A Sales Tax Is Better at Promoting Healthy Diets

than the Fat Tax and the Thin Subsidy

October 9, 2019

Appendix

A Proof of Proposition 1

To derive the effects of the policy parameters on consumption, we totally differentiate

the system of first-order conditions (11) and (12) with respect to H∗, F ∗, τF , σ and τC .

The total differential results in the following system of equations1:

(a11 a12a21 a22

)︸ ︷︷ ︸

=J

(dH∗

dF ∗

)=

(0 −VZpGα VZpGα

VZpF 0 VZpF

) dτFdσ

dτC

, (A.1)

where

a11 ≡∂2NetU

∂H2= UCCC

2H − 2UC`βCH + UCCHH + U``β

2 + VZZ p̃Gα2pG < 0, (A.2a)

a12 ≡∂2NetU

∂H∂F= (UCCCH − UC`β)CF + UCCHF + VZZ p̃GαpF , (A.2b)

a21 ≡∂2NetU

∂F∂H= (UCCCH − UC`β)CF + UCCHF + VZZ p̃FαpG, (A.2c)

a22 ≡∂2NetU

∂F 2= UCCC

2F + UCCFF + VZZ p̃FpF < 0. (A.2d)

In order to simplify the remaining analysis, we use the functional forms of U(C, `, Z)

and C(H,F ) to derive the following expressions:

CHF = (1− ρ)CHCFC

, (A.3a)

1Before deriving the total differential, we use the government’s budget constraint and (9) to expressZ as Z = I − pFF − pGαH.

1

CHH = −CHFF

H, CFF = −CHF

H

F, (A.3b)

UCC = −ηUCC, UC` = η

UC`, (A.3c)

U` =η

1− ηUCC

`, U`` = −ηUCC

`2. (A.3d)

Using Equations (A.2a)-(A.2d) and the expressions (A.3a)-(A.3d), we can calculate the

determinant of the matrix J , which is given by

|J | = a11a22 − a12a21

= −VZZUCC

[η(βCF )2ππτC +

(1− ρ)CHCF (p̃FF + p̃GαH) (pFF + pGαH)

HF

+ηβC

`

(βCF (pFπ + p̃FπτC ) + pF p̃F

βC

`

)]+U2CCHCFη(1− ρ)

HF

(1 +

βH

`

)2

> 0.

(A.4)

In simplifying Equation (A.4), we used the condition C = CHH + CFF which follows

from the constant returns property of the consumption basket C. Moreover, the term π

in Equation (A.4) denotes the opportunity cost of leisure and is positive by assumption,

i.e., π = (p̃FCH/CF − p̃Gα)/β > 0, while πτC = π(τF = σ = 0)/(1 + τC) and equals

the lowest value that π can take (since π is increasing in τF and σ according to (13a)

and (13b)) and, thus, πτC > 0. Note that the matrix J is the Hessian matrix and the

second-order conditions require a11 < 0, a22 < 0, and |J | > 0, which is unambiguously

satisfied according to Equations (A.2a), (A.2d) and (A.4). In order to derive the effects

of tax changes on optimal consumption, we apply Cramer’s rule. Consider first a change

in the fat tax dτF 6= 0. It has the following impact on consumption:

dF ∗

dτF=

1

|J |

∣∣∣∣∣ a11 0

a12 VZpF

∣∣∣∣∣ =VZpFa11|J |

< 0, (A.5)

dH∗

dτF=

1

|J |

∣∣∣∣∣ 0 a12VZpF a22

∣∣∣∣∣ = −VZpFa12|J |

(A.6)

= −VZpF

[UCCFC

(CH(1− ρ)− ηCH − ηβC/`) + VZZ p̃GαpF]

|J |> 0, if ρ > ρ.

According to Equation (A.5), the fat tax unambiguously lowers unhealthy consumption

F ∗. On the other hand, Equation (A.6) contains only positive terms with the exception

of the term containing (1−ρ). If ρ = 1, i.e. if the elasticity of substitution εC = 1/(1−ρ)

is infinite, then this term is zero and dH∗ > 0. On the other hand, if H and F are nearly

perfect complements, i.e., ρ → −∞, then the negative term dominates and dH∗ < 0.

2

Define the value of ρ that satisfies dH∗ = 0 as ρ and the corresponding elasticity of

substitution as εC = 1/(1− ρ). Then for εC > εC , healthy consumption is increasing in

the fat tax and for εC < εC , it is decreasing.

Next, we derive the effects of a change in the thin subsidy. Applying Cramer’s rule

to the system of equations (A.1), we get

dH∗

dσ=

1

|J |

∣∣∣∣∣ −VZpGα a120 a22

∣∣∣∣∣ = −VZpGαa22|J |

> 0, (A.7)

dF ∗

dσ=

1

|J |

∣∣∣∣∣ a11 −VZpGαa21 0

∣∣∣∣∣ =VZpGαa12|J |

(A.8)

=VZpGα

[UCCFC

(CH(1− ρ)− ηCH − ηβC/`) + VZZpGαp̃F]

|J |< 0, if ρ > ρ.

According to Equation (A.7), the thin subsidy unambiguously raises healthy consump-

tion. Equation (A.8) is proportional to the negative of Equation (A.6) if τF = σ = 0.

For positive τF and σ, (A.8) again behaves qualitatively opposite to (A.6). Thus, there

exists a value ρ̂, such that (A.8) is positive, zero or negative if ρ is smaller than, equal

to or larger than ρ̂, where ρ̂ = ρ for τF = σ = 0. Thus, we have proven the second part

of Proposition 1.

Lastly, we consider a change in the sales tax. It results in the following consumption

changes:

dH∗

dτC=

1

|J |

∣∣∣∣∣ VZpGα a12VZpF a22

∣∣∣∣∣ =VZ [pGαa22 − pFa12]

|J |(A.9)

=VZ|J |

[UCCFC

(ηCFβπτC +

ηpFCβ

`− CH(1− ρ)

(αpG

H

F+ pF

))+VZZp

2FαpG(τF + σ)

]> 0, if ρ > ρ̃, τF = σ = 0,

dF ∗

dτC=

1

|J |

∣∣∣∣∣ a11 VZpGα

a21 VZpF

∣∣∣∣∣ =VZ [pFa11 − pGαa12]

|J |(A.10)

= −VZ|J |

[UCC

(ηCHCFβπτC +

ηCβCF`

(pFCHCF

+ βπτC

)+ ηpF

(Cβ

`

)2

+CHCF (1− ρ)

(αpG + pF

F

H

))+ VZZp

2Gα

2pF (τF + σ)

]< 0, if τF = σ = 0.

Equation (A.9) is positive if the term in brackets is greater than zero. The term in

the last row of (A.9) is zero if there is no fat tax or thin subsidy. Moreover, πτC > 0

from Equation (13c). The only negative term is the one containing (1− ρ). Thus, if H

3

and F are perfect substitutes (ρ = 1), then dH∗ > 0. On the other hand, if the two

types of meals are nearly perfect complements (ρ → −∞), then dH∗ < 0. Define the

degree of substitutability that satisfies dH∗ = 0 as ρ̃ and the corresponding elasticity of

substitution as ε̃C = 1/(1− ρ̃). Then, a sales tax raises healthy consumption if εC > ε̃C ,

leaves it unchanged if εC = ε̃C and lowers H∗ if εC < ε̃C . Its effect on F ∗ is negative if

τF = σ = 0, as in this case all terms in Equation (A.10) are negative. �

B Proof of Proposition 2

The effect of the policy instrument x = τF , σ, τC on the weight gain is given by

dS

dx= ξ

(δdF ∗

dx+ ε

dH∗

dx

), x = τF , σ, τC . (B.1)

Since each policy instrument may simultaneously increase H∗ and lower F ∗, the sign

of Equation (B.1) is ambiguous. We calculate the effect of τF on obesity by inserting

Equations (A.5) and (A.6) in (B.1). We derive after some manipulation

dS

dτF= ξ

VZpF|J |

{−UCC

[η

(CH +

βC

`

)(δ

(CH +

βC

`

)− εCF

)+ CHCF (1− ρ)

(δF

H+ ε

)](B.2)

+VZZ p̃Gα(δpGα− εpF )

}< 0, if δ

(CH +

βC

`

)> εCF ∧ δpGα > εpF .

The right-hand side of Equation (B.2) consists of two terms in the first row and one

term in the second row. Only the second term in the first row containing (1 − ρ) is

unambiguously negative. A sufficient condition for dS/dτF < 0 is that the other two

terms are also negative. This proves the first part of Proposition 2.

Next, we derive the impact of a change in the thin subsidy. We insert Equations

(A.7) and (A.8) in (B.1) and derive after some manipulation

dS

dσ= ξ

VZpGα

|J |

{UCC

[ηCF

(εCF − δ

(CH +

βC

`

))+ CHCF (1− ρ)

(δ + ε

H

F

)](B.3)

+VZZ p̃F (δpGα− εpF )

}< 0, if δ

(CH +

βC

`

)> εCF ∧ δpGα > εpF ∧ ρ = 1.

Equation (B.3) consists of one nonnegative term in the first row (containing (1 − ρ))

and two other ambiguous terms. The ambiguous terms are negative under the same

conditions as in Equation (B.2). A further sufficient condition for dS/dσ < 0 is that

the nonnegative term is zero, i.e. ρ = 1 and εC →∞. Thus, we have proven the second

4

part of Proposition 2.

In order to prove the last part, we insert Equations (A.9) and (A.10) in (B.1). The

resulting expression can be simplified to

dS

dτC= ξ

VZ|J |

{UCC

[(εCF − δ

(CH +

βC

`

))(ηCβpF

`+ ηCFβπτC

)− CHCF (1− ρ)·

·(δ + ε

H

F

)(pGα + pF

F

H

)]− VZZpFpGα(τF + σ)(δpGα− εpF )

}< 0,

if δ

(CH +

βC

`

)> εCF ∧ τF = σ = 0. (B.4)

According to Equation (B.4), if there is no fat tax and no thin subsidy, then only one of

the conditions sufficient for (B.2) and (B.3) to be negative, is sufficient for dS/dτC < 0.

This proves the last part of Proposition 2. �

C General production function

This section extends the model by assuming a more general production function. We

replace the Leontief production function for H (defined in Equation (2)) by the CES

function

H = (ψkkν + ψGG

ν)1ν , (C.1)

where ψk, ψG > 0 and ν ≤ 1. The elasticity of substitution between time and ingredients

is 1/(1 − ν). The goods are perfect complements if ν → −∞ and perfect substitutes

if ν → 1. The main model considers only the case of perfect complements. Prior to

solving the model, we redefine the opportunity cost of time. Using Equation (3) and

the expression F = (C − CHH)/CF , we can rewrite (3) as

p̃FCF

C +H

k

(p̃GG

H− CHCF

p̃F

)k + Z = I + TR. (C.2)

Using furthermore the time constraint k = T − `, we can express the price of leisure as

π ≡ H

k

(CHCF

p̃F − p̃GG

H

). (C.3)

Note that in the special case of the Leontief production function (2), H/k = 1/β and

G/H = α, such that π from (C.3) coincides with its value in the main text.

5

The consumer solves the following maximization problem:

max`,G,Z

NetU = U(C, `, Z)− bS(F,H) s. t. (C.4)

k = T − `, (C.5)

F =I + TR− p̃GG− Z

p̃F. (C.6)

The three first-order conditions are given by

∂NetU

∂`= U` − UCCHHk + bξεHk = 0, (C.7)

∂NetU

∂G= UCCHHG − UCCF

p̃Gp̃F− bξ

(HG − δ

p̃Gp̃F

)= 0, (C.8)

∂NetU

∂Z= VZ − UCCF

1

p̃F+ bξ

1

p̃F= 0. (C.9)

To prove Proposition 1, we follow the same steps as in Appendix A. First, we totally

differentiate the first-order conditions (C.7)-(C.9) with respect to `,G, Z and the tax

instruments (as in Appendix A, we insert the government’s budget constraint in the

consumer’s budget constraint prior to differentiation): a11 a12 a13a21 a22 a23a31 a32 a33

︸ ︷︷ ︸

=J

d`∗

dG∗

dZ∗

=

0 0 0

b21 b22 b23b31 0 b33

dτF

dσ

dτC

, (C.10)

where

a11 ≡∂2NetU

∂`2= UCC(CHHk)

2 − 2UC`CHHk + UCCHHH2k + U`` + (UCCH − bξε)Hkk

= UCC(CHHk)2 − 2UC`CHHk + UCCHHH

2k + U`` +

U`Hk

Hkk < 0, (C.11a)

a12 ≡∂2NetU

∂`∂G= (U`C − UCCCHHk)

(CHHG − CF

pGpF

)− UCHk

(CHHHG − CHF

pGpF

)− (UCCH − bξε)HkG

= (U`C − UCCCHHk)

(CHHG − CF

pGpF

)− UCHk

(CHHHG − CHF

pGpF

)− U`Hk

HkG,

(C.11b)

a13 ≡∂2NetU

∂`∂Z= − 1

pF[(U`C − UCCCHHk)CF − UCCHFHk] , (C.11c)

a21 ≡∂2NetU

∂G∂`= (U`C − UCCCHHk)

(CHHG − CF

p̃Gp̃F

)− UCHk

(CHHHG − CHF

p̃Gp̃F

)

6

− (UCCH − bξε)HkG

= (U`C − UCCCHHk)

(CHHG − CF

p̃Gp̃F

)− UCHk

(CHHHG − CHF

p̃Gp̃F

)− U`Hk

HkG

(C.11d)

a22 ≡∂2NetU

∂G2= UCC

(CHHG − CF

p̃Gp̃F

)(CHHG − CF

pGpF

)+ UC

(CHHHG − CHFHG

(p̃Gp̃F

+pGpF

)+ CFF

p̃Gp̃F

pGpF

)+ (UCCH − bξε)HGG

= UCC

(CHHG − CF

p̃Gp̃F

)(CHHG − CF

pGpF

)+ UC

(CHHHG − CHFHG

(p̃Gp̃F

+pGpF

)+ CFF

p̃Gp̃F

pGpF

)+U`Hk

HGG < 0, (C.11e)

a23 ≡∂2NetU

∂G∂Z=

1

pF

[UCCCF

(CHHG − CF

p̃Gp̃F

)+ UC

(CHFHG − CFF

p̃Gp̃F

)],

(C.11f)

a31 ≡∂2NetU

∂Z∂`= − 1

p̃F[(U`C − UCCCHHk)CF − UCCHFHk] , (C.11g)

a32 ≡∂2NetU

∂Z∂G=

1

p̃F

[UCCCF

(CHHG − CF

pGpF

)+ UC

(CHFHG − CFF

pGpF

)],

(C.11h)

a33 ≡∂2NetU

∂Z2= VZZ +

UCCC2F + UCCFFp̃FpF

< 0. (C.11i)

The terms on the right-hand side of (C.10) are given by

b21 ≡ −∂2NetU

∂G∂τF= [UCCF − bξδ]

[− p̃GpF

p̃2F

]= −VZ p̃GpF

p̃F< 0, (C.12a)

b22 ≡ −∂2NetU

∂G∂σ= [UCCF − bξδ]

[−pGp̃F

]= −VZpG < 0, (C.12b)

b23 ≡ −∂2NetU

∂G∂τC= [UCCF − bξδ]

p̃FpG − p̃GpFp̃2F

=VZpGpF (τF + σ)

p̃F≥ 0, (C.12c)

b31 ≡ −∂2NetU

∂Z∂τF= [UCCF − bξδ]

[−pFp̃2F

]= −VZpF

p̃F< 0, (C.12d)

b33 ≡ −∂2NetU

∂Z∂τC= [UCCF − bξδ]

[−pFp̃2F

]= −VZpF

p̃F< 0. (C.12e)

We continue by simplifying Equations (C.11a)-(C.11i), where we use the expressions

(A.3a)-(A.3d) together with the following expressions regarding the production function

(C.1):

HkG = (1− ν)HkHG

H, Hkk = −HkG

G

k, HGG = −HkG

k

G. (C.13)

7

The simplification of the aij terms gives the following results:

a11 = −UCC

[η

(C

`+ CHHk

)2

+ (1− ρ)CHCFH2k

F

H+ (1− ν)

HGGηC2

Hk(1− η)`

], (C.14a)

a12 =UCC

[η

(C

`+ CHHk

)(CHHG − CF

pGpF

)+ (1− ρ)CHCFHk

(F

HHG +

pGpF

)− (1− ν)

HGηC2

H(1− η)`

], (C.14b)

a13 =UCCpF

[−ηCF

(C

`+ CHHk

)+ (1− ρ)CHCFHk

], (C.14c)

a21 =UCC

[η

(C

`+ CHHk

)(CHHG − CF

p̃Gp̃F

)+ (1− ρ)CHCFHk

(F

HHG +

p̃Gp̃F

)− (1− ν)

HGηC2

H(1− η)`

], (C.14d)

a22 = −UCC

[η

(CHHG − CF

p̃Gp̃F

)(CHHG − CF

p̃Gp̃F

)+ (1− ρ)CHCF

F

H

(HG +

H

F

pGpF

)(HG +

H

F

p̃Gp̃F

)+ (1− ν)

HGkηC2

HG(1− η)`

],

(C.14e)

a23 =UCCFCpF

[η

(CHHG − CF

p̃Gp̃F

)− (1− ρ)CH

(HG +

H

F

p̃Gp̃F

)], (C.14f)

a31 =UCCp̃F

[−ηCF

(C

`+ CHHk

)+ (1− ρ)CHCFHk

], (C.14g)

a32 =UCCFCp̃F

[η

(CHHG − CF

pGpF

)− (1− ρ)CH

(HG +

H

F

pGpF

)], (C.14h)

a33 = VZZ −UC

Cp̃FpF

[ηC2

F + (1− ρ)CHCFH

F

]. (C.14i)

First, we show that the determinant |J | is negative, as required by the second-order

conditions. After some tedious calculations, we derive

|J | = −U3C(1− ν)HGη

2(1− ρ)CHC2FT

p̃2FpF (1− η)`3GF

[π`+

p̃FCF

C +`

k(p̃FF + p̃GG)

]+ VZZU

2C ·

·{

(1− ρ)CHCFηF

H

[1

`

(HG +

HpGFpF

)+HkpGFpF

] [1

`

(HG +

Hp̃GF p̃F

)+Hkp̃GF p̃F

]+

(1− ν)HGkη

H(1− η)`G

[η

(C

`+CFp̃F

πτC

)(C

`+CFp̃F

π

)+(1− ρ)CHCF

F

H

(H

k+GHpGkFpF

)(H

k+GHp̃GkF p̃F

)]}< 0. (C.15)

8

We can now calculate the effect of the fat tax on at-home meals H∗. Applying the

Cramer rule, we get

dH∗

dτF= −Hk

d`∗

dτF+HG

dG∗

dτF(C.16)

=1

|J |

−Hk

∣∣∣∣∣∣∣0 a12 a13

−VZpGpFp̃F

a22 a23

−VZpFp̃F

a32 a33

∣∣∣∣∣∣∣+HG

∣∣∣∣∣∣∣a11 0 a13a21 −VZpGpF

p̃Fa23

a31 −VZpFp̃F

a33

∣∣∣∣∣∣∣

=VZU

2C

|J |p̃F

{(1− ν)HGη

Gk(1− η)`

[(1− ρ)CHCFH − ηCF

(CHH + k

C

`

)]+

(1− ρ)CHCFηH2G

`2

}− VZ p̃GpFVZZUC

|J |p̃FC

[(1− ρ)CHCFH

2k

pGpF

−η(C

`+ CHHk

)(HkCF

pGpF

+HGC

`

)− (1− ν)HGηC

2

(1− η)`k

].

All the negative terms in (C.16) are proportional to (1−ρ). They vanish for ρ→ 1 and

tend to infinity for ρ→ −∞. Hence, there exists a value ρ such that dH∗/dτF > (< 0)

for ρ > (<)ρ.

To derive the change in F ∗, we use the budget constraint and apply Cramer’s rule:

dF ∗

dτF=

1

pF

[−pG

dG∗

dτF− dZ∗

dτF

](C.17)

=1

|J |pF

−pG∣∣∣∣∣∣∣a11 0 a13a21 −VZpGpF

p̃Fa23

a31 −VZpFp̃F

a33

∣∣∣∣∣∣∣−∣∣∣∣∣∣∣a11 a12 0

a21 a22 −VZpGpFp̃F

a31 a32 −VZpFp̃F

∣∣∣∣∣∣∣

=VZU

2C

|J |p̃F

{(1− ν)HGη

H(1− η)`

[(1− ρ)CHCF

HF

kG+ηk

G

(CH

H

k+C

`

)2]

+(1− ρ)CHCFηH

2GF

H`2

}+VZpGp̃GVZZa11|J |p̃F

< 0.

According to (C.17), the effect of the fat tax on the demand for away-from-home food

is unabiguously negative. This concludes the proof of Proposition 1 (i).

To prove Proposition 1 (ii), we follow the same steps and derive the following results:

dH∗

dσ= −VZpGU

2C

|J |p̃FpF

{(1− ν)HGη

k(1− η)`

[(1− ρ)CHCF

H

F+ ηC2

F

]+

(1− ρ)CHCFηHGH

F`

(1

`+Hk

H

)}− VZpGVZZUC

|J |C

[(1− ρ)CHCFH

2k

pGpF

−η(C

`+ CHHk

)(HkCF

pGpF

+HGC

`

)− (1− ν)HGηC

2

(1− η)`k

], (C.18)

9

dF ∗

dσ=

VZU2CpG

|J |p̃FpF

{(1− ν)HGη

H(1− η)`

[ηCF

(CH

H

k+C

`

)− (1− ρ)CHCF

H

k

]−(1− ρ)CHCFηHG

`

(1

`+Hk

H

)}+VZp

2GVZZa11|J |pF

. (C.19)

There is one negative term in (C.18) which is proportional to 1− ρ. If ρ is sufficiently

large, the term becomes small and dH∗/dσ > 0. Otherwise, dH∗/dσ is undetermined.

All the positive terms in (C.19) are proportional to (1− ρ). Hence, dF ∗/dσ is positive

for ρ→ −∞ and negative for ρ→ 1. Hence, a critical value ρ′ exists, such that dF ∗/dσ

is negative (positive) if ρ > (<)ρ′.

Analogously, we derive the effect of τC on the demand for H and F :

dH∗

dτC=

VZU2C

|J |p̃F

{(1− ν)HGη

(1− η)`

[(1− ρ)CHCF

(H

Gk+Hp̃GF p̃F

)− ηCF

G

(CFHπ

kp̃F+C

`

)]+

(1− ρ)CHCFηHG

`

[1

`

(HG +

Hp̃GF p̃F

)+Hkp̃GF p̃F

]}+(τF + σ)

VZpGpFVZZUC|J |p̃FC

[(1− ρ)CHCFH

2k

pGpF

−η(C

`+ CHHk

)(HkCF

pGpF

+HGC

`

)− (1− ν)HGηC

2

(1− η)`k

]+(τF + σ)

VZpGpFU2C

|J |p̃F 2pF

[(1− ρ)CHCFηHGH

F`

(1

`+Hk

H

)+

(1− ν)HGη

(1− η)k`

(ηC2

F + (1− ρ)CHCFH

F

)], (C.20)

dF ∗

dτC=

VZU2C

|J |p̃F

{(1− ν)HGη

H(1− η)`

[ηk

G

(CH

H

k+C

`

)(CFHπ

kp̃F+C

`

)+(1− ρ)CHCF

F

G

(H

k+HGp̃GFkp̃F

)]+

(1− ρ)CHCFηHGF

H`

[1

`

(HG +

Hp̃GF p̃F

)+Hkp̃GF p̃F

]}−(τF + σ)

VZpG|J |p̃F

{VZZa11pG −

U2C

p̃F

[(1− ρ)CHCFηHG

`

(1

`+Hk

H

)+

(1− ν)HGη

H(1− η)`

(ηCF

(CHH

k+C

`

)+ (1− ρ)CHCF

H

k

)]}. (C.21)

When τF = σ = 0, the terms in the third to sixth rows in (C.20) vanish. The only

negative terms in the first and second rows of (C.20) are proportional to (1 − ρ).

Following the same steps as in part (i) of Proposition 1, it follows that for τF = σ = 0,

there exists ρ̃, such that the demand for at-home food increases (declines) following an

increase in τC if ρ > (<)ρ̃. It is futhermore easy to see that in the case τF = σ = 0, all

10

terms in (C.21) are negative and dF ∗/dτC < 0. This concludes the proof of Proposition

1.

To prove Proposition 2, we follow the same steps as in Appendix B. Using Equation

(B.1), we derive the following effect of τF on the weight gain S:

dS

dτF= ξ

VZU2C

|J |p̃F

{(1− ρ)CHCFηH

2G

H`2(δF + εH) +

(1− ν)HGη

H(1− η)`· (C.22)

·[ηk

G

(CH

H

k+C

`

)(δ

(CH

H

k+C

`

)− εCF

H

k

)+

(1− ρ)CHCFH

kG(δF + εH)

]}−ξVZZ p̃GUC|J |p̃FC

{(1− ν)HGηC

2

(1− η)`k

(δpG

G

H− εpF

)+ (1− ρ)CHCFH

2kpG

(F

Hδ + ε

)+η

(CHHk +

C

`

)[HkpGk

H

(δ

(CH

H

k+C

`

)− εCF

H

k

)+CHG

`

(δpG

G

H− εpF

)]}< 0, if δ

(CH

H

k+C

`

)> εCF

H

k∧ δpG

G

H> εpF .

The conditions for a negative dS/dτF in (C.22) are identical to the conditions in (B.2)

when one substitutes β for H/k and α for G/H in the case of a Leontief production

function.

The effect of the thin subsidy on the gain weight is derived analogously and equals

dS

dσ= ξ

VZU2CpG

|J |p̃FpF

{−(1− ρ)CHCFηHG

F`(δF + εH)

(1

`+Hk

H

)+

(1− ν)HGη

H(1− η)`· (C.23)

·[ηCF

(δ

(CH

H

k+C

`

)− εCF

H

k

)− (1− ρ)CHCFH

Fk(δF + εH)

]}−ξVZpGVZZUC

|J |pFC

{(1− ν)HGηC

2

(1− η)`k

(δpG

G

H− εpF

)+ (1− ρ)CHCFH

2kpG

(F

Hδ + ε

)+η

(CHHk +

C

`

)[HkpGk

H

(δ

(CH

H

k+C

`

)− εCF

H

k

)+CHG

`

(δpG

G

H− εpF

)]}< 0, if δ

(CH

H

k+C

`

)> εCF

H

k∧ δpG

G

H> εpF ∧ ρ→ 1.

Similarly to the fat tax case, the conditions for a negative dS/dσ in (C.23) are analogous

to the conditions in (B.3).

Lastly, the impact on S of a change in the sales tax is the following:

dS

dτC= ξ

VZU2C

|J |p̃F

{(1− ρ)CHCFηHG

H`

[1

`

(HG +

Hp̃GF p̃F

)+Hkp̃GF p̃F

](δF + εH) +

(1− ν)HGη

H(1− η)`·

·[ηk

G

(CFHπ

kp̃F+C

`

)(δ

(CH

H

k+C

`

)− εCF

H

k

)

11

+(1− ρ)CHCFH

kG(δF + εH)

F p̃F +Gp̃GF p̃F

]}+(τF + σ)ξ

VZpG|J |p̃F

εpF

{VZZUC|J |C

[(1− ρ)CHCFH

2k

pGpF

−η(C

`+ CHHk

)(HkCF

pGpF

+HGC

`

)− (1− ν)HGηC

2

(1− η)`k

]+

U2C

p̃FpF

[(1− ρ)CHCFηHGH

F`

(1

`+Hk

H

)+

(1− ν)HGη

(1− η)k`

(ηC2

F + (1− ρ)CHCFH

F

)]}−(τF + σ)ξ

VZpG|J |p̃F

δ

{VZZa11pG −

U2C

p̃F

[(1− ρ)CHCFηHG

`

(1

`+Hk

H

)+

(1− ν)HGη

H(1− η)`

(ηCF

(CHH

k+C

`

)+ (1− ρ)CHCF

H

k

)]}< 0, if δ

(CH

H

k+C

`

)> εCF

H

k∧ τF = σ = 0. (C.24)

Equation (C.24) is negative under analogous conditions to Equation (B.4) in Appendix

B. This concludes the proof of Proposition 2 in the case of a general production function.

D Derivation of price elasticities

In this Section, we derive the price elasticities reported in Table 4. We use the estimates

of Okrent and Alston (2012), which are shown in Table D.1.

The category “FAFH and Alcohol” includes food-away-from-home and alcoholic

drinks. However, Okrent and Alston (2012) report that alcoholic drinks represent only

15.5% of the expenditures in that category. For that reason, we assume that this

category corresponds to food of type F in our model. Thus, the own-price elasticity of

the category “FAFH and Alcohol” gives εFF . There are six food-at-home categories,

which together constitute our food of type H. To aggregate their own- and cross-price

elasticities, we use their expenditure shares as weights. These shares are reported by

Okrent and Alston (2012), and we show them in Table D.2. In the first row of Table

D.2, we report the average share of food category i in expenditures in the period 1998-

2010, as reported by Okrent and Alston (2012). In the second row, we normalize the

sum of the expenditure shares to equal unity, such that we can calculate the share

of each category within the food group H. Thus, the value 0.152 for the normalized

expenditure share of cereals and bakery says that 15.2% of the overall expenditures for

H are spent on cereals and bakery. Use index i, j = 1, . . . , 6 to denote the six food-at-

home categories and denote as sj the normalized expenditure share of category j. We

12

Table D.1: Own-price and cross-price elasticities

With respect to price of

Elasticity ofdemand fora

Cerealsandbakery

Meat andeggs Dairy

Fruitsand veg-etables

NonalcoholicBever-ages

OtherFAHb

FAFHandalcoholc

Cereals andbakery

−0.58(0.25) 0.05(0.12) 0.36(0.08) −0.31(0.13) −0.09(0.15) 0.25(0.29) 0.16(0.34)

Meat andeggs

0.03(0.07) −0.31(0.17) 0.02(0.05) 0.11(0.09) 0.08(0.04) 0.26(0.11) 0.17(0.32)

Dairy 0.49(0.10) 0.06(0.12) −0.05(0.09) −0.03(0.11) −0.16(0.07) −0.44(0.16) 0.23(0.30)

Fruits andvegetables

−0.30(0.13) 0.18(0.15) −0.02(0.08) −0.79(0.19) 0.02(0.09) 0.58(0.20) 0.24(0.39)

Nonalcoholicbeverages

−0.21(0.32) 0.31(0.16) −0.25(0.12) 0.05(0.20) −0.65(0.39) 0.65(0.46) −0.05(0.51)

Other FAH 0.15(0.18) 0.27(0.11) −0.19(0.07) 0.36(0.12) 0.18(0.12) −0.98(0.30) 0.45(0.34)

FAFH andalcohol

0.03(0.07) 0.05(0.11) 0.03(0.04) 0.05(0.08) −0.01(0.05) 0.14(0.11) −0.71(0.38)

Nonfood −0.02(0.01) −0.05(0.01) −0.02(0.00) −0.02(0.01) −0.01(0.00) −0.04(0.01) −0.06(0.04)

a Source: Okrent and Alston (2012). Standard errors in parentheses.b FAH = Food-at-home.c FAFH = Food-away-from-home.

calculate the remaining food products elasticities according to the following formulae:

εHH =6∑i=1

(6∑j=1

sjεji

), (D.1)

εHF =6∑j=1

sjεjF , (D.2)

εFH =6∑j=1

εFj, (D.3)

where εij denotes the elasticity of food category i with respect to the price of category

j.

The elasticity of non-food products with respect to the price of F εZF is taken

directly from Table C1. The second non-food price elasticity is derived according to

εZH =6∑j=1

εZj, where j = 1, . . . , 6 denotes the at-home food categories. The resulting

expressions are reported in Table 4.

13

Table D.2: Own-price and cross-price elasticities

Cerealsandbakery

Meat andeggs Dairy

Fruitsand veg-etables

NonalcoholicBever-ages

OtherFAH Total

Absoluteexpenditure sharea

1.66 1.21 2.88 1.69 0.75 2.76 10.95

Normalizedexpenditure share

0.152 0.263 0.11 0.154 0.068 0.252 1

a Source: Okrent and Alston (2012).

E Calibration: tables and figures

Figure E.1: Average kcal per meal at- and away-from-home for male and female individ-

uals with 95% confidence intervals. Source: Author’s calculations based on NHANES

2009− 2010 (CDC, 2010).

14

Figure E.2: Average kcal per meal at- and away-from-home for different BMI groups of

male and female individuals with 95% confidence intervals. Under- and normalweight if

BMI less than 25, overweight if BMI ∈ [25, 30) and obese if BMI ≥ 30. The under- and

normalweight individuals are represented in one category due to the small number of

underweight individuals in NHANES. Source: Author’s calculations based on NHANES

2009− 2010 (CDC, 2010).

15

Figure E.3: Average kcal per meal at- and away-from-home for different age groups of

male and female individuals with 95% confidence intervals. Source: Author’s calcula-

tions based on NHANES 2009− 2010 (CDC, 2010).

16

Figure E.4: Average kcal per meal at- and away-from-home for different income groups

of male and female individuals with 95% confidence intervals. Income is defined as “ratio

of family income to poverty threshold” (variable INDFMPIR in NHANES), where a

value 1 indicates income equal to the poverty threshold. Source: Author’s calculations

based on NHANES 2009− 2010 (CDC, 2010).

Table E.1: Revenue Neutral Introduction of a Fat Tax and a Thin Subsidy

τF = 10%, σ = 6.7%

Male Female

Consumption, weight and

utility effectsa:

%∆F : −12.1 −6.44

%∆H: 3.31 1.28

%∆W : −3.24 −1.39

CV ($/day): 0.0096 0.034

EB/kcal (cent/kcal): 0.048 0.057a Source: Author’s calculations.

17

Table E.2: Sensitivity Analysis: εC = 2 and εC = 5

τF = 10% σ = 10% τC = 10%

Male Female Male Female Male Female

A. εC = 2

Consumption, weight andutility effectsa:%∆F : −4.21 −2.19 −0.74 −0.23 −3.49 −1.96%∆H: 0.54 0.17 1.38 0.56 −0.8 −0.38%∆W : −1.67 −0.68 0.85 0.36 −2.52 −1.04

CV ($/day): 0.4 0.35 −0.55 −0.46 0.95 0.81EB/TR: 0.022 0.011 0.007 0.0027 0.0099 0.0054EB/kcal (cent/kcal): 0.026 0.034 − − 0.019 0.025

B. εC = 5

Consumption, weight andutility effectsa:%∆F : −13.85 −7.85 −3.89 −1.5 −10.34 −6.44%∆H: 2.76 1.13 2.34 0.82 0.51 0.32%∆W : −4.64 −2.04 0.11 0.095 −4.86 −2.16

CV ($/day): 0.38 0.34 −0.56 −0.46 0.94 0.81EB/TR: 0.078 0.042 0.011 0.004 0.021 0.013EB/kcal (cent/kcal): 0.03 0.041 − − 0.02 0.03a Source: Author’s calculations.

Fat Tax

Sales Tax

2 3 4 5ϵC

0.018

0.020

0.022

0.024

0.026

0.028

EB/kcal

(a)

Fat Tax

Sales Tax

2 3 4 5ϵC

0.020

0.025

0.030

0.035

0.040

EB/kcal

(b)

Figure E.5: Excess burden per kcal reduction in consumption of men (Panel (a))

and women (Panel (b)) as functions of the elasticity of substitution εC , measured in

cent/kcal.

18

Fat Tax

Sales Tax

2 3 4 5ϵC

0.02

0.04

0.06

0.08

EB/TR

(a)

Fat Tax

Sales Tax

2 3 4 5ϵC

0.01

0.02

0.03

0.04

EB/TR

(b)

Figure E.6: Excess burden per one dollar of tax revenues for men (Panel (a)) and

women (Panel (b)) as functions of the elasticity of substitution εC .

(a) (b)

Figure E.7: Excess burden per kcal reduction in consumption of men (Panel (a)) and

women (Panel (b)) as functions of the kcal per meal δi and εi, where the domains of δi

and εi are defined by their 95% confidence intervals. The upper (red) plane represents

the fat tax and the lower (blue) plane the sales tax.

19

(a) (b)

Figure E.8: Excess burden per one dollar of tax revenues for men (Panel (a)) and

women (Panel (b)) as functions of the kcal per meal δi and εi, where the domains of δi

and εi are defined by their 95% confidence intervals. The upper (red) plane represents

the fat tax and the lower (blue) plane the sales tax.

20