Wheat is widely regarded as the most essential agricultural crop globally and is consumed daily across cultures. While the quantity of wheat consumed varies by country due to cultural and economic factors, its role as a staple food remains consistent worldwide (Statista, 2023). The significance of wheat is well recognized, and experts universally consider it a strategic agricultural commodity (Statista, 2023; Pourmokhtar, 2019). In 2022, the global area under wheat cultivation reached 219 million hectares, with total production amounting to 808 million tons. India accounted for the largest share of harvested area, covering 30 million hectares (14%), while China led in production volume, producing 138 million tons (17%) the highest among all countries (FAO, 2023; Sheikh et al., 2025).

In 2023, Afghanistan ranked 26th among 126 wheat-producing countries, with a cultivated area of 1.8 million hectares and a total production of 3.8 million tons (FAO, 2023). Notably, wheat accounts for approximately 89% of total grain consumption in Brazil, underscoring its global importance. In Afghanistan, wheat bread is the staple food, and the government has made continuous efforts in recent years to boost domestic wheat production. However, wheat yields in the country are highly dependent on rainfall and weather conditions. According to a recent report by the Food and Agriculture Organization of the United Nations (FAO), the drought that began in late 2020 was identified as the primary driver of the country's food crisis and widespread food and nutritional insecurity. Experts have described this as the worst drought in decades, with projections indicating it would persist across all 34 provinces until at least mid-2022. Statistics from 2021 further reveal that 25 out of 34 provinces remained affected by drought, resulting in a 20% decline in agricultural production compared to the previous year. The lack of a reliable and systematic irrigation infrastructure has consistently been cited as one of the major constraints to agricultural productivity and farmer satisfaction in the region (Ministry of Agriculture, Irrigation and Livestock of Afghanistan, 2021; Sheikh et al., 2026).

Water scarcity and limited arable land are among the most significant barriers to the development of Afghanistan's agricultural sector (AFSANA, 2022; Zia et al., 2026). In an effort to achieve self-sufficiency in wheat production, the Afghan government has launched a five-year national initiative known as the Wheat Sector Development Program (WSDP). This program is being implemented in 16 major wheat-producing provinces, as well as in a decentralized manner across all 34 provinces, with the goal of increasing national wheat production to 7 million tons. The program aims to enhance production efficiency by improving cultivation and irrigation practices. According to a 2020 report from the Ministry of Agriculture, Irrigation and Livestock (MAIL), Afghanistan's population was estimated at approximately 32.9 million, with an annual wheat requirement of 6.49 million tons. In 2023, the country produced 4.3 million tons of wheat, marking a 13% increase compared to the previous year. Approximately 80% of this wheat is grown on irrigated land, while the remaining 20% is cultivated under rainfed conditions. Producing one kilogram of wheat requires around 1,000 liters of water. Based on this estimate, wheat production in 2020 consumed approximately 4,080 million cubic meters of water (MAIL, 2020). Fig. 1 illustrates the geographical distribution of wheat production across the country. 

Fig. 1: Geographical location of Afghanistan in terms of irrigated and rainfed wheat production.

The area covered in this study is Nangarhar Province, located in eastern Afghanistan and recognized as one of the five largest provinces in the country. According to the most recent statistics published in 2022, Nangarhar has approximately 150,000 hectares of irrigated land and 34,000 hectares of rainfed land. Wheat cultivation accounts for 55.72% of the total cultivated area in the province. Given the strategic importance of wheat in this region, it is essential to enhance both the efficiency of wheat production and the productivity of the associated input factors. This necessitates identifying the production relationships and analyzing the composition and interactions of various production inputs. Rather than employing a production function, this study uses a cost function approach. While the production function is commonly used to assess production conditions and to estimate key indicators such as input elasticities, functional coefficients, and returns to scale the cost function, as a dual of the production function, offers several distinct advantages:

• First, the cost function is linearly homogeneous of degree one in input prices, which means that homogeneity in the production process is not required. For example, if all input prices increase by a factor of k, the total cost will also increase by the same factor.
• Second, using input prices rather than physical quantities is advantageous. In estimating a production function, the presence of multiple independent variables often introduces multicollinearity, complicating the estimation process. In contrast, the cost function substitutes prices for physical input quantities, and input prices are generally less collinear. Therefore, the cost function provides a more robust and reliable framework for analysis (Ray, 1982).

The foundational exploration of cost function properties was initiated by Hotelling, (1932). This work was later expanded by Samuelson, (1947) who introduced the concept of the “factor price frontier” Although several economists contributed to the development of this field most notably McElroy, (1975) a significant breakthrough came with Shephard's (1953) introduction of a dual relationship between cost and production functions. His analysis was grounded in the properties of convex sets, as developed by Fenchel, (1951) and marked a major advancement in economic theory. Since then, numerous studies have examined the application of cost functions in economic research. The following are among the most notable studies relevant to the current research:'

Jahani and Asghari, (2005) estimated a translog cost function to analyze the cost structure and production of rainfed wheat in East Azerbaijan Province, Iran. Their findings indicated that chemical fertilizers and seeds function as complementary inputs, as do machinery and labor. Furthermore, the substitution elasticity measure developed by Mori Shima, (MSE) showed that the elasticity between labor-related inputs specifically chemical fertilizers and consumable seeds is greater than one, suggesting a strong substitutive relationship between these inputs. Keskin et al. (2010) employed the seemingly unrelated repeated regression method to estimate the cost and demand functions for tomatoes and cucumbers in the Uzundere region of Turkey. Their findings revealed that the gross profit from cucumbers exceeded that of tomatoes, with labor input constituting the largest proportion of total production costs. Additionally, the results indicated low price elasticity of input demand for cucumbers and similarly low elasticity of inputs with respect to price changes in tomato production.

Chabot and Dorosh, (2007) analyzed wheat prices, market dynamics, and food security in Afghanistan. Their study found that large-scale food aid inflows had no significant downward effect on domestic wheat prices until mid-2003. However, following the 2003 harvest, ongoing food aid deliveries led to an approximate 15% reduction in producer prices. Furthermore, the research emphasized that, given the considerable potential for rehabilitating irrigation infrastructure, there was a strong basis for increasing domestic wheat production and thereby reducing dependence on imports. Pourmokhtar and Kadirzadeh, (2013) used the cost function instead of the production function to analyze production technology. The sample size was 257 water wheat farmers in Kurdistan province. The data for the crop year 2011-2012 were collected through a survey, interviews with farmers, and questionnaires. Samples were selected using a multistage random cluster sampling method. The results showed that the translog cost function had a good fit to the researched data, and structural hypotheses such as constant return to scale (CRS), homogeneity, and homothetic of production technology were not significant. Land input accounted for the largest share of production costs. According to the results obtained from Allen's elasticity of substitution, except for labor inputs with poison and water with fertilizer, the rest were of substitution type. All own-price elasticities of demand were obtained as small as one, so the demand for all inputs was inelastic.

Zha and Ding, (2014) used the translog cost function to estimate the production factor contribution equations for China's electricity industry. According to the results, among the three production factors, energy had the least sensitivity to price. In addition, the estimation results of four types of elasticity (cross-elasticity of demand, elasticity of substitution of Morishima, elasticity of substitution of Allen and McFadden) showed that there is a possibility of significant substitution between energy and capital, while energy and labor are poor substitutes for each other. In addition, the findings indicate that more capital should be invested to reduce energy consumption in the electricity industry of the country. Rigi and Shahraki, (2017) estimated the water demand function of the industrial sector in Zahedan, Iran based on the translog cost function. In this study, cost-share equations were estimated using the iterative seemingly unrelated regression (ISUR) approach. The results showed that water demand is inelastic because the price elasticity calculated for water is less than one (-0.07). In addition, the values calculated for Allen Ozawa and Morishima's elasticity indicate a strong substitution relationship between water as a production input and machinery (6.69) and buildings (1.30). On the other hand, there was weak substitutability between water and land inputs (0.65) and a complementary relationship between water and labor (-0.43).

Aliahmadi et al. (2018) used the SUR method to estimate the translog cost function of water demand for wheat in Sistan. Required data were collected from 150 wheat farmers. The results showed that all the coefficients of the variables in the water cost share model were significant, except for the hired labor force. Because of the low coefficient of determination in the cross-sectional data, the coefficient of determination in the estimated model for the wheat crop was 0.61, which represents a good fit of the model. Was. In addition, the absolute value of the price elasticity of water demand for wheat is greater than one, which indicates that it is possible to control the water demand in this region by adopting pricing policies that influence production inputs other than water. Du et al. (2019) employed a translog cost function to compare the impact of operational patterns on peach and cherry production costs by estimating the elasticity of substitution between and among inputs. They found that the own-price elasticity of all input factors was negative, while substitution relationships existed between labor and land, labor and fertilizer, fertilizer and manure, and manure and pesticide. This indicates that Beijing's agricultural sector is labor intensive, while fertilizers and pesticides are rarely used. 

Kamruzzaman et al. (2021) used the translog cost function to evaluate the economic costs of broiler production in the Dhaka, Rajshahi, Mymensingh, and Chittagong regions of Bangladesh. A total of 210 questionnaires were completed. The findings showed that broiler farming incurred most of its costs from its operating input, mainly feed. Broiler farming was financially profitable, but the performance of Mymensingh division was comparatively low, arising from a high unit cost of production and low unit price selling. The net return was the highest in the Dhaka division, while the Rajshahi division showed the highest ratio of returns on investment. However, in terms of cost (variable) and net return of broiler farming, no significant differences were observed among the study areas. The values of own price elasticity for feed, chicken price, and labor price were negative and inelastic, at -0.00249, -0.05718, and -0.13101, respectively. In addition, a complementary relationship was found between feed and day-old chicks and between feed and labor, while day-old chicks and labor were substitutes. The study also revealed that cross-price elasticity was highly inelastic, and changes in the prices of inputs did not result in massive changes in the quantity of other inputs demanded for broiler farming. Rohani et al. (2021) examined the factors affecting the four dimensions of sustainable agricultural development in Khorasan Razavi Province using Seemingly Unrelated Regression Equations (SURE). A sample of 398 farmers was selected through a two-stage random cluster sampling method, and data were gathered via questionnaires administered in 2019. The study found average index values for social, environmental, economic, and political sustainability at 0.55, 0.47, 0.41, and 0.32, respectively. Participation in training programs, interest in agriculture, and job satisfaction showed significant positive correlations with various aspects of agricultural sustainability. Moreover, membership in farmers' cooperatives was significantly linked to political sustainability; land ownership was associated with social and political dimensions; and land consolidation correlated with economic sustainability. Based on these findings, the authors recommend expanding extension training programs focused on sustainable agriculture. Given the emphasis on managerial sustainability, policymakers should prioritize investment in agricultural infrastructure, enhancement of decision-making processes, empowerment of agricultural organizations, enactment of supportive legislation, and promotion of sustainable employment. Additionally, establishing an efficient marketing system to reduce post-harvest losses and production costs is essential to stabilize farmers' incomes.

Abbasian et al. (2022) used cross-sectional data for the 2018-2019 crop years to estimate the price of water for mangoes and to estimate its demand with an emphasis on environmental inputs. To this end, the actual price of water is determined by the residual method, and the demand function is estimated using the translog cost function and the equations of the contribution of inputs to cost. The results support the good fit of the model used for the cost function of mangoes in the county studied. The results for the coefficients in Chabahar County indicated that water cost has a positive relationship with the prices of manure, water, seedlings, and crop yield and a negative relationship with the prices of pesticides and chemical fertilizers.  Based  on  the  results  of  the  water  demand  function,  water  is  a  substitute  for manure, chemical fertilizer, and seedlings, revealing the impact of water use management  and  economic  valuation  on  improving  the  use  of  other  environmental  inputs (pesticides,  manure,  and  chemical  fertilizers)  and  seedlings,  as  well  as  the  water  itself,  in  mango production in this region. Policies such as optimal pricing of inputs, including pesticides,  manure,  chemical  fertilizers,  and  seedlings,  are recommended  to  curb  the  resulting  environmental pollution. Roshandeh et al. (2022) estimated the water demand function in rice production using the translog cost function combined with the seemingly unrelated regression (SUR) method. The data for this study were collected during the 2017–2018 cropping season through 314 questionnaires administered to rice farmers in Gorgan County, using stratified random sampling. The results indicated that most coefficients were statistically significant at the 5% level, and Allen's partial elasticities exhibited the expected signs, reflecting an inverse relationship between water price and water demand. Specifically, the own-price elasticity of water demand was estimated at -0.19, implying that a 1% increase in water price leads to a 0.19% decrease in water demand. This suggests that water demand is price inelastic, meaning that increases in water prices result in relatively small reductions in water consumption. Based on these findings, the study recommended the implementation of complementary policies such as enhancing water use efficiency, adopting modern irrigation technologies, and formulating effective water pricing strategies.

Merdan and Comakli, (2025) analyzed the factors influencing the import demand for wheat and corn in Türkiye using the Seemingly Unrelated Regression (SUR) Method. The results of the study highlight that, among the explanatory variables in the wheat import model, only the one-year lagged value of wheat imports and the exchange rate are statistically significant. Specifically, a 1% increase (or decrease) in wheat imports from the previous year leads to a more than 0.80% increase (or decrease) in current wheat imports. In contrast, variables such as per capita national income, wheat import price, total wheat consumption, and the one-year lagged value of wheat production were found to be statistically insignificant in determining the quantity of wheat imports. Thus, the exchange rate and past wheat imports are the key drivers of wheat import demand. Similar findings were observed in the analysis of corn import demand. In the corn import model, the study found that only total corn consumption was statistically significant. A 1% increase (or decrease) in total corn consumption results in a more than 0.70% increase (or decrease) in corn imports. Other variables, including per capita national income, exchange rate, corn import price, lagged value of corn production, and the one-year lagged value of corn imports, were statistically insignificant. Based on these findings, the study concludes that wheat imports are primarily influenced by the previous year's wheat import levels and exchange rates, whereas corn imports are mainly driven by total domestic corn consumption.

Most studies that employ translog cost functions use the elasticity of production factors to measure the magnitude of the rebound effect. Input  demand  functions  can  be  derived from  two  methods:  deriving  the  profit function  in  relation  to  the  price  of  inputs (Hotelling's  lemma)  or  deriving  the  cost function  in  relation  to  the  price  of  each input  (Shephard's  lemma).  In  the  first method,  the  direct  demand  function  is obtained;  in  the  second  method,  the indirect (conditional) demand functions for the  inputs  are  obtained.  However, if the output function has an increasing return to scale (IRS) feature, the input demand function should be estimated by minimizing the cost function. However,   if  the  output  function  has  a decreasing return to scale (DRS) function by maximizing  the  function,  profit  can  be obtained  as  a  function  of  input  demand (Sadrzadeh et al., 2013).  Given  that  In  the  agri-cultural  sector,  farmers  try  to  minimize  production costs (Y) by choosing a specific combination of  production  inputs  (X).  

Based on the duality theory, the production structure of an industry can be studied using both production and cost functions. Therefore, the production function has a minimum cost function as a secondary system. Therefore, all technical relationships between production levels and factors depend on the production function, as reflected in the cost function (Varian, 1992). The main inputs of agricultural production include capital (K), labor (L), and other input factors (M). Thus, the production function can be expressed as (Du et al., 2019).

There is a duality relationship between the production and cost functions, and the cost of production is given below.

Where Q represents the total output; C is the total cost; and PK, PL, and PM represent the prices of capital, labor, and other input factors, respectively. We construct the minimum cost function corresponding to equation (3). To reflect the relationship between the factor price and input factor under the condition of cost minimization.

To obtain a cost function, it is necessary that the company's budget is minimized owing to limited technology, or that the production company is maximized owing to budget constraints. These two orders are equivalent. Finding a solution to this problem is equal-price technical substitution with a negative relative price of production inputs. Assuming a particular technology, it would be simple (at least conceptually) if the cost function was achieved by minimizing the cost relation. On the other hand, with the assumption of the cost function, it is possible to achieve a production function and production technology if such a cost function has been invented. If any concept is defined according to the characteristics and properties of the production function, it has a corresponding definition of the cost function according to these characteristics and properties, and vice versa (Diewert, 2023). Various forms can be used to estimate the cost function, such as the Translog function, Leontif, and Second Degree, and errors in the selection form of the function are always one of the sources of specification error in econometrics. Gujarati, number of parameters, simplicity of interpretation and calculations, good fit, power of generalization, and prediction, among the important criteria in determining the model of economic analysis, are superior for experimental works (Gujarati, 2022).

The Translog cost function is the most common form of cost function used in related studies to examine the cost structure of agricultural products (Ray, 1982؛ Amor, 2022). With the selection of the translog cost function, according to the case of Shephard's lemma, it is possible to estimate the conditional demand functions of the inputs within the framework of the systemic equations, and seemingly unrelated regressions are restricted. This is because the function is limited because it does not allow the possibility of substitution between inputs. Additionally, allowing for a variable return to scale with changing production levels is necessary for a U-shaped average cost function. Based on the criteria of selection, superior, and wide application, the translog cost function and the overall shape of the function will be in the form of relations (4) and (5). 

Where C is the cost, Q is the output, Pi is the price of factor i, Pj is the price of factor j, and   is the coefficient to be estimated. The symmetry condition implies that ( ), another restriction on the parameter estimates, is that the cost function must be homogenous of degree one in input prices given Q. This implies the following restrictions on equation (5):

The translog cost function can be estimated directly, and gains in efficiency can be obtained by estimating the optimal, cost-minimizing input demand equations and transforming them into cost share equations. By logarithmically differentiating Equation (4) with respect to input prices and employing Shephard's lemma (duality between production and cost function), the following cost-share equation is obtained: According to Shephard's lemma, we can obtain an input cost-share function. The general form of the input share demand function is given by equation (6) (Roshandeh et al., 2022; Moss et al., 2003):

Defining the cost sharesas  , it follows that (Du et al., 2019).

Where Si represents the cost share of input i.

The third condition, which refers to the concept of alignment in the direction of changes in input prices and costs, requires that the fulfilled share of each input in the total cost of production for all numerical observations be greater than zero (Du et al., 2019; Roshandeh et al., 2022).

To satisfy the fourth condition, the matrix derivatives of the quadratic cost function   should be a negative semi-definite matrix; that is, the slope of the input demand function should be negative relative to the input price It is also necessary to test the structural hypotheses of the cost function, including the homothetic production function, constant returns to scale, and Cobb-Douglas production function (Kuroda, 1987; Gervais et al., 2006; Pourmukhtar and Kadirzadeh, 2013):

The translog cost function does not have special interpretation coefficients, and most of the elasticity is discussed. The elasticity of substitution shows the sensitivity of one variable to changes in another (Greer, 2012). In this study, it is particularly important to obtain substitution coefficients. For this purpose, we calculated the Allen partial elasticity of substitution (AES). This type of elasticity, which is named Allen-Uzawa elasticity of substitution, is used to group each pair of inputs in terms of substitution and complementarity. In other words, Allen's cross-elasticity of substitution shows the degree of substitution between two inputs. If the algebraic value of Allen's cross-elasticity of substitution is positive, there is a substitution relationship between two inputs, and if the algebraic value of Allen's cross-elasticity of substitution is negative, there is a complementary relationship between two inputs. It is expected that the signs of this type of elasticity will be negative because the demand for any normal good has an inverse relationship with its price. Allen's partial elasticity was used to interpret the relationship between substitution or complementarity between product inputs (Sorrell, 2008).

Price elasticity, which changes with the share of inputs, is calculated as follows: The input-factor demand analysis uses Allen-Uzawa cross substitution elasticity and own price elasticity of demand   to measure the responses of demand to price changes. The elasticity, as mentioned above, can be estimated using the parameters of the input share function equations (Du et al., 2019).

Where Si and Sj are the shares of the ith and jth input, respectively. Here, a positive value of APES between i and j indicates a substitute relationship between i and j, and a negative value implies a complementary relationship between these two variables. The APES relationship between two variables assesses the response of one variable for quantity demand to the change in price of other related variables by keeping the output constant; however, other quantities of different factors can vary. 

In this study, the structure of the wheat product cost function was obtained regionally and in a cross-sectional manner based on statistical data obtained from the completion of 162 questionnaires from 22 districts of Nangarhar province. 

Table 1 summarizes the economic information obtained from the averaging of the supplementary questionnaires regarding wheat products.

Table 1: Economic indicators of wheat production per hectare in Nangarhar province Afghanistan.

Source: research findings

In this study, the translog cost function was selected as the superior form. Considering that the translog cost function is similar to its production function in a linear logarithmic form and has the necessary flexibility to present the results, it has been used for estimation (Laukkanen & Nauges, 2011). After estimating the function, the correlation coefficient between the disturbance components was checked, which indicated a correlation between them? In the following, the translog cost function is estimated by the repeated unrelated regression method for the wheat crop of Nangarhar province, and the results are presented in Table 2.

Table 2: Estimation results of total translog cost function of wheat crop in Nangarhar Province, Afghanistan.

Source: research findings

According to Table 2, most of the coefficients are significant with high confidence (more than 95%). It should be noted that the logarithm of the labor price and the logarithm of the production quantity have a negative and significant effect, and the logarithm of the seed price, the logarithm of the water price, and the logarithm of the fertilizer price have a positive and significant effect on the cost of wheat production in Nangarhar Province.

Table 3 shows the partial elasticity of Allen substitution. As it can be seen, own partial elasticities of Allen inputs are negative, which indicates that according to the theory, there is an opposite relation-ship between the price and the amount of demand. The absolute value of Allen's own partial elasticity for labor, water, and machinery inputs is more than one, which indicates that these inputs are elastic; that is, with a one percent increase in the price of inputs, their demand decreases by more than one percent. According to the results of Allen's own partial elasticity, among the studied inputs, water input has the highest sensitivity, and seed input has the least sensitivity of the demand amount to its price. Based on the results of Allen's partial cross elasticity, the inputs of machinery–labor force and machinery-water are more than one, indicating the existence of a strong relationship between the above inputs. 

The relationship between water input and seeds is complementary and substitutes for other inputs (labor, machinery, and fertilizer). Regarding substitute inputs, it can be stated that by reducing the price of these inputs, the demand for water will decrease, and, as a result, water waste will be avoided. Regarding complementary input, it can be said that with an increase in seeds, water consumption also increases.