Potential impacts of transportation infrastructure improvements to maize and cassava supply chains in Zambia

Utilizing the theoretical framework described in section 2.1, we implement a supply chain model for Zambia spanning the years 2010 to 2020. The temporal resolution of the model is the monthly time period (t). The spatial resolution of the model is the district scale, representing the markets (m, n) within the model. We treat each individual district as a market catchment and study the behavior of agents in each district. Additionally, we incorporate international trade in our model between Zambian districts and a lumped 'Rest of World' entity. This means that 'model export' estimates exports from districts to the 'Rest of World'; 'model imports' are estimates of how much the 'Rest of World' sends to each district in Zambia.

The model focuses on two staple crops: maize and cassava (c). Each crop is treated as a homogeneous agricultural good in every market following the standard assumption in modeling staples within developing countries (Porteous 2019, Sotelo 2020). Other goods are collectively represented as an outside numeraire good.

In each market m, the behavior of four agents is vital: a representative producer, a representative consumer, a representative public trader, and a representative private trader. We assume the behavior of the representative producer is tractable, which means that the production of crops is treated as exogenous to the model. Incorporating external production into the model serves as a simplification of real-world dynamics. This simplification aligns with our objective to ascertain whether it is possible to enhance food security by lowering trade costs within the current production conditions. Within a market, the representative consumer and producer engage in transactions involving the buying and selling of crops at the district wholesale price. The representative public trader refers to the government crop procurement organization, the FRA. The representative private trader has the ability to purchase goods from market m and sell them in another market n, provided that there exists a transportation link between the two markets. The behavior of the representative consumer, public trade, and private trade is modeled by extending the fundamental rational expectations framework of Williams et al (1991) for each market.

The supply chain model must adhere to the market clearing condition, which ensures that the supply and demand in each market are in equilibrium. This condition is expressed by equation (1):

$$\begin{align} H_{cmt}+F_{cmt}+S_{cm,t-1} = Q_{cmt}+S_{cmt} +\sum_{n\neq m}T_{cmnt}. \end{align} \tag{ 1 }$$

In equation (1), the left side represents the total supply available to the private trader in each market m. The supply consists of three components: local production (H_cmt ), FRA releases (F_cmt ), and staple carried over from the last period's storage ($S_{cm,t-1}$). On the right side of equation (1), private traders decide how to allocate this total supply. They decide the amount to be sold for local consumption (Q_cmt ), the quantity to be stored for the next period (S_cmt ), and the volume to be traded with other markets (T_cmnt ).

The private trader strategically allocates supply to maximize their profit each month, considering the storage cost, trade cost, current price ⁶ , time value of money, and rational expectations of future price. The mathematical expression for this profit maximization problem is shown in equation (2)

$$\begin{align} \max_{S_{cm{t}^{^{\prime}}}, T_{cmn{t}^{^{\prime}}}} \mathbb{E}_{t} \left[ \sum_{{t}^{^{\prime}} = t}^{\infty} \frac{-P_{cm{t}^{^{\prime}}}\left(H_{cm{t}^{^{\prime}}}+F_{cm{t}^{^{\prime}}}\right) +P_{cm{t}^{^{\prime}}}Q_{cm{t}^{^{\prime}}} - k_{m}S_{cm{t}^{^{\prime}}} +\sum_{n} \left(P_{cn{t}^{^{\prime}}}T_{cmn{t}^{^{\prime}}}-\kappa_{c}\tau_{mn}\left |T_{cmn{t}^{^{\prime}}} \right | \right) }{ \left(1+r_{{t}^{^{\prime}}} \right)^{{t}^{^{\prime}}-t}} \right] \end{align} \tag{ 2 }$$

where $\mathbb{E}_{t}$ represents the mathematical expectations operator, conditional on the information available at time t. k_m represents the per-unit storage cost, r_t is the real interest rate at time t, κ_c is the trade cost parameter for different crops, and τ_mn denotes the relative trade cost between market m and n.

Replacing Q_cmt with equation (1) and taking the first-order condition with respect to T_cmnt , we obtain the spatial arbitrage condition ⁷ , as shown in equation (3)

$$\begin{align} \left[P_{cmt}+\kappa_{c}\tau_{mn}-P_{cnt}\unicode{x2A7E}0\right] \perp \left[T_{imnt}\unicode{x2A7E} 0\right]. \end{align} \tag{ 3 }$$

Taking the first-order condition with respect to S_cmt yields the temporal arbitrage condition, as depicted in equation (4):

$$\begin{align} \left[P_{cmt} +k_{m}-\frac{E_{t} \left[P_{cm,t+1}\right]}{1+r_t} \unicode{x2A7E} 0 \right]\perp \left[S_{cmt} \unicode{x2A7E} 0\right]. \end{align} \tag{ 4 }$$

The representative consumers decide their monthly consumption of each crop and the numeraire good to maximize their utility given their budget constraint. We employ a two-stage budget procedure to estimate the demand for each good, assuming weak separability of consumer preferences for each crop compared to outside goods. In the first stage, consumers decide the allocation of their total expenditure between staples and outside goods. We assume a quasilinear utility function for the numeraire good, meaning that the first stage utility function is linear with respect to the outside goods and is strictly concave in the crop composite as illustrated by equation (5)

$$\begin{align} U_{mt} = D^{\epsilon }_{m}N^{\epsilon }_{mt}\frac{Q^{1+\frac{1}{\epsilon}}_{mt}}{1+\frac{1}{\epsilon}} + X_{mt} \end{align} \tag{ 5 }$$

where, U_mt represents consumer utility in market m at time t. D_m is a constant parameter for each market, and N_mt represents the population. Q_mt refers to the consumption of the crop composite. X_mt represents the consumption of the numeraire. ε indicates the elasticity of demand for the staple composite.

In the second stage, the price of each crop and total expenditure on staples determine the consumption of each crop. The preferences for individual crops follow a constant elasticity of substitution (CES) model, as shown in equation (6)

$$\begin{align} Q_{mt} = \left[ \sum_{c\in C_{m}}\alpha^{\frac{1}{\sigma}}_{cm}Q^{\frac{\sigma-1}{\sigma}}_{cmt} \right]^{\frac{\sigma}{\sigma-1}} \end{align} \tag{ 6 }$$

where σ and α_cm are demand parameters indicating the elasticity of substitution and share parameters, respectively. The sum of the share parameters for crops in each market is normalized (i.e. $\sum_{c\in C_{m}}\alpha_{cm} = 1$).

In summary, the behavior of representative consumers can be expressed by equation (7)

$$\begin{align} \begin{array}{ll@{}ll} \text{maximize}_{\left\{Q_{cmt}\right\}_{c\in C_{m}},X_{mt}} & U_{mt} &\\ \text{subject to}& \sum_{c\in C_{m}}P_{cmt}Q_{cmt}+X_{mt}\unicode{x2A7D} Y_{mt}.\end{array}\end{align} \tag{ 7 }$$

The representative consumers maximize their utility function subject to their income constraints. The inverse demand function for each crop in each market is obtained using the method of Lagrange multipliers, resulting in equation (8). The detailed derivation of equation (8) is provided in the supporting information (SI). 

$$\begin{align} P_{cmt} = \frac{{\alpha }_{cm}^{1/\sigma }}{Q_{cmt}^{1/\sigma }}\times \frac{Q_{mt}^{1/\sigma+1/\epsilon }}{{\left(D_{m}N_{mt}\right)}^{1/\epsilon }}. \end{align} \tag{ 8 }$$

The FRA was assigned the responsibility of acquiring, storing, and subsequently distributing stocks in the market during periods of food scarcity and other types of disasters that lead to price fluctuations (Mukumba et al 2018). FRA follows a 'price-band' scheme to keep prices within a narrow range, which have not proven successful especially in the case of abundant or meager harvests. In each district, the FRA offers to purchase maize for the government stockpile ⁸ at a 'floor price', with the purchase target limited by financial constraints and local production. Generally, the FRA sells maize at subsidized prices during food stresses to large millers , which are often located in large cities, such as the capital city of Lusaka. The most vulnerable groups, for example the rural poor, could lose out from the differing purchase and release activities of FRA in urban and rural areas (Mason and Myers 2013). We decide to estimate the FRA as a separate entity in the model, rather than applying the general method for modeling the price band policy, due to these concerns. The annual district-level FRA purchase data is obtained from the FRA office and we take it as given (Food Reserve Agency 2021). We identify the locations of prominent millers and model the FRA maize flows to the major millers as a multi-source, multi-destination transportation problem using the linear programming, formulated as equation (9)

$$\begin{align} \begin{array}{ll@{}ll} \text{minimize} & \displaystyle\sum\limits_{i = 1}^{s}\sum\limits_{j = 1}^{d} \tau_{ij} x_{ij} &\\ \text{subject to}& \displaystyle\sum\limits_{j = 1}^{d}x_{ij} \unicode{x2A7D} a_{i}, \; &i = 1 ,\dots, s\\ & \displaystyle\sum\limits_{i = 1}^{s} x_{ij} \unicode{x2A7E} b_{j},\; &j = 1 ,\dots, d \\ & \displaystyle x_{ij}>0 \end{array} \end{align} \tag{ 9 }$$

where the objective function aims to minimize the cost of trade between FRA stock i and millers j. The per unit transportation cost is denoted by τ_ij and the shipment amount is represented by x_ij . Given the absence of rerouting, mass conservation has been accounted for by the two constraints: (1) the total outgoing shipment must not exceed the FRA stock in that market (a_i ). The annual district-level FRA purchase data is provided by the FRA office. The initial stock for 2009 is assumed to be zero, and subsequent FRA stocks for each market (a_i ) are calculated year-by-year. (2) The total incoming shipment must meet or exceed the demand of millers (b_j ). The demand for each market is determined based on the daily capacity of local large milling plants. In our model, we focus solely on the aggregate flow and not individual truck deliveries, so we do not need to include an upper threshold. Subsequently, we estimate the monthly FRA purchase and release amounts $(F_{cmt})$, assuming that the purchases occur during the harvest month and the stocks are released evenly over the lean months.

The monthly crop price series are collected over a span of 132 months, covering 72 markets from January 2010 to December 2020 (Central Statistical Office of Zambia 2020). The price series for maize and cassava exhibit incompleteness. Maize has only 7364 out of 9504 (77%) possible observations, while cassava has only 3293 out of 9504 (35%) possible observations, indicating a partial representation of the market dynamics for these crops. The original price series are wholesale prices in the local currency of Zambia (Kwacha, ZMK). To facilitate comparison and analysis, we convert the price series to USD/kg ⁹ using the monthly USD-ZMK exchange rate deflated by US Consumer Price Index to the base year 2010. Across all time periods and markets in Zambia, the average price of maize was 0.215 USD kg⁻¹, and the average price for cassava flour/meal was 0.631 USD kg⁻¹.

Figure 2 illustrates the substantial variation in crop price across districts and over time. This spatial and temporal variability in price is a key factor driving regional trade. The significant price dispersion indicates incomplete or costly arbitrage. Seasonal price variations are significant for both maize and cassava, therefore, a more detailed supply chain model at a finer time-scale could provide insights for storage management.

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International cassava trade is not considered in our model due to its minimal amount. The maize price for the Rest of World market is assumed to be the Johannesburg market price (SAFEX). The assumption is based on the fact that the majority of agricultural imports to Zambia originate from South Africa, which is a nearby maize-surplus country ¹⁰ . The average price for SAFEX maize is 0.202 USD kg⁻¹.

The annual national population is obtained from the UN Population Division (United Nations and Social Affairs 2022). To allocate the national-level population data to specific districts, we utilize the district-level population in 2010 from the Zambia population census (Central Statistical Office of Zambia 2019). We assume that the percentage of the national population residing in each district remained constant throughout the study period and that there are no changes in population within a given year. This approach enables us to derive monthly population data at the district level.

Population density varies spatially throughout Zambia, ranging from 3 people per km² in Mufumbwe to 3681 people per km² in Lusaka. The average population of Zambia during the study period is 15.9 million, representing a 35% increase from 2010 to 2020.

The annual estimates of district-level maize production are obtained from the Crop Forecast Survey (CFS) (Central Statistical Office of Zambia 2021). Zambia is divided into four distinct agro-ecological zones (FAO 2021), each with varying climate conditions that result in different harvest times. Accordingly, we assign May as the harvest month of maize for districts in Region I, June for districts in Region IIa, January for districts in Region IIb, and April for districts in Region III. Consequently, production entries exist only for the assumed harvest month, while the remaining months have a value of zero.

The annual district-level cassava production data is derived from the CFS, with missing data recorded in the years 2006, 2019, and 2020. To address the missing cassava production, linear regression is utilized for interpolation. Given the versatile nature of cassava harvests, which can occur in any season depending on local grain reserves and prices, we assume a January harvest for cassava across all years. This assumption is made based on the prevalent practice of harvesting cassava during the lean season to mitigate food insecurity.

Figure 3 displays the distribution of per capita production across different regions of Zambia. The southern regions exhibit higher levels of per capita maize production compared to the northern regions. Notably, the district of Mpongwe records the highest production at 789 kg per capita, while Lusaka reports the lowest production at 0.81 kg per capita. According to Sitko et al (2011), Zambians consume an estimated 85 to 140 kg of maize per person annually. The variations in production levels among districts emphasize the significance of trade as a means to address the disparities.

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Figure 4 presents the historical data depicting the government's maize procurement through the FRA over time. It is evident that FRA purchases have decreased over the years, resulting in an expanded role for private traders in the market.

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We estimate the amount of maize released in areas with large millers as described in section 2.1.5. Notably, the percentage of FRA purchases relative to total maize production in Zambia has exhibited significant fluctuations on an annual basis, with higher levels observed during election years, such as 2011 and 2015. Furthermore, recent data suggests that FRA purchase prices have been relatively lower than the national average wholesale price, which may have contributed to the decline in FRA purchases. This lack of competitiveness in procurement prices has led to an increasing flow of surplus production through private channels. This observation reinforces the need for our supply chain model to encompass both public and private traders.

The elasticity of demand for the staple composite (ε) has been estimated in the literature (Roberts and Schlenker 2013), which was used in previous regional maize trade models in Sub-Saharan Africa (Porteous 2019). In line with these studies, we adopt a value of ε = −0.066. Furthermore, We assume Cobb–Douglas preferences (σ = 1) for crop consumption, as commonly employed in equilibrium models (Pohit 1997, Hosoe 2004). Cobb–Douglas preferences assume constant expenditure shares across crops, which are represented by the share parameters (α_cm ). This study delves into district-level share parameters, which means that we do not explicitly model individual household behavior. However, we do include household-level data in our study through their aggregated characteristics: (a) composition of rural and urban population by district; and (b) the per capita monthly expenditure on maize and cassava for rural and urban stratum for each district. The expenditure refers to total estimated value of consumption from purchase, self-production, and gift specifically. This expenditure is derived from the 2010 Living Conditions Monitoring Survey, a stratified, nationally representative survey conducted by Zambia's Central Statistical Office (CSO) (Central Statistical Office of Zambia 2010). Leveraging these share parameters and average food prices in each district, we calculate the price and consumption composites. Subsequently, we straightforwardly estimate the demand shifter parameter as $D_{m} = \frac{1}{T}\sum_t \frac{Q_{mt}}{N_{mt}P_{mt}^{\epsilon}}$.

The monthly real interest rate (r) is used as a measure of the time value of money. We obtain the r from the World Bank (World Bank 2022), which is estimated by adjusting the lending rate for inflation. For maize, we utilize the storage cost data from Porteous (2019), which is reported as k = 6.919 USD/metric ton. However, since cassava can be harvested year-round once the roots have matured, we assume zero storage costs for cassava.

To estimate the relative trade cost τ_mn , we utilize price data for sugar and Zambia's transportation network. We adopt the approach that states,'If a homogeneous commodity can only be produced in one region, the difference in prices (of that commodity) between the origin region and any other consuming region equals the cost of trading between the two regions'. This approach has been previously applied to India (Donaldson 2018) and Peru (Sotelo 2020) due to lack of crop transport data. In Zambia, over 90% of sugar is produced by Zambia Sugar Plc, whose factories operate in Mazabuka, located in the southwest of the country (Kalinda et al 2014). By using trade cost represented by the price difference of sugar between the origin (Mazabuka) and destinations ($\mbox{ln}p_d-\mbox{ln}p_o$), we associate the relative trade cost with the effective distance, aiming to estimate the relative trade cost for other market pairs, specified in equation (10)

$$\begin{align} \tau_{mn} = \mathrm{e}^{\mathbb{E}\left[\beta_0+\beta_1 \ln{d_e}\right]}-1 \end{align} \tag{ 10 }$$

where effective distance (d_e ) represents a weighted distance along the 'lowest-effective distance route' between origin and destination, using any available modes of transportation. We estimate the parameters β₀ and β₁ through ordinary least squares regression. p denotes the wholesale sugar price and the subscripts d and o represent the destination and origin.

The effective distance d_e can be expressed as:

$$\begin{align} d_e = \alpha_\mathrm{rail}\times d_\mathrm{rail}+\alpha_\mathrm{main}\times d_\mathrm{main}+\alpha_\mathrm{informal}\times d_\mathrm{informal}. \end{align} \tag{ 11 }$$

The weighted ratio vector ($\alpha_\mathrm{rail}$, $\alpha_\mathrm{main}$, $\alpha_\mathrm{informal}$) is determined using nonlinear least squares through grid search. We find the α vector that minimizes the sum of squared residuals for $\ln{p_d}-\ln{p_o} =$ $\beta_0 + \beta_1 \ln{d_e}+ \mathrm{error}$ by recomputing the lowest-effective distance routes for each trail.

To construct the transportation network of Zambia, we utilize the OpenStreetMap database (OpenStreetMap contributors 2022) as detailed in the supplementary information (SI). The weights used to calculate the shortest path are determined by multiplying the weight ratio α by the physical length of the road or railway. Specifically, $\alpha = \alpha_\mathrm{rail}$ for railways, $\alpha = \alpha_\mathrm{main}$ for trunk, primary and secondary roads, $\alpha = \alpha_\mathrm{informal}$ for tertiary and unclassified roads. We fix $\alpha_\mathrm{rail}$ as 1 and vary $\alpha_\mathrm{main}$ and $\alpha_\mathrm{informal}$ until the minimum root-mean-square error (RMSE) is obtained. The workflow for calculating the relative trade cost between districts within Zambia is summarized in figure 5.

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We assume that the optimal α vector and corresponding β derived from sugar prices also apply to maize and cassava. To obtain the per unit (additive) trade cost for each crop and trading pair, we multiply the τ_mn with κ_c . The trade cost multiplier κ_c for each crop c is determined through grid searching, aiming to achieve the best fit between the trade model price output and observed prices.

Trade costs are crucial parameters calibrated in this study. The grid search yields an α vector of $\alpha_\mathrm{rail} = 1, \alpha_\mathrm{main} = 2, \alpha_\mathrm{informal} = 22$, indicating that transportation via the main road system incurs higher costs compared to the railway. Additionally, transport through informal roads, which typically connect smaller towns and villages, involves significantly higher costs than through the main roads that link larger towns. The unit trade cost in Zambia ranges from 0.032 to 0.72 USD km⁻¹t⁻¹. The average trade cost for maize was 0.092 ± 0.062 USD km⁻¹t⁻¹. According to the USDA's 2016 Grain Truck and Ocean Rate Advisory Report (USDA 2016), the national average long-haul grain transportation cost for the United States in 2015 was 0.033 USD km⁻¹t⁻¹. It is not surprising to see higher trade costs in Zambia compared to the United States, considering the less developed transportation infrastructure in Zambia compared to the United States. Moreover, a high degree of spatial heterogeneity in unit trade costs is observed, resulting from uneven infrastructure development. The railway serves as a vital north–south artery through central Zambia, complemented by a dense distribution of main roads that surround it. As a result, these areas experience relatively lower transport costs.

Figure 7 presents that both private and public flow of maize in Zambia concentrates near urban districts, including prominent cities such as Lusaka, Kitwe, Ndola, Kabwe. The trade is also flourishing around the Nakonde region situated near international ports. The FRA plays a crucial role in the management of food distribution by actively purchasing surplus food from various districts within the Eastern Province. These surplus food supplies are then efficiently transported to the central hub in Lusaka, where they are further distributed to meet the demands of the population. It is worth noting that the major food flows observed in Zambia often coincide with the presence of the country's extensive railway network. The railway network serves as a critical artery, allowing for the swift and cost-effective transportation of agricultural products from surplus-producing districts to the central hub and beyond. The annual trade flow results are presented in supporting information (SI)

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Because there is no empirical information on intra-regional trade in Zambia to validate our model results against, we compare our regional trade flow output with another modeled regional flow developed by Porteous (2019) for Sub-Saharan Africa. We aggregate our district-level flow data to ensure consistency with the coarse spatial scale of Porteous (2019)'s ¹² . The comparison is displayed in figure 8. Note that Porteous (2019) presents results for 2003 to 2012, so we show the overlapping years in figure 8.

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The two model results are of the same order of magnitude (the width of the lines in figure 8 represents the mass of flow). Both (Porteous 2019) and our results observes maize transport from southwestern Zambia to Lusaka market shed and from Solwezi market shed to Kitwe market shed. Maize flows from in-surplus areas to places where demand is higher or actively served as international export hubs. Another important flow observes in our model result is the two-way flow between Mongu market shed and Lusaka/Solwezi market shed. This is mainly due to the different maturity times of maize in different regions in our model. The Mongu region, despite its low production, produced some irrigated maize around January, and this costly maize flows to the Lusaka and Solwezi regions where consumption is high. Most of the maize in other regions matures between May and July, and the cheap maize flow back to the Mongu region.

In our analysis, we compare the flow results of international trade (trade between Zambia's districts and the aggregated 'Rest of World' node) with the BACI dataset, which is an international trade database for the product-level. The BACI datasets utilize raw data from UN COMTRADE database and address inconsistent declarations of exporters and importers (Gaulier and Zignago 2010). Figure 9 presents a comparison of Zambia's international import and export of maize between the BACI datasets and our model's output.

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Both the BACI datasets and our model results confirm that Zambia is a net maize exporter, as the volume of international exports surpasses the international imports. The model's export results are generally higher than the BACI export for most years. This disparity can be attributed to the fact that the BACI dataset only tracks formal trade between countries, whereas African countries have a long history of unreported informal trade. In certain years, our model estimates exports to be considerably higher than the BACI records, even up to 15 times greater volumes. These instances of low formal BACI exports often coincide with periods of government-imposed export bans, which are not explicitly restricted in the model. However, export bans in Zambia exist even though there is a surplus of domestic production. Taking the year 2013 as an example, despite normal production levels and the FRA releasing a large quantity of stocks that year, the government maintained an export ban. The absence of a viable domestic market might compel private traders to export food to other countries leading to the high international export amount as shown in our model, indirectly supported by past news reports of grain traders requesting the government to reverse the ban on exporting maize (Lusakatimes 2013). Furthermore, our price results closely approximate survey prices, implying the persistence of informal and/or illegal exports during export ban periods to maintain price levels.