Study design

Our study is designed in four phases. First, following Chambers and Pieralli (Reference Chambers and Pieralli2020) and Plastina et al. (Reference Plastina, Lence and Ortiz-Bobea2021), we apply a nonparametric technique to estimate TFP using a combination of climate variables and conventional input and output variables from crop production values to estimate an output-oriented DEA-Malmquist index (Fare et al., Reference Fare, Grosskof, Norris and Zhang1994). Secondly, we assess the impact of climate variables on TFP change by directly controlling for climate variability in the production frontier technology. Thirdly, we compare the results of steps 1 and 2 to determine possible bias in TFP change as a result of the omission of weather effects from the production technology. Finally, we made two more sets of TFP estimates to test the individual effects of the two climate variables (temperature and precipitation) on TFP growth and its components (TECCH, TECH and SECH). We drew an analogy to show how accounting for these variables together or separately may change TFP estimates across African countries and regions. The main goal of this exercise is to quantify the effects of climate shocks and demonstrate why climate variables should not be disregarded in the measurement and decomposition of agricultural TFP change in Africa.

DEA-Malmquist index specification

The DEA-Malmquist index is a nonparametric analytical technique that calculates the production frontier using a linear programming system. It is common practice to utilize the Malmquist index technique as a traditional TFP measure since it enables the decomposition of TFP change into components with respect to the frontier (i.e., a technical change index, an index of technological change and a scale efficiency change) (Nondo and Jaramillo, Reference Nondo and Jaramillo2018; Ding and Zhang, 2021; Bernard et al., Reference Bernard, Song, Hena, Ahmad and Wang2022). The index takes both the border's movement and the separation between each production entity and the frontier into account. The frontier may be defined as nations (DMUs) that have embraced best-practice technology as a result of advancements. The Malmquist index's usage in growth studies is fairly simple and convenient because it assesses productivity increases over time without imposing prior assumptions (Pathak, Reference Pathak2019).

Empirical specification

Consider agricultural inputs m as a function of both physical market inputs represented by x and climate factors denoted by c, with m = x + c, for each country j at time t. This method allows for the calculation of the climate-inclusive TFP index (C _TFP), which is based on m inputs and y output, where y represents the country j's total crop production at time t. In other words, the two climate variables (i.e., temperature and precipitation) are jointly accounted for alongside the conventional inputs when the m variable is used in the production technology. As stated in the study's design, we examine both the combined effects of these variables and their individual effects (see Equations 1, 2 and 3). As a result, we derived a typical TFP index (N _TFP), a precipitation control index (P _TFP) and a temperature control index (H _TFP). According to the productivity accounting approach, productivity can be measured in relation to two periods (t and t + 1). Thus, using the method of Chambers and Pieralli (Reference Chambers and Pieralli2020), the productivity index for t + 1 is then calculated with regard to t, the base period:

(1)$$C_{TFP_{t, t + 1}} = \left({\displaystyle{{y_{\,jt}} \over {m_{\,jt}}}} \right)/\left({\displaystyle{{y_{\,jt + 1}} \over {m_{\,jt + 1}}}} \right){\rm \;}$$

The C _TFP represents the combined effects of the climate-inclusive measure of TFP growth. The other three measures of TFP, as described above, can be shown mathematically as follows:

(2)$$N_{TFP_{t, t + 1}} = {\rm \;}\left({\displaystyle{{y_{\,jt + 1}} \over {x_{\,jt + 1}}}} \right)/\left({\displaystyle{{y_{\,jt}} \over {x_{\,jt}}}} \right)$$

The N _TFP is a measure of TFP change that does not take into account climate input variables in production technology. y _jtdenotes total crop production output y for the country in year t. While x _jt represents the conventional market and material inputs used by country j for crop production in time t, this is the typical TFP estimate that is normally used to explain productivity growth trends and patterns in Africa. In this study, however, it is used as a reference base for our climate control productivity estimates:

(3)$$P_{TFP_{t, t + 1}} = {\rm \;\;}\left({\displaystyle{{y_{\,jt + 1}} \over {x_{\,jt + 1} + p_{\,jt + 1}}}} \right)/\left({\displaystyle{{y_{\,jt}} \over {x_{\,jt} + p_{\,jt}}}} \right){\rm \;}$$

P _TFP is the TFP estimate that exclusively controlled for the effect of precipitation in the production technology. The below expression captures the effect of temperature, H _TFP:

(4)$$H_{TFP_{t, t + 1}} = \left({\displaystyle{{y_{\,jt + 1}} \over {x_{\,jt + 1} + h_{\,jt + 1}}}} \right)/\left({\displaystyle{{y_{\,jt}} \over {x_{\,jt} + h_{\,jt}}}} \right){\rm \;}$$

Our goal is to estimate the agricultural productivity of Africa in a nonparametric framework while accounting for natural inputs and to contrast the results with a conventional productivity approximation that ignores climate considerations. Instead of creating a weather productivity index like Chambers et al. (Reference Chambers, Pieralli and Sheng2020), we create climate-inclusive indices to demonstrate how the omission of natural weather inputs may impact the measurement and decomposing of agricultural TFP in Africa. Following Fare et al. (Reference Fare, Grosskof, Norris and Zhang1994), the Malmquist output-oriented distance function for the production technology is estimated, based on the constant return to scale (CRS) assumption (Bernard et al., Reference Bernard, Song, Hena, Ahmad and Wang2022):

(5)$$MP_t^{Ocrs} = \{ {( {y_{\,jt}, \;m_{\,jt}{\rm \;}} ) \colon m( {c_{\,jt}x_{\,jt}} ) {\rm \;that\;produces\;}y} \} $$

Equation (5) accounts for the stochastic nature of climate variability in agricultural production technology (Chambers and Pieralli, Reference Chambers and Pieralli2020). The Malmquist output distance function at period t is given in Equation (6) (Fare, et al., Reference Fare, Grosskof, Norris and Zhang1994; Henderson and Russell, Reference Daniel and Henderson2005):

(6)$$D_t^0 ( {y_{\,jt}, \;{\rm \;}x_{\,jt}, \;c_{\,jt}} ) = {\rm min}\{ {\theta \colon ( {y_{\,jt}, \;{\rm \;}x_{\,jt}, \;c_{\,jt}} ) \in P_t^{Ocrs} } \} {\rm} = [ {{\rm max}\{ {\theta \colon ( {y_{\,jt}, \;{\rm \;}x_{\,jt}, \;c_{\,jt}} ) \in P_t^{Ocrs} } \} } ] ^{{-}1}$$

Under the framework of Equation (6), productivity growth at time t is calculated by dividing the observed output y _jt, by the highest output that could be produced from the observed total input, which includes climate, at time t:

(7)$$\eqalign{& D_t^0 ( {y_{\,jt}, \;\;x_{\,jt}, \;c_{\,jt}} ) = \displaystyle{{y_{\,jt}} \over {( {y_{\,jt}, \;\;x_{\,jt}, \;c_{\,jt}} ) }} \cr & {\rm or} \cr & D_t^0 ( {y_{\,jt}, \;\;m_{\,jt}} ) = \displaystyle{{y_{\,jt}} \over {( {y_{\,jt}\;m_{\,jt}} ) }}} $$

where m _jt = x _jt +  c _jt = climate-inclusive input bundle:

• x _jt = traditional inputs (land, labor, capital and materials)

• c _jt = climate variables (temperature and precitation)

We can now decompose a climate-inclusive productivity index for a given country from period t to period t + 1 using Equation (7), as described by Fare et al. (Reference Fare, Grosskof, Norris and Zhang1994):

(8)$$D_t^0 ( {y_{t + 1, {\rm \;\;}}m_{t + 1}, \;y_{t, }{\rm \;}m_t} ) = \left[{\displaystyle{{d_t( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_t( {m_t, \;y_t} ) }}\displaystyle{{d_{t + 1}( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_{t + 1}( {m_t, \;y_t} ) }}} \right]^{1/2{\rm \;\;\;\;}}$$

The function$\;d_t^0 ( {y_{t + 1, \;\;}m_{t + 1}, \;y_{t, }\;m_t} )$ indicates the distance between periods t and t + 1technology. The expressions (m _t+1, y _t+1) and (m _t, y _t) are aggregate input and output vectors of two periods, t and t + 1, respectively. The period t technology is denoted by the first component in the bracket d _t(m _t+1, y _t+1)/d _t (m _t, y _t), while the period t + 1 technology is represented by the second component d _t+1(m _t+1, y _t+1)/d _t+1(m _t, y _t) (Bernard et al., Reference Bernard, Song, Hena, Ahmad and Wang2022). Each numerator and denominator $\;\;D_t^0 ( {m_t, \;y_t} ) , \;\;D_{t + 1}^0 ( {m_t, \;y_t} )$,$D_t^0 ( {m_{t + 1}, \;y_{t + 1}} )$, and $D_{t + 1}^0 ( {m_{t + 1}, \;y_{t + 1}} )$ represents a distance function that, when compared, measures the effectiveness of a country's productive technology throughout two different periods, t and t + 1. To estimate the Malmquist index, these functions must first be linearly estimated through the DEA technique. D ⁰ is the geometric mean of the distance function. D ⁰ > 1 signifies TFP growth during the specified time frame (t and t + 1). D ⁰ < 1 denotes a drop in productivity, whereas D ⁰ = 1 denotes stagnation (Caves et al., Reference Caves, Christensen and Diewert1982). For simplicity, m _t is used to represent the aggregate value of the climate-inclusive input bundle (c _t + x _t), where c _t is a measure of two climatic variables (temperature and precipitation), and x _t measures the conventional input (labor, capital, material). Equation (8) can be further decomposed into two separate efficiency components (efficiency change, denoted by EFCH, and technical efficiency change, denoted by TECH):

(9)$$EFCH = \displaystyle{{d_t( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_t( {m_t, \;y_t} ) }}$$

(10)$$TECH = \left[{\displaystyle{{d_t( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_{t + 1}( {m_{t + 1}, \;y_{t + 1}} ) }}\displaystyle{{d_t( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_{t + 1}( {m_{t + 1}, \;y_{t + 1}} ) }}} \right]^{1/2}$$

The climate-inclusive TFP index (C _TFP) can be expressed as below using Equations (10) and (11) (Fare et al., Reference Fare, Grosskof, Norris and Zhang1994):

(11)$$C_{TFP} = \displaystyle{{d_{t + 1}( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_t( {m_t, \;y_t} ) }}\left[{\displaystyle{{d_t( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_{t + 1}( {m_{t + 1}, \;y_{t + 1}} ) }}\displaystyle{{d_t( {m_{t + 1}, \;y_{t + 1}} ) } \over {d_{t + 1}( {m_t, \;y_t} ) }}} \right]^{1/2{\rm \;\;\;}}$$

The component outside the bracket in (11) is the technology used between periods t and t + 1. It represents a measure of efficiency change (EFCH). This happens when adjustments to the ratio of actual outputs allow a nation to advance toward a higher productivity frontier. It is frequently referred to as ‘technology catch-up’ because it indicates a nation's effective utilization of productive technology, such as land, labor and capital, under advantageous agro-climatic circumstances. The second element measures technological advancement (TECCH). It embodies the potential movement of a productive technology toward the frontier (Nondo and Jaramillo, Reference Nondo and Jaramillo2018; Ding and Zhang, Reference Ding and Zhang2021). Changes in technological know-how, adjustments to the total amount of input used, and adjustments to the observable patterns of the climate may all be responsible for the potential shift in the technological frontier (Chambers and Pieralli, Reference Chambers and Pieralli2020). In Africa, where agriculture is reliant on nature for the production of crops and livestock, the effect of the non-market input (e.g., climate variability) on agricultural productivity is even more profound.

Variable selection and data sources

The gross value of 162 crop commodities, $1000 at a constant 2015 global average, is our single output variable (crop_output). Our input matrix includes land (crop_land), labor (Agr_labor), capital (Agr_capital), agricultural machines (Agr_machine), fertilizer (Fert_Agr) and irrigation land (tot_IrriArea). Climate controls include the annual mean of precipitation (Prcip_Amean) and temperature (Temp_Amean). All factors were selected based on earlier research (Kumar and Raj Gautam, Reference Kumar and Raj Gautam2014; Liang et al., Reference Liang, Wu, Chambers, Schmoldt, Gao, Liu, Liu and Sun2017; Njuki et al., Reference Njuki, Bravo-Ureta and O'Donnell2018; Sridhara et al., Reference Sridhara, Gopakkali, Manoj, Patil, Paramesh, Jha and Prasad2022). The number of economically active adults (males and females) who work mostly in agriculture is used to calculate the labor force. Cropland refers to the area used for crops per 1000 ha. The capital variable represents the value of net capital stocks at constant 2015 prices of $1.000. The service flows of capital equipment and farm stocks of farm machinery, which are measured in millions of metric tons, serve as representations of agricultural machinery. The fertilizer variable is measured by total N, P₂O₅ and K₂O nutrients from inorganic fertilizers and N from organic substances. Irrigation is the total area currently equipped for irrigation, at 1000 ha. Data for all of the above variables were obtained from the US Department of Agriculture (USDA) Economic Research Service database, updated as of October 2021. We used this database because it was designed to measure global agricultural TFP (Baráth and Fertő, Reference Baráth and Fertő2020; Plastina et al., Reference Plastina, Lence and Ortiz-Bobea2021). Its primary source is the FAOSTAT database (Food and Agriculture Organization Statistics). The temperature and precipitation variables are the annual means of historical time series data recorded under the Coupled Model Intercomparison Projects (CMIP6_SSP119). The Shared Socio-economics Pathways (SSP1-1.9) is a climate scenario driven by different socio-economic assumptions selected to promote climate models for CMIP6. The CMIP6 is an expansion over CMIP5 which aims to keep warming below 1.5°C above pre-industrial levels by 2100. It was initiated following the Paris Agreement, during which nations committed to continuing their attempts to keep global warming to 1.5°C. Table 1 provides a statistical summary of these variables. A detailed explanation of these variables and their sources is provided in the Appendix (Table A1).

Table 1. Variable statistics

According to the data in Table 1, the average annual mean temperature (2000–2019) was 24.55°C, with a standard deviation of 2.95 and a minimum and highest mean of 17.6 and 29.38, respectively. Precipitation has an annual average mean of 997.17 mm and a standard deviation of 635.69 mm, with the minimum and maximum annual precipitation being 22.5 and 2785.33 mm, respectively. To better understand the trend and distribution of these variables across Africa, we created five regional diagrams, as shown in Figures 1–5.

Fig. 1. Distribution of temperature and precipitation in Africa: Eastern Africa, SSA (2000–2019).

Fig. 2. Distribution of temperature and precipitation in Africa: Southern Africa, SSA (2000–2019). Source: Authors’ calculation.

Fig. 3. Distribution of temperature and precipitation in Africa: Central Africa, SSA (2000–2019).

Fig. 4. Distribution of temperature and precipitation in Africa: Northern Africa, SSA (2000–2019).

Fig. 5. Distribution of temperature and precipitation in Africa: Western Africa, SSA (2000–2019).

These regional graphs illustrate the annual variations of countries' temperatures and precipitation in each region, as well as how the distribution changes over time. Most nations in Eastern, Southern and Central Africa, for example, are now seeing increases in temperature and precipitation (see Figs. 1–3). In Northern Africa, many countries saw an irregular pattern of increases in temperature for most of the last two decades (Fig. 4). In Western Africa, on the other hand, the annual average temperature is decreasing while precipitation is increasing (Fig. 5). Differences in the distribution of these climate variables are important in explaining the vulnerability of Africa's agricultural industry to climate change.