2.1 Exact Affine Stone Index demand system

We estimate the approximate EASI implicit Marshallian demand system developed by Lewbel and Pendakur (Reference Lewbel and Pendakur2009), which is derived from consumer theory and expresses budget shares in linear forms. The EASI log of cost function $C(.)$ for the exact model for $J$ goods, is expressed in terms of utility $u$, a $J$- vector of log prices $p$, a $J$- vector of utility coefficients $b_1$ and a $J$- vector of errors $\epsilon$, as

(1)\begin{equation} C(p,u,\epsilon) = u + p^\prime (b_0 + b_1 u + rz) + \frac{1}{2} p^\prime Ap + p^\prime \epsilon, \end{equation}

where $A$ is a $J \times J$ matrix of parameters with elements $[A]_{jk} = a_{jk}$, and $z$ is an $r \times J$ matrix of control variables including household demographics and geographical identifiers. Applying Shephard's Lemma to the cost function, we obtain Hicksian demands, and via duality, the vector of Marshallian budget shares in implicit form:

(2)\begin{equation} w = b_0 + b_1 y + rz + Ap + \epsilon, \end{equation}

where $y$ is interpreted as a measure of implicit utility and the log of real household expenditure. Engel curves are modelled as approximately linear, while the $\epsilon$ terms are random utility parameters reflecting individual unobserved preference heterogeneity. Solving for $y$ by equating the cost function $C(p,u,\epsilon )$ to the log of nominal household expenditure $x$ and substituting for $w$ yields the following expression, with $p^\prime w$ equal to the exact log of the Stone price index (Stone, Reference Stone1954):

(3)\begin{equation} y = x - p^\prime w + \frac{1}{2} p^\prime Ap. \end{equation}

This expression, however, relies on the matrix of estimable parameters $A$, and is endogenous due to its dependence on the budget shares $w$. We therefore estimate the approximate model (Lewbel and Pendakur, Reference Lewbel and Pendakur2009), with $\tilde {y} = x - p^\prime \bar {w}$, and deflate the household's nominal expenditure on food and fuelFootnote ³ $x$ by the consumer price index, whose commodity weights are derived from national household expenditure surveys (UBOS, 2018).

The compensated price elasticity of the budget share for good $j$ in response to an increase in the price of good $k$, is obtained from equation (2) as follows (Lewbel and Pendakur, Reference Lewbel and Pendakur2009):

(4)\begin{equation} e_{w_j,p_k}^c = \frac{a_{jk}}{w_j}, \end{equation}

where $w_j$ is the observed budget share of good $j$. In the empirical estimation, we use the average observed budget share only of individuals that consume good $j$ to compute the elasticity. To obtain price elasticities of demand, we note that the cross-price budget share elasticities equal the corresponding price elasticities of demand. The own-price elasticities are related to the budget share elasticities as follows:Footnote ⁴

(5)\begin{equation} e_{j,k}^c = e_{w_j,p_k}^c - 1, \end{equation}

where $e_{j,k}^c$ is the compensated price elasticity of demand for good $j$ with respect to the price of good $k$.

The expenditure elasticities of demand are derived as follows:

(6)\begin{equation} e_{j,x} = \frac{b_{1j}}{w_j} + 1. \end{equation}

We then derive the uncompensated (Marshallian) price elasticities of budget shares from the Slutsky equation in budget share form, as follows:

(7)\begin{equation} e_{j,k}^u = e_{j,k}^c - e_{j,x} w_k. \end{equation}

2.2 Econometric method

We estimate the EASI demand system via the Tobit Type I model for each budget share equation, with a lower limit of zero and an upper limit of one. A significant proportion of households in our sample report zero consumption expenditure for individual items (see table 1), which motivates the use of a corner solution response model (Wooldridge, Reference Wooldridge2010). The latent budget share equation for good $j$ and household $i$, based on equation (2), is

(8)\begin{equation} w_{ij}^{*} = b_{0j} + b_{1j} \tilde{y}_i + \sum_{k=1}^{J} a_{jk} p_{kd} + \sum_{l=1}^{L} r_{lj} z_l + \tilde{\epsilon}_{ij}, \end{equation}

where $\tilde {\epsilon }_{ij} \sim N(0,\sigma _\epsilon ^2)$, with the corresponding Tobit Type I model

\[ w_{ij} = \max(0,w_{ij}^{*}), \]

where $w_{ij}$ is the observed budget share, $\tilde {y}_i$ is the natural logarithm of household real expenditure (nominal expenditure on food and fuel $x$, deflated by the consumer price index), $p_{kd}$ are log prices for commodity $k$ and district $d$, $z_l$ is an $L$-vector of controls (these include the region of residence, rural/urban area and household size to capture economies of scale within the household), and $\tilde {\epsilon }_{ij}$ is a normally distributed error term. We subsequently estimate average marginal effects to derive the elasticities. In online appendix D, we discuss an alternative method to estimate the demand system, and perform robustness checks.

Table 1. Expenditure shares for the full sample (% of expenditure on food & fuel)

Note: The sample sizes for items with positive reported expenditures in column 3 are distinct for each item and hence not reported.

2.3 Welfare impacts

The first-order (FO) welfare effect of a price change is the additional expenditure incurred by households for the original consumption bundle to be affordable. In budget share form, the FO effect is expressed as

(9)\begin{equation} FO = \sum_{j=1}^K w_j^0 \frac{\Delta p_j}{p_j^0}, \end{equation}

where $w_j^0$ and $p_j^0$ are the initial budget share and price for good $j$, respectively, with expenditures aggregated across the $K$ goods that exhibit price increases. For goods unaffected by carbon pricing (in this study, charcoal and firewood), the welfare effect is zero.

To account for household demand changes in response to relative price hikes and shifts within the consumption basket, we analyse the second-order (SO) welfare effects. Because SO calculations account for substitution effects, they are typically smaller than FO calculations, which overestimate welfare losses. Banks et al. (Reference Banks, Blundell and Lewbel1996) derive SO approximations to the welfare loss of an indirect tax on a single item in the consumption basket. Renner et al. (Reference Renner, Lay and Greve2018) extend this framework to multiple simultaneous price changes through a SO Taylor series expansion of the expenditure function. We follow their approach to compute SO welfare effects, or the compensating variation, which measures the amount of compensation households require to achieve their initial utility levels at the new set of prices. As a proportion of household expenditure, the approximate SO welfare effect is

(10)\begin{equation} SO \approx \sum_{i=1}^J w_i^0 \frac{\Delta p_i}{p_i^0} + \frac{1}{2} \sum_{i=1}^J \sum_{j=1}^J w_i^0 e_{i,j}^c \frac{\Delta p_i}{p_i^0} \frac{\Delta p_j}{p_j^0}. \end{equation}

The first term represents the FO effect, while the second term measures changes in consumption due to multiple price changes, captured by the compensated price elasticities of demand ($e_{i,j}^c$) and initial expenditure shares ($w_i^0$). As the carbon pricing policy leaves the prices of biomass (charcoal and firewood) unaffected, any cross-price effects involving solid fuels will result in no additional welfare losses, despite non-zero demand changes.

2.4 Changes in final consumption

We compute the total change in quantity demanded for each item, in response to a carbon price, based on the estimated uncompensated price elasticities of demand. Consider a Marshallian demand function $f_i$ for commodity $i$, as a function of the $J$-vector of prices $q$,Footnote ⁵ and total household expenditure $x$,

\[ f_i = f_i (q_1,q_2,\ldots q_i,\ldots q_J;x). \]

The proportionate change in quantity demanded is then approximately equal toFootnote ⁶

(11)\begin{equation} \frac{\Delta f_i}{f_i} \approx \sum_{j=1}^J e_{f_i,q_j}^u \frac{\Delta q_j}{q_j}. \end{equation}

Estimates of the change in quantity demanded for each group $i$ are thus linear approximations, particularly in the event of significant price increases, resulting in large values for $\frac {\Delta q_j}{q_j}$. Rather than modelling nonlinear price changes, we adopt a parsimonious approach to obtain precise estimates of the price elasticities of demand.

We additionally compute the change in nutrient intake due to carbon pricing, across the population. We first recalculate budget shares with respect to households’ total food expenditure, and use these modified budget shares as weights for the demand changes derived in equation (11) for the $k$ food groups.Footnote ⁷ We then calculate the percentage change in the consumption of calories, protein and micronutrients due to carbon pricing. We follow Ecker and Qaim (Reference Ecker and Qaim2011) and analyse changes in the demand for five micronutrients: iron, vitamin C, riboflavin, folate and vitamin B12, which we subsequently aggregate, for tractability in presenting results. Using a food composition table for Uganda from HarvestPlus (Hotz et al., Reference Hotz, Abdelrahman, Sison, Moursi and Loechl2012), we first individually calculate the average level of nutrients $n_i$ for calories (kcal), protein (grams) and micronutrients (grams) in each food category $i$ per kilogram, using food item-specific consumption shares as weights for the broad food category. We then obtain the percentage change in total nutrient intake $\frac {\Delta N}{N}$ due to the carbon price as follows:

(12)\begin{equation} \frac{\Delta N}{N} \approx \frac{\sum_{i=1}^k n_i w_{f_i} \frac{\Delta f_i}{f_i}}{\sum_{i=1}^k n_i w_{f_i}}, \end{equation}

where $w_{f_i}$ is the modified expenditure share for food group $i$, for a total of $k$ food groups.

3.1 Expenditure and price data

Household expenditure data are drawn from the Uganda National Household Survey (UNHS), conducted by the Uganda Bureau of Statistics (UBOS), for the latest year available, 2016–17. The UNHS is nationally representative, with a sample size of 15,912 households for an estimated population of 43 million (UBOS, 2018). We exclude 2 per cent of observations with missing data on total household expenditure (on food and energy), resulting in a sample of 15,682 observations. The survey comprises detailed consumption modules on expenditures incurred for food items and energy sources, with respective weekly and monthly recall periods. Following Deaton and Zaidi (Reference Deaton and Zaidi2002), we scale expenditures to annual levels. For each item, we aggregate the expenditures incurred on market purchases and the self-reported expenditure values of goods produced at home, collected from villages/forests or items received in-kind. This also applies to charcoal and firewood. Households report the monetary value of the self-collected biomass, and we utilise this information while evaluating the non-purchased charcoal and biomass. Self-consumption is substantial in Ugandan households, with 35 per cent of food consumption from seasonal harvests by farmers, particularly for cereals and pulses (UBOS, 2018). Similarly, over 90 per cent of firewood consumed as fuel within homes is collected by household members from nearby forests and village plots (UBOS, 2018). While over 60 per cent of households report using firewood for cooking, nearly 30 per cent also use charcoal for cooking (UBOS, 2018), highlighting the strong dependence on solid fuel use.

Individual food items are categorised into commodity groups based on likely complementarities between items (for example cereals may complement vegetables, fruits, or meat), in order to satisfy the assumption of weak separability between different commodity groups. We thus model demand for 12 consumption groups including four energy sources – electricity, kerosene, charcoal, and firewood, and eight food categories – cereals; fruits; vegetables; meat & fish; milk & eggs; cheese, oils & fats; alcohol & tobacco; and other food & drink. These food categories thus reflect the different broad food groups of a standard diet as well as potential complementarities between food groups. They further reflect broad consumption patterns in Uganda, where cereals and vegetables comprise a significant proportion of the food basket (see table 1).

The demand system excludes other non-durable and durable goods due to lack of commodity-level price data. However, the non-durable items considered in the present analysis account for 65 per cent of households’ mean total spending. We further exclude transportation from the demand system estimation due to inadequate expenditure and price data,Footnote ⁸ as well as potential simultaneity of transport prices and other commodity prices. However, the microsimulation of a carbon pricing policy also includes an implicit tax on emissions generated from the transportation of fossil-based energy souces (electricity and kerosene) and all food items in the country.Footnote ⁹

To curtail the influence of outliers on the price elasticities of demand, we trim the top and bottom 1 per cent of observations of the real expenditure distribution for estimating elasticities, resulting in a final sample of 15,324 households, covering 96 per cent of the Ugandan population. Nevertheless, we utilise the full sample of households across the expenditure distribution to compute changes in welfare, commodity demand and nutrient intake.

Data on commodity prices are drawn from two sources. First, energy prices for kerosene, biomass and food items are obtained from the Uganda National Panel Survey (UNPS), conducted by the World Bank's LSMS for the closest year available, 2015–16. We rely on the World Bank Group's UNPS survey for data on energy prices as the UNHS, 2016–17, does not contain price data for biomass (charcoal and firewood) and contains sparse price data for kerosene, whereas the UNPS survey contains substantial data on energy prices at the household level. Data on energy prices are essential to estimate elasticities in our analysis.

We construct average annual prices for each of the 12 commodities in the demand system, by taking simple averages of the item-specific market prices reported by households, at the district – rural/urban level. This ensures significant spatial heterogeneity in prices, while annual averages help mitigate seasonal influences on biomass prices. Aggregating prices at the district level further helps control for potential endogeneity of prices due to household production, and mitigates concerns of quality differentiation in unit prices (Capéau and Dercon, Reference Capéau and Dercon2006). For districts with missing price data in the UNPS, we apply the corresponding national, annual average price for each commodity, to complete the sample. We then inflate all commodity prices to 2016–17 levels using the consumer price index for Uganda, to match prices to household expenditures.Footnote ¹⁰

Second, we obtain national electricity tariffs for Uganda from the Electricity Regulatory Authority (ERA).Footnote ¹¹ These are available at a quarterly frequency, and the price structure involves a two-block tariff, at a threshold consumption level of 15 kWh units per month. While the UNHS 2016–17 does not contain data on the quantity of electricity consumed, corresponding data from the UNPS, 2015–16, suggest that around 90 per cent of households with reported expenditures for electricity consumed more than 15 units per month. Since we cannot assign the tariff based on the household's electricity consumption, we only apply the higher block tariff as a uniform price for electricity for all households in Uganda.

3.2 CO2 emissions data

To analyse the distributional welfare impacts of a national carbon price, we combine the household-level expenditure survey with the Global Trade Analysis Project (GTAP 10) database for 2014, which provides carbon dioxide emissions levels for 65 sectors, based on an environmentally-extended multiregional input-output table (Aguiar et al., Reference Aguiar, Chepeliev, Corong, McDougall and van der Mensbrugghe2019). We map the 140 consumption items of the UNHS to 16 corresponding sectors in GTAP 10 (documented in the online appendix, table A8).

We rely on the GTAP database for information on indirect CO2 emissions entailed in cultivation and transport of commodities, since alternative databases such as Exiobase and World Input-Output Database (WIOD) do not provide sector-level data on CO2 emission intensities for Uganda. To the best of our knowledge, the GTAP is the only database with country- and sector-specific data on emission intensities for a number of low-income countries, including Uganda. The GTAP database covers emissions from key polluting sectors such as the oil, cement, agriculture and forestry sectors, for Uganda. The Government of Uganda specifically targets reducing emissions from the petroleum, energy and forestry sectors, in its updated Nationally Determined Contribution (NDC), submitted to the UNFCCC in 2022 (Ministry of Water and Environment, 2022).

For our welfare analysis, we draw on sector-level emission intensity data for broad food groups and energy sources such as kerosene and electricity. Data on CO2 emission intensities for solid fuels (charcoal and firewood) are drawn from the forestry sector of GTAP. GTAP 10 improves upon previous versions by specifically incorporating the electricity mix in Uganda and other countries, which leads to more accurate emission intensity data. The GTAP database has been applied to the Ugandan context (Ismail et al., Reference Ismail, Lubwama, Kirabira and Sebbit2023), and to cross-country studies on carbon pricing (Steckel et al., Reference Steckel, Dorband, Montrone, Ward, Missbach, Hafner, Jakob and Renner2021) and poverty alleviation in LMICs (Bruckner et al., Reference Bruckner, Hubacek, Shan, Zhong and Feng2022).

The carbon intensity for a specific item can either be attributed to direct emissions due to energy consumption (electricity and kerosene), or indirect emissions from the transportation of commodities (Renner et al., Reference Renner, Lay and Greve2018). We analyse both direct and indirect emissions for all commodity groups modelled in the demand system. We first compute the weighted carbon intensities for the 12 categories of the demand system based on the mean consumption shares of individual items within each of the 12 categories from the UNHS. This yields a household-level dataset with expenditures and budget shares for various commodity groups, and corresponding weighted carbon intensities for each category. Subsequently, we calculate percentage price changes for all commodities owing to a carbon price of USD40/tCO2.

3.3 Descriptive statistics

Summary statistics for expenditure shares, annual household expenditure on food and energy, and prices used in estimation of the EASI demand system are presented in table 1. The mean budget shares (column 1) suggest that households’ consumption baskets for food and fuel are largely composed of vegetable consumption, followed by cereals, meat & fish, and other foods, while energy comprises less than 5 per cent of expenditure. A substantial proportion of households further report zero expenditures for various items. For instance, only around 16 per cent of households report positive expenditures on electricity (column 4), with 84 per cent of the sample reporting zero spending on electricity, which is largely due to lack of widespread access to the electric grid (IEA, 2020).Footnote ¹² For the sub-sample of electricity users (column 3), the average budget share is 5.7 per cent, compared to mean budget shares of 0.6 per cent for kerosene, 2.6 per cent for charcoal and 3.6 per cent for firewood, for the respective users.

Mean expenditure shares and standard deviations for all items by expenditure tercile – rural/urban groups, are presented in the online appendix, table A1. Households’ budget shares for different energy items by terciles based on the household's total expenditure, and rural/urban areas, are displayed in online appendix figure A1. The share of biomass in households’ total energy expenditure declines with income, from 84 per cent for the average rural household in the bottom tercile to 55 per cent for the average urban household in the top third of the total expenditure distribution. Similarly, the average budget share for kerosene declines from 15 to 3 per cent across expenditure terciles and rural/urban areas. The share of electricity in households’ energy mix, however, rises sharply from 1 per cent on average among rural households in the bottom tercile to 42 per cent on average among urban households in the top expenditure tercile.

Summary statistics for commodity prices are presented in table 2. We observe significant price variation for most items, in particular kerosene, charcoal, meat & fish, and cheese, oils & fats, which ensures reliable econometric identification.

Table 2. Summary statistics for item-specific prices

Note: Prices are in Ugandan shillings (UgX) per unit.