) Z : u = 0 Z : u = 0 . . Z : u = 0 62582 . Z : u = 0 62582 . Table 3: All potentially optimal pointswhere the left hand side addresses the revenue at point 9 and the righthand side addresses the revenue at point 13. The solution to the equation is α = 4 .
62, which is beyond the global upper bound of 1 for the tax rate. Thus,this break point can be disregarded. We perform the same calculations for allpairs of the potentially optimal pricing strategies and derive all potentiallyoptimal tax rates α in interval [0 , Table 4: All potentially optimal tax rates α ≤ α ≤
1, the maximalutilities of the company appear only in three price points: (22) Coca-Colais priced at 4.7 CU, Coca-Cola Zero is priced at 5.47 CU; (25) Coca-Colais priced 5.63 CU, Coca-Cola Zero is priced at 5.47 CU; (26) Coca-Cola ispriced at 9.75 CU, Coca-Cola Zero is priced at 5.47 CU. Moreover, points25 and 26 are optimal only in α = 0, while point 22 is optimal on the entireinterval [0; 1].Enumerating over all potentially optimal tax rates listed in Table 4, wederive that the maximum social welfare of 156037 CU is achieved with α = 1,when the company sets the prices 4.7 CU for Coca-Cola and 5.47 CU forCoca-Cola Zero. In this paper we develop a model for coordinating the interests of the gov-ernment, companies and heterogeneous consumers. The model is based ona sequential game represented by a three-level mathematical program. Wedesign an algorithm efficiently solving the program, i.e., obtaining a sociallyoptimal solution in time polynomial in the input size of the problem.Surprisingly, for the case known in the literature [Holtrop et al., 2017], weobtain that the optimal sugar tax rate maximizing the commonly used socialwelfare function is equal to 100%. Furthermore, the real prices for Coca-Cola and Coca-Cola Zero are greatly underestimated compared to the pricesthat maximize the company (Coca-Cola) revenue. Moreover, the revenuemaximizing prices with and without taxation are exactly the same. Thisphenomena might be caused by either oversimplification of the social welfarefunction, or by oversimplification of the consumer behavior model, or byinefficiency in the market caused by irrationality of the players, e.g., firmsunderpricing their products, or by a combination of the above factors. In thisway, the approach proposed in the paper can be used in different contexts inorder to benchmark the social welfare and also in order to check reasonabilityof the models/utilities of the players.
Acknowledgment
We express our gratitude to Vladimir Kovalenok for the help in developingthe example. 15 eferences [Ball et al., 2015] Ball, K., McNaughton, S. A., Le, H. N., Gold, L.,Ni Mhurchu, C., Abbott, G., Pollard, C., and Crawford, D. (2015). In-fluence of price discounts and skill-building strategies on purchase andconsumption of healthy food and beverages: outcomes of the supermar-ket healthy eating for life randomized controlled trial–.
The Americanjournal of clinical nutrition , 101(5):1055–1064.[Bernoulli, 2011] Bernoulli, D. (2011). Exposition of a new theory on themeasurement of risk. In
The Kelly Capital Growth Investment Criterion:Theory and Practice , pages 11–24. World Scientific.[Desai and Ratneshwar, 2003] Desai, K. K. and Ratneshwar, S. (2003). Con-sumer perceptions of product variants positioned on atypical attributes.
Journal of the Academy of Marketing Science , 31(1):22–35.[FAO and WHO, 2017] FAO, IFAD, U. W. and WHO (2017). The state offood security and nutrition in the world.
Building climate resilience forfood security and nutrition , page 201.[Ferreira et al., 2007] Ferreira, I., Van Der Horst, K., Wendel-Vos, W., Kre-mers, S., Van Lenthe, F. J., and Brug, J. (2007). Environmental corre-lates of physical activity in youth–a review and update.
Obesity reviews ,8(2):129–154.[Geliebter et al., 2013] Geliebter, A., Atalayer, D., Flancbaum, L., and Gib-son, C. D. (2013). Comparison of body adiposity index (bai) and bmiwith estimations of% body fat in clinically severe obese women.
Obesity ,21(3):493–498.[Glanz et al., 2012] Glanz, K., Bader, M. D., and Iyer, S. (2012). Retailgrocery store marketing strategies and obesity: an integrative review.
American journal of preventive medicine , 42(5):503–512.[Holtrop et al., 2017] Holtrop, N., Cleeren, K., Geyskens, K., and Verhoef,P. (2017). The impact of nutritional health claims on sku choice. In
EMAC Conference .[Mead and Richerson, 2018] Mead, J. A. and Richerson, R. (2018). Packagecolor saturation and food healthfulness perceptions.
Journal of BusinessResearch , 82:10–18. 16Minkov et al., 2015] Minkov, N., Schneider, L., Lehmann, A., andFinkbeiner, M. (2015). Type iii environmental declaration programmesand harmonization of product category rules: status quo and practicalchallenges.
Journal of Cleaner Production , 94:235–246.[Morales, 2005] Morales, A. C. (2005). Giving firms an e for effort: Consumerresponses to high-effort firms.
Journal of Consumer Research , 31(4):806–812.[Okada, 2005] Okada, E. M. (2005). Justification effects on consumer choiceof hedonic and utilitarian goods.
Journal of marketing research , 42(1):43–53.[Rettie and Brewer, 2000] Rettie, R. and Brewer, C. (2000). The verbal andvisual components of package design.
Journal of product & brand man-agement , 9(1):56–70.[Smith et al., 2013] Smith, W. K., Gonin, M., and Besharov, M. L. (2013).Managing social-business tensions: A review and research agenda forsocial enterprise.
Business Ethics Quarterly , 23(3):407–442.[Solomon et al., 2012] Solomon, M., Russell-Bennett, R., and Previte, J.(2012).
Consumer behaviour . Pearson Higher Education AU.[Stevenson and Ingwersen, 2012] Stevenson, M. J. and Ingwersen, W. W.(2012). Environmental product claims and life cycle assessment.