Water access, farm productivity, and farm household income: Sri Lanka's Kirindi Oya irrigation system

2.1. Production function

Most households in the study area grow rice as the major crop on their farms, and rice is the primary user of irrigation water. Consequently, we estimate a single-output production function for rice:

(1)

where y is the output of rice per hectare, x is the vector of input allocations to rice, (i = 1, 2,…, n) and (i = 1,…, m) are purchased inputs, i = m + 1,…, n are inputs with constrained supply, and z is a vector of environmental and policy variables.

Holding the environmental and policy variables constant, the producer's decision problem is given by

(2)

where p is the output price, r_i is the price of a purchased input, x_i is the quantity used of a variable input, is the level of a constrained input, and λ_i is a Lagrange multiplier. This leads to first-order conditions:

(3)

Choice of functional form for production analysis is a particularly important consideration. Previous researchers have used a wide variety of functional forms. For example, agricultural production functions related to rice farming in other irrigation systems in the Sri Lankan Dry Zone were previously estimated by Abeygunawardena (1986) using a linear functional form and by Wijayaratne (1986) and Jegasothy et al. (1990) using a quadratic functional form. The Cobb–Douglas is the most widely used functional form in agricultural production and has been used in a variety of applications ranging from farm level (Bakhshoodeh and Thompson, 2001) to international comparisons (Barker et al., 1985). It continues to be widely used (e.g., Crost et al., 2007; DiFalco et al., 2008; Ferreira et al., 2007; Wei, 2007). This functional form imposes the strong assumptions that a constant percentage change in inputs leads to a constant percentage change in output at all levels of inputs and that the elasticity of substitution between inputs is 1. Less restrictive alternatives to the Cobb–Douglas, such as the translog and quadratic, could provide a more accurate description of production technology. Yet, the Cobb–Douglas functional form has several advantages over these more “flexible” functional forms, including parsimony of parameters, convenience for inference, and perhaps more importantly, it tends to be globally well behaved (Antle and Pingali, 1995). This last property is particularly important in simulating production at different levels of inputs, which is an important objective of our study. Since the translog nests the Cobb–Douglas functional form, we estimated the translog functional form to model rice yield and tested for the Cobb–Douglas as a special case.

Rice production for the jth producer can be expressed using the stochastic translog production function as a quadratic function in logarithms of output and input quantities. The Cobb–Douglas is linear in logarithms of the same variables. Considering all inputs to be potentially variable except land, we specify the production function as a yield function:

(4)

where y is the rice output per hectare, x is an input quantity, D is a dummy variable, c is a control variable, α and β are parameters to be estimated, e is the stochastic disturbance term, i and j index inputs, k indexes the dummy variables, and l indexes the control variables. To reduce clutter, the season subscript has been suppressed. Variables used in estimating the production function in our study include six productive input variables—water, fertilizer, labor, machinery, chemicals, and seed. They also include intercept dummy variables for subarea, season, and season × subarea, and slope dummy variables for interactions of subarea and season with each productive input. Managerial ability control variables include the household head's education level expressed as number of years of schooling, years of farming experience, and a dummy variable (not in logarithms) for attainment of 10 years of schooling. These variables are included as proxies for unobserved managerial ability. Distance from distribution canal to the field inlet of irrigation is included as a proxy control variable for the quality of irrigation supply.

We estimate the translog production function using pooled ordinary least squares (OLS) with time fixed effects. We use this model to conduct the nested test for the Cobb–Douglas functional form that we find is not rejected. We then estimate the Cobb–Douglas functional form using the same estimator and perform robustness checks with this functional form using the random effects panel data estimator with time fixed effects and an OLS estimator for each season's production function estimated independently. Lacking firm-level price data, we are unable to account explicitly for the endogeneity of the input choices. Thus, the possibility of bias in the reported estimates due to regressor endogeneity remains.1

2.2. Economic welfare indicators

Following Just et al. (2004), we consider two widely used economic welfare indices for producers—profit and producer surplus. In the long run, producer surplus for the industry is equal to the sum of producer profits. Consequently, profit, defined as gross revenue minus total costs, is an obvious candidate as a measure of producer economic welfare for the firm seeking to maximize profit. However, profit serves as an appropriate measure only in certain cases. An alternative to profit as an economic welfare measure is producer's net revenue, defined as the excess of total revenue over variable costs, which is short-run producer surplus. This measure, also known as quasi-rent (R), is generally considered to be more appropriate in the context of smallholders who have difficulty valuing fixed costs, especially land.

We compute net revenue from rice farming for each cultivation season through farm budgeting. We define farm income as net revenue generated from activities within the family farm. To obtain household income, we also include nonfarm income. We conduct a series of simulations on alternative water allocation policies based on these budgets and the estimated parameters of the production function.

4.1. Input use patterns and production relationships

Per hectare means, standard deviations, minima, and maxima of farm size as well as variables used in the production function estimation are reported in Table 1 for each cultivation season and subarea. Also reported in the table are mean differences of the variables between subareas and between seasons and results of t-tests for comparison of their differences. t-tests indicate significant interseasonal and inter-subarea differences in means of rice yields, water quantities applied, and machinery used per hectare. They also show significant inter-subarea differences in means of seed and labor used and significant interseasonal differences in means of chemicals used per hectare. Mean quantity of irrigation water applied per hectare is higher in the Maha season than in the Yala Season for both subareas and is higher in the OIA subarea than in the NIA subarea in both seasons. Farmers in both subareas use higher levels of chemicals during the Maha season. This is due to preventive measures taken due to the higher incidence of pests and diseases during the wetter Maha season. OIA farmers use less labor and more rice seed and machinery than NIA farmers during both seasons indicating substitution of machinery and seed for labor in both seasons. Further, correlation analysis of the farm-level data indicates a strong negative relationship (−0.531 and −0.526 for Maha and Yala seasons, respectively) between quantities of labor and machinery.

These data were analyzed to determine relationships between input intensities and rice production per hectare. The initial model was formulated as a Cobb–Douglas production function and estimated with all observations using pooled OLS. Variance inflation factors were calculated as multicollinearity diagnostics. High multicollinearity was evident in the data. Three variables with highest variance inflation factors (each in excess of 1,000) that were not statistically significant were deleted. They were the season dummy, subarea dummy × log of fertilizer, and season dummy × log of fertilizer variables. The parameter estimates for the model estimated with the remaining variables are reported in the first column of Table 2.

Thirteen of the 22 independent variables were statistically significant at the 0.10 level. All of the inputs except chemicals and seed had a statistically significant positive impact on yield. The estimated production function exhibited diminishing returns to scale with a scale elasticity of 0.82. The area dummy (OIA) was statistically significant as were most of its interaction terms. It had a significant interaction effect with labor, machinery, and chemicals. In each of these cases, the OIA had a lower partial elasticity of production than did the NIA. The season dummy (Maha) was significant as an interaction term with area, labor, and machinery. All three had positive impacts indicating that the partial elasticities of production were higher for both labor and machinery in the Maha (wet) season than in the Yala (dry) season. The significantly positive coefficients on farming experience and the completed schooling dummy variable indicate that yields increased with farming experience and with attainment of 10 years of education. Only seed and the indirect variables of years of education and distance to the distributor canal had no significant impact on yields.

We conducted statistical tests for heteroskedasticity and the current Cobb–Douglas specification with the three deleted variables. The test statistics are reported in Table 2. We found no evidence of heteroskedasticity nor of model misspecification based on these tests. Neither null hypothesis was rejected.

We also estimated the production function using the translog functional form. This form was the second-order Taylor expansion for which the Cobb–Douglas was the first-order expansion. It included the same variables as in the Cobb–Douglas production function plus the 21 quadratic terms on input quantities. An F-test was conducted to determine whether all 21 coefficients on the quadratic terms were jointly zero. With a test statistic of 1.29, the P-value was 0.18, and the hypothesis that the true functional form was Cobb–Douglas was not rejected against the alternative that it was translog.

As robustness checks on our pooled OLS estimation of the Cobb–Douglas production function, we include three additional models. They included the same specification estimated with a random effects panel data estimator and separate OLS estimates for each season.

The production function was reestimated using panel data procedures because the data set included observations for the same producers in two seasons. Based both on the Hausman test and our interest in determining whether there were systematic differences in the production function for the two subareas, we used the random effects estimator. The fixed effects estimator prevented our keeping the group cross-sectional dummy variables in the specification. The parameter estimates for this model are reported in the second set of columns in Table 2. To allow for the possibility of heteroskedastic errors, robust standard errors were computed. Fewer parameters were statistically significant, and the parameter estimates were very different from the OLS estimates. For example, the scale elasticity for the panel data estimates was 0.22, little more than a quarter of that from the OLS estimates and much lower than most agricultural production function estimates in which all inputs are treated as variable. Therefore, the panel data estimates are not considered reliable and will not be used to draw economic inferences.2

A separate production function was estimated by OLS for each season. They differed from the model estimated by pooled OLS in deleting all season interaction variables and only using observations from one season. The same procedures were followed as in the pooled OLS estimation. Multicollinearity diagnostics were calculated first and variables deleted with the highest variance inflation factors that were not statistically significant. That resulted in deletion of the subarea dummy × log of fertilizer from both models as well as subarea dummy × log of water, subarea dummy × log of machinery, and subarea dummy × log of seed from the Yala Season model. Ten of 16 variables were statistically significant at the 0.10 level in the Maha season model and nine of 13 variables were significant in the Yala model. Returns to scale was estimated to be 0.96 in the Maha season and 0.70 in the Yala season, thus bracketing the pooled OLS estimate and being much higher than the panel data estimate. The subarea dummy was statistically significant in both seasons. The only input variables that were statistically significant in both seasons were water, labor, and Subarea dummy × log of labor. In all models, the estimated partial production elasticity was greatest for water. As in the other models, completed schooling and the log of farming experience were statistically significant in these models. These model estimates will be used as robustness checks to the pooled OLS production function estimates for selected economic implications.

4.2. Marginal value productivities of inputs

The marginal productivity of each input was calculated at the data means for each subarea and cultivation season using the pooled OLS estimates and the individual season OLS estimates of the production function. All calculated marginal products were positive. Each was also declining with increasing level of input use. Thus, our estimated production function was consistent with the theory of price-taking profit-maximizing firms.

Marginal value products (MVPs) were obtained by multiplying marginal products by average product prices. They are presented in Table 3 together with their respective opportunity costs (OCs) for the potentially constrained inputs of water and labor and the subsidized input of fertilizer. We used the current value of net benefits realized by inland capture fishery when reservoir water is not released for irrigation as the OC of water. Wage rates for hired labor and the subsidized price of fertilizer were used as the OCs of labor and fertilizer, respectively.

The ratio of MVPs and OCs are much larger than unity for water and fertilizer, suggesting that an additional unit of water or fertilizer in rice production will generate additional revenues higher than the OC of the respective resource. In contrast, the MVP/OC ratio for labor is less than unity in all four cases, indicating that the true OC for additional labor is much less than the wage rate.

This information provides important policy insight for allocation of inputs. Since there is no established market price for water, the calculated MVP for water can be used as an estimate of the true value of water at the prevailing market price for rice.

The MVP/OC ratio for fertilizer calculated from the pooled OLS parameter estimates is very large, with all estimates greater than 10. Farmers currently pay a subsidized price for the allocated fertilizer. It can be inferred that farmers are likely to apply more fertilizer (until MVP equals OC) even if they are not subsidized. The agronomically maximum realizable farm level yield for the irrigation district is considered to be about 10.0 metric tons per hectare. The mean yield on sample farms in each subarea and season is less than 75% of this expected yield plateau. Thus, there appears to be considerable opportunity to increase yield and net returns from rice through economic application of more fertilizer.

As goodness of fit and statistical tests are satisfactory, the pooled OLS estimate of the production function is judged suitable to predict output due to changes in input combinations within the range of our data. We use this estimated model to simulate the effects of applying different quantities of water on rice yields.

The same qualitative implications about marginal value productivities and their relationship to input OCs were obtained from the individual season production functions, the results of which are also reported in Table 3. The estimated MVPs for water and labor were similar to those from the pooled OLS estimates. The estimates for fertilizer showed greater difference between the two seasons.

4.3. Economic welfare implications of differential access to water

Farm budgeting techniques were used to compute gross revenue, net revenue, and profit from rice for representative farms at the data means. Their averages and standard deviations by subarea and season are reported in Table 4.

In preparing rice budgets, household-owned inputs were valued at market rental/sales values for the respective season and area, and the OC of capital per season was considered to be 4.5% of all cash costs, to reflect the ongoing short-term market borrowing rate of about 18% per annum. Family labor was imputed at the wage rate incurred by the household during the respective season. Based on the convention used in standard farm budgeting, 5% of net revenue was imputed as returns to management. Land rental practices varied greatly within and across subareas. Precise land rental rates were difficult to assess. Based on information from farmer leaders, land rental values per hectare per season were imputed at SLR 20,000 for OIA and 15,000 for NIA.

Representative farm annual household incomes were calculated for each subarea using the calculated rice net revenue, the mean share of rice income in farm income for the sample in each subarea, the mean share of farm income in household income for the sample in each subarea, and the mean farm size. Calculated means and standard deviations of household annual net revenue from rice, net farm income, and household income are also reported in Table 4. Each of these indicators of economic welfare is considerably higher in OIA than in NIA, and statistical tests based on the farm-level data indicate that they are significantly higher.

The survey revealed important differences in the distribution of household income in each subarea between rice, farm, and nonfarm income. Shares of rice income in annual household income varied from 10% to 100% and its average was greater in the more established farming subarea, OIA (44%), than in NIA (38%). However, it is in the skewness of the distribution of rice to household income that the greatest differences in income composition between subareas are evident. The distribution of OIA households is slightly skewed to lower percentages of household income from rice, while the distribution of NIA households is strongly skewed toward higher percentages.

As evidenced from the survey and household budgeting, water availability and rice income was associated. This finding was corroborated by our production function estimates that revealed large positive impacts of additional water. Using the pooled OLS estimates of the production function, the effect of water use on net rice income was simulated per hectare for each individual farm for both seasons by changing water quantities applied while holding all other input quantities at current levels. We first simulated net revenues from rice for all farms while holding water quantities applied at the current OIA mean. Water quantities were then decreased to the current NIA mean, while all other input levels were kept at current levels (see Table 5). A paired sample t-test indicated that net rice revenues are higher when water levels are set at the OIA mean, and the differences are highly significant. It is anticipated that annual household income will also be increased if the quantity of water applied per hectare is increased. In the next section, we examine the effects on farm household income of alternative interventions to allocate additional water.

4.4. Economic implications of alternative augmented water allocation plans

The Veheragala Reservoir Project, a project under construction during the survey period to increase water available through trans-basin diversions, did not anticipate increasing the irrigated area in the KOISP. Therefore, additional water would be available for distribution only among existing farms.

Our simulated interventions target farms that currently use quantities of water less than the minimum level defined by alternative allocation criteria, each of which is described in Table 6. The simulations focus only on the distribution of additional water supplied by the new project and do not consider reallocation of initial water rights. Therefore, water quantities applied and thus yields on farms that currently use higher quantities of water than implied by the allocation criteria remain unchanged.

When water is allocated based on economic criteria, water quantity is allocated among subarea–season strata so that their mean MVPs for water are the same when all available water is exhausted. The agronomic criterion used in other plans ensures that all farms receive sufficient water to meet minimum agronomic requirements and then additional water, if any, is allocated according to the economic criterion.

The pooled OLS estimated production function is used to predict yields at new water levels for each farm assuming that quantities of other inputs, except fertilizer in one simulation, remained at current levels. Water is allocated to each farm until all farms in the respective subarea receive the minimum water quantities as defined above.

We consider four alternative water allocation plans. All plans are constrained by the availability of 50 million cubic meters (MCM) of additional water at the reservoir (or 35 MCM at the farm level).

In Plan A, the allocation of additional water is based solely on the economic criterion of equalizing MVP of water in all receiving subarea–season combinations. This criterion generates maximum economic value from the additional water as a Pareto optimum. Water quantity is defined as the level of water at which MVPs for receiving additional water are equated when the total additional water is exhausted.

Plan B deviates from Plan A as the latter ensures that all farms receive sufficient water to meet minimum agronomic requirements for crop evapotranspiration (ET) and percolation losses. This criterion is applied to all farms even if a subarea–season stratum has a sufficiently low mean MVP that it does not receive additional allocations based on the economic criterion. Based on Dimantha and de Alwis (1985) and Guerra et al. (1998), we define the minimum water levels as 10,250 and 10,750 m³/ha, respectively, for the Maha and Yala seasons. After the minimum water requirement is met for each farm, any additional water available is allocated based on the economic principle of Plan A.

In addition to these two basic water allocation plans, we consider two additional plans to highlight policy implications. Plan C represents a water allocation plan based on Plan B criteria with farmers applying fertilizer at rates recommended by the Sri Lanka Department of Agriculture. It supports the government's objective of increasing crop productivity by promoting fertilizer application. Plan D is a “business as usual” plan to compare net benefits if KOISP managers allocate additional water in the same proportions by subarea and season as the current allocation. All of the plans maintain the (strong) assumption that, with more water available, households do not change any other production decision, other than changing fertilizer rates in Plan C. Thus, we assume that households do not reallocate labor away from nonfarm jobs, hire more labor, or increase seed, chemical, or machinery input use when they receive additional water. Thus, our estimates of income effects are lower bound estimates since these inputs are not permitted to adjust to their optimal levels.

The pooled OLS estimated production function was used to predict yields at new water levels for each farm. Yields for farms that receive additional water allocations would increase. Yields for other farms would remain unchanged. Consequently, the marginal product of water for the subarea–season combinations receiving water will decrease. With constant output price, decreasing marginal productivity implies that MVP also decreases as the quantity of an input increases. Compared to current water quantities and yields, the incremental water quantities and resulting yield increases from each of the four alternative water allocation plans are presented in Table 7.

Under the current water allocation, mean water use per hectare shows notable differences between subareas and between seasons (left column, top panel). Per hectare water use in the highest water using subarea–season stratum (OIA-Maha) is a quarter higher than in the lowest water using stratum (NIA-Yala). However, annual total water used by both subareas is almost equal (middle panel) because there are more irrigated hectares in NIA.

When the economic criterion is used to allocate additional water (Plan A), OIA would receive just a little more than 6% of the additional water available. Further, it would not receive any additional water during the Maha season because the available water quantity is exhausted before reaching the current low MVP of farms in that stratum. In Plan B, OIA receives additional water during both seasons, but only to meet the minimum agronomic requirements on each farm. In Plan C with rice fertilized at Department of Agriculture recommended rates, the allocation to OIA is the same as in Plan B but there is minor redistribution in NIA between seasons. In both Plans B and C, the OIA receives about 14% of the additional water available. Under Plan D, the allocations of additional water are proportional by subarea and season to the current allocation, so each subarea receives approximately the same amount.

MVPs of water in each subarea and season are also reported in Table 7 (within parentheses in the top panel). When the mean water quantity increases, the MVP of water decreases because the marginal product decreases. MVPs of water are equated in Plan A for all subarea–season strata receiving additional water and in Plan B for both NIA seasons. The agronomic constraint provides sufficient additional water for OIA that the MVP in both seasons is less than for NIA. Because of the additional fertilizer applied in Plan C, the MVPs in all subareas and seasons increase over Plan B.

Mean current yields and predicted yield increases for each stratum are shown in the bottom panel of the table. Except when fertilizer is also increased (Plan C), highest incremental yields in NIA for both seasons occur with Plan A (for which the economic criterion is used). In all other plans, a part of the water is reallocated based on other criteria, and incremental yields for NIA are less.

The average physical and value productivities (APP and AVP) of water can be computed from the information presented in Table 7. The APPs range from 0.61 kg/m³ for OIA in the Yala season to 0.71 kg/m³ for NIA in the Maha season. These estimates are in the middle of the range of 0.2–1.2 kg/m³ reported by Bouman et al. (2007) based on agronomic trial data from China, India, Malaysia, and the Philippines. They are also within the range of 0.2–0.9 kg/m³ reported by Murray-Rust et al. (1999) for subsystems of the Sri Lanka GalOya Irrigation System. They are slightly higher than the range of global average water productivity estimates for rice of 0.15–0.60 kg/m³ and those for China and Southeast Asian countries of 0.4–0.6 kg/m³ reported by Rosegrant et al. (2002).

Using average prices for rice (see Table 3), the U.S. exchange rate (see Table 1) during the survey period, and the India average exchange rate during 2007 (2.689 SR/Indian rupee), our AVP estimates range from $0.09 to $0.11/m³ (3.7–4.4 Indian rupees/m³). These estimates are on the low end of the range of $0.10–$0.74/m³ reported by Hussain et al. (2007) for South Asia and Africa and a little lower than the range of 4.5–7.8 Indian rupees/m³ reported by Kumar et al. (2008) for the Western Punjab and Eastern Uttar Pradesh regions of India. Thus, although our OLS estimates of the production function exceed our panel data estimates, they provide physical and value productivity estimates that are well within the range of other relevant literature.

The predicted yield increases are used to compute predicted aggregate increases in net revenue from each plan. The predicted aggregate increases in net revenue and the percentage share of the increases are reported by subarea–season stratum, season, and subarea in Table 8.

The distribution of increased net revenue among subareas and among seasons differs the most with Plan A. The share of increases in aggregate net revenue accruing to the NIA subarea and to the Yala season is also highest with Plan A.

Without increasing use of any input other than water, total annual increase in net revenue for the water system (KOISP row) is highest under both Plans A and B. A difference is evident only at the second decimal place. Net benefits are expected to increase an additional 14% by adding fertilizer in addition to water. Comparing Plans B and C, we find that nearly 9/10 of the increased net revenue in OIA accrues because of increased fertilizer use and one-tenth because of the higher MVP of water when fertilizer is increased. However, the magnitude of net revenue increases due to application of more fertilizer is higher in NIA for both seasons because initial yields are lower and incremental fertilizer quantities are higher. Increases in net revenue are somewhat more equally distributed among subareas and seasons in Plan C, and therefore, if implemented will contribute to a continuation of current income disparities between subareas.

Plan D (status quo) generates a lower increase in annual net revenue than any of the alternatives. Increases in net revenue are 7% higher under Plans A and B and 22% higher under Plan C than under Plan D, all of which demonstrate that the current water allocation criterion is suboptimal at the margin. Its distribution of additional water is also much more favorable to OIA and thus would contribute to a continuation of initial income disparities.

We report the distributional effects of additional water on household income for each plan in Table 9 using both the pooled OLS and individual season estimates of the production function. Distributional effects are noted by measuring the average additional household income for households grouped into five income quintiles prior to the intervention. Income group 1 represents the quintile of households with the lowest initial income.

Under each of the intervention plans, the additional water will have considerably higher relative impact on households with lower incomes and thus is expected to have a positive impact in reducing income inequality. In fact, the order of increases by income groups is exactly the opposite of the ordering of initial income. This finding is robust both across plans and across production function estimators. Both production function estimators suggest that Plan C benefits the lowest income quintile the most and Plan D benefits it the least. With some ties, the same finding applies to the next lowest and the middle income quintiles. The highest income quintile is unaffected by plan selection. Consequently, not only would the status quo Plan D provide the smallest aggregate benefit from the additional water, it would also have least impact on reducing income inequality.