A referee report should include the following: • A summary of the research paper • A recommendation to the journal editor as to whether the work should be published, whether it would benefit from revisions, or whether it should be rejected due to a lack of substantive intellectual contribution • A discussion of the research’s contribution to the field’s current understanding of the topic • Analysis of the methodological approach, especially the identification of shortcomings or areas lacking clear explanation The cover letter is essentially a synopsis of the referee report and is a great exercise in the concise presentation of your thoughts.Corrective tax design in oligopoly
Martin O’Connell†∗ and Kate Smith∗

Job Market Paper
This version: January, 2020. Most recent version available here.
Correspondence: martin o@ifs.org.uk.
We study the design of taxes aimed at limiting externalities in markets
characterized by differentiated products and imperfect competition. In such
settings policy must balance distortions from externalities with those associated with the exercise of market power; the optimal tax rate depends on
the nature of external harms, how the degree of market power among externality generating products compares with non-taxed alternatives, and how
consumers switch across these products. We apply the framework to the topical question of taxes on sugar sweetened beverages. We use detailed data
on the UK market for drinks to estimate a model of consumer demand and
oligopoly pricing for the differentiated products in the market. We show the
welfare maximizing tax rate leads to welfare improvements over 2.5 times as
large as that associated with policy that ignores distortions associated with
the exercise of market power.
Keywords: externality, corrective tax, oligopoly
JEL classification: D12, D43, D62, H21, H23, L13
Acknowledgements: The authors would like to gratefully acknowledge financial support from the European Research Council (ERC) under ERC2015-AdG-694822, the Economic and Social Research Council (ESRC) under
the Centre for the Microeconomic Analysis of Public Policy (CPP), grant
number RES-544-28-0001, and the British Academy under pf160093. Data
supplied by TNS UK Limited. The use of TNS UK Ltd. data in this work
does not imply the endorsement of TNS UK Ltd. in relation to the interpretation or analysis of the data. All errors and omissions remain the responsibility
of the authors.

Institute for Fiscal Studies and University College London.
One-fifth of all consumer spending is undertaken in markets subject to taxes, at least
in part, aimed at altering behavior to limit externalities.1 Many of these markets
are characterized by the presence of large multi-product firms that are likely to
exercise substantial market power. For instance, soft drink markets, the subject
of new taxes in several jurisdictions, are dominated by Coca Cola Enterprises and
PepsiCo. Distortions associated with the exercise of market power have important
implications for corrective tax design. Buchanan (1969) points out that efforts
to fully correct for externalities are only justified in conditions of competition; in
imperfectly competitive environments price is already in excess of marginal cost and
externality correcting policy that fails to take account of this can reduce welfare.
However, the bulk of the long literature on the design of corrective taxes, dating
back to Pigou (1920), assumes a perfectly competitive environment.
Our contribution in this paper is to study the design of taxes levied on externality generating products in markets characterized by product differentiation,
strategic firms and imperfect competition, and to undertake a substantive empirical
application to the taxation of sugar sweetened beverages. We write down a simple
optimal tax model that shows how patterns of consumer substitution, positive pricecost margins and strategic price re-optimization affect the optimal corrective tax
prescription. In our empirical application we estimate a detailed model of consumer
demand and oligopoly price competition in the market for non-alcoholic drinks, and
compute the optimal sugar sweetened beverage tax. We show that despite substantial price-cost margins on these products, there is nonetheless a case for levying a
positive tax rate, in part, because in equilibrium consumers switch to other imperfectly competitive products. Nevertheless, the optimal rate lies below the rate a
planner that ignores distortions associated with the exercise of market power and
aims at full internalization of externalities would set, and results in substantially
larger welfare gains.
We consider a setting in which there are many differentiated products available
to consumers. The consumption of one set of products generates an externality (in
proportion to some specific product attribute), while the remaining set generate no
external costs. The products are supplied by a set of (potentially multi-product)
firms that derive market power from the imperfectly substitutable nature of the
products in the market. A social planner sets a linear tax on the externality gen1
Spending on alcohol, tobacco, soft drinks, fuel, and motoring (all of which are subject to
some kind of excise duty in the UK – Levell et al. (2016)) accounts for 24% of spending recorded
in the UK’s consumer expenditure survey (Living Costs and Food Survey (2017)).
erating product attribute, with the aim of improving welfare. To focus on the
interaction between externality correction and imperfect competition we assume
the planner sets the tax rate to maximize economic efficiency in the market; the
planner does not have a redistributive motive, nor a revenue raising constraint.2
If the market was perfectly competitive, the optimal rate would be equal to the
marginal external cost (if homogeneous across consumers, as in Pigou (1920)) or,
when there is heterogeneity in marginal externalities, the optimal rate would be
equal to a weighted average of marginal external costs (as in Diamond (1973)).
Under imperfect competition the optimal tax rate equals the traditional corrective component minus an adjustment for the distortion associated with the exercise
of market power. In a market with just one product supplied by a monopolist, the
optimal rate is equal to the marginal externality minus the equilibrium price-cost
margin on the product. In a two product market (where one product is associated with externalities and one is not), the planner cares both about achieving an
efficient level of total consumption, and achieving allocative efficiency across the
two products. A higher equilibrium price-cost margin on the externality generating
product acts to reduce the optimal rate, while a higher margin on the substitute
(untaxed) product acts to increase it. The extent to which the margin on the nonexternality generating product raises the optimal rate depends on how strongly
the tax shifts consumption towards it from the taxed product; in the limit, if consumption switches one-for-one between the products, the optimal tax rate equals
the marginal externality minus the difference in equilibrium price-cost margins between the two products. With many differentiated products, switching within the
set of taxed products, as well as the specific alternative products that consumers
switch most strongly towards, also influences the optimal rate.
We use the framework to study the taxation of sugar sweetened beverages. Consumption of these products is strongly linked to diet related disease, which creates
externalities through increased societal costs of funding both public and insurance
based health care.3 In recent years, motivated by public health concerns, a number of countries and localities have introduced taxes on these products; as of May
2019, 41 countries and 7 US cities had some form of sugar sweetened beverage tax
Sandmo (1975) shows that in the face of a revenue raising constraint, an efficiency maximizing planner that can set a linear tax on each product in the economy will set a tax rate on
an externality generating good that entails a Pigovian component plus a distortionary Ramsey
component. Kopczuk (2003) shows this additivity property holds under much more general conditions, including when there are redistributive motives. See Bovenberg and Goulder (2002) for a
thorough review of work on how the interaction between corrective taxes and other distortionary
taxes changes the Pigovian tax prescription and can limit the effectiveness of externality correcting
For a survey of the evidence see Allcott et al. (2019b).
in place (GFRP (2019)). The market for these products is characterized by large
multi-product firms that offer strongly branded products and are likely to enjoy
significant market power. To implement our optimal tax framework requires estimating own- and cross-price demand elasticities between products in the market,
and the equilibrium price-cost margins on these products (both for products subject
to the tax and for substitutes).
We use longitudinal data on purchases of non-alcoholic drinks that households
bring into the home and that individuals consume while on-the-go. Most empirical
studies of sugar sweetened beverage taxes do not cover purchases made on-the-go,
yet they are an important part of the market.4 We obtain demand elasticities by
estimating a model of consumer choice among the differentiated products in the
drinks market (in the broad spirit of Berry et al. (1995)). We model preferences
over key product attributes as random coefficients, allowing the coefficient distributions to depend on consumer age, income and a measure of total dietary sugar.
The overall preference distribution takes the flexible form of a mixture of normal
distributions, relaxing functional form restrictions otherwise imposed on product
demand curves.5,6
Following a long tradition in the empirical industrial organization literature
we treat price-cost margins as unobservable (see Bresnahan (1989)), using our demand estimates and the equilibrium conditions of an oligopoly pricing game to infer
marginal costs (as, for instance, in Nevo (2001)). Our estimates suggest that, on
average, prices are around double marginal costs, though there is considerable variation in price-cost margins across products. In particular, small pack sizes typically
have larger price-cost margins (per liter) than bigger sizes. Our demand estimates
suggest consumers switch more strongly away from large sizes in response to a tax,
meaning a higher tax rate drives up the average margin among taxed products,
which plays an important role in determining the optimal rate. The empirical demand and supply model allows us to simulate, in equilibrium, consumer substitution
patterns and product level margins, and serves as an important input into solving
for the optimal tax rate.
An exception is Dubois et al. (2019), who focus on modeling on-the-go demand for drinks.
In particular, the flexible preference distribution helps relax curvature restrictions on demands. As highlighted by Weyl and Fabinger (2013), demand curvature is one important determinant of how equilibrium prices respond to tax changes.
A potential threat to the validity of our demand estimates is the presence of neglected dynamics. In particular there is evidence in the US that consumers stockpile soft drinks (Hendel
and Nevo (2013), Wang (2015)). We provide evidence that stockpiling is much less relevant in the
UK context; when consumers purchase on sale they tend to switch brands or pack type, with no
evidence of significant changes in the timing of purchase.
We find, for reasonable levels of the externality from sugar consumption, that the
optimal tax on the sugar in sweetened beverages is positive . However, the optimal
rate lies below the Pigovian rate that would be optimal under perfect competition.
It also lies below the rate a planner that ignores distortions associated with the
exercise of market power and aims at full internalization of externalities would set.
The optimal tax rate lies below the rate aimed at full internalization of externalities
due to the existence of substantial price-cost margins for sugar sweetened beverages.
The weighted average margin for these products is actually increased with the tax,
as people switch to smaller sizes with higher margins, and firms raise margins by
increasing prices by more than the tax. However, the optimal rate is positive in
part because consumers switch towards alternative drinks products also supplied
We use an estimate of the public health costs from sugar sweetened beverage
consumption to calibrate the marginal externality, however, there is considerable
uncertainty over the size of this parameter. We show how varying the externality affects the optimal tax rate; at all positive levels of the externality, ignoring
distortions associated with the exercise of market power when setting tax leads
to substantially lower welfare than under optimal policy. If externalities are also
generated by substitute goods that contain sugar, a tax on the sugar in sweetened
beverages will be less effective at combating externalities. We show that the existence of untaxed externalities leads to a reduction in the optimal rate of around
20%. We also show that the optimal rate increases modestly in the extent to which
externalities are concentrated among those with high overall dietary sugar.
To show how the degree of market power exercised by firms influences the potential welfare gains from levying a tax on externality generating products, we
simulate the optimally set tax under counterfactual firm ownership structures. A
more competitive market structure leads to welfare gains (in the absence of tax), as
increases in consumer surplus swamp reductions in firm profitability and increased
externalities. In addition, a tax on externality generating goods leads to larger
welfare gains under more competitive market structures, suggesting that there is
complementarity between competition and corrective tax policy.
We compare the performance of the optimal tax on sugar to a number of alternatives. The welfare gains associated with the optimal rate are 2.5 times as large
as tax policy set by a planner that ignores the distortions associated with the exercise of market power. Almost all jurisdictions that have introduced taxes on sugar
sweetened beverages do so on a volumetric basis, rather than in proportion to sugar
content. We find that an optimally set volumetric tax achieves only 60% of the
welfare gains achieved by the optimally set sugar tax rate. Some localities, notably
Philadelphia, apply a volumetric tax to both artificially and sugar sweetened beverages as a revenue raising measure; we show that it is much more costly in welfare
terms to raise revenue with this instrument compared to a tax levied only on sugar
sweetened beverages.
We contribute to a small but growing literature that uses empirically rich treatments of markets to evaluate how imperfect competition affects fundamental tax
design questions. Fowlie et al. (2016) use a dynamic oligopoly model of a homogeneous goods market (for concrete) and show that carbon abatement policy aimed at
full internalization of social costs is welfare reducing, whereas policy that explicitly
recognizes distortions associated with the exercise of market power has the potential
to improve welfare. A set of recent papers study optimal commodity taxation in
the differentiated product market for liquor. Miravete et al. (2018a, 2018b) show
the peak and shape of the Laffer curve associated with an ad valorem tax rate depends on the strategic pricing behavior of distillers, and quantify welfare gains that
would be realized if government instead set product specific taxes/prices. Conlon
and Rao (2015) show existing “post and hold” price regulations facilitate collusion
and lead to allocative inefficiencies, and substantial consumer welfare gains would
be realized by replacing them with a higher level of taxation. A number of papers
consider optimal subsidy design in health insurance markets in which providers exercise market power, (see Tebaldi (2017), Polyakova and Ryan (2019) and Einav
et al. (2019)) and show targeted subsidies engender equilibrium pricing responses
and spillovers to non-targeted groups.
Our work also relates to a rapidly growing literature studying sugar sweetened
beverage taxation. One set of papers use data covering the introduction of taxes
to estimate the impact on prices and/or purchases.7 A second set of papers use
estimates of consumer demand to simulate the introduction of taxes similar to
those used in practice.8 Like us, Allcott et al. (2019a) study the optimal design
of a tax on sugar sweetened beverages. They consider a perfectly competitive
environment in which a social planner with a preference for redistribution sets a
tax on sugar sweetened beverages along with a non-linear labor tax. They find
evidence of larger internalities among low income households, which, all else equal,
See, for instance, Bollinger and Sexton (2018) and Rojas and Wang (2017) who study the
Berkeley tax, Seiler et al. (2019) and Roberto et al. (2019) who study the Philadelphian tax, and
Grogger (2017) who study the Mexican tax. For a full survey of the recent literature see Griffith
et al. (2019).
These papers include Bonnet and Réquillart (2013), Wang (2015), Harding and Lovenheim
(2017) and Dubois et al. (2019).
increases the optimal rate set by a planner with preferences for redistribution.9 Our
work complements theirs by focusing on the impact of imperfect competition on tax
design, while abstracting from issues of redistribution.
The dominant paradigm in modern public economics is the use of sufficient
statistics to assess the welfare consequences of policy reforms (Chetty (2009)). This
is the approach taken by Jacobsen et al. (2018) to quantify the welfare loss associated with the inability to levy product-specific Pigovian taxes. It is also used by
Ganapati et al. (2019) to measure incidence of input taxes in imperfectly competitive markets. In our setting, the welfare effects of changing the tax rate depend on
the switching between, and price-cost margins for, a large set of differentiated products. To estimate these we specify a model of demand and supply in the market.10
This enables us to estimate elasticities and price-cost margins for disaggregate products, and allows us to simulate the effect of non-local tax changes, and therefore
recover the optimal tax rate. To provide evidence that our empirical model successfully captures behavior in the market, we use quasi-experimental variation on
price changes resulting from the recent introduction of the UK’s soft drinks tax to
validate our estimated model.
The rest of this paper is structured as follows. In Section 2 we consider the design
of corrective taxes in markets, such as that for drinks products, in which firms set
prices above marginal costs. Section 3 describes the UK market for drinks and the
micro panel data we use on purchase decisions made for consumption outside as well
as in the home. In Section 4 we present our empirical model of consumer demand
and firm pricing competition. Section 5 presents our empirical tax results. A final
section draws together the implications of our results and concludes.
Corrective tax design in imperfect competition
Our aim is to highlight how distortions associated with the exercise of market power
influence the efficiency maximizing rate of tax on externality generating products.
We consider a market that comprises a set of differentiated products, a subset of
which have externalities associated with their usage. The products are provided
by firms who set their prices under conditions of imperfect competition. We be9
Gruber and Koszegi (2004) and O’Donoghue and Rabin (2006) also consider the design of
internality correcting taxes.
An important difference between our setting and that in Ganapati et al. (2019) is that we
model a market in which asymmetric product differentiation and multi-product firms are central. This means tax incidence cannot be expressed as a function of a small number of sufficient
gin by considering a stylized market in which there are just two products, before
generalizing the analysis to a market with many products.
We consider a social planner whose task it is to set a tax rate for the externality
generating goods. The planner’s objective is to maximize efficiency. We abstract
from possible redistributive motives, focusing instead on how imperfect competition
alters the optimal externality correcting tax prescription.11
A two product market
Set-up. Consider a market that comprises two products, j = {1, 2}. Consumer
i, facing prices, p = (p1 , p2 ), chooses how to allocate her income, yi , between the
two products and a numeraire good (which represents expenditure outside of the
market of interest). We assume consumers have preferences that are quasi-linear
and can be represented by the indirect utility function Vi (p, yi ) = yi + vi (p), and
denote consumer level demand for product j by qij (p). The quasi-linear preference
structure means that a price change for either product does not induce any income
effects. This assumption is reasonable when focusing on a market that accounts for
a small share of total consumer spending,12 and it enables us to focus on a planner
that seeks to maximize economic efficiency.
Each unit of product 1 consumed creates an externality. We initially assume the
externality is homogeneous across individuals and denote it by φ. Product 2 is a
substitute for product 1; its consumption does not create any externalities. A social
planner chooses the rate of tax, τ , to set on product 1. Both products are supplied
imperfectly competitively at constant marginal cost; the equilibrium prices are such
p 1 − τ − c1 = µ 1
p 2 − c2 = µ 2 ,
Under perfect competition and when the planner can set a non-linear labor tax, redistributive
motives do not influence optimal commodity taxes as long as differences in consumption patterns
across the income distribution are driven purely by income differences and consumers are utility
maximizing (Saez (2002)). Jaravel and Olivi (2019) show that this extends to an economy characterized by imperfect competition. Kaplow (2012) shows that accompanying externality correcting
taxes with a distribution-neutral adjustment to the income tax system can offset the redistributive
effects of the corrective taxes across the income distribution.
In general, the own price effect on demand for good j follows the Slutsky equation ij =
p q
ij + jy ij eij , where ij and hij are the Marshallian and Hicksian own-price elasticities of demand,
p q
and eij is the income elasticity. For a small budget share good jy ij ≈ 0, meaning ij ≈ hij and
preferences are approximately quasi-linear.
where cj denotes the marginal cost and µj denotes the equilibrium price-cost margin
for product j (per unit, for instance liter, of consumption). The equilibrium prices
and margins depend both on the rate of tax levied on product 1 and the marginal
costs of both products, as well as whether the products are supplied by a monopolist
or duopolists.13 For notational simplicity we suppress this dependence.
We assume that the numeraire is competitively supplied, and its consumption
does not generate any externalities. We relax this assumption when we empirically
implement our results in Section 5.
Optimal policy. We consider a social planner that chooses the rate of tax to
maximize total welfare, which equals the total consumer surplus from participation
in the market, v(p), minus total externalities plus tax inclusive profits. Tax inclusive
profits on product 1 are given by (p1 − c1 )q1 and are equal to the sum of net profits
(p − τ − c1 )q1 and tax revenue, τ q1 . The planner’s problem is:
max v(p) − φq1 + (p1 − c1 )q1 + (p2 − c2 )q2 .
The optimal tax rate, τ ∗ , is implicitly defined by:


τ = φ − µ1 − µ2 ×

where dτj = ∂pj1 dp
+ ∂pj2 dp
is the derivative of equilibrium consumption of product

j with respect to the tax. We expect dq
< 0 and, as the goods are substitutes, dτ dq2 dq2 1 > 0. We refer to the expression dτ / − dq
as the switching ratio; it captures

the extent to which any reduction in consumption of the externality generating
product induced by a marginal increase in the tax rate is redirected towards the
substitute good (taking account of the equilibrium pricing responses).
When the two products are supplied competitively (so µj = 0 for j = {1, 2}
regardless of the level of τ ) the optimal policy is a Pigovian tax (τ ∗ = φ). Whenever
the products are supplied under imperfect competition, the optimal tax rate is equal
to the Pigovian rate plus an adjustment for non-competitive pricing.
Under imperfect competition it is instructive to consider two special cases. First,
suppose demands for the two products are independent (i.e. so qj (p1 , p2 ) = qj (pj )


, is zero, and the
for j = {1, 2}). This implies the switching ratio, dq

equilibrium prices of the two goods are independent of one another. In this case the
For instance, if the two products are supplied by separate firms that compete in a Bertrand
∂qj (p)
game µj = −qj (p)/ ∂p
. Solving the two optimal pricing equations yields equilibrium prices
(p1 (τ ), p2 (τ )) (where we suppress the dependence of prices on marginal cost), and associated
margins (µ1 (τ ), µ2 (τ )).
optimal tax rate is (implicitly defined by) τ ∗ = φ − µ1 , product 1 is priced at the
efficient level, p1 = c1 + φ, and the equilibrium price of product 2 is left unaffected
by the tax. Second, suppose instead there is no switching in or out of the market,
so in response to price changes consumers only reallocate their demand between

/ − dq
= 1. In this case the optimal tax rate
the two products, which implies dq

is τ = φ − (µ1 − µ2 ) and the difference in equilibrium prices of the two products
is p1 − p2 = (c1 − c2 ) + φ. The tax achieves an efficient allocation (of the fixed
consumption level) between the two products.

In practice, dq
/ − dq
is likely to lie somewhere between 0 and 1; the imperfect

competition adjustment to the Pigovian tax rate partly reflects how policy changes
total consumption in the market and partly how it influences the allocation of consumption across the two products. To see this, note that we can re-write equation


. The more
(2.2) as τ ∗ = φ − [(1 − SR) µ1 + SR (µ1 − µ2 )], where SR := dq

strongly the reduction in equilibrium quantity of product 1 from a marginal change

in the tax rate is directed to product 2 (i.e. the closer dq

is to 1), the more

weight is placed on the difference in equilibrium margins of the two goods.
Many differentiated products
In practice, corrective taxes are typically used in markets in which there are many
differentiated products. To the extent that there is variation across the equilibrium
price-cost margins of these products and in whether their consumption generates externalities,14 this will influence the optimal tax prescription. In addition, it matters
whether the tax is levied directly on the product characteristic that is associated
with externalities, or whether the tax is levied on a per unit basis. For instance,
a tax on sugar sweetened beverages can either be levied directly on sugar, or on a
volumetric (i.e. per liter) basis.
Suppose there are many products j = {1, . . . , J}. A subset of products, j ∈ S,
contain an attribute that is associated with an externality, where zj denotes the
amount of the attribute in product j, while for the remaining products, j ∈
/ S
(which we denote by the set j ∈ N ), zj = 0. Consider a tax levied on z. The
products are supplied in an imperfectly competitive environment with equilibrium
prices satisfying:
pj − τ zj − cj = µj ∀j ∈ S
pj − cj = µj ∀j ∈ N .
For instance, in the case of sugar sweetened beverages, a given amount of consumption of a
product with 10g of sugar per 100ml, all else equal, is likely to be associated with more externalities
than one with 5g sugar per 100ml.
In Appendix A we show that in this case the optimal tax rate can be expressed
as follows:
Proposition 1. Define: (i) the derivative of the total equilibrium quantity of the
set of externality generating products with respect to the tax as dQ
= j∈S dτj , (ii)

the share that product j ∈ S contributes to this derivative as wjS = dτj / dQ
, (iii)

the analogous expressions for the set of products that do not generate externalities
dqj dQN
/ dτ ), and (iv) the derivative of the total
(i.e. dQ

equilibrium quantity of the externality generating attribute with respect to the tax
rate as dZ
= j∈S zj dτj . The optimal tax rate is then implicitly defined by:

τ ∗ = φ − dZ dQS

wjS µj −
wjN µj ×




This expression generalizes the optimal tax formula in the two good case (equation (2.2)). Now the rate depends on the weighted average price-cost margin among
the sets of externality and non-externality generating products. As the tax rate
varies, the average margin term may vary for two reasons – (i) firms may reoptimize their prices, changing product level price-cost margins, and (ii) consumers, in
equilibrium, may switch differentially away from/towards products with different
equilibrium margins. The many product optimal tax expression also depends on
the ratio of the marginal change in equilibrium quantity of the externality generating attribute and equilibrium quantity of the externality generating goods with
dQS 15
). This term results from the tax being levied
respect to the tax rate (i.e. dZ

on the externality generating product attribute rather than volumetrically on the
externality generating products (see Appendix A for the expression for a volumetric tax). All else equal, the more effective is the tax at lowering consumption of
attribute z relative to consumption of the z containing products, the higher is the
optimal rate.
Heterogeneity in externalities. Marginal externalities may be heterogeneous,
either because externalities depend non-linearly on an individual’s total intake of
the externality generating attribute, or because, conditional on consumption, some
individuals’ intake is more problematic than others. Let φi denote the marginal ex15
In the case of a tax on the sugar in sweetened beverages, this captures the ratio of the
marginal change in sugar consumption with respect to a small change in the tax over the marginal
change in liters of sugar sweetened beverage consumption with respect to the tax.
ternality for individual i.16 Now the planner must trade-off setting a tax rate that is
too high for those that create relatively small marginal externalities, and one that is
too low for those that generate high externalities. In this case, the externality component in equation (2.3), φ, is replaced by the weighted average (across consumers)
marginal externality, i ωi φi , where the weight, ωi , is the contribution of individual
i to the marginal change in the equilibrium quantity of the externality generating
characteristic with respect to a marginal change in the tax rate (see Appendix A
for the full expression). The more strongly those whose marginal consumption is
most socially costly respond to the tax, then the more effective will be the tax in
correcting for externalities and, all else equal, the higher will be the optimal rate.
The expression i ωi φi takes a similar form as the optimal externality correcting
tax with heterogeneous externalities in a perfectly competitive market, derived in
Diamond (1973). However, in an imperfectly competitive environment, the weights
ωi incorporate the equilibrium pricing response of firms in the market.
Broader externalities. In some circumstances a policymaker may be restricted
to set a tax on a subset of externality generating products, perhaps due to some
political constraint.17 In this case, the corrective component in equation (2.3), φ,
, where dZdτ denotes the marginal reduction in the
is scaled by the ratio dZdτ dZ

externality generating characteristic from taxed and untaxed products associated
denotes the marginal reduction in the
with an increase in the tax rate, and dZ

externality generating characteristic from taxed products only (the full expression
is provided in Appendix A). If, in equilibrium, a marginal increase in the tax rate
induces switching from the taxed to untaxed goods that create an externality, then

dZ A dZ S
< 1, and, all else equal, the optimal tax rate is lower. dτ dτ Full externality internalization. A policymaker may choose to ignore the distortions associated with the exercise of market power, aiming instead at full externality internalization. One approach to doing this is to set a Pigovian tax, τ = φ. Doing this fails to recognize that equilibrium quantities in the market are already below the competitive level, and, as pointed out by Buchanan (1969), can actually be welfare reducing. Even if the policymaker is willing to ignore this and aims at full externality internalization (relative to the zero tax market equilibrium), the pricing 16 When externalities are a non-linear function of intake of attribute z, the total externality P i) individual i creates is Φ(Zi ), where Zi = j zj qij . In this φi denotes dΦ(Z dτ . 17 A leading example is when a good can be imported tax-free (see Fowlie et al. (2016) who study greenhouse gas emissions leakage due to imported concrete). In the case of sugar sweetened beverage taxes, some legislators have argued for a broadening of the base to cover other sources of dietary sugars (for instance, see House of Commons Health Committee (2018)). 11 response of firms can undermine the Pigovian policy. For instance, suppose there is just one product in the market and that equilibrium pass-through of a Pigovian tax is 150%; the tax leads to a price increase in excess of the marginal externality and an exacerbation of market power concerns, as the equilibrium price-cost margin for the taxed good increases. The policymaker can mitigate this issue by adjusting the Pigovian tax rate by the inverse of the equilibrium pass-through rate – setting τ so that τ = φρ , where ρ is the pass-through rate defined as the change in the equilibrium consumer price divided by the tax). In Appendix A we formalize the problem a planner solves when aiming for full externality internalization, relative to the zero tax equilibrium quantities, and show that tax policy will depend on the weighted average pass-through rate across all taxed products, as well as the equilibrium margin adjustment on non-taxed alternatives. Internalities. Corrective taxes are sometime justified on the basis of the presence of internalities – costs consumers impose on themselves by making choices that fail to maximize their underlying utility. Internalities may arise for many reasons including consumer self-control problems, incorrect beliefs and inattention. Our framework accommodates internalities that lead to consumer welfare taking the P form vi (p) − ϕi j zj qij , where ϕi can be interpreted as the marginal internality. We show in Appendix A that if demand is generated from a discrete choice random utility model and internalities arise from consumers over-estimating their underlying preference for a particular attribute (z) when making consumption decisions, the expression for consumer welfare will take this form. 2.4 Empirical implementation We apply our framework to the topical issue of the taxation of sugar sweetened beverages. We estimate consumer demand and firm competition in the UK market for non-alcoholic drinks; the model allows us to simulate equilibrium quantities (allowing for the endogenous response of prices) and price-cost margins for any given tax policy. We calibrate two key parameters over which there is considerable uncertainty: the magnitude of externalities from sugar sweetened beverages, and the degree of market power outside the drinks market. Our analysis assumes that firms compete in their price setting, but hold fixed the portfolio of products they offer and non-price features of these products. A tax that is levied directly on the sugar content of products potentially incentivises firms to reduce the sugar content of some of their products to reduce tax liability (though this will depend on how this changes production costs and the strength of 12 consumer preference for sugar). We return to this point when discussing our results in Section 6. 3 The drinks market We model behavior in the UK market for drinks. Our market definition includes all chilled or ambient non-alcoholic beverages with the exception of water and unsweetened milk. Figure 3.1 shows a classification of drinks that we use to refer to different sets of products throughout the rest of the paper. We refer to one subset of the drinks as soft drinks. These include carbonates, fruit concentrates and sports and energy drinks. Soft drinks can be further divided into sugar sweetened beverages and diet (or artificially sweetened) beverages. We refer those drinks that are not soft drinks as sugary alternatives. These include fruit juice and flavored milk; they are generally exempt from sugar sweetened beverage or soft drinks taxes. Figure 3.1: Drinks classification *drinks refers to all non-alcoholic drinks with the exception of water and unsweetened milk. 3.1 Externalities from sugar sweetened beverages There is considerable evidence that consumption of sugar sweetened beverages increases the risk of developing a number of diseases.18 Sugar sweetened beverages are high in sugar and the sugar is in liquid form; this means it is digested quickly, which leads to spikes in insulin and a higher propensity to develop type II diabetes. Calories consumed in liquid form are also less likely to sate appetites, which means people are less likely to compensate for their intake with reduced calories from other sources and thus consumption of these drinks leads to weight gain. Sugar sweetened beverage intake is also associated with increased blood pressure and a higher risk of cardiovascular disease, as well as causing tooth decay. 18 Allcott et al. (2019b) provide a useful summary of the evidence. The Scientific Advisory Committee on Nutrition (2015) provide a thorough review of the medical literature. 13 The higher disease burden associated with sugar sweetened beverages leads to costs borne by people other than the person consuming the products (i.e. externalities). A central source of externalities are raised public costs of funding heath care systems. These can result from higher taxpayer costs of publicly funded systems and from increased premiums in insurance based systems. For instance, in the UK it is estimated that the costs of treating obesity and related conditions added £5.8 billion in 2006-07 to the costs of public health care provision (Scarborough et al. (2011)). Wang et al. (2012) estimate that a 15% reduction in sugar sweetened beverage consumption in US would lead to a $17.1 billion saving in heath care costs over 10 years; a portion of this saving would be realized by the consumer themselves; however, this portion is likely to be small (for instance, Cawley and Meyerhoefer (2012) estimate 88% of the US medical costs of treating obesity are borne by third parties). These externalities have led many governments, including the UK (Scientific Advisory Committee on Nutrition (2015)), to specifically target reductions in the intake of sugar sweetened beverages. There is also concern about high levels of added sugar (including from foods) in diet more broadly. The World Health Organization recommends average intake of added sugars should not exceed 10% of total dietary energy (World Health Organization (2015)), while the UK has adopted the more stringent target of 5%.19 In our analysis of a sugar sweetened beverage tax, we allow for the possibility that the nature of externalities from sugar sweetened beverages interact with broader dietary sugar, and we consider the implications for the optimal tax on these products if there are externalities created by switching to other markets. 3.2 Purchase data We use micro data on the grocery purchases of a sample of consumers living in Great Britain (i.e. the UK excluding Northern Ireland). The data contain information on household level purchases for home consumption (“at-home”), as well as purchases made by individuals for consumption outside of the home (i.e. “on-the-go”). Onthe-go consumption is an important part of soft drink intake – accounting for 30% of total soft drink consumption and 40% of total sugar consumption from soft drinks.20 Our data are collected by the market research firm Kantar and comprise two parts: 19 These targets are in fact stated in terms of “free sugars”, which are similar to added sugar but also include naturally occurring sugars in fruit juices and honey. 20 Based on our calculations using the National Diet and Nutrition Survey, an individual level dietary intake survey representative of the UK population. 14 the Kantar Worldpanel covers the at-home segment of the market and the Kantar On-The-Go Survey covers the on-the-go segment. The Kantar Worldpanel contains details of all the grocery purchases (including food, drink, alcohol, toiletries, cleaning produce and pet foods) that are made and brought into the home by a representative sample of just over 30,000 British households from January 2008 to December 2012. Participating households use a hand held scanner to record all grocery purchases at the UPC level (i.e at the disaggregate level at which items are barcoded). Households participate in the survey for several months, and the data contain detailed information on the UPCs they buy (including brand, flavor, size and nutrient composition), the store where the transaction took place, and transaction level prices. The Kantar On-The-Go Survey is based on a random sample of just under 3000 individuals drawn from the Worldpanel households. Using a cell phone app, individuals record purchases of food and drinks at the UPC level made on-the-go from shops and vending machines (the data do not cover bars and restaurants). The data contain details of the item they purchased, as well as transaction store and price, from June 2009 to December 2012. Individuals aged 13 and upwards are included in the sample. 3.3 Consumers We use the term consumer to refer to households in the at-home segment, and individuals in the on-the-go segment. In our empirical demand model we incorporate observed and unobserved heterogeneity in consumer preferences. We allow observed heterogeneity across the at-home or on-the-go segments, as well as allowing preferences to vary depending on consumer age and with a measure of the total sugar in the consumer’s diet in the preceding year. This allows us to capture any differences in demand behavior along dimensions over which marginal externalities from sugar sweetened beverage intake might vary. Table 3.1 shows the groups into which we place consumers. In the at-home segment we split households based on whether there are any children (people aged under 18) in the household or not. In the on-the-go segment we separate individuals aged 30 and under from those aged above 30. We also differentiate between those with low, high or very high total dietary sugar. This measure is based on the household’s (or, for individuals in the on-the-go sample, the household to which they belong) share of total calories purchased in the form of added sugar across all grocery shops in the preceding year. We classify those that purchase less than 10% of their calories from added sugar (corresponding to meeting the World Health 15 Organization’s guideline) as “low dietary sugar”, those that purchase between 10% and 15% as “high dietary sugar”, and those that purchase more than 15% of their calories from added sugar as “very high dietary sugar”. Table 3.1: Consumer groups No. of consumers % of sample 7499 11930 7291 3561 8382 5185 17 27 17 8 19 12 240 576 381 601 1319 757 6 15 10 16 34 20 At-home segment (households) No children, low dietary sugar No children, high dietary sugar No children, very high dietary sugar With children, low dietary sugar With children, high dietary sugar With children, very high dietary sugar On-the-go segment (individuals) Under 30, low dietary sugar Under 30, high dietary sugar Under 30, very high dietary sugar Over 30, low dietary sugar Over 30, high dietary sugar Over 30, very high dietary sugar Notes: Columns 2 and 3 show the number and share of consumers (households in the at-home segment, individuals in the on-the-go segment) in each group, respectively. If consumers move group over the sample period (2008-12) they are counted twice, hence the sum of the numbers of consumers in each group is greater than the total number of consumers. Dietary sugar is calculated based on the share of total calories from added sugar purchased in the preceding year; “low” is less than 10%, “high” is 10-15% and “very high” is more than 15%. Households with children are those with at least one household member aged under 18. 3.4 Firms, brands and products In Table 3.2 we list the main firms that operate in the drinks market and the brands that they own. We focus on the principal brands in the market; these comprise over 75% of total spending on non-alcoholic drinks in both the at-home and the on-thego segments.21 The firms Coca Cola Enterprises and Pepsico/Britvic dominate the market, having a combined market share exceeding 65% in the at-home segment and close to 80% in the on-the-go segment. Each of these firms owns several well recognized and long established brands, including some soft drinks and fruit juice brands. The most popular single brand is Coke (also known as Coca Cola), which accounts for over 20% of the at-home and 36% of the on-the-go market segment. 21 The brands include all soft drinks brands with more than 1% market share in either segment, as well as the main fruit juice and flavored milk brands. For some brands, there are only a very small number of transactions in one of the two segments of the market; we therefore omit these brands from the choice sets in that segment. 16 In addition to the main branded products, we include store brands in our analysis; these are popular in the at-home segment. The majority of soft drinks brands are available in sugar sweetened (“regular”) and artificially sweetened (“diet” and/or “zero”) variants. In Table 3.3 we list the variants available for each brand. Among the regular variants there is variation in sugar content across brands – many of the carbonates have around 10g of sugar per 100ml, with some of the fruit flavored soft drinks (such as Oasis and Vimto) having less sugar per 100ml. This variation in sugar content means a tax levied directly on sugar will have different implications to one levied volumetrically (i.e. per liter of product sold). Brand-variants can be purchased in different sizes for two reasons: (i) the availability of different pack sizes (or UPCs), and (ii) the purchase of multiple units. For instance, a consumer may choose to purchase one 2l bottle of Diet Coke, or a pack of 6×330ml cans, or two 2l bottles, and so on. Purchases of multiple units of the same brand-variant most commonly involve 2, or sometimes 3, units of the same pack (or UPC) and are typically a consequence of multi-buy offers. Multi-buy offers in the UK market are long running, so the set of UPCs for which multiple units are popular is broadly stable over time. We incorporate the choice consumers make over size into our model of demand. Specifically, we define products as brand-variant-size combinations, and we model the consumer’s choice of product from a discrete set of alternatives. For each brandvariant, the set of possible sizes includes both the available pack sizes (i.e. UPCs) and the most common multiple unit purchases of UPCs.22 Table 3.3 shows, for each brand-variant, the number of sizes available to consumers in the at-home and on-the-go segments. For instance, Diet Coke is available in 10 sizes in the at-home segment, and two sizes in the on-the-go segment.23 On-the-go sizes are always designed as a single serving, while at-home sizes are typically multi-portion. 22 Specifically, we include a size option corresponding to multiple units of a single UPC if that UPC-multiple unit combination accounts for at least 10,000 (around 0.2%) of transactions. This means that for over 75% of transactions of branded products, we accurately model the choice over number of units to purchase. 23 These are, in the at-home segment, 1.25l and 2l bottles, multi-packs of 330ml cans containing 6, 8, 10 and 12 cans, two- and three- unit purchases of 2l bottles, and two-unit purchases of 6-pack and 8-packs of cans; and, in the on-the-go segment, a 500ml bottle and 330ml can. 17 Table 3.2: Firms and brands Market share (%) Firm Brand Type Coke Capri Sun Innocent fruit juice Schweppes Lemonade Fanta Dr Pepper Schweppes Tonic Sprite Cherry Coke Oasis Soft Soft Fruit Soft Soft Soft Soft Soft Soft Soft Robinsons Pepsi Tropicana fruit juice Robinsons Fruit Shoot Britvic fruit juice 7 Up Copella fruit juice Tango Soft Soft Fruit Soft Fruit Soft Fruit Soft JN Nichols Ribena Lucozade Lucozade Sport Vimto Barrs Price (£/l) At-home On-the-go At-home On-the-go Soft Soft Soft Soft 33.0 20.4 3.1 2.1 1.7 1.7 1.2 1.1 1.0 0.8 – 33.6 10.7 10.1 6.1 2.6 1.6 0.9 0.8 0.8 7.6 3.3 3.1 1.1 1.6 59.1 36.4 – 1.6 – 5.3 3.4 – 2.8 4.0 5.6 20.0 – 11.6 3.8 0.8 – 1.7 – 2.2 12.7 3.4 6.4 2.9 – 0.86 1.08 2.03 0.44 0.79 0.75 1.22 0.77 0.96 – 2.09 – 7.09 – 2.10 2.08 – 2.08 2.17 2.15 1.09 0.64 1.62 1.49 2.17 0.70 1.74 0.66 – 1.93 3.63 2.83 – 1.88 – 1.73 1.69 1.11 1.15 1.06 2.20 2.37 2.22 – Irn Bru Soft 0.6 2.6 0.61 1.93 Merrydown Shloer Soft 2.0 – 1.79 – Red Bull Red Bull Soft 0.2 3.5 3.67 5.27 Muller Frijj flavoured milk Milk – 1.4 – 1.90 Friesland Campina Yazoo flavoured milk Milk – 0.8 – 1.95 Store brand soft drinks Store brand fruit juice Soft Fruit 21.2 13.1 8.1 0.0 – – 0.62 1.05 – – Coca Cola Enterprises Pepsico/Britvic GSK Store brand Notes: Type refers to the type of drinks product: “soft” denotes soft drinks, “fruit” denotes fruit juice, and “milk” denotes flavored milk. The fourth and fifth columns display each firm and brand’s share of total spending on all listed drinks brands in the at-home and on-the-go segments of the market; a dash (“–”) denotes that the brand is not available in that segment. The final two columns display the mean price (£) per liter for each brand. 18 Table 3.3: Brands, sugar contents and sizes Sugar Number of sizes Firm Brand Variant (g/100ml) At-home On-the-go Coca Cola Enterprises Coke Diet Regular Zero Regular Regular Diet Regular Diet Regular Diet Regular Diet Regular Diet Regular Diet Regular Diet Regular Diet Regular Diet Max Regular Regular Diet Regular Regular Diet Regular Diet Regular Regular Diet Regular Regular Diet Regular Diet Regular Diet Regular Regular Diet Regular Regular Regular Diet Regular Regular 0.0 10.6 0.0 10.9 10.7 0.0 4.2 0.0 7.9 0.0 10.3 0.0 5.1 0.0 10.6 0.0 11.2 0.0 4.2 0.0 3.2 0.0 0.0 11.0 9.6 0.0 10.3 9.9 0.0 10.8 0.0 10.1 3.5 0.0 10.8 11.3 0.0 3.6 0.0 5.9 0.0 8.7 9.1 0.0 10.8 10.8 9.5 0.0 10.3 10.4 10 9 7 3 4 2 2 2 2 2 2 2 2 2 2 2 2 – – 6 6 5 6 5 4 2 2 2 2 2 1 2 3 2 4 3 1 1 3 4 1 1 3 – 1 – – 4 2 2 2 2 2 – 1 – – 1 2 1 2 – – – 2 1 2 1 1 – – 2 2 2 1 1 – – 1 2 – – 2 1 2 2 1 1 – – 2 2 – 1 1 1 1 – – – Capri Sun Innocent fruit juice Schweppes Lemonade Fanta Dr Pepper Schweppes Tonic Sprite Cherry Coke Oasis Pepsico/Britvic Robinsons Pepsi Tropicana fruit juice Robinsons Fruit Shoot Britvic fruit juice 7 Up Copella fruit juice GSK Tango Ribena Lucozade Lucozade Sport JN Nichols Vimto Barrs Irn Bru Merrydown Red Bull Shloer Red Bull Muller Friesland Campina Store brand Frijj flavoured milk Yazoo flavoured milk Store brand soft drinks Store brand fruit juice Notes: The final two columns displays the number of sizes of each brand-variant in the at-home and on-the-go segments of the market; a dash (“–”) denotes that the brand-variant is not available in that segment. 19 3.5 Choice sets and price measurement Table 3.3 summarizes the full set of products available to consumers in each market segment. However, the set of products available to a consumer on a particular day, as well as the price vector they face, will depend on the retailer that they visit. At-home segment The median household undertakes a grocery shop once a week. We define a “choice occasion” as any week in which a household purchases groceries, and model what, if any, drink a household purchases on a choice occasion.24 We observe households for an average of 36 choice occasions each year, and in total, we have data on 3.3 million at-home choice occasions. On around 42% of choice occasions, a household purchases a drink, with the average time between drink purchases being 12 days. Households select one brand-variant (as defined by columns 2 and 3 of Table 3.3) on 60% of choice occasions on which drinks are purchased. On choice occasions in which a household chooses multiple (typically 2 or 3), we assume that (conditional on household specific preferences) these purchases are independent (for instance, because they are bought for different household members). For each choice occasion we observe the retailer in which the purchase was made and the exact price paid. Table 3.4 lists retailers and the share of drinks spending that they account for in each segment. In the at-home segment, four large national supermarket chains account for almost 90% of spending, with the remaining expenditure mostly being made in smaller national retailers. Each of these retailers offers all brands, with some variation in the specific sizes available in each retailer. We model the decision consumers take over what to purchase from the available set of products in the retailer they visit, taking their choice over which retailer to shop with as given. This assumption is common in models of consumer good choice.25 In our context this assumption is reasonable. On the median choice occasion a consumer visits one retailer, and expenditure on non-alcoholic drinks comprises a small share (4%) of total grocery expenditure. Retailer choice is likely to be driven by proximity of nearby stores and overall preferences for grocery outlet 24 We focus on households that record making regular purchases; this excludes transactions (accounting for less than 2% of the total number) made by households who record making fewer than 10 shopping trips a year. We also focus on households who record making at least one non-alcoholic drink purchase. 25 An exception is Thomassen et al. (2017), who show that switching across supermarkets can influence pricing incentives for aggregated grocery goods (e.g. meat, dairy etc.). 20 (which we control for in demand). In practice, the majority of consumers’ drinks expenditure (over 70%) is made in the retailer they more frequently visit. The four main retailers in the UK implement national pricing policies.26,27 This means that if we observe a transaction price for a particular UPC in a store belonging to one of the retailers, Tesco say, we know the price that consumers shopping in other Tesco stores at the same time faced for that UPC. Using the large number of transactions in our data we can construct the price vector households faced in each retailer in each week. For the smaller retailers we construct a mean transaction price for a product as a measure of the price faced by consumers. Table 3.4: Retailers Expenditure share (%) at-home on-the-go Large national chains of which: Tesco Sainsbury’s Asda Morrisons 87.0 19.9 34.7 16.8 19.8 15.7 – – – – Small national chains 10.7 16.4 Vending machines 0.0 9.2 Convenience stores in region: South Central North 2.3 54.6 – – – 13.6 15.5 25.5 Notes: Numbers show the share of total non-alcoholic drink expenditure, in the at-home and onthe-go segment, made in each retailer. On-the-go segment The natural periodicity for on-the-go purchases is at the daily level.28 In this segment we define a choice occasion as any day on which the individual buys a cold beverages (including bottled water). We observe individuals for an average 26 The supermarkets agreed to implement national pricing policies following a Competition Commission investigation into supermarket behavior (Competition Commission (2000)). 27 Close to uniform pricing within retail chains has been documented in the US, see, for instance, DellaVigna and Gentzkow (2019) and Hitsch et al. (2017). 28 As in the at-home segment, we focus on individuals who record regularly, dropping less than 3% of total transactions that are made by those who record fewer than 5 purchases each year. 21 of 44 choice occasions each year, and in total, we have data on 286,576 on-the-go choice occasions. On 60% of choice occasions individuals choose to buy one of the products listed in Table 3.3, and on 90% of these choice occasions they buy only one product.29 The large four supermarkets are less prominent in the on-the-go segment, collectively accounting for less than 20% of on-the-go spending on drinks (see Table 3.4). This, coupled with the fact that the single portion cans and bottles are similarly priced across the large four supermarkets, motivates their aggregation into one composite retailer. The majority of transactions in the on-the-go segment are in local convenience stores. This means that for these choice occasions, unlike in the at-home segment, we do not observe the price of non-selected products in consumers’ choice sets. Therefore, in the case of convenience stores, for all options in consumer choice sets we use a mean monthly price, where the price is constructed using all convenience store transactions in each of three regions (the south, central, and north regions of the UK). Dependence across the at-home and on-the-go segments We model consumer choice for at-home consumption and for on-the-go consumption separately.30 A concern with this is that recent at-home household purchases influence decisions that individuals make on-the-go (for instance, a recent at-home purchase may make an individual less likely to buy while on-the-go). We check for evidence of such non-separabilities across the at-home and on-the-go segments. Specifically, for individuals in the on-the-go sample we test whether recent purchases of drinks by their household in the at-home segment influences either their propensity to purchase drinks or the quantity they buy, finding no evidence of such dependence. Full details are provided in Appendix B. 3.6 Price variation The vector of prices that a consumer faces when making a purchase varies across time and retailers. Here we describe this variation and in Section 4.2 we discuss how it allows us to identify the key parameters driving consumer demand behavior. 29 On the rare case when they buy multiple products (usually 2 or 3) we treat these as independent purchases. 30 When constructing market level demand we weight each segment such that their share of total sugar from sugar sweetened beverage matches that in the National Diet and Nutrient Survey (an individual level dietary intake survey, representative of the UK population). 22 The at-home segment is characterized by products that are sold in multi-portion sizes, and it is dominated by retailers that have national pricing policies. An important source of price variation is promotions (i.e. price reductions), which differ in their timing, duration and depth, both across UPCs and retailers. In the drinks market promotions are either multi-buy offers (for instance, a discount for purchasing 2 of the same UPC), or ticket price reductions (when a UPC has a temporarily low price). 30% of the transaction in our data are a multi-buy offer, 20% a ticket price reduction. We provide a graphical example of each promotion type in Figure 3.2, which shows the price for two specific UPCs over the most recent year of our data for two different retailers (Tesco and Sainsbury’s). Panel (a) shows price series for a 2l bottle of Coke. In both retailers, (with the exception of one week in Tesco) 1 unit of a 2l bottle is priced at £2. However, over most of the year each retailer runs a multi-buy offer, where 2 bottles can be purchased at a discounted per bottle price, though the depth of discount varies both over time and between retailers.31 Panel (b) shows price series for a pack of 12×330ml cans of Coke. This UPC does not have a multi-buy offer, but is reasonably frequently subject to a ticket price reduction. Figure 3.2: Examples of price variation for Coke options (b) 12x330ml cans 5 Price per unit (£) 4 4.5 1 3.5 1.2 Price per unit (£) 1.4 1.6 1.8 2 (a) 2l bottle 0 10 20 Tesco: 1 unit Tesco: 2 units Week 30 40 50 0 Sainsbury's: 1 unit Sainsbury's: 2 units 10 20 Week 30 40 50 Tesco: 1 unit Sainsbury's: 1 unit Notes: Panel (a) shows the weekly price series for a 2l bottles of Coke in Tesco and Sainsbury’s when either one unit or two units are purchased. Prices are expressed per unit. Panel (b) shows the weekly price series for a pack of 12x330ml cans of Coke in Tesco and Sainsbury’s when one unit is purchased. In the examples in Figure 3.2 average prices are similar across the two retailers, but the time path of price changes is different. This is true more generally. To illustrate this we compute measures of price stability suggested by DellaVigna and 31 In our demand model we treat one 2l bottle and two 2l bottles of Coke as different options. 23 Gentzkow (2019). First, we calculate the yearly absolute log price difference.32 This entails, for each product, retailer and year, computing average log price (where the average is taken across weeks), and then computing the deviation in this for each retailer pair. The median of these deviations is 8 log points, indicating a relatively low level of cross-sectional differences in average prices across retailers. Second, we calculate the weekly correlation of log prices. To do this we obtain the residuals from regressing log prices on product-year fixed effects for each retailer. For each product we compute the correlation in residuals between each retailer pair. The median of these correlations is 0.13, indicating that the degree of co-movement in prices over time across retailers is low. In addition, no retailer sets systematically low or high prices – among the big four retailers, across product-weeks (for products that are branded and available in multiple retailers), Asda is the cheapest retailer most (for 27% of product-weeks) and the most expensive the least (for 17%) amount of time, and Sainsbury’s is cheapest the least (for 22%) and most expensive the most (for 31%) amount of time. A concern with relying on price variation from promotions to estimate demand is that households respond to them by intertemporally switching their purchases (i.e stocking up during sales) and hence failing to model this behavior will result in an overestimate of own-price elasticities (Hendel and Nevo (2006a)). A number of papers have documented evidence of stockpiling in the US market for soft drinks (see Hendel and Nevo (2006b), Hendel and Nevo (2013), Wang (2015)). Although we cannot rule out that there may be some stockpiling underlying transactions in our data, the evidence for it is much less clear than in the US. Specifically, UK households purchase drinks, on average, more than twice as frequently (every 12 days on average) as those in the US (see Hendel and Nevo (2006b)), and when a household does purchase on sale there is no meaningful change in the timing of purchases.33 Instead, we find that sales are associated with switching across pack types (e.g. cans to bottles), brands and sizes. One reason why stockpiling is less prevalent in the UK, compared with the US, is that the relatively long running nature of UK price promotions create less incentives to stockpile. For instance, for each of the soft drinks products summarized in Table 3.3, the average time between a price change of 25% or more is 10 weeks, whereas in the US prices can fluctuate by large amounts from week to week (see an archetypical example in Figure 1 of 32 DellaVigna and Gentzkow (2019) instead compute the quarterly absolute log price difference. Given the relatively long running nature of promotions in the UK we choose instead to do this at the yearly level. 33 Hendel and Nevo (2006b) find, in the US, buying soft drinks on sale is associated with an average reduction in the time from previous purchase of 3 days, and an increase to the next purchase of 2.5 days. We find changes of 0.23 and 0.14 respectively. See Appendix B. 24 Hendel and Nevo (2013)). A second reason is that transport and storage costs in the UK are likely to be much higher, with the average size of UK homes around half of those in US, and vehicle ownership rates 25% lower.34 In the on-the-go segment only 20% of spending is done in the big four supermarkets, with around 55% of expenditure occurring in conveniences stores. Price promotions are less common in this segment, with price variation driven by regional differences in price in convenience stores, and differences in convenience store prices with national retailers and vending machines. 4 Estimating demand and supply To implement our optimal corrective tax framework, we need to know how consumers switch across disaggregate products in response to price changes and the level of and how firms, in response to tax, adjust the price-cost margins on these products. We estimate a model of consumer demand in the drinks market using a discrete choice framework in which consumer preferences are defined over product characteristics (Gorman (1980), Lancaster (1971), Berry et al. (1995)). This approach enables us to model demand and substitution patterns over the many differentiated products in the market, while incorporating rich preference heterogeneity crucial to capturing realistic substitution patterns. We identify firms’ unobserved marginal costs by coupling our demand estimates with the equilibrium conditions from an oligopoly pricing game (Berry (1994), Nevo (2001)). 4.1 Consumer demand We model which, if any, drink product a consumer (indexed i) chooses on a choice occasion. We treat the decisions that households make in the at-home segment and individuals make in the on-the-go segment separately, allowing for the possibility that preferences vary on each type of choice situation, but for notational parsimony we suppress a market segment index. We index the drink products by j = {1, . . . , J}. The products vary by brand, which we index by b = {1, . . . , B}, size, indexed by s = {1, . . . , S}, and whether or not they contain sugar (for instance, the brand Coke comes in Regular, Diet and Zero variants). The consumer chooses between the available drinks products, and choosing not to buy a drink, which we denote by j = 0. The set of products 34 The mean floor space of UK homes in 2008 was 85m2 , while in 2009 in the US it was 152m2 (UK Government (2018)). In 2014 the US had 816 vehicles per capita (U.S. Department of Energy (2019)), in 2017 the UK had 616 (ACEA (2019)). 25 available to the consumer, as well as the prices they face, depends on which retailer they visit – we index retailers by r and denote the set of available drink options in retailer r by Ωr . Consumer i in period t, with total period income or budget yit , solves the utility maximization problem: V (yit , prt , xt , it ; θi ) = max ν(yit − pjrt , xjt ; θi ) + ijt . j∈{Ωr ∪0} (4.1) where prt = (p1rt , . . . , pJrt ) is the price vector faced by the consumer, xjt are (nonprice) characteristics of product j and xt = (x1t , . . . , xJt ) (note p0 = 0 and x0 = 0); θi is a vector of consumer level preference parameters; and it = ( i0t , i1t , . . . , iJt ) is a vector of idiosyncratic shocks. The function ν(.) captures the payoff the consumer gets from selecting option j. Its first argument, yit − pjrt , is spending on the numeraire good – i.e. spending outside the drinks market. We assume that preferences are quasi-linear, so yit − pjrt enters ν(.) linearly. This means that yit differences out when the consumer compares different options; we therefore suppress the dependency of ν(.) on yit . We assume that ijt is distributed i.i.d. type I extreme value. Under this assumption the probability that consumer i selects product j in period t, conditional on prices, product characteristics and preferences, is given by: σj (prt , xt ; θi ) = exp(ν(pjrt , xjt ; θi )) P , 1 + j 0 ∈Ωr exp(ν(pj 0 rt , xj 0 t ; θi )) (4.2) and the consumer’s expected utility is given by: v(prt , xt ; θi ) = ln X exp{ν(pjrt , xjt ; θi )} + C, (4.3) j∈Ωr where C is a constant of integration. Specification details Let d = (1, . . . , D) index the consumer groups shown in Table 3.1. We assume that the payoff function ν(.) for consumer i belonging to consumer group d(i) and for product j belonging to brand b(j) and of size s(j) takes the form: (1) (2) ej + γd(i) x ν(.) = −αi pjrt + βi x ejt + ζd(i)b(j)s(j)rt , where (1) (2) (3) (4) (5) ζd(i)b(j)s(j)rt = ξd(i)b(j)s(j) + ξd(i)b(j)r + ξd(i)b(j)t + ξd(i)s(j)r + ξd(i)s(j)t . 26 We allow for consumer specific preferences for price (i.e. the marginal utility of e(1) e(1) income) and a subset of product characteristics denoted by x includes a j ; x j constant, which captures a preference for drinks versus not buying them, dummy variables indicating whether the product has positive sugar content that is less than 10g or equal to or more than 10g per 100ml, dummy variables indicating if the product is a cola, lemonade, store brand soft drink or fruit juice, and an indicator for whether the size is large.35 These individual level preferences play a key role in (2) allowing the model to capture realistic substitution patters across products. x ejt is a measure of the stock of advertising for the product in the current period;36 we allow the effect of advertising to vary across consumer groups. ζd(i)b(j)s(j)rt denotes a set of consumer group specific shocks to utility. These include: brand-size effects, which control for unobserved consumer preferences that are time-invariant; brand- and size-retailer effects, which capture the possibility that, on average, consumer preferences over brand and size differ across retailers; and brand- and size-time effects, that control for shocks to demands through time. We model the consumer specific preferences, (αi , βi ) as random coefficients. We specify the distribution for αi as log-normal and βi as normal, both conditional on consumer group d. The overall random coefficient distribution is a mixture of normal distributions.37 As well as enabling us to capture realistic patterns of substitution across products, inclusion of rich unobserved heterogeneity also adds flexibility to the curvature of market demand (see Griffith et al. (2018)), which is important for recovering realistic patterns of pass-through (Weyl and Fabinger (2013)). 4.2 Identification Our key identification assumption is that, conditional on our demand controls (including those for unobserved product attributes), the residual price variation is exogenous (and, in particular, the shocks to consumer’s payoff functions, ijt , are i.i.d.). The main form of price variation we exploit is differential time series variation across retailers. 35 Defined as larger than 2l in the at-home segment or 500ml in the on-the-go segment. We measure monthly TV advertising expenditure in the AC Nielsen Advertising Digest. We compute product specific stocks based on a monthly depreciation rate of 0.8. This is similar to the rate used in Dubois et al. (2018) on similar data in the potato chips market. 37 The means (conditional on d) of the constant, cola, lemonade, store brand, fruit juice and (1) large random coefficients are collinear with ξd(i)b(j)s(j) . We normalize them to zero. We allow for correlation (conditional on d) between the preferences for non-alcoholic drinks and sugar. 36 27 (1) Our demand controls include brand-size effects, ξd(i)b(j)s(j) ; these control for timeinvariant unobserved characteristics that vary across brands and sizes. For instance, consumers may value one brand over a second for reasons not fully controlled for by observed product characteristics; failure to control for this would likely result in correlation between ijt and prices. By interacting brand with size effects we allow for the possibility that strength of unobserved brand effects vary across product sizes (and pack types). Numerous brand-sizes are available in both sugar sweetened and diet variants. We control for the amount of sugar per 100ml in a product in e(1) the option characteristic vector, x j . We are therefore able to identify the mean (as well as standard deviation) of the consumer group specific preferences for sugar (based on the restriction that the impact of sugar on utility does not vary across brands). (3) The time (quarterly) varying brand effects, ξd(i)b(j)t , control for shocks to national level demands for each brand. We additionally control for time varying size effects (5) ξd(i)s(j)t , which capture any tendency through time for demands for larger versus small sizes to fluctuate. As discussed in Section 3.2, the large four retailers that dominate the market have national pricing policies; the time varying effects help control for national level shocks to demand that could be correlated with these (2) prices. In addition, we control (through x ejt ) for product level advertising, which will capture the effect on demand of the (overwhelmingly national) advertising in the UK drinks market.38 For convenience stores we use mean regional prices. We include region-quarter varying drinks effects in demand to control for the possibility of regional shocks to demand for drinks. (2) (2) We also control for brand-retailer effects, ξd(i)b(j)r , and size-retailer effects, ξd(i)s(j)r . These capture the possibility that either the prominence of products belonging to different brands, or of large versus small sizes, may vary across retailers. They also capture average differences in consumer brand and size preferences across retailers. An important restriction we make is the absence of retailer-time shocks to product demands that correlate with price setting.39 As outlined in Section 3.6 average prices across retailers are similar, but co-movement in prices is low, with, for instance, the use of price promotions not synced across retailers. We assume that this create randomness in the prices faced by consumers that is not a consequence of retailers anticipating time varying demand shocks that differ for their consumers compared to those in other retailers. Given the national nature of much retailing, 38 Note targeted price discounts through use of coupons – common in the US (see Nevo and Wolfram (2002)) – is not a feature of the UK market. (2) (3) (4) (5) 39 The (ξd(i)b(j)r , ξd(i)b(j)t , ξd(i)s(j)r , ξd(i)s(j)t ) effects control for all pairwise interactions between (b, s, r, t) but not higher order interactions. 28 pricing and advertising in the UK drinks market, and the absence of targeted price offers and coupons, we believe that this assumption is reasonable. 4.3 Supply model We model price competition among the firms operating in the UK drinks market. We assume that they simultaneously set prices to maximize profits in a NashBertrand game, abstracting from modeling retailer-manufacturer interactions. This outcome can be achieved by use of non-linear vertical contracts (see Villas-Boas (2007), Bonnet and Dubois (2010)).40 In Section 5.4 we show how our optimal tax results are influenced by different supply-side models. Let pm = (p1m , . . . , pJm ) denote the prices that drinks firms set in market m, where markets are temporal (and defined as quarters).41 Market demand for product j is given by: Z qjm (pm ) = σj (pm , xm ; θi )dF (θ)Mm , where Mm denotes the potential size of the market.42 We denote the marginal cost of product j in market m as cjm .43 We index the drinks firms by f = (1, . . . , F ) and denote the set of products owned by firm f by Jf . Firm f ’s total variable profits in market m are Πf m (pm ) = X (pjm − cjm )qjm (pm ). (4.4) j∈Jf We assume firms engage in Bertrand competition and that the prices we observe in the data are the Nash equilibrium outcome of this game, and thus they satisfy the 40 Non-linear contracts with side transfers between manufacturers and retailers allow them to reallocate profits and avoid the double marginalization problem. Bonnet and Dubois (2010) show evidence of price equilibria in the French bottled water market consistent with use of non-linear contracts. 41 In the supply model we average over price variation within a quarter, as this is likely to reflect random price promotions strategies rather than fundamentals of demand or supply. Specifically, let M denote the set P of (r, t) pairs observed in market m, the market price for product j is defined as pjm = (|M|)−1 (r,t)∈M pjrt . 42 Mm is the potential number of non-alcoholic drinks transactions in market m, it differs from the true market size due to inclusion in the demand model of the option to purchase no drinks. 43 Note, in Section 2 we express quantity in terms of units (i.e. liters) and prices and marginal costs per liter. Here we express quantity as number of transactions and price and marginal cost per transaction. The difference is one of convenience rather than substance, multiplying qjm by the size of the product and dividing pjm and cjm by the size of the product transforms the variables into their analogues in Section 2 without changing the nature of the firms’ problem. 29 set of first order conditions: ∀f and ∀j ∈ Jf qjm (pm ) + X j 0 ∈J (pj 0 m − cj 0 m ) f ∂qj 0 m (pm ) = 0. ∂pjm (4.5) From this system of equations we can solve for the implied marginal cost, cjm , for each product in each market. Counterfactual market equilibrium. When solving for the optimal tax rate we need to solve for the associated counterfactual market equilibrium. Denote the set of sugar sweetened beverages by S and the total sugar content of option j ∈ S by zj (noting that for j ∈ / S zj = 0). If some tax rate τ , levied on the sugar in sweetened beverages, is in place the set of first order conditions are: ∀f and ∀j ∈ Jf qjm (p0m ) + X (p0j 0 m − τ zj 0 − cj 0 m ) j 0 ∈Jf ∂qj 0 m (p0m ) = 0. ∂pjm For any τ , we can solve the system of equations to obtain the vector of counterfactual equilibrium prices, p0m = (p01m , . . . , p0Jm ).44 Solving for the optimal tax rate also requires us to compute the derivative of 0 the equilibrium price vector with respect to the tax rate, dpdτm . To obtain this we differentiate the first order conditions with respect to the tax rate and solve the resulting system of equations. For details see Appendix E. 4.4 Demand estimates We estimate the demand model outlined in Section 4.1 using simulated maximum likelihood.45 The estimated coefficients exhibit some intuitive patterns; those with more added sugar in their diets (based on their purchases in the preceding year) have stronger preferences for high sugar drinks products, and those with below median income are more sensitive to price, have stronger preferences for soft drinks and weaker preferences for fruit juice. The variance parameters of the random coefficients are all significant both statistically and in size, indicating an important 44 Note that for the set of store brand products, we do not model price re-optimization – for store brand sugar sweetened beverages we assume pass-through of any tax is 100%, and for store brand diet beverages we assume consumer prices remain unchanged. 45 We allow all parameters to vary by consumer group and estimate the choice model separately by groups. In the at-home segment, for each group, we use a random sample of 1,500 households and 10 choice occasions per household; in the on-the-go sample we use data on all individuals in each group and randomly sample 50 choice occasions per individual, weighting the likelihood function to account for differences in the frequency of choice occasion across consumers. We conduct all post demand estimation analysis on the full sample. 30 role for unobserved preference heterogeneity. We report the coefficient estimates in Appendix C. The estimated preference parameters jointly determine our demand model predictions of how consumers switch across products as prices change. The model generates a large matrix of market level own- and cross-price demand elasticities; in Table 4.1 we summarize the market level own- and cross-price elasticities. The mean own-price elasticity is around -2.5, though with significant variation around this: 25% of products have own-price elasticities with magnitude greater than 2.8, a further 25% of products have own-price elasticities with magnitude less than 1.8. The distribution of the cross-price elasticities exhibits a high degree of skewness, with the mean being equal to the 75th percentile. This reflects consumers being significantly more willing to switch between products close together in product characteristic space. Table 4.1: Summary of own- and cross-price elasticities No. elasticities Own-price Cross-price Percentile per market Mean 25th 50th 75th 175 18757 -2.431 0.013 -2.795 0.003 -2.434 0.007 -1.765 0.014 Notes: In each market there are 175 own-price elasticities (one for each product) and 18757 crossprice elasticities (between product pairs available either in the at-home or on-the-go segment). Numbers summarize the distribution of market elasticities based on the most recent year covered by our data (2012). Table 4.2 illustrates consumers’ tendency to switch between similar products by showing product level elasticities associated with a price change for the single portion sizes of Coke Regular and Coke Diet. It shows the impact on demand for each of the single portion sizes of Coke and Pepsi, and the mean elasticities for other sugar sweetened and diet beverages, and for fruit juice and flavored milk. The table illustrates a number of intuitive patterns: (i) consumers are more willing to switch across cola products of the same variety (sugar vs. non-sugar) than they are to alternative drinks; (ii) consumers are more willing to switch between products of the same size than they are to different sizes; (iii) consumer substitution from sugary varieties of Coke to sugary non-cola drinks (both sugar sweetened beverages, fruit juice and flavored milk) is stronger than it is from Diet Coke. Similar patterns are present for multi-portion products. In Appendix C we report product level own and cross-price elasticities for popular products in the at-home and on-the-go segments. 31 Table 4.2: Select elasticities for Coke Coke Regular 330 500 Regular 330 500 Diet 330 500 -1.66 0.46 0.14 -2.24 0.10 0.08 0.02 0.16 Pepsi Other drinks Diet 330 500 Zero 330 500 Regular 330 500 Diet 330 500 Max SSBs 330 500 Diet Fruit Flav. juice milk 0.12 0.10 0.04 0.25 0.14 0.10 0.04 0.25 0.44 0.39 0.13 0.86 0.11 0.11 0.04 0.25 0.13 0.11 0.04 0.26 0.07 0.19 0.01 0.06 0.05 0.13 0.07 0.20 -1.60 0.10 0.28 -2.63 0.37 0.24 0.10 0.56 0.09 0.07 0.02 0.15 0.33 0.30 0.10 0.53 0.31 0.26 0.10 0.53 0.02 0.05 0.05 0.18 0.02 0.04 0.01 0.04 Notes: Numbers show the mean price elasticities of market demand (for products listed in top row) in the most recent year covered by our data (2012) with respect to price changes for the single portion sizes of Coke Regular and Coke Diet (shown in first column). “Other drinks” exclude Coke and Pepsi and are means over products belonging to each set. In Table 4.3 we summarize the effects of increasing the price of all sugar sweetened beverages by 1%. The resulting reduction in demand (in liters) for sugar sweetened beverages is 1.48% (i.e. our estimates correspond to an own-price elasticity for sugar sweetened beverages of 1.48%). The diversion ratio (defined as the percentage of the reduced sugar sweetened beverage demand that is diverted to each group of substitute products) is 27.3% for diet drinks and 6.7% for alternative sugary drinks. The percent change in expenditure on non-alcoholic drinks is 0.05% – the price increase leads to a modest increase in drinks expenditure. 95% confidence intervals are given in brackets.46 Table 4.3: Switching from sugar sweetened beverages Own-price elasticity of Diversion ratio Elasticity of sugar sweetened beverages Diet beverages Sugary alternatives drinks expenditure -1.48 27.3% 6.7% 0.05 [-1.52, -1.43] [26.8%, 28.1%] [6.4%, 7.0%] [0.04, 0.07] Notes: We simulate the effect of a 1% price increase for all sugar sweetened beverage products. Column 1 shows the % reduction in volume demanded of sugar sweetened beverages, columns 2 and 3 shows how much of the volume reduction is diverted to diet beverages and sugary alternatives and column 4 shows the percent changes in total non-alcoholic drinks expenditure. Numbers are for the more recent year covered by our data (2012). 95% confidence intervals are given in square brackets. The optimal tax formula, given by equation (2.3), partly depends on how much any reduction in the equilibrium quantity of taxed drinks induced by a marginal 46 To calculate the confidence intervals, we obtain the variance-covariance matrix for the parameter vector estimates using standard asymptotic results. We then take 100 draws of the parameter vector from the joint normal asymptotic distribution of the parameters and, for each draw, compute the statistic of interest, using the resulting distribution across draws to compute Monte Carlo confidence intervals (which need not be symmetric). 32 tax increase is shifted to non-taxed substitutes. The diversion ratios suggest a significant amount of demand for sugar sweetened beverages will be diverted to nontaxed drinks, while the elasticity of total non-alcoholic drinks expenditure indicates only a modest degree of switching from the numeraire. However, these diversion ratios and elasticities summarize the demand effects at observed prices. The optimal tax formula depends on changes in equilibrium quantities (which depend on supply responses) and are evaluated at the optimal tax rate. We fully incorporate this when we solve for the optimal tax rate in Section 5. 4.5 Supply estimates We use the first order conditions of the firms’ profit maximization problem (equation (4.5)) to solve for the marginal cost of each product. This enables us to compute the equilibrium price-cost margins (which we express per liter) and price-cost mark-ups (margin over price) at the observed market equilibrium (where no sugar sweetened beverage tax is in place). In Table 4.4 we summarize the distribution of (observed) prices, costs, margins and mark-ups across products. The average mark-up is 0.55 (price is around double marginal cost), though there is considerable variation around this.47 In Appendix C we show mean marginal costs, margins and mark-ups by brand. Equilibrium price-cost margins play an important role in determining the optimal tax policy. All else equal, the higher (lower) are margins on externality (nonexternality) generating options, the lower (higher) will be the optimal tax rate on the externality product attribute. At observed prices, the (unweighted) average margin across sugar sweetened beverages is 0.78, while it is 0.76 across alternative drinks. How these margins adjust in equilibrium with the tax, and how consumers switch within the two sets of options and between them is important in determining the optimal tax rate. 47 This broadly accords with evidence from accounting data, with gross margins in this market being reported to be around 50-70% (see Competition Commission (2013)). 33 Table 4.4: Summary of costs and margins Percentile Mean 25th 50th 75th 1.44 0.67 0.77 0.55 0.83 0.31 0.43 0.41 1.16 0.61 0.56 0.50 1.96 0.83 0.98 0.67 Price (£/l) Marginal cost (£/l) Price-cost margin (£/l) Price-cost mark-up (Margin/Price) Notes: We recover marginal costs for each product in each market. We report summary statistics for the most recent year covered by our data (2012). Margins are defined as price minus cost and expressed in £ per liter; mark-ups are margins over price. In Figure 4.1 we show how observed prices, marginal costs, and equilibrium price-cost margins vary with product size. There is strong non-linear pricing; in per liter terms smaller products are, on average, more expensive. Average marginal costs are broadly constant across the size distribution, with the exception of small single portion sizes (i.e. with sizes no larger than 500ml), which, on average, have higher costs. Price-cost margins are declining in size – the average margin (per liter) is more than twice as large for the smallest options compared with the largest. This turns out to be important in driving the optimal tax rate, as one way consumers respond to the tax is by shifting their basket of taxed products towards small, high margin sizes. 0.50 Price, cost, margin (£/l) 1.00 1.50 2.00 2.50 Figure 4.1: Price-cost margins, by product size (0,0.5] (0.5,1] Price (£/l) (1,2] (2,3] Products within size range (l) Marginal cost (£/l) (3,4] (4,∞] Margin (£/l) Notes: We group products by size. The figure shows the mean price, cost, and margin (all expressed in £/l) across products within each size range. Numbers are for the more recent year covered by our data (2012). 34 4.6 Model validation We use data on the price changes of non-alcoholic drinks following the introduction of the UK’s Soft Drinks Industry Levy (SDIL) in 2018 to validate our empirical model of the market. We use a weekly database of UPC level prices and sugar contents for drinks products, collected from the websites of 6 major UK supermarkets, that cover the period 12 weeks before and 18 weeks after the introduction of the tax.48 The UK’s tax is levied per liter of product, with there being a lower rate of 18p/liter for products with sugar contents of 5-8g/100ml and a higher rate of 24p/liter for products with sugar content > 8g/100ml. We use an event study
approach to estimate the price changes for the sets of products subject to each rate
and for the set of drinks products not subject to the tax – full details are provided
in Appendix D. We find evidence that the tax was slightly overshifted, with price
increases of 26p/liter for products subject to the higher rate and 19p/liter for products subject to the lower rate (implying average pass-through rates of 105-108%),
with no change in the price of untaxed products. We simulate the effect of the tax
using our estimated model of supply and demand. Figure 4.2 shows the estimated
price changes in the data (grey markers) for the high and low tax groups (the figure
for untaxed products is shown in the Appendix D), and the predicted price changes
using our model. The predicted price changes from the model are very close to the
observed price changes.
The supermarkets are the big four – Tesco, Asda, Sainsbury’s and Morrisons – as well as
smaller national chains Iceland and Ocado. We are grateful to the University of Oxford for
providing us with access to these data, which were collected as part of the foodDB project.
Figure 4.2: Comparison of model predictions with event study
Change in price per litre
Change in price per litre
(b) Lower rate treatment group
(a) Higher rate treatment group
Price change (model)
Mean effect (data): 0.19
Mean effect (model): 0.20
Tax: 0.18
Mean effect (data): 0.26
Mean effect (model): 0.26
Tax: 0.24
Price change (data)
Price change (model)
Price change (data)
Notes: Grey markers show the estimated price changes (relative to the week preceding the introduction of the tax) for the set of products subject to the higher and lower rates. Full details are
given in Appendix D. 95% confidence intervals shown. The blue line shows the value of the tax,
and the red line shows the predicted price changes from our estimated model of the UK drinks
These patterns are broadly consistent with the literature that uses ex post evaluation methods to estimate the effects of sugar sweetened beverage taxes on prices;
for example, the Philadelphian tax was found to be fully passed through to prices
(Seiler et al. (2019), Cawley et al. (2018)), and in Mexico the tax was fully to slightly
more than fully passed through to prices (Grogger (2017), Colchero et al. (2015)).
An exception is Berkeley, where pass-through of the tax is estimated to be statistically insignificant or low (e.g. Bollinger and Sexton (2018)). A likely reason for low
tax pass-through in Berkeley is, given the small geographical area in which the tax
is operation, consumers can readily avoid the tax through cross-border shopping.
Corrective tax results
In this section, we embed our estimated empirical model of supply and demand in
the drinks market into the tax design framework set out in Section 2 to solve for
the optimal tax rate on sugar sweetened beverages and its effect on prices, purchases, and welfare. We also consider the tax’s performance relative to alternative
tax instruments, and how its performance is affected by the structure of the market.
We repeat, for reference, the implicit formula for the optimal rate of tax levied on
the externality generating product attribute (the sugar in sweetened beverages),
equation (2.3), with the exception that we split out the effect of switching to the
set of untaxed drinks products from the effect of switching to the numeraire good:

τ =
− dZ dQS

wjS µj

wjN µj


Market power in drinks market

µ̃ ×


Numeraire good market power
Here we denote the externality term by φ̃; the precise form this takes will depend on
whether there is heterogeneity in marginal externalities and whether externalities
arise from consumption of untaxed goods. We use µ̃ to denote the mark-up on the
to denote the marginal effect of the tax on equilibrium
numeraire good, and dX

consumption of the numeraire good.49
Our demand and supply model allows us to simulate, for any tax rate, the degree
of switching between drinks products and from drinks to the numeraire, and the
equilibrium price-cost margins on products in the drinks market. However, it does
not provide us with information on the marginal externalities nor on the mark-up
on the numeraire good. We use existing evidence to calibrate these parameters,
and first describe how the patterns of consumer switching and firms’ endogenous
margin adjustment affect the optimal tax rate. We then show how the optimal rate
and associated components of welfare vary with the calibrated parameters.
Baseline calibration
In Section 3.1 we summarize the well-established evidence that relates consumption
of sugar sweetened beverages to non-trivial externalities. However, placing a numerical value on the marginal externality associated with an extra gram of sugar from
these products is challenging. We begin by considering a marginal externality of
£4.00 per kg of sugar that is associated with sweetened beverages (which translates
to approximately 1.3 pence per ounce of sugar sweetened beverage). This value is
similar to what is implied by epidemiological estimates of the impact on health care
costs (e.g. Wang et al. (2012)).50 In this case φ̃ = φ = 4. Below we show how the

Consumption of the numeraire good is given by X = i yi − j pij qij . See Appendix E
for how the planner’s problem is modified to accommodate the numeraire good.
Wang et al. (2012) estimate that a 15% reduction in consumption of sugar sweetened beverages among US adults aged 24-65 would result in health care costs savings of $17.1 billion over 10
years. Converting this to savings per person, per kg of sugar and adjusting for differences in the
cost of providing health care in the UK implies an externality of roughly £4 per kg of sugar.
optimal rate varies with the magnitude, degree of heterogeneity, and source of the
externalities from sugar consumption.
The optimal rate also depends on the degree to which there is market power
associated with the numeraire good (which represents what consumers switch towards when lowering their drinks expenditure),51 and the direction and strength
of consumer switching towards the numeraire good. We calibrate the mark-up on
the numeraire good using an estimate for the UK economy wide mark-up from
De Loecker and Eeckhout (2018). Their estimate implies µ̃ = 0.4; the average
mark-up on drinks products, based on our estimates, is around 30% higher than
this.52 Below we show how the optimal rate depends on the value of the numeraire
Optimal tax rate
Under our baseline calibration of the marginal externality function and the pricecost mark-up on the numeraire good, the optimal rate of tax on the sugar in sweetened beverages is £1.74 per kg of sugar.53 This results in non-trivial price increases
for the taxed products (of 14% on average). However, the optimal tax rate lies well
below the rate that would be optimal under perfect competition (a Pigovian tax
of £4 per kg of sugar). It also lies below the rate that would be set by a planner
that ignores distortions associated with the exercise of market power would set: a
planner that takes the allocation in the absence of tax as a benchmark and aims
for full externality internalization relative to this baseline, would set a tax rate of
£3.64 per kg of sugar.54
The reason why the optimal rate lies below the rate aiming at full internalization
of externalities is the existence of positive price-cost margins for sugar sweetened
beverage products. This is reflected in the optimal tax formula by the weighted
average margin term,
j∈S wj µj . This expression reflects both the equilibrium
product level price-cost margins set by drinks firms on the taxed products (i.e. µj )
and, through the weights, switching within the set of taxed products. In particular,
Note that as we are free to normalize the price of the numeraire to 1, µ̃ can equivalently be
interpreted as the numeraire price-cost margin or mark-up.
De Loecker and Eeckhout (2018) adopt the convention of measuring mark-ups as price over
marginal cost, and estimate that this is 1.68 in the UK economy. This corresponds to a markup defined as margin over price on the numeraire of around 0.4. The average of our estimated
mark-ups on drinks is 0.55.
We run the optimal tax analysis using the most recent year covered by our data (2012).
A planner aimin…
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