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a INERA, 03 B.P. 7192 Ouagadougou 03, Burkina Faso
b Department of Agricultural Economics, Purdue University, West Lafayette, IN 47907 USA
lowenberg-deboer{at}agecon.purdue.edu
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ABSTRACT |
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 TOPNOTES ABSTRACT INTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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Abbreviations: NPV, net present value • FOC, first order conditions • SOC, second order conditions • NGO, non-governmental organization • FCFA, African Financial Community Franc
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INTRODUCTION |
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TOPNOTESABSTRACTINTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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In both industrialized and developing countries, the spacing of the conservation barriers has been based mainly on physical criteria, such as slope, soil characteristics, and rainfall patterns (FFTC, 1995; FAO, 1988; Schwab and Frevert, 1985). In mechanized agriculture the width of equipment is often a determining factor in the spacing of conservation barriers. The yield effects of windbreaks as a function of spacing are well documented (Kort, 1988), but methods for incorporating that yield response into the design of windbreaks are not well developed.
Initial economic evaluation of these conservation techniques focused on the value of benefits over the life of the physical or biological barrier for a fixed spacing. As information on crop response has accumulated, a few studies have looked at the effect of the spacing between parallel barriers on economic returns. In Nebraska, Brandle et al. (1992) compared windbreak spacing of 193.6, 128.0, and 66.5 m. They found that the more widely spaced windbreaks provided higher returns, mainly because they required less land to be taken out of crop production. Shively (1996) estimated maize response to hedgerow spacing in the Phillippines as a quadratic function. He found that the spacing required to maximize either yield or profit is somewhat wider than the spacing that minimizes erosion.
On the Central Plateau of Burkina Faso, constructing rock bunds on the contour is one of the most effective techniques for reducing erosion and increasing water infiltration while improving yields. About 24% of arable land in Burkina Faso is severely degraded (Kambou et al., 1994). Most of this degraded land is on the Central Plateau, where population pressure has led farmers to cultivate marginal lands and reduce fallow periods. Kaboré et al. (1994) estimate that on average fields protected by rock bunds have 11% higher sorghum [Sorghum bicolor (L.) Moench] yields than unprotected fields. Wright (1985) reported a 47% increase in average sorghum and millet [Pennisetum glaucum (L.) R. Br.] yields with bund spacings varying from 10 to 50 m. Vlaar (1992) reports doubling of yields in some cases when rock bunds are constructed. Hulugalle et al. (1990) and Maatman et al. (1998) indicate that the impact of rock bunds is substantially enhanced when combined with other and water conservation techniques such as tied ridges and zaî (an intensive manure management method).
The objective of this paper is to outline a method for determining the net present value (NPV) maximizing spacing for rock bunds and to provide an application of this method using the data gathered by the Water, Soil Fertility, Irrigation and Mechanization Program (ESFIMA) of the National Institute of Agricultural Research (INERA) of Burkina Faso. The method is intended to be used in the design stage of conservation projects by planners who determine the spacing to be used by project staff when they are involved in building barriers and in training farmers to do their own construction. Yield response is estimated as a function of bund spacing. Calculus is used to optimize bund spacing. The NPV maximization is assumed as a first approximation of farmer objectives.
Several methods are used for rock bund construction in Burkina Faso but all are based on the principle of slowing rainfall runoff with rock barriers. Water is held on the field longer, increasing infiltration. Sediments are deposited behind the bund. On the Central Plateau rock bunds are usually preferred to earth bunds because they are permeable. Water is trapped behind earth bunds and may flood crops. Rock bunds slow runoff, but allow some water to pass. Kaboré et al. (1994) provide an overview of the primary rock bund construction techniques in Burkina Faso.
This paper is divided into two primary sections: the theoretical optimization model and use of that model with data from the Kirsi site on the Central Plateau in Burkina Faso. The final section includes conclusions and implications for research.
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Theoretical model |
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TOPNOTESABSTRACTINTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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Assume the yield response to distance upslope from a well constructed rock bund is:
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This relationship is assumed for distances 0 to M. Beyond M, the yield is constant at:
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The economic variable of interest is the spacing of the bunds. The average yield over a parcel protected by rock bunds might be estimated as the average over all distances represented. If the spacing is less than M, the approximation would be:
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Because the yield response to distance is assumed to be continuous, the units of distance can be made very small. The variable X can be written as the integral over the distance from 0 to S:
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This integral has a closed form solution:
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Economic Benefits
The NPV maximization approach is necessary because rock bunds can have effects over many crop seasons. Farmers everywhere have multiple objectives. In Burkina Faso these objectives include maximizing profits and NPV, managing risk, maintaining food self-sufficiency, and protecting the environment. This analysis uses NPV as a first approximation of the farmer's objectives. Profit maximization in any one season would not tell the whole story. By discounting future cost and benefit flows in NPV, all flows can be put in terms of value at the time of bund construction. Risk, food self-sufficiency and environmental protection are important, but beyond the scope of this analysis. Typically, science proceeds in small increments. In economics it is common to first solve the simpler deterministic problem of maximizing profits or NPV before tackling more complex risk and environmental issues. The general form of the NPV would be:
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The FCFA exchange rate was about 500 FCFA/US in 1995. The term (10000/S) is the number of meters of rock bund per hectare.
Theoretically, all of the variables might change over time. Prices might trend up or down. Bund yield effect and maintenance may change as the bund ages. To focus on the economics of bund spacing it is useful to consider the case of constant expected prices, costs and coefficients. The NPV takes the form:
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. The term W is often called an annuity factor. It gives the present value of a uniform series of benefits over a given period (Barry et al., 1983).
The interior extreme maximum and minimum points occur when the first derivative is equal to zero. These stationary points may be either maximums or minimums:
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The first order conditions (FOC) can be interpreted by noting that the first term, WP[(a1/2) + (a22/3)S], is negative in the neighborhood of the NPV-maximizing spacing. The term represents the present value of the marginal decline in the value of production as spacing is increased. The second term, W x K x 10000/S2, is the present value of marginal decrease in the cost of maintenance as spacing is increased. The third term is the decrease in construction cost as spacing is increased.
As it is stated, optimization requires that at the spacing that maximizes NPV the value of yield loss from increased spacing will be exactly offset by the gain from reduced maintenance and construction cost. Because of the form of the response function (i.e., yield declines as the variable input is increased), the signs of the marginal product and marginal cost terms are the opposite of the usual crop response case, but the interpretation is the same.
The FOC are necessary, but not sufficient conditions for a maximum. The second order conditions (SOC) check the second derivatives around the stationary point. If slope is decreasing, the point is a maximum:
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If the response takes the form in Fig. 1, the first term in the SOC is positive and the stationary point of the NPV expression will be a maximum only if the cost terms are large relative to the quadratic term (WPa22/3).
Solving the FOC for S will provide the interior solution for NPV-maximizing bund spacing. A complete analysis also should check the corner solution of no rock bunds and the yield given by Eq [2].
This analysis assumes that no crop area is lost by building rock bunds. For the Central Plateau that is a reasonable simplifying assumption. The rock bunds are narrow and with manual labor crops can be planted very close to the stones. For those cases in which bunds are stabilized with grass strips or hedge rows, it is necessary to multiply the yield in the NPV expression by the proportion of land that remains in crop production (1 - w/S, where w = the width of the grass band or hedge row). In this case, there will be a third cost term in the FOC reflecting land removed from production that would increase optimal spacing.
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Empirical application |
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TOPNOTESABSTRACTINTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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P1: One bund along the lower edge of the site.
P2: A bund along the lower edge and another spaced at 50 m upslope.
P3: A bund along the lower edge, one at 33 m, and one at 66 m.
P4: A bund along the lower edge, one at 25 m, another at 50 m, and a fourth bund at 75 m.
In 1993 and 1994, yields were measured at the following distances upslope from the lower edge of the site for all spacings: 4, 12, 21, 29, 37, 46, 54, 62, 71, 79, 87, and 96 m. In 1992, P1, P2, and P4 yields were measured at 6, 19, 31, 44, 56, 69, 81, and 96 m, while the P3 yields were measured at 6, 27, 39, 60, 72, and 93 m. In 1993 and 1994, two yield measurements were taken for each spacing. In 1992, two measurements were taken for P1, P2, and P4, but 3 yield measurements were taken for P3.
The local variety of sorghum was planted and traditional agronomic practices were used. No fertilizer was applied.
Response Function
A quadratic response function was estimated with dummy variables for seasons and for the location on a given slope (Table 1)
. The dummy variables are specified so that the coefficients give the difference between the mean yield for a given category and the overall mean. For example, the 1993 dummy variable coefficient is the difference between the mean yield in 1993 and the mean for the 3 yr data period. A t test with the null hypothesis that the coefficient is zero indicates whether mean yields for the year or location differ from the overall average. The location dummy was tested because of a hypothesis that having one or more bunds upslope may have beneficial effects because it protects crops from flooding and sedimentation. The estimation assumed that all distances in the data set were less than M (i.e., that the bund had some effect at all distances for which yields were measured.).
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The quadratic terms are positive in both models, but not statistically significant at conventional 5 or 1% levels. The estimated quadratic coefficients will be retained for the economic analysis because econometric theory strongly encourages model specification based on previous knowledge of the process being modeled. The regression statistical tests assume that the functional form is known. Econometric theory discourages ad hoc models based on the particular data set being analyzed because such models are difficult to interpret and because the significance levels in conventional statistical tests do not take into account errors that may occur in the iterative process of model development.
Alternative models were estimated for linear, square root and logarithmic functional forms. Judging from the R2, the alternative models did not provide a better fit than the quadratic.
Optimization
The NPV of returns to sorghum production (V) was maximized with respect to bund spacing for three scenarios using estimated bund construction and maintenance costs. Sensitivity testing was done on the opportunity cost of labor and the discount rate. Only results for Model 2 are reported; spacing estimates for model one are very similar.
The NPV is calculated for returns to family resources with traditional production techniques, including local varieties with seed saved by the farmer, no chemical fertilizer, and all family labor. Thus, all variable inputs are supplied out of family resources and the variable cash costs term (C) is zero. A more complete analysis would jointly optimize bund spacing with fertilizer-improved varieties and other inputs. Unfortunately, such interactions can not be tested with available data; the Kirsi trial used traditional inputs.
Bund construction costs vary widely depending on the type of transportation and how labor is organized. In this section optimatization is carried out for three cost examples to show how optimal bund spacing would vary with cost.
Community Land Management Projects
The usual objective of such projects is to protect a substantial portion of the land in a given village or watershed. Because large quantities of rock are required they must often be transported some distance. Local people work together to gather stones and build the bunds in community workdays. People may not be working on their own land. Typically, the community workdays are full days (roughly 8 h) to make efficient use of trucks and other equipment.
Farmer with NGO Assistance
The farm family is working to protect its own fields, but rocks must be transported from a distance. In the case cited by Kaboré et al. (1994), the non-governmental organization (NGO) helped with training in contour tracing and with truck transportation. To use the truck efficiently, full-day workdays are planned for hauling.
Farmer with Rocks Nearby
The farm family is working to protect its own fields using rocks found close to the fields. In the case studied by Kaboré et al. (1994), the farmer laid the contours himself and transported the rock with a wheelbarrow that he already owned. This is a relatively rare situation, but it is included to show the full range of costs levels. The farm family does the heavy work of stone gathering, transporting, and laying during the cooler early morning hours.
The cost estimates per meter of bund are shown in Table 2 assuming farm family labor has an opportunity cost of 50 FCFA/hr in the dry season (Lowenberg–DeBoer et al., 1994). A sensitivity test will be done assuming the legal minimum wage of 143 FCFA/hr. The cost estimate for the community land management projects is higher mainly because of greater labor use. The Vlaar labor estimate gives a range of 80 to 160 person days/ha. Table 2 estimates assume 8 h per day and the midpoint of the range. Kaboré et al. hypothesize that the greater labor use in community projects is related to the lack of motivation when individuals are not working on their own fields and the full day work schedule. Most bund construction occurs during the hot season when temperatures may reach 45°C in the afternoon and work efficiency is low. It should be noted that this cost estimate does not include the expenditures on the survey team that traces the contours or administrative costs for the project. Surveying and administrative costs are potentially important in estimating the overall net benefit, but they are fixed costs that do not change much with different bund spacings. Similarly, the cost of the earthen bund upslope and to the sides of the Kirsi site was not included in the cost analysis because such protection bunds are not usually used on farms, and if they were used they would be a fixed cost that would not affect spacing.
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The farmer with rocks close to the field has the lowest cost. In the case cited by Kaboré et al., the farmer used tools already on the farm and traced the contours without assistance. It is hypothesized that the labor times are low because the farm family is working only on their own land and only in the cooler early morning hours.
Parameters needed for the optimization include the sorghum price, the discount rate, annual maintenance costs, and the useful life of rock bunds. The average harvest time market price of sorghum in the area was 59 FCFA/kg during the period 1992 to 1994. The annual maintenance costs are assumed to be 5 FCFA/m. The useful life of rock bunds has not been well studied; for the purpose of these examples, T in Eq. [6] is assumed to be 10 yr.
The appropriate discount rate depends on the decision maker's opportunity cost of capital. Lowenberg–DeBoer et al. (1994) indicate that private opportunity cost of capital for Burkinabé farmers is at least 50% annual. The high discount rate has been linked to poorly developed financial institutions and a chronic shortage of capital. A 10% cost of capital was used as the social discount rate for the donor and NGO-assisted scenarios, reflecting the great availability and lower cost capital in the industrialized countries. The discount rate sensitivity test assumed that the social discount rate was 6% and the farmer rate 100% annually. For the social rate this was based on the argument that the rate should reflect a long-term social discount rate. For the farmer discount rate sensitivity test, 100% was chosen based on studies that indicate that some Sahelian farmers have opportunity costs of capital of 100% annually or more (Lowenberg–DeBoer et al., 1994).
Optimization was carried out using the FOC described above and a Quattro Pro spreadsheet (Corel, Ottawa, Canada). The Quattro Pro command /TOOLS/SOLVE was used to solve the FOC for the NPV-maximizing distance. The SOC are negative for all the solutions discussed here.
Results are given for the three cost scenarios above (Table 3) . The NGO scenario is considered both from the social cost angle and the farmer's private cost perspective. The NGO social cost analysis includes all costs and a 10% baseline discount rate. The NGO farmer cost analysis includes only the farm family labor and used a 50% baseline discount rate.
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If the opportunity cost of labor is 143 FCFA/hr, with baseline discount rates, optimal spacing rises to 33 m for the scenario of the farm with rocks nearby. Under the NGO scenario bund construction is beneficial for society at large, but not the individual farmer. For community projects bund construction cannot be justified by economic benefits alone.
Except for the scenario of a field with rocks nearby, these spacings are greater than the approximately 30-m spacing that is suggested based on physical criteria alone. This is consistent with the usual production economics result that less of an input (in this case meters of rock bund per hectare) is used when maximizing economic returns than when maximizing physical yield.
Breakeven construction costs and spacing were calculated using the spreadsheet. For the baseline social cost scenarios the maximum construction cost is 191 FCFA per meter of bund and the spacing at this cost would be 47 m. For the baseline farmer cost scenarios, the maximum construction cost would be 61 FCFA per meter and the optimal spacing would be 47 m. Discount rate sensitivity testing modifies the maximum allowable construction cost, but the spacing at those maximum costs remains 47 m. With the 6% discount rate the community project could support a cost up to 229 FCFA/m. At the 100% discount rate the maximum construction cost for farmers is 31 FCFA/m.
Figure 2 shows how NPV changes as spacing is increased. The level of the NPV curve is higher for the community and farm/NGO scenarios because of their lower discount rate. The NPV surfaces beyond a 30 m spacing are relatively flat and as a consequence the NPV difference between a 30 m and the 47 m maximum spacing (suggested by the breakeven analysis) is relatively small. Because of the lower labor productivity in the community scenario, the NPV maximizing spacing is wider than for the other scenarios.
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Conclusions |
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TOPNOTESABSTRACTINTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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This analysis suggests that the NPV-maximizing spacing of rock bunds on the Central Plateau of Burkina Faso depends on the type of construction, transport cost, and how labor is organized. Because of high labor inputs, yield increases do not cover bund construction costs for the community project scenario; environmental and other benefits not considered here may still justify this type of project.
For the conditions at the Kirsi site, the optimal spacing for a farmer who can construct bunds with farm tools from rocks available close to the field is between 23 and 45 m. The more common case of a farmer who works with an NGO to trace contours and transport rock has a NPV-maximizing spacing of 30 to 43 m. Except for the community scenario, the NPV surface rises sharply for initial increases in distance up to about 30 m, but it is relatively flat in the neighborhood of the maximum. This suggests that in the Kirsi case not much is lost by using the 30 m spacing based on physical criteria if farmers are working on their own fields with or without the help of an NGO. Those planning rock bund projects using community labor should consider wider spacing.
Further research in this area should relax some of the stringent assumptions made in this first approximation. Risk and environmental aspects should be incorporated in the theoretical model. For Burkina Faso an effort should be made to include in the analysis the age of the rock bunds and soil characteristics. Interactions of bunds with fertilizer, improved genetics and other inputs need to be examined. Empirical results need to be derived for a range of slopes, soil conditions, and rainfall patterns. With these additional research results it would be possible to develop extension information that would guide bund spacing according to local conditions, including the cost of construction, slope, and soil type.
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NOTES |
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TOPNOTESABSTRACTINTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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Received for publication June 1, 1999.
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TOPNOTESABSTRACTINTRODUCTIONTheoretical modelEmpirical applicationConclusionsREFERENCES |
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