Demand model estimation
We adopt a hierarchical approach to the estimation of the price elasticities which requires a number of separability assumptions to keep the dimensions of the estimated models manageable. Foods are grouped into nutritionally meaningful categories and according to their embedded emissions (see Fig. 1 in appendix). The model at the top level represents the household’s decision to allocate overall food expenditure between broad food categories including drinks (see Table 5). Next, seven models represent the household’s decision to allocate expenditure between the dairy & eggs through to the alcohol categories. Below this, the starches, fats and beverages categories each have a model explaining the budget allocation between a further disaggregation, for example, into coffee and tea & cocoa drinks. A Quadratic Almost Ideal Demand System (QUAIDS) (Deaton and Muellbauer 1980; Banks et al. 1997) is estimated for each of the eleven demand systems each of which has m equations. The QUAIDS was chosen because it allows for non-linear Engel curves which means expenditure elasticities can vary with expenditure levels. The estimation method we employ is detailed in Tiffin and Arnoult (2010) and has subsequently been applied by others including Kasteridis et al. (2011). Our accounts for censoring in the expenditure shares (see Online Resource 1) by adapting the model to incorporate infrequency of purchase (Cragg 1971; Blundell and Meghir 1987). A system of probit and demand equations is estimated
$$ \left(\begin{array}{c}\hfill {y}^{*}\hfill \\ {}\hfill {s}^{*}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {X}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {X}_2\hfill \end{array}\right)\left(\begin{array}{c}\hfill {\beta}_1\hfill \\ {}\hfill {\beta}_2\hfill \end{array}\right)+\left(\begin{array}{c}\hfill u\hfill \\ {}\hfill v\hfill \end{array}\right) $$
(1)
where y* is a vector of latent probit variables, s* is a vector of latent expenditure shares (see also Section 3.2), X
1 is a vector of constants, X
2 is a matrix of prices and household expenditure. Expenditure shares are expressed as follows
$$ {S}_{ih}^{*}={\alpha}_i+{\displaystyle \sum_{i=1}^m{\gamma}_{ij} \ln \left({p}_{jh}\right)+{\beta}_i\left(\frac{e_h}{a(p)}\right)}+\frac{\lambda }{b(p)}{\left\{ \ln \left(\frac{e_h}{a(p)}\right)\right\}}^2,i=1,\ldots m $$
(2)
where
$$ a(p)= \exp \left({\alpha}_0+{\displaystyle \sum_{i=1}^m{\alpha}_i \ln \left({p}_{ih}\right)+\frac{1}{2}{\displaystyle \sum_{i=1}^m{\displaystyle \sum_{j=1}^m{\gamma}_{ij} \ln \left({p}_{ih}\right) \ln \left({p}_{jh}\right)}}}\right),b(p)={\displaystyle \underset{i=1}{\overset{m}{\varPi }}{p}_i^{\beta i}} $$
(3)
P
ih
is the price faced by household h for food category i, e
h
is food expenditure of household h, and α and β are the coefficients to be estimated. Adding up, symmetry, homogeneity and concavity are imposed in the estimation. Because the covariance matrix is singular we drop one of the share equations, obtaining estimates that are invariant to which equation is dropped (Barten 1969). Food categories are aggregated using the EKS quantity index (Elteto and Koves 1964; Szulc 1964), which is a multi-lateral version of the superlative Fisher Ideal index, which is used to compute the implicit price index. A superlative index (Diewert 1976) offers some mitigation towards the concerns over the potential endogeneity of prices.
We use Markov Chain Monte Carlo (MCMC) methods to sample from the posterior distribution of the parameters. MCMC methods are commonly used to sequentially draw values from an approximation to the posterior density which is proportional to the product of the likelihood function and prior distributions of the parameters. Because the distribution of a sampled draw depends only on the last value drawn, the draws form a Markov Chain and under the right conditions these draws converge to represent draws from the full posterior. We use Gibbs sampling to draw subsets of parameters using their conditional posterior distributions (Gelman et al. 2014).
The data is censored and we therefore use the infrequency of purchase model (Cragg 1971; Blundell and Meghir 1987) which separates the discrete buying decision from the continuous decision on how much to buy. Accordingly, two types of latency arise. Both stem from the fact that observed purchases differ from consumption when households consume from stocks. The first type of latency arises where a purchase is observed and it must be deflated to yield an estimate of the latent quantity consumed:
$$ {q}_{ih}^{*}={q}_{ih}{\Phi}_{ih} $$
(4)
where q
ih
is the observed purchase and \( {\Phi}_{ih}=P\left({q}_{ih}>0\right)=P\left({y}_{ih}^{*}>0\right)=\Phi \left({x}_{1h}{\beta}_{1i}\right) \) is the probability that a purchase is made. Using the estimate of latent consumption the latent share is obtained:
$$ {s}_{ih}^{*}=\frac{p_{ih}{q}_{ih}^{*}}{{\displaystyle {\sum}_{i\in C}}{p}_{ih}{q}_{ih}^{*}} $$
(5)
where C is the set of observations that are not censored. Where food is consumed from stocks but no purchase is observed the second type of latency occurs. We treat this as an incomplete data problem and replace the censored values using data augmentation (Tanner and Wong 1987). Thus, where household h makes no purchase of food category i the observed zero is replaced with a latent share s
ih
* drawn from the conditional distribution:
$$ {s}_{ih}^{*}\Big|{\beta}_1,{\beta}_2,\varSigma, D\sim N\left({\mu}_{ih},{V}_i\right) $$
(6)
where μ
ih
and V
i
are the conditional mean and conditional variance respectively and D is observed and latent data. Observed shares are scaled using an adapted Wales and Woodland (1983) procedure as described in Tiffin and Arnoult (2010) to ensure that adding up holds.
We draw 12,000 MCMC samples and discard the first 2000 draws to ensure convergence. The remaining 10,000 draws taken from the posterior provide the basis for inference. As convergence diagnostics we use trace plots and the Geweke test (Geweke 1992) which compares the means from the first and second half of the Markov chain.
Unconditional price elasticities (Online Resources 2) are computed for each group (Edgerton 1997) as the mean of the household specific elasticities within the group (Online resource 3). These assume that a price change of one of the food categories changes food expenditure available to all other food groups. The elasticities are used to compute the changes in food consumption. Changes in nutrient intake are computed using the corresponding nutrient conversion factors provided by the LCF survey.
Computing food related emissions
Previous studies have computed food related emissions based on food purchases. This is problematic when the frequency of purchase differs across socio-economic groups. Consider two households that eat steak once a week and which are representative of two different socio-economic groups. One household shops for steak once every three weeks and the other shops once a month. The level of emissions that we compute depends on which households happened to shop in the two week period in which the data is collected. The latent quantities which are computed in the course of estimating the model offer a solution to the problem. Because these have been adjusted taking into account the probability that a purchased occurred in the survey period, a true representation of emissions can be obtained. For cases where no purchase is made and the latent share is obtained using Eq. 6, the latent share may be negative. Where this occurs we assume that consumption is zero and therefore no emissions arise. Overall, emissions g
ih
are:
$$ {g}_{ih}=\left\{\begin{array}{ccc}\hfill {q}_{ih}^{**}{F}_i\hfill & \hfill if\hfill & \hfill {s}_{ih}^{*}>0\hfill \\ {}\hfill 0\hfill & \hfill if\hfill & \hfill {s}_{ih}^{*}\le 0\hfill \end{array}\ \right. $$
(7)
where
$$ {q}_{ih}^{**}=\frac{s_{ih}^{*}{\displaystyle {\sum}_{i\in C}}{p}_{ih}{q}_{ih}^{*}}{p_{ih}} $$
(8)
and F
i
is the carbon conversion factor of food category i.
Determining the tax rate and proportional price increases
Table 2 shows the tax rate for each food category and the resultant proportional price changes for Scenario A which imposes a tax on all foods according to their emission content. Because the administrative costs of taxing foods with low emissions would be disproportionately high, we also simulate a tax only on foods with above average levels of emissions (Scenario B).
Table 2 Average tax rate (in pence/gramme) and resultant proportional price changes (in %)
The tax rate is computed using the agriculture marginal abatement cost curve (MACC) provided by the Department for Environment, Food and Rural Affairs (Moran et al. 2008) which suggests that investment of £24.10/tCO2e (£28.41/tCO2e, 2011 prices) can reduce UK agricultural GHG emissions by 16.2 %, and from the Stern Review’s (Stern 2007) social cost of carbon which is calculated as £22–£26/tCO2e (2011 prices) emitted to maintain global atmospheric concentration of carbon dioxide equivalents at 450–550 ppm. The marginal external cost is assumed to be £2.841/tCO2e per 100 g of product (Moran et al. 2008). The next most cost-effective abatement strategy suggested by the MACC (Moran et al. 2008) is £205.40/tCO2e at 2011 prices and is prohibitive. The tax rate chosen is therefore a compromise between the practicality of abatement and societal costs. To obtain the tax rate we multiply the carbon conversion factor by the marginal external cost of emissions of £2.841/tCO2e. The proportional price changes are obtained by comparing the average price per 100 g of food with the new price after the tax has been added. This is done for each food category and also for each group because average prices differ between groups. Households that buy cheaper products of a given food category and therefore have lower average expenditure per food unit, experience larger proportional price increases, and vice versa. For example, SEC4 tends to have the smallest unit values for many food categories and therefore experiences the largest proportional price increases in most cases. By contrast, SEC1 has the smallest proportional price increases as people working in higher managerial & professional functions are expected to have a higher income (see Table 4) and spend more on food (see Table 5) which is likely to translate into them buying more expensive products.
The impact of the tax also depends on the proportion of food expenditure that a specific socio-economic group devotes to food categories that are emission intensive and therefore attract a higher tax rate. The expenditure shares show that the OTHER group tends to have the largest expenditure shares for emission intensive food categories such as milk, pork, beef, lamb, other meat and therefore its tax burden will be higher. By contrast, SEC1 has larger expenditure shares for less emission intensive foods such as fresh fruit, fresh vegetables, and fruit juice, and both SEC1 and SEC2 have smaller expenditure shares for meat and dairy foods. Thus, higher SECs will be less affected by the tax because they buy relatively more foods with low emission intensity while SEC4 will be most affected because it consumes relatively more emission intensive foods and experiences larger proportional price increases.