Stochastic Frontier Model for the Evaluation of the Sustainability of Urban Gardens in Puebla, Mexico
Abstract
1. Introduction
Conceptualization of Sustainability Assessment in Urban Gardens
2. Materials and Methods
2.1. Place of Research
2.2. Econometric Modeling Framework
- ln(Yi) = logarithm of vegetable yield (Swiss chard, spinach, lettuce, tomato Var. ‘Rio Grande’) in kg/ha for producer i.
- ln(ALi) = logarithm of the amount of labor in workdays/ha for producer i.
- ln(Fi) = logarithm of the amount of fertilizers in kg/ha for producer i.
- ln(Wi) = logarithm of the amount of water in m3/ha for producer i.
- (vi − ui) = error term composed of two elements:
- vi = random component capturing measurement errors and other factors.
- ui = non-negative random component representing the inefficiency level of producer i, following a truncated normal distribution.
2.3. Population and Sample
2.4. Estimation of Nutrient Balance in Physical Terms
- Ext Nuti,j = total nutrient extraction for vegetables per cultivated area.
- i = nutrient extraction (nitrogen, phosphorus, potassium, calcium, magnesium, sulfur).
- j = conventional agriculture or urban garden.
- cei,j = extraction coefficient for each nutrient.
- Prodj = vegetable production per cultivated area.
- Rep Nuti,j = total nutrient replenishment.
- i = nutrients.
- j = cultivated area.
- ferti,j = nutrients supplied by fertilization per hectare.
- suspj = surface area in cultivated hectares.
- CT = total nutrient replenishment cost.
- Qi = quantity of nutrient i to be replenished.
- Pi = price per unit of nutrient i.
- n = total number of evaluated nutrients (nitrogen, phosphorus, potassium, etc.).
2.5. Water-Use Efficiency
- EV = economic value of water for agricultural use in both production systems ( MXN/m3)
- P = selling price of the crops
- Qwith water = productivity with sufficient water
- Qwithout water = productivity under water restrictions
- Cwith water = production costs with sufficient water
- Cwithout water = production costs under water restrictions
- Virrigation water = volume of water used (m3/ha)
2.6. Stochastic Frontier Model
- ln(Yi) = vegetable production i (kg/ha)
- ln(ALi) = labor used in vegetable i production (workdays/ha)
- ln(Fi) = fertilizers applied in vegetable i production (kg/ha)
- ln(Wi) = water used in vegetable i production (m3/ha)
- vi = random error
- ui = technical inefficiency term
- Ci = total production cost
- Yi = obtained production
- Pi = input cost
- α = parameters of the cost function
- ηi = error term, where: ηi = vi − ui.
- “vi“ represents a random error component;
- “ui“ represents a non-negative error component, capturing inefficiency.
2.7. Estimates of the Environmental Cost Function Using Stochastic Frontiers
- ln(Yi) = Represents the natural logarithm of the income per hectare of producer i. It is the dependent variable. It indicates the economic productivity that the producer obtains after deducting the production costs (MXN/ha).
- lnSj = Natural logarithm of the cultivated surface per hectare of producer i. This factor indicates how much space is used for cultivation (ha).
- lnLAi = Natural logarithm of the cost of the day’s wage in pesos per hectare of producer i. This factor measures the labor effort applied in monetary terms and how it influences net income. (MXN/ha)
- lnFi = Natural logarithm of the cost of fertilizers in pesos per hectare of producer i. It represents the cost of the chemical or organic inputs used to improve soil fertility and their impact on income (MXN/ha).
- lnRi = Natural logarithm of the adjusted yield in kilograms per hectare for producer i. It indicates the amount of product that was obtained, adjusted for possible losses or improvements, and how it influences profitability (kg/ha).
- ei = Idiosyncratic error. It represents the variations in the dependent variable (Yi) that are not explained by the independent variables of the model. Specifically, in this production and econometrics model, ei considers random effects and unobserved influences that may affect the result of each observation.
- β0, β1, β2, β3, β4 = Estimated coefficients.
- ln(Yi) = natural logarithm of net income in pesos per hectare of producer i (MXN/ha).
- lnSi = natural logarithm of cultivated area in hectares of producer i (ha).
- lnLA = natural logarithm of the cost of daily wages per hectare of producer i (MXN/ha).
- lnFi = natural logarithm of the cost of fertilizers per hectare of producer i (MXN/ha).
- lnCNBi = Natural logarithm of the cost of nutrient balance for producer i. This indicator measures the cost associated with the replacement of nutrients extracted by vegetables, considering the use of fertilizers and other inputs necessary to maintain soil fertility. It represents a key environmental cost in agricultural systems (MXN/ha).
- lnCWi = Natural logarithm of the cost of water used per hectare for producer i. This value considers the cost of the water resource based on the volume of water used in production and its contribution to net income. It includes water tariffs and other related costs (MXN/m3).
- lnRi = natural logarithm of the adjusted yield in kilograms per hectare for producer i (kg/ha).
- ei = idiosyncratic error.
- ln(Yi) = natural logarithm of net income in pesos per hectare for producer i (MXN/ha).
- lnSi = natural logarithm of cultivated area in hectares for producer i (ha).
- lnLA = natural logarithm of daily wage cost per hectare for producer i (MXN/ha).
- lnFi = natural logarithm of fertilizer cost per hectare for producer i (MXN/ha).
- lnRi = natural logarithm of adjusted yield in kilograms per hectare for producer i (kg/ha).
- ei = idiosyncratic error.
- ln(Yi) = natural logarithm of net income in pesos per hectare of producer i (MXN/ha).
- lnSi = natural logarithm of cultivated area in hectares of producer i (ha).
- lnLA = natural logarithm of the cost of daily wages per hectare of producer i (MXN/ha).
- lnFi = natural logarithm of the cost of fertilizers per hectare of producer i (MXN/ha).
- lnCNBi = natural logarithm of the cost of nutrient balance for producer i (MXN/ha).
- lnCWi = natural logarithm of the cost of water used per hectare of producer i (MXN/m3).
- lnRi = natural logarithm of the adjusted yield per hectare for producer i (kg/ha).
- lnPAi = Natural logarithm of the average price of agroecological products for producer i. This value reflects the price at which products grown using agroecological practices (urban garden) are sold, offering an insight into their economic competitiveness compared to conventional products (MXN/kg).
- lnPCi = Natural logarithm of the average price of conventional products for producer i. This value represents the price at which products obtained through traditional agricultural practices are sold, functioning as a reference point in the price comparison (MXN/kg).
- ei = idiosyncratic error.
3. Results and Discussion
3.1. Economic Assessment of Nutrient Balance in Different Agricultural Systems
3.2. Technical and Economic Efficiency and Yield in Different Agricultural Systems
3.3. Stochastic Frontier Analysis in Different Agricultural Systems
3.4. Cost–Benefit and Profitability Indices Using Stochastic Frontier in Different Agricultural Systems
3.5. Net Revenues Using Stochastic Frontiers in Different Agricultural Systems
3.6. Importance and Implications of the Results
3.7. Limitations and Recommendations
4. Conclusions
5. Practical Recommendations
- For municipal policymakers, incentivize the adoption of water-efficient technologies and facilitate regulated access to vacant land for agricultural use.
- For urban and peri-urban producers, prioritize high-value crops such as Swiss chard, which showed better performance under urban conditions, and implement precision irrigation to enhance water efficiency.
- For urban planners, integrate urban gardens into zoning and land-use regulations as multifunctional spaces that promote food security and environmental resilience.
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Test | Crops | Statistic Value | 95% CI | p-Value | Interpretation |
---|---|---|---|---|---|
Likelihood Ratio | Tomato | 8.234 | [7.012, 9.456] | 0.012 * | Significant |
Swiss chard | 6.892 | [5.678, 8.106] | 0.018 * | Significant | |
Spinach | 5.456 | [4.234, 6.678] | 0.028 * | Significant | |
Lettuce | 7.123 | [5.901, 8.345] | 0.015 * | Significant | |
Breusch–Pagan | Tomato | 1.892 | [1.234, 2.550] | 0.169 | Homoscedasticity |
Swiss chard | 1.934 | [1.276, 2.592] | 0.165 | Homoscedasticity | |
Spinach | 1.876 | [1.218, 2.534] | 0.172 | Homoscedasticity | |
Lettuce | 1.912 | [1.254, 2.570] | 0.168 | Homoscedasticity | |
Jarque–Bera | Tomato | 0.876 | [0.456, 1.296] | 0.645 | Normal |
Swiss chard | 0.892 | [0.472, 1.312] | 0.632 | Normal | |
Spinach | 0.864 | [0.444, 1.284] | 0.649 | Normal | |
Lettuce | 0.883 | [0.463, 1.303] | 0.638 | Normal |
Production System | Crops | |||||||
---|---|---|---|---|---|---|---|---|
Tomato | Swiss Chard | Spinach | Lettuce | |||||
kg/ha * | MXN/ha + | kg/ha * | MXN/ha + | kg/ha * | MXN/ha + | kg/ha * | MXN/ha + | |
Urban Garden | 10,204 | 71,428.56 | 10,204 | 71,428.56 | 10,204 | 71,428.56 | 10,204 | 71,428.56 |
Conventional Agriculture | 772 | 18,072.52 | 615 | 14,384.85 | 490 | 11,666.90 | 415 | 9906.05 |
Reference Parameters (Literature) | 850 | 11,704.50 | 410 | 5645.70 | 320 | 4406.40 | 290 | 3993.30 |
Production System | Crops | |||||||
---|---|---|---|---|---|---|---|---|
Tomato | Swiss Chard | Spinach | Lettuce | |||||
m3/ha * | MXN/m3 + | m3/ha * | MXN/m3 + | m3/ha * | MXN/m3 + | m3/ha * | MXN/m3 + | |
Urban Garden | 2,755 | 130.63 | 1,286 | 721.19 | 1,286 | 232.73 | 1,082 | 201.82 |
Conventional Agriculture | 6,600 | 11.48 | 4,800 | 4.17 | 3,900 | 7.18 | 3,500 | 12.00 |
Reference Parameters (Literature) | 6,371 | 11.50 | 4,250 | 5.18 | 3,280 | 9.15 | 3,750 | 12.00 |
Crops | Production System | ||||||||
---|---|---|---|---|---|---|---|---|---|
Urban Garden | Conventional Agriculture | Reference Parameters (Literature) | |||||||
ET | EE | Performance (kg/ha) | ET | EE | Performance (kg/ha) | ET | EE | Performance (kg/ha) | |
Tomato | 0.80 | 0.82 | 40,111.11 | 0.94 | 0.80 | 61,600.00 | 0.91 | 0.76 | 39,800.00 |
Swiss chard | 0.90 | 0.78 | 47,155.56 | 0.85 | 0.78 | 34,208.00 | 0.86 | 0.72 | 35,000.00 |
Spinach | 0.74 | 0.71 | 20,755.56 | 0.63 | 0.73 | 17,860.00 | 0.71 | 0.68 | 20,000.00 |
Lettuce | 0.62 | 0.83 | 21,918.52 | 0.60 | 0.82 | 21,100.00 | 0.92 | 0.81 | 32,500.00 |
p-value | 0.04 * | 0.01 * | 0.02 * | 0.03 * | 0.05 * | 0.03 * | 0.04 * | 0.03 * | 0.03 * |
Crops/System | Urban Garden | Conventional Agriculture | Reference Parameters (Literature) | |||
---|---|---|---|---|---|---|
(C/R) | (PI) | (C/R) | (PI) | (C/R) | (PI) | |
Scenario 1 (without environmental costs, mixed prices) | ||||||
Tomato | 8.37 | 725.80 | 4.82 | 382.40 | 4.23 | 323.50 |
Swiss chard | 25.64 | 2485.20 | 3.28 | 228.70 | 4.12 | 312.80 |
Spinach | 9.25 | 824.90 | 2.42 | 142.80 | 3.31 | 231.50 |
Lettuce | 7.12 | 612.30 | 4.56 | 356.20 | 7.45 | 645.80 |
Scenario 2 (with environmental costs, mixed prices) | ||||||
Tomato | 7.54 | 654.20 | 4.15 | 315.60 | 3.76 | 276.40 |
Swiss chard | 23.42 | 2342.60 | 2.74 | 174.30 | 3.65 | 265.40 |
Spinach | 8.63 | 763.10 | 2.23 | 123.50 | 3.04 | 204.70 |
Lettuce | 6.68 | 568.40 | 4.18 | 318.70 | 6.94 | 594.20 |
Scenario 3 (without environmental costs, conventional prices) | ||||||
Tomato | 3.12 | 212.40 | 4.82 | 382.40 | 4.23 | 323.50 |
Swiss chard | 4.15 | 315.60 | 3.28 | 228.70 | 4.12 | 312.80 |
Spinach | 3.56 | 256.80 | 2.42 | 142.80 | 3.31 | 231.50 |
Lettuce | 5.14 | 414.50 | 4.56 | 356.20 | 7.45 | 645.80 |
Scenario 4 (with environmental costs, conventional prices) | ||||||
Tomato | 2.82 | 182.60 | 4.15 | 315.60 | 3.76 | 276.40 |
Swiss chard | 3.87 | 287.40 | 2.74 | 174.30 | 3.65 | 265.40 |
Spinach | 3.37 | 237.20 | 2.23 | 123.50 | 3.04 | 204.70 |
Lettuce | 4.82 | 382.30 | 4.18 | 318.70 | 6.94 | 594.20 |
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Muñoz-Nuñez, E.; Romero-Arenas, O.; Silva Gómez, S.E.; Rueda Luna, R.; Munguía Pérez, R.; Huerta-Lara, M. Stochastic Frontier Model for the Evaluation of the Sustainability of Urban Gardens in Puebla, Mexico. Urban Sci. 2025, 9, 164. https://doi.org/10.3390/urbansci9050164
Muñoz-Nuñez E, Romero-Arenas O, Silva Gómez SE, Rueda Luna R, Munguía Pérez R, Huerta-Lara M. Stochastic Frontier Model for the Evaluation of the Sustainability of Urban Gardens in Puebla, Mexico. Urban Science. 2025; 9(5):164. https://doi.org/10.3390/urbansci9050164
Chicago/Turabian StyleMuñoz-Nuñez, Elimelec, Omar Romero-Arenas, Sonia Emilia Silva Gómez, Rolando Rueda Luna, Ricardo Munguía Pérez, and Manuel Huerta-Lara. 2025. "Stochastic Frontier Model for the Evaluation of the Sustainability of Urban Gardens in Puebla, Mexico" Urban Science 9, no. 5: 164. https://doi.org/10.3390/urbansci9050164
APA StyleMuñoz-Nuñez, E., Romero-Arenas, O., Silva Gómez, S. E., Rueda Luna, R., Munguía Pérez, R., & Huerta-Lara, M. (2025). Stochastic Frontier Model for the Evaluation of the Sustainability of Urban Gardens in Puebla, Mexico. Urban Science, 9(5), 164. https://doi.org/10.3390/urbansci9050164