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Working paper

Sugar Taxes and Changes in Total Calorie Consumption: A Simple Framework (WP 16/06)


This paper demonstrates the potential importance, when considering total calorie intake, of allowing for the substitution effects of imposing a selective tax on a commodity having a high sugar content, when non-taxed commodities exist and also have relatively high calorie content. A framework is presented which allows the elasticity of calorie consumption with respect to a price change to be derived. This brings out the role of relative budget shares, relative calorie content of goods and relative prices to be clearly seen, along with own- and cross-price elasticities. Their absolute values for each commodity group are not required. It is demonstrated that the focus of attention needs to be much wider than a simple concentration on the own-price elasticity of demand for the commodity group for which a sumptuary tax is envisaged.


I am grateful to Sarah Hogan for drawing my attention to a number of useful papers, and Matt Cowan for comments on an earlier version of this paper.


The views, opinions, findings, and conclusions or recommendations expressed in this Working Paper are strictly those of the author(s). They do not necessarily reflect the views of the New Zealand Treasury or the New Zealand Government. The New Zealand Treasury and the New Zealand Government take no responsibility for any errors or omissions in, or for the correctness of, the information contained in these working papers. The paper is presented not as policy, but with a view to inform and stimulate wider debate.


1 Introduction

It is widely recognised that high sugar consumption is linked to obesity, type-2 diabetes and cardiovascular disease. In view of the large proportion of sugar consumption attributed to non-alcoholic drinks, especially among young people, this has led to pressure for the imposition of a high sumptuary tax rate on sugar-sweetened beverages (SSBs), commonly referred to as a ‘sugar tax'. For example, in New Zealand, Ni Mhurchu et al. (2014, p. 96) suggested that, ‘a 20% tax on carbonated drinks could be a simple, effective component of a multifaceted strategy to tackle New Zealand's high burden of diet-related disease'. Indeed, such a ‘sugar tax' has received most policy emphasis, although increased obesity can be attributed to a wide range of causes; see, for example, Cutler et al. (2003).

Arguments for a sugar tax obviously rely on, among other things, establishing a consequent significant reduction in consumption. This is complicated by a range of problems. For example, surveys usually measure expenditure at the household rather than indivdual level. The health status of those responding to a price increase is not generally known, and higher responses may come from healthy consumers rather than the target population group. Surveys typically measure expenditure rather than consumption, which may be spread over a period of time longer than the survey period. Combined with this last point is the fact that the price actually paid per unit is to a large extent endogenous, in view of the price variations in the market. Furthermore, even within the group of SSBs, there is considerable heterogeneity, and a failure to allow adequately for quality variations can lead to a substantial bias in the estimation of elasticities, as shown by Gibson and Kim (2013). Substitution towards lower-priced SSBs that are higher in calorie content may mean that a tax on SSBs could have harmful rather than beneficial effects.

In addition, emphasis is often placed only on the own-price elasticity of demand for SSBs.[1] Substitution towards other non-taxed goods that are high in calories can also take place, reducing or even eliminating any direct reduction in the consumption of SSBs.[2] However, greater recognition is being given to this feature, which arises from the relative price change produced by the tax. Fletcher et al. (2010) found that the US tax on soft drinks was ‘completely offset' by substitution towards other high-calorie drinks, so that ‘the revenue generation and health benefits of soft drink taxes appear to be weaker than expected' (2010, p. 973). The US taxes are relatively low, and they mention the possibility that there may be nonlinear or threshold effects for higher tax rates. However, Fletcher et al. (2014) later reported that they found no evidence of such effects.

Using data for Mexico, Aguilar et al. (2016) found full (and sometimes more than 100 per cent) shifting for a tax on sweetened drinks but 66 per cent shifting for a tax on other high calorie foods.[3] As a result of substitution combined with differential price changes, they found that 'total calories consumed from all products captured in our dataset did not change' (2016, p. 3). This led to the conclusion that, 'the taxes introduced are unlikely to have the desired effect of reducing obesity' (2016, p. 14). The change in relative prices arose because 'suppliers of products with more calories per unit passed less of the tax on to consumers', so that, 'the relative price of calories per unit actually decreased' (2016, p. 16).

Other evidence has been somewhat mixed. Smith et al. (2010) found a small degree of substitution in the US, using elasticities evaluated at mean values, thereby reducing but not eliminating the effects of a tax on SSBs. Lin et al. (2010), again using US data, observed substitution, but the cross-price elasticities were nevertheless quite small. Briggs et al. (2013) found evidence in the UK for substitution but reported small cross-price elasticities. In a different, but related, context Nhoaham et al. (2009, p. 1330) found that in New Zealand fruit and vegetables were gross complements with milk, cheese and fats, so that a subsidy for fruit and vegetables may not achieve its objective.[4]

Given a comprehensive set of own-price and cross-price elasticities for well-defined commodity groups, along with initial prices and quantities demanded, it is not difficult to work out the overall effect on the demand for each good as a result of a specified change in relative prices resulting from an indirect tax change (with suitable assumptions regarding the extent to which any tax is passed on to consumers in the form of higher unit prices). With information on the calorie content per unit of each good, the effect on total calorie consumption could then be evaluated. However, such extensive information is rarely available. It is therefore useful to construct a simple framework for examining the effect on total calorie consumption of a selective tax, rather than simply the consumption of SSBs. Furthermore, the overall effect on calorie consumption needs to consider the relative importance of the taxed group in total calorie consumption. A concentration on the proportional reduction in consumption of the taxed group only may overstate the effectiveness in achieving the ultimate objective.

The aim of the present paper is therefore to construct such a framework. The basic relationships are set out in Section 2, using a simple model in which there are just three types of good, two of which have a high calorie content. It is shown that the effectiveness of a sugar tax can be examined using just one own-price elasticity and one cross-price elasticity, and three fundamental ratios. These ratios are of relative budget shares, relative prices and relative calorie values of the groups with high calorie content. A convenient expression is obtained for the elasticity of total calorie consumption with respect to a change in the price of one of the goods. In the absence of empirical estimates, illustrative examples are given in Section 3. These examples illustrate the value of the framework in making it easy to consider the sensitivity of results to crucial variables. However, the analysis takes the relevant demand elasticities as given, so that it does not begin by specifying a form of utility function. Hence, no attempt is made to consider the wider question of how to evaluate the costs (including excess burdens) and benefits (including any externalies) of such a tax policy. Brief conclusions are in Section 4.


  • [1] The way in which anticipated 'direct' demand reductions resulting from an SSB tax are calculated may also lead to an overstatement. Typically an elasticity is used to obtain the proportional change in demand,


    , following a proportional change in price of


    , using a 'point' elasticity, η, combined with an (implicit) linearisation of the demand function, such that




    η. However, suppose the elasticity (rather than the slope) is thought to be constant, so that the demand curve is log-linear. In this case the appropriate formula is




    η - 1. The difference is likely to be negligible for very small price changes, but proponents of an SSB tax usually argue for at least a 20 per cent ad valorem rate. For example, with


    = 0.2, an elasticity of η = -1.1 gives


    η = -0.22 and




    η - 1 = -0.1817. If initial demand is 2 million units, the former overstates the demand reduction by 76,555 units. A 'higher' elasticity of -1.2 produces percentage reductions in demand of 24 and 19.65 for the two approaches, with the linearisation approximation overstating the reduction in demand by almost 87,000 units.



    (1 +p˙).


    (1 +p˙).
  • [2] In a wide-ranging review of food taxes, Jeram (2016, p. 28-30) stressed problems arising from quality change within the taxed group and substitution towards other (untaxed) high calorie goods.
  • [3] Colchero et al. (2015) also found full or over-shifting of the SSB tax for Mexico.
  • [4] Ni Mhurchu et al. (2013) found cross elasticities were mixed for New Zealand, with some groups being complements, but they used very broad categories. Nghiem et al. (2011) examine demand elasticities for foods in Australia and New Zealand but the categories used are not helpful in the context of SSBs.


2 A Simple Model

This section examines the elasticity of calorie consumption with respect to a change in the price (arising from a tax change) of a commodity with high sugar content, in a situation where a substitute exists which also has high calorie content but remains untaxed. Given the relationship between weight change and calorie intake, a link can then be made from the tax (and price) change to weight change.[5]

2.1 Total Calorie Consumption

For simplicity, suppose consumption can be divided into three goods. Goods 1 and 2 are high in calories per unit, while good 3 has no calories. Consumption of good i is denoted qi. Define the coefficient, γi, as the calorie content per unit of consumption of good i. Hence the calories attributed to good i, defined as ci, are:



Total calories arising from consumption are denoted by C. Since, by definition, γ3 = 0, this is given by:



Consider a change in the price of good 1. This can arise from an ad valorem tax at the rate, τ, on good 1 only. On the strong assumption that the tax is fully shifted to consumers, this leads to a proportional change in the price of the good given by dp1/p1 = τ.

The resulting change in total calorie consumption is given by:



This can be written as:



Define the price elasticity of good i with respect to a change in the price of good j as ei,j, so that:



and define the elasticity of total calorie consumption with respect to the price of good 1, ηC,p1, using:



Using (5) and (2), and multiplying (4) by p1/C, (6) becomes:



Let si denote the share of good i in total calorie consumption. Then, by definition:



Substituting (8) into (7) gives:



This result shows that the total elasticity, ηC,p1, is a calorie-share-weighted sum of the own-price elasticity e1,1 and the cross-price elasticity, e2,1. The latter measures the proportional change in the demand for good 2 resulting from a unit proportional change in the price of good 1. In the simple case where both γ2 and γ3 are zero, then ηC,p1 = e1,1, since γ1 = 1: this elasticity is usually the only focus of attention when considering the effectiveness of a sugar tax. Where a substitute good exists that is also high in calories (γ2 > 0) then, even if the cross-price elasticity, e2,1, is zero, the proportional reduction in total calorie consumption is ηC,p1 = s1e1,1 and is clearly less than e1,1.

Equation (9) shows that the total effect of a tax on good 1 can be obtained with knowledge only of the initial calorie shares and the two elasticities, e1,1 and e2,1. If, as is likely, goods 1 and 2 are gross substitutes, the consumption of good 2 increases when the price of good 1 increases, so that e2,1 > 0. The own-price elasticity, e1,1, is of course negative.


  • [5] A general framework was also presented by Schroeter et al. (2008), who examined a food demand model involving maximisation of a utility function. The ‘arguments' of the function include different food types (differing by calorie content) along with body weight. The latter is affected by exercise as well as total calorie intake. Utility is maximised subject to the budget constraint, involving food costs along with the cost of exercise (though a time constraint was not included). However, their focus was actually on the relationship between body weight, W, exercise and food consumption, which was used to provide an elasticity decomposition of weight change with respect to the price of the high-calorie food. Writing the general function, W = W


    , straightforward total differentiation of W gives


    = ∑ i



    . In general, let ηa,b denote the elasticity of a with respect to a change in b. Then it is easily seen that: ηW,p1 =



    = ∑ i



    = ∑ iηW,xiηxi,p1. Here ηW,p1 is a total elasticity, while all others are partials (although Schroeter use partial derivatives throughout). The authors examined evidence relating to the various elasticities.

    (x ,x ,...)  1  2.

    dW- dp1.

    ∂W- ∂xi.

    ∂xi- ∂p1.

    p1 W-.

    dW dp1.

    ( xi∂W )   W-∂xi.

     ( p1∂xi)   xi∂p1-.


2.2 Calorie Shares

Consider the determination of the si values. Let wi denote the budget share of good i, so that:



and if total expenditure is defined as yi = ∑ i=13p iqi, the budget shares are simply:



Hence the quantity, qi, can be written as qi = wiy / pi, and substituting into (8) gives:





Clearly it is only necessary to obtain an expression for s1, since s2 = 1 -s1. Substitution and rearrangement gives:



Hence, conveniently, only the relative price, p1 / p2, is needed rather than absolute prices, and the relative calorie content,


, is needed rather than absolute amounts.

γ2 γ1.


2.3 The Three Ratios

Equation (13) indicates the importance of three relative values. Clearly





, and




is simply,


, the ratio of calories contributed by good 2 to those of good 1. However, it is useful to retain the form above to show the separate role of relative budget shares, relative prices and relative calorie values.

(   )  ww2    1.

(  )  pp1   2.

q2q- 1.

(  )   γ2γ1-.

(  )  ww21.

(  )  pp12.



Substitute (13) into (9) and rearrange to get:







= ∏ k=13r k, with r1 =


and so on, the effect on ηC,p1 of a change in any of the ratios can be expressed as:

(  )  γ2  γ1.

(   )   w2   w1.

(   )   p1   p2.

γ2 γ1.




Hence, since e1,12,1 > 0,


> 0 and ηC,p1 unambiguously rises - and therefore in absolute terms become smaller - as each of the three ratios, rk, increases. Furthermore, as expected, ηC,p1 is larger in absolute terms for a higher budget share, w1, of the taxed good relative to the untaxed good, and for a relatively higher calorie content.

∂η∂Cr,p1    j.


Equation (14) can also be used to consider the size of the cross-price elasticity, e2,1, required such that any direct effect on calorie intake arising from e1,1 is completely eliminated by the substitution towards untaxed goods. Letting


denote the absolute value of the own-price elasticity of the taxed good, it can be shown that ηC,p1 > 0 so long as:





The higher the relative price of the substitute, the smaller its budget share relative to the taxed good, and the higher the calorie content of the taxed good relative to the substitute, the greater is the chance that the direct effect of a tax will outweigh the indirect effect arising from the cross-price elasticity. Of course, in the unlikely case where the second untaxed good is a gross complement, that is, its sign is negative, the indirect effect reinforces the direct effect.

3 Illustrative Examples

To give some idea of the potential effect of allowing for the second good that is both a substitute for the first (taxed) good and is relatively high in calories per unit, suppose that the absolute value of the own-price elasticity,


, is equal to 0.8. This is within the range reported in a number of studies of sugar-sweetened beverages. Suppose also that the relative price, p1 / p2 = 0.8, so that the price per unit of good 2 is higher than that of good 1. The order of magnitude of budget shares appears to vary substantially between countries and demographic groups. For present purposes, suppose good 2 has a budget share of w2 = 0.03.


Figure 1: Variation in ηC,p1 with e2,1


Figure 1: Variation in η<sub>C,p<sub>1</sub></sub> with e<sub>2,1</sub>.

Figure 1 shows the variation in ηC,p1 with e2,1 for different combinations of w1, the budget share of the taxed good, and of the ratio γ2 / γ1. From equation (14) these profiles are obviously linear, and the relevant elasticity ηC,p1 increases - that is, it decreases in absolute terms - as the cross-price elasticity, e2,1, increases. Clearly, the higher the budget share and the relative calorie content of the taxed good 1, the greater is the effectiveness of the tax in reducing calorie consumption. But for all combinations shown in Figure 1, the effectiveness of the tax is substantially reduced as the cross-price elasticity increases. In view of the limited empirical information about this elasticity, further empirical work is warranted.

4 Conclusions

The aim of this paper has been to demonstrate the potential importance of allowing for substitution effects in considering the effectiveness of imposing a tax on a commodity (or group) having a high sugar content, when non-taxed commodities exist and also have relatively high calorie content. A framework was presented which allows the role of relative budget shares, relative calorie content of goods and relative prices to be clearly seen. Importantly, in determining the elasticity of total calorie consumption with respect to changes in the price of taxed SSBs, the absolute amounts (of prices, budget shares, and calorie content) for each commodity group are not required. It was demonstrated that the focus of attention needs to be much wider than a simple concentration on the own-price elasticity of demand for the commodity group for which a sumptuary tax is envisaged.


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