THE ANALYSIS OF RISK IN IRRIGATIONPROJECTS IN DEVELOPING COUNTRIES

By IAN LIVINGSTONE AND ARTHUR HAZLEWOOD

This article deals with an issue that arises in the planning of irrigation schemesin developing countries: the volume of water a scheme should be designed toutilize when the volume of water available varies widely from year to year.'

The constraint on agricultural development can be taken to be the amount ofwater available for irrigation development, rather than the supply of land. Asone standard text puts it,

'in the Near and Far East there are many regions where there is more landsuitable for irrigation than there is water available for irrigation. Thereforewe must use the available water resources as intensely as possible, as long asit is economic to do so.'2

The issue arises because rainfall, and therefore the flow of the rivers, is muchgreater in some years than in others.3 A scheme which has the capacity, in termsof the area of land cleared and provided with canals, distribution channels andother irrigation infrastructure, to make full use of the water available in goodyears will suffer from water deficiency with insufficient water to irrigate the wholearea adequately in years of low water supply. The larger the land area developedfor irrigation the larger the investment cost and the greater the frequency of yearsin which water is deficient. The smaller the capacity of the scheme, the smallerthe investment cost and the smaller the risk of water deficiency, but also thesmaller the ability to utilize the larger volumes of water available in good years.With a large capacity investment is 'wasted' in dry years; with a small capacitywater is 'wasted' in wet years.

In other guises the issue under discussion is a familiar one in public economics:it is the 'capacity margin problem' which faces planners in, for instance, the elec-tricity supply industry, in which there is fluctuating demand for power (as com-pared to fluctuating supply of water in our case) and it is necessary to trade offreduced capital costs for smaller capacities against the consequences of excessdemand, load-shedding and unsatisfied customers. As Rees states, in this situation

1 The argument of the article was developed in the course of a study of the developmentpotential of the Usangu Plains of Southern Tanzania which has been carrkd out by the authors,and some use is made below of data from that study. See also an earlier article by the authors,'Complementarity and Competitiveness of Large- and Small-Scale Irrigated Farming: ATanzanian Example', BULLETIN, August 1978.

2 E. Kuiper, Water Resources Project Economics, Butterworth, 1971, p. 120.The discussion assumes run-of-the-river irrigation and no inter-year water storage by the

use of dams,21

22 BULLETIN

'the problem of investment planning resolves itself into that of determiningthe appropriate size of the capacity margin.'4

Whereas in the electricity case there are awkward difficulties involved in valuingthe effects of non-supply to consumers, in terms of consumer's surplus, in theirrigation case losses of potential output are in principle readily measurable incash terms, simplifying the problem considerably in this respect at least.

For determining the capacity margin in irrigation projects, there is a loosely-applied but widely recognized rule-of-thumb in engineering practice which recom-mends that the volume of water assumed for planning purposes should be thatwhich can be expected to be available in eighty years out of a hundred. This isreferred to as the 'one-in-five year' rule and indicates a recommended degree ofcaution: only in one year out of five will the water be insufficient for the installedcapacity.

The 'one-in-five year' rule is not set out in engineering manuals as a firm andprecise recommendation. But 'engineers are pragmatists and design according toempirical formulas that experience has shown to be reasonably safe',5 and therule, or something approximating to it, does appear to be informally applied on awide scale and most water and irrigation engineers seem familiar with it.6 It isargued in this article that such a rule provides an optimal solution only in par-ticular circumstances, and that its general application is inconsistent with thestandard net present value criterion for evaluating alternative investment pro-jects. This provides an interesting example, therefore, of conflict between 'tech-nical' and 'economic' efficiency, and one which is of extreme importance in invest-ment decisions all over the world, not merely a useful class-room illustration. Wealso compare the optimum solution for a large mechanized farm with that forpeasant farmers, and find that it is likely to be rational for peasants to follow aless cautious policy than a large farm.

IIThe determination of the area irrigable with a particular flow of water is

largely a technical matter. Given the combination of crops7 to be grown, and theSee the excellent discussion of the capacity margin problem' in the electricity supply

industry in R. Rees, Public Enterprise Economics, 1976, Chapter 9.Maas et al., Design of Water Resource Systems, p. 126.A Ministry of Agriculture Bulletin on irrigation uses illustratively 'the seasonal demand

in the fifth driest year in 20 years. . .', which is equivalent to a 75-80 per cent year, comparedwith the 'eighty per cent year' of the 'one-in-five year' rule. (Ministry of Agriculture, Bulletin202, Water for Irrigation, Appendix G, p. 75.) In A. K. Biswas, ed., Systems Approach toManagement, pp. 165-66, it is stated that 'water users are concerned with what is termed asafe or firm yield' and for the accompanying hypothetical flow sequence it is calculated that'the mean probability of a yield of 2 million m3 being exceeded is 0.80'. A Ministry of OverseasDevelopment engineer has said (private communication) 'The one-in-five dry year flow is nota standard assumption; I have seen a report which estimates rainfall at 80 per cent probabilityand river flow at 90 per cent for the same project. 80 per cent is frequently adopted at anearly stage of project investigation merely because the data are inadequate to make a betterestimate.' The difference between 80 and 90 per cent is, of course of no significance from thepoint of view of our own argument, merely representing a marginally higher degree of caution,and the statement as a whole lends support to the belief that the one-year-in-five rule, orsomething very like it, is a commonly used role of thumb.

The main crop we have in mind, and the crop already grown under irrigation on a limitedscale in Usangu, is paddy.

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 23

detailed planting practice, the water requirement for proper plant growth can beprescribed. When account is taken of the rainfall, evaporation rates, and losses inthe distribution system, the gross requirement for irrigation water per hectare ineach month can be determined. By dividing this water requirement into thevolume of water available, the maximum number of hectares irrigable in eachmonth of the year is obtained. Unless the time-pattern of water requirementcorresponds closely to the time pattern of water availability, the maximum areairrigable will be widely different in different months. The constraining month isthat with the smallest irrigable area, and that is the maximum area that can besuccessfully irrigated. The fact that a scheme of this size would leave surpluswater in other months is immaterial: a bigger area could not be irrigated becauseof insufficient water in at least one critical month. Which month is limiting willdepend on the pattern of water flow in the river, which will generally be highlyseasonal in tropical countries, and the equally seasonal pattern of water require-ment for the crop. The better the 'fit' of water availability to water demand, thegreater the area irrigable.8

The area irrigable will, of course, vary from year to year with the variations inwater supply in the critical month. The figures of the water available in thecritical month over a run of years can be arranged in the form of a frequency orprobability distribution. This distribution will be positively skewed, with a longtail representing exceptional years of heavy rainfall and high river flow whichmay be many times the average.9 The question is, on which level of water avail-ability should the planned irrigable area be based: the mean value, the medianvalue, the most frequently occurring or modal value, or which? The one-in-fiveyear rule would base the area to be irrigated on the twenty-percentile river flow,generally yielding a significantly smaller planned area than either the mean or themodal flow.

IIIThe nature of the problem is illustrated in Fig. 1. Along the horizontal axis

we plot 4, the irrigable area. On the vertical axis (4) represents the probabilityof the water available being just sufficient to supply such an area. Since theirrigable area is assumed proportional to the water available (in the constrainingmonth), the shape of the frequency distribution is identical with that for riverflows.'0 This frequency distribution will start some way from O along the hori-zontal axis, since the water flow in any year is unlikely to fall below some positiveminimum level. If this minimum level is sufficient to provide for some 'guaran-

8 Although in reality there will be erratic variations in the seasonal pattern of water supply.we are assuming that the pattern of water availability within years is constant and are con-cerned only with erratic variability in supply from year to year. The limitations of thisassumption are referred to later (pp. 29-30).

The degree of kurtosis will vary from river to river depending on the magnitude of theoccasional extremely high flows.

A difficulty facing all analysis of irrigation in most less developed countries is the poorquality of the data and particularly the short run of years for which they are available, makingit difficult to fit a theoretical frequency distribution.

o

o

4

P(q)A

Fig. 1. Diagrammatic illustration of the risk problem

teed' hectarage, g, then the distribution will start a distance g along the horizontalaxis. It will simplify the discussion considerably if we measure the planned irri-gable area, 4, from this point, rather than from O, shifting the vertical axis to O'in a simple transformation, since the maximization problem is unaffected by thischange. Measuring from O', q20 corresponds to the twenty-percentile, q0 to themode, and q50 to the median. If, for simplicity, we assume zero variable costs ofcultivation and complete divisibility (no economies of scale) in land development,total costs will equal a4, where 4 is the planned hectarage and a the land develop-ment costs per hectare, and can e represented in the diagram by a vector fromthe origin O'. If v=.y is the revenue per hectare, p being the price per ton of thecrop, and y the yield in tons, then total revenue TR would equal vq, if waterwere sufficient for the whole planned area.

Let us assume further that if the water available in a particular year is less thanthe requirement for the entire planned area a proportionately reduced area iscultivated." If more than the necessary water is available the excess is in effectwasted, as 4 represents the maximum cultivable (i.e. prepared) area. The expectedvalue of profits for any given planned area 4 will therefore equal

n r r r 1E VP4 = [v4

Jp(q) dg + y

Jqß(q) dg] - a4 (1)

t=i 1 o

the first term being revenue in years when water is sufficient or more than sufficientfor an area 4 and the second term revenue in years when water supply is less thanthis, where the irrigation works have a life of n periods during which the valuesof y and 4 are constant.

The authors are grateful to the referee for comments which have permitted some im-provement in mathematical presentation in this section.

24 BULLETIN

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 25

The 'optimum' 4 which maximizes net proceeds is thus obtained by differen-tiating the above expression with respect to the choice variable 4 and is given by'2

t9(EVPq)-. =v±$P(q)dq_a=O (2)

or, y f p(q) dq_a (3)t=1

Here the integral represents the upper tail of the distribution above 4, that is,the probability of obtaining sufficient water to irrigate 4. Multiplied by v4 thisis the value of expected proceeds, so that the left-hand side of (3) represents theexpected return per hectare. Condition (3) thus states simply that the expectedreturn per hectare should equal the expected development cost per hectare,assumed constant here.

This can be explained further with reference to Fig. 1. In the bracket of (1),the first term represents proceeds in years when water is sufficient for the wholearea. Choosing a larger 4 means more revenue in each such year, but this ismultiplied by a lower probability (smaller tail). The second term of the bracketrepresents revenue in years when water is insufficient for the full acreage to beplanted. A larger 4, say 42 instead of 4, in the figure, permits an additional por-tion of the water distribution (shaded) to be used. However, the possibility ofadditional revenue from being able to take advantage of greater water supply insome years has to be balanced against an increment, iNKa ¿4, in capital costs.

IV

Once the irrigation system has been constructed, in a year in which watersupply is low in relation to the capacity of the system, the water deficiency canin principle be met either

at the 'extensive margin', by reducing the area cultivated while main-taining an optimum rate of water application, implying a reduction inoutput proportional to any reduction in water availability; orat the 'intensive margin', by a lower rate of water application on a givenarea, resulting in lower yields per hectare; orby some combination of (A) and (B).

This last is the most likely response, but does not involve any different theoreticalfeatures from the other two, and can be neglected for expositional purposes.'3

12 Since

. { f p(q) dq}= f (q) dqp()and

{ f qpq13 Reducing the area irrigated at the extensive margin will not be a straightforward matter,

since shortage of water will only become apparent in the course of the growing season and willnot be known ex ante. For this reason a simultaneous 'retreat' along both extensive and in-tensive margins is likely as the farmer or farm manager attempts to assess the information

26 BULLETIN

In the case of (B), variable costs, and therefore total costs, will be the same irres-pective of actual water availability in any year. In the case of (A), fixed costswill be as for the planned area, with variable costs depending on the actual areacultivated each year. It will be seen that the illustration in Section III referredto case (A) but with zero variable costs.

We can now establish the conditions for the validity of the one-in-five ruletaking into account the possible responses to a water deficiency set out above. Acomparison will be made of 4 q20, as for the one-in-five rule, and 4 = q50, a plannedarea based more adventurously on a fifty-per-cent year calculation.

Model A: Extensive margin.With 4=q20, water would be sufficient for the full area four years out of five(yielding revenue of 4vq20). In one year out of five water supply would be deficient,though not necessarily zero: for simplicity we may assume a quantity half waybetween the optimal level and zero (equivalent to, out of two years, one at zeroand one at the high level), yielding a revenue of -vq20. Total revenue 'every' fiveyears would be vq20, that is to say, the expected value of proceeds would be0.9v q20 per annum. If variable costs are assumed to be zero, and ignoring deprecia-tion, then total costs would be aq20 and 'profits' over, say, a 10-year period wouldbe (9va)q20.

If 4=q50, then water would be sufficient in 5 years out of 10, yielding out of a10-year period revenue of 5vq50. In the other five years, assuming a simple sym-metry of the distribution, water can be taken as adequate to irrigate areas equalto zero, -q50, jq50, q50, and totalling 2vq50. Total revenue in any 10-yearperiod would then equal 7vq50, or 0.7vq50 per annum. Total costs would be aq50for this larger planned area, and 'profits' over a 10-year period (7v - a)q50.

The question is, in what circumstances is q20 the better choice, and when isq50 the better choice? Also, is there any reason why q20 should as a general rulebe the best choice? The planned area q50 will be bigger than q20 by an amountwhich depends on the skewness of the water supply distribution: in general wecan say q50=kq20 where k is a constant determined by the degree of kurtosis. Ifwe take k = 3 for illustrative purposes,14 then profits for 4 = q50 will be 3(7v - a)q20and the difference in profits over the alternative choice will be

¿=3(7va)q20(9va)q20=(12v-2a)q20

From this we can see that will be greater or less than zero according as

available to him at successive points in time. Some indication that an extensive margin doesexist is given in the following statement in a leading text on rice cultivation in the Tropics:'The hazards of drought and flood in paddy cultivation in Asia are very great and the annualloss of crop attributable to these causes is considerable, although frequently obscured bythe current practice of basing yields per acre on the area harvested rather than on the areaplanted.' D. H. Grist Rice, 4th Edition, London, 1965. p. 77.

14 From the data obtained for a number of rivers in the Usangu Plains it is clear that a valuefor k of about 2 or 3 is likely.

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 27

6va or (v/a) -k.'5 For general values of k, z would be positive or negativeaccording as

V>,. k-1a<7k-9

Since z may be positive or negative q20, the one-in-five solution, is not invariablyoptimal, and either q or q50, or some other point, may be the optimal plannedcapacity. The outcome depends on the relative values of y, the value of outputper hectare, and a, the land development costs per hectare. The greater y in rela-tion to a, the larger will be the optimum capacity. With a high value of outputper hectare, and/or low land development costs, it would be worth incurring thecosts of establishing a large capacity for the sake of the good years, even thoughthe larger the developed area the greater the frequency of years in which waterdeficiency would make irrigation of the whole area impossible.

Model B: Intensive margin, i.e. variable levels of water application.In considering response at the intensive margin we shall use rounded figures whichare derived from data for the Chimala River in the Usangu Plains of Tanzania.January is the constraining month which determines the maximum area irrigable,and the water requirement per hectare is taken as 2.4 litre/secs. The one-in-fiveyear river flow is 1600 litre/secs, giving a maximum irrigable area of 667 ha. Themedian year flow is 3200 litre/secs, yielding a maximum area of 1333 ha. Thereforek = 2.

The first need is to consider the relation between levels of water applicationand the yield per hectare. The optimum level of water application for rice inmany parts of the world appears to be 6 acre/feet for a season,16 but data aredifficult to find regarding the precise response function of yields to suboptimallevels of application. It will suffice, however, to deal with this in percentageterms on a rather hypothetical basis. Information suggests that the relationshipis non-linear with, say, a 25 per cent fall in the level of application producing a20 per cent fall in output per hectare, and a 50 per cent fall in application yieldinga 75 per cent fall in output per hectare. These figures are implied in the responsefunction depicted in Fig. 2 which is drawn to approximate closely the shape of asimilar function used for expository purposes in the standard irrigation text men-tioned earlier.'7

To simplify matters we shall in this case assume a normal distribution forwater flows.'8 Since the 20-percentile q20= 1600 and the median q,0 =3200, this

15 In practice if a smaller area were cultivated in years of water deficiency, there would bea saving in variable costs even though capital costs would be unaffected. This does nothowever, make any substantial difference to the argument, and requires only minor modifica-tion of the calculations. If these are re-worked with expected net revenue per hectare equalto 0.9 (vm) and 0.7 (vni) respectively, where m=variable costs, we obtain the result¿0 according as (vm)/a*.

16 See D. H. Grist, Rice, 4th ed., London, 1965, p. 34.17 Kuiper, Op. cit., Fig. 7.11, p. 122.18 It is convenient to use a normal distribution because the percentile values can be taken

from the standard tables, and for this part of the argument the kurtosis of the distribution isnot relevant.

28

100

wou

80Bo-

60

40

owo'o 20Cwo

BULLETIN

I I I I I r

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Litre-secs per hectare

Fig. 2. Response function for yields in relation to level of water application.

implies a standard deviation of 1901 litre/secs. Fig. 3 shows a normal distributionwhich has been subdivided into five sections of 20 per cent.

Out of any 10 years we could expect two years' flows to fall within each ofthe five subdivisions, and these can be assumed without any significant loss of

Fig. 3. A normal distribution of water flows.

generality to be the 5-, 15-, 25-, 35-, 45-, 55,- 65-, 75-, 85-, and 95-percentiles.Given a 1901, the values of the percentiles can be deduced as in Table 1.

TABLE 1

Percen file Values of River FlowPercenlileValue (litre/secs)Percentage of time value

is exceeded

X5 X15 X25 X35 X45 X55 X65 X.5 X85 X9572 1230 1918 2468 2962 3438 3972 4482 5170 6328

95 85 75 65 55 45 35 25 15 5

20% 20% 20% 20% 20%

I I I I I I

O X5 X15 X25 X35 X45IX55 X65 X75 X85 X95 X (River flows

1,600 3,200 in litre/secs)

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 29

It follows that if 4 = q20, water can be expected to suffice for optimal applica-tion in eight years out of ten. In the other two years water availability can betaken to be 1230 litre/secs and 72 litre/secs. Spread over a planned area of667 ha. this implies applications of

1230 721.85 and =0.11 htre/secs per hectare.

Reading from Fig. 2 these imply output equal to 83 per cent of the maximumproduction value per hectare in the first case, and nil production in the other.Expected total revenue over a 10-year period would therefore be 8.83 times thevalue of output in a non-deficient year, that is, 8.83vq20.

Since in this model the full area is always cultivated, variable costs will be thesame every year irrespective of the water available and will in effect become fixedso that total costs

TC = (a + m)4

where again a is land development costs per hectare and m cultivation costs perhectare. Thus for 4= q20, 10-year profits will equal

P = [8.83v - (a + m)]q20.

If 4 = q50 =1333 ha., water available will be adequate for the full area in fiveyears out of 10. Expected flows in the other 5 years can be taken as 72, 1230,1918, 2468 and 2962 litre/secs. Over 1333 ha. these give applications of 0.05,0.92, 1.44, 1.85, and 2.22 litre/secs per ha. From Fig. 2, these yield respectivelynil, nil, 54, 83, and 97 per cent of maximum output per ha. Thus expected revenueover 10 years will equal

(5 + O + O + 0.54 + 0.83 + 0.97) vq50 = 7.34vq50

and total profitsP = [7.34v - (a + m)]q50

Thus the extra profits from the larger 4 will equal= [7.34v - (a + n)]q50 [8.83 (a + m)]q20.

Here k=2 and q50 =2q20, so that

= [5.85v - (a + m)]q20and

V Iaccording as a+m<5.85

This is an almost identical result to that for Model A with k=2, where (v/a) isequal to -.

These results need to be qualified. It will be recalled that irrigable areas werecalculated on the assumption of a given water requirement in the particular con-straining month, with year-to-year fluctuations of rainfall and water flow beingrelated to that figure. There will, however, in practice be considerable intra-seasonal variability of rainfall as well as inter-seasonal variation. This leadsespecially to problems of timing which in paddy cultivation can be crucial, althoughthere is some flexibility, so that lags of one or two weeks within a regular pattern

30 BULLETIN

would not affect our analysis. More seriously, intra-year fluctuations could alterthe identity of the constraining month. This is related to the point made earlier(p. 25, n. 13) regarding the limited information available to the farmer ex anteabout the supply of water, involving a need constantly to re-interpret theinformation as it emerges regarding water supply at successive points intime. These factors will clearly complicate the decision regarding 4, the plannedirrigated area, but given that a choice must be made in principle, in the face ofuncertainty, the foregoing analysis retains its validity.

V

In the calculations above no discounting was applied over the hypothetical10-year period employed. Strictly speaking we should, in comparing the twoplanned hectarages, discount the expected values of revenue over 10 years toobtain a figure for net present value. Using Model A, and assuming land develop-ment costs incurred in Year O yield potential revenues in Years 1 to 10, then for4 = q20,

0.9vq20NPVq20= (l+r) aq20

while for 4 = q50,

NPVq50=

O.7vqaq90

Clearly, which is the bigger of the two depends upon the same elements as in theundiscounted case: discounting does not affect the result, since expected values areequal in all years, and therefore the one-in-five rule does not necessarily yield theoptimal planned hectarage and its application could be at odds with the ordinarynet present value criterion for appraising investment projects.

VI

The above argument has assumed throughout the maximization of expectedvalues on a probabilistic basis. This can be taken as representing risk-neutralbehaviour since the extent of year-to-year fluctuations does not affect the selectionof 4, given total (in the last calculation discounted) expected values. As indicatedin Section I, the risks being assumed are risks of water deficiency, and thus failureto obtain the 'target' income each year,19 risks which increase the greater the sizeof the planned irrigable area. This is the same as the conventional trade-off inrisk theory: a relatively small planned area will lead to the same level of revenuein most years, while a large 4 will produce very much wider fluctuations in revenue,so that a higher mean value of net earnings must be set against greater uncertaintyas measured, for instance, by the standard deviation.

1 An additional risk which exists if there is a series of dry years is of salinity in the surfacesoil owing to lack of sufficient water application to permit leaching (downward percolation ofwater towards the ground water table) to keep the soil free of a salt build-up. See Kuiper,o. cil., p. 123.

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 31

It is perhaps worth setting out more explicitly the nature of the increasedrisk resulting from a larger planned area. With a larger area the amplitude ofrevenue fluctuations would be increased by higher revenues in years of abundantwater, but the revenues in the other years would be no smaller than they wouldbe with a smaller planned area: the revenue peaks are higher, but the troughs arenot lower.20 This follows from the fact that when water is insufficient for thewhole planned area, the area actually planted could in principle be reduced inproportion with the reduction in water supply, giving the same revenue as wouldhave been obtained from a smaller planned area. Revenue is the same, thoughcosts of course are higher.

The problem arises for a single large farm or scheme, of course, only when thescheme is large compared to the water available: a relatively small scheme wouldbear little or no risk of water deficiency. For peasant producers it arises when thetotal area under peasant irrigation reaches a significant size in relation to water

Fig. 4. Risk level as a function of planned irrigated area.

supply. An increment in planned irrigable area represents in this case the decisionof a marginal peasant producer to 'enter the industry' by establishing an addi-tional farm of, say, one hectare in the face of the marginal level of risk of waterdeficiency. Assuming equitable distribution of water, the entry of a marginalproducer would increase risk equally for all existing producers.

Figure 4 illustrates a hypothetical risk function i=F(), giving the risk of

20 The situation as illustrated in the following diagram:

R'

r AC'RAC

O Time

Increasing the developed land area increases average costs from AC to AC' and revenue fromRR to R'R', with R'R' nowhere below RR. Although this may increase profits on average,greater instability is attached to earnings, with occasional losses.

32 BULLETIN

water deficiency (risk level) as a function of the planned irrigated area.2' Thefunction is directly derivable from Fig. 1, the risk of water deficiency, r, beingequal in that figure to

I p(q)dq.Jo

It will be observed that, in Fig. 4, lT =0.2 for 4= q20, lT=0.5 for 4=q50, etc.In discussing response to risk, particularly as between large farms and peasant

producers, a careful distinction must be made. Given the values of u and a inModel A a 'risk-neutral' solution will yield a particular value of 4( = 4, say) asoptimal: that which maximizes the net present value of expected net proceedsirrespective of the magnitude of the year-to-year fluctuations. Preference formore stable income or for avoiding losses in occasional years even at the cost of alower average income implies a degree of risk-aversion and would lead to a choiceof 4 below 4rn,. Distinct from this situation is that in which the selection of alarger or a smaller 4, the selection that is of more or less risk of water deficiency,is the consequence of differences in the value of u and/or a. With the same wateravailability, a larger y and/or a smaller a will result in a larger q being selected,without there being any difference in risk aversion in the previous sense, butimplying a willingness to accept wider fluctuations of income and possibility oflosses in particular years.22

For a large mechanized farm a will be relatively high, and will produce acautious policy, even with risk-neutrality. A degree of risk aversion in the firstsense would induce the decision-maker to select a still smaller 4, that is a smaller4 than that indicated by maximum NP V.23 The one-in-five rule incorporatedinto the technical design of irrigation schemes is in fact justified by engineers asa necessary 'factor of safety' in the face of risk. It needs to be re-emphasized,however, that such a factor should be determined, not on technical, but on eco-nomic and social grounds. The financial position of a large mechanized farm, forinstance, may not permit it to accept a possibility of serious water deficiency,involving financial losses, more than very occasionally. Even if the farm is State-run (as is likely to be the case in Tanzania), with ready access to credit, the managerwill be under personal pressure to demonstrate success and may not wish to runthe risk of serious setbacks, even if these are likely to be compensated in futureyears.24'25

21 See Rees, op. cil., p. 146, Fig. 9.2, for a similar risk function used in his analysis ofelectricity supply.

22 These need not be financial losses, but losses calculated using imputed values for familylabour for construction or cultivation.

23 In the example elaborated by Kuiper, o. cit., it is worth noting that no mention is madeof the assumptions made in this regard, which amount to risk neutrality in our sense. Wemay note also that the example is cast entirely in terms of the intensive margin, withoutreference to the extensive margin; and that costs are taken as $80 per acre without distinctionas to fixed and variable costs.

24 Against this, however, it should be noted that if a government is assessing an irrigationscheme (whether for large or small-scale production) in social cost-benefit terms, it shouldin determining planned area apply a social rate of discount which is risk-neutral and reflects'the law of large numbers', thus selecting

25 With positive variable costs in Model A, (o - m) /a , and we may note that the higherare variable costs, n, the lower the numerator y - m and the more cautious the policy likely

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 33

It is different for peasants and for planned schemes of village irrigation withpeasant production. Here it is a question of the risks cultivators are prepared toassume and which it is proper to impose on them. If peasant farmers are poor,with low incomes and food supply, they are likely to attach a high value to extraoutput,26 even if this is available only in better years, particularly if developmentcosts comprise mainly family labour, possibly off-peak labour at that, and aretherefore relatively low. Stability of income could mean chronic near-starvation;with an irrigable capacity to take advantage of years of abundant water, they areable to eat properly in at least some years. A high degree of caution, such as isrepresented by the one-in-five year rule, may be appropriate in heavily capital-intensive irrigation schemes, with loans to be serviced and repaid, and with otherheavy costs. But in a scheme to provide the irrigation infrastructure for peasantsit would be quite another matter to deny cultivators the full benefits of the yearsof higher river flows so as to reduce the risk of their doing relatively worse in otheryears. The producers would not, in the years of low water, make use of the fullarea in which investment in land development and capital works had been made.But it would not be a beneficial form of stabilisation to cut off the potential peaksof prosperity, depriving the cultivator of larger crops in the years of plentifulwater so that he did not have so far to fall in the years when water was scarcer.After all, a very large part of agriculture in developing countries would not passa test of viability based on the worst year in five.

Interestingly enough, there is some evidence that peasant cultivators, in theUsangu Plains at least, are prepared to accept high risk of wasted effort in someyears to reap the rewards of the good years. Comparison of the total areas actuallyirrigated by peasants along a few rivers indicated that these were larger than theareas which, according to formal measures of water availability, could be properlyirrigated by the run-of-the-river, other than in exceptionally good years. It shouldbe admitted that the water flow data on which this conclusion is based are shaky,as are the estimates of cultivated area, and that, as pointed out elsewhere, ßartof the difference may have an alternative explanation.27 Nor does the selectionof a riskier option by peasants than by a large farm necessarily mean that peasantproducers have a lower degree of risk aversion, which would be measured by adifference between and It is simply that if it costs them less to prepare anadditional hectare for possible cultivation it may be worth their while doing so,even if the additional production is erratic, compared with a large, mechanizedfarm: and similarly if a higher valuation is placed on extra output. It is neverthe-

to be adopted: a higher sa has the same effect as a lower y. This provides a further possibleexplanation of a more conservative policy by large farms using mechanized methods of cul-tivation, with relatively high variable costs.

26 It may appear odd that output destined in part for 'subsistence' is given a higher value,compared with marketed output from a large mechanized farm: national income accountsgenerally attach a low value to subsistence output. This is, however, a weakness in nationalaccounts. It must be noted, on the other hand, that in y =p .y a high value for p in the caseof peasants may be offset by a lower yield, y.

27 The maximum irrigable area is calculated on the basis of water availability in a singleconstraining month, usually January for peasants in Usangu. By using 'left over' water inFebruary with later planting, however, peasants who would otherwise lack water, could addto the area under irrigation. See A. Haziewood and I. Livingstone, op. cit.

34 BULLETIN

less of interest that peasant producers might, in the face of the same objectiveconditions of water availability, select a larger irrigable area associated with morewidely fluctuating incomes, taking risks because they are poor, in apparent con-trast with the risk-reducing 'optimizing peasant'.28

Such a behaviour pattern is not, in fact, necessarily inconsistent with the usualpicture of the risk minimizing peasant.29 A cautious rather than profit maxim-izing policy results in the latter case from the danger of drought leading to dis-astrous loss of income, and possibly of land, and even to starvation.30 In thesituation discussed here, failure of water supplies may produce starvation, butpeasant producers cannot avoid this by developing a smaller area. Indeed, asdiscussed presently, if possibilities of storing the crop exist, a larger area couldproduce extra food in good years to be used as a reserve against subsequent cropfailure.

The question of full utilization of capital works arises sharply where the authori-ties supply certain basic irrigation infrastructure in support of village irrigationby peasant producers: it may be thought that peasants give too little weight tothe risk of years of little water, and that it would be wasteful to provide an infra-structure which would be fully used only intermittently. However, it must bekept in mind that an intermittent utilization of capital equipment is not in itselfa sign of excessive investment: the economic rate of return may still be high enoughto justify the investment, even though the capital equipment is not fully utilizedat all times. (How profligate is the community's expenditure on beds if the factthat most of them are unoccupied all day is a sign of malinvestment!) The eco-nomic return from the large area it is possible to irrigate in years of high water,years of which advantage could not be taken if the irrigation system had beenconstructed on a smaller scale, may well justify investment in the larger system,even though it is fully utilised only in some years.

We defined our investment decisions problem as a 'capacity margin' problemof the public utility type. As pointed out, by Rees, this in turn is

'... essentially an inventory problem. In general a firm would confront theproblem of uncertain demand by determining a capacity level and a level of

28 See M. Lipton, 'The Theory of the Optimising Peasant', Journal of Development Studies,1969.

29 Against this, it was a feature of the area (possibly true of some other parts of Tanzania)that for reasons of taste peasant producers insisted on growing maize as a rain-fed crop, ratherthan millet (sorghum), despite the marginality of the area for maize and the much higher risksattached to its cultivation, specifically risks of the disaster type. In certain other parts ofEast Africa, millet is quite acceptable as a food crop. Generalizations regarding the riskaversion of peasant producers clearly need to be treated with caution.

Thus the decision to accept greater fluctuation of income and thus uncertainty bydeveloping additional irrigable area would not be affected by the 'survival algorithm' asoutlined by Weeks. In the latter's exposition the 'ability' to take risks depends on the farmer'sdistance from some 'disaster level': but here the farmer's situation in bad years is not affected,according to our assumptions. See J. Weeks, 'Uncertainty, Risk, and Wealth and IncomeDistribution in Peasant Agriculture', Journal of Development Studies, 1970-71.

Compare the distinction made in a recent book by Roumasset, in which it is argued that inrespect of peasant producers risk should be measured by the probability of disaster, not by thevariance of income. Clearly both types of uncertainty are important. J. A. Roumasset,Rice and Risk: Decision Making Among Low-Income Farmers, North Holland Publishing Co.,1976.

RISK IN IRRIGATION PROJECTS IN DEVELOPING COUNTRIES 35

inventory of finished goods, where the latter would absorb the uncertain-ties.'3'

The problem in the case of electricity supply is that. electricity can only be "stored" in the form of the capital equipment

required to generate it, so that the inventory must consist of capacity, andnot output. In general, this is a more expensive way of holding inventorythan is the type of storage facility required for most goods'

In the case of irrigation, in contrast, two types of storage, in addition to capa-city, are possible: storage of water and storage of the Crop. Storage of waterthrough the construction of dams is a ready solution by irrigation engineers toreduce risk of water deficiency and maintain a high level of utilization of preparedcapacity: construction of dams is, however, expensive and in many cases pro-hibitively so. However, storage facilities for the crop may also make it possibleto realise the benefits of a larger irrigation system and can be considered a partialsubstitute for storage of water. Thus if the capacity of the irrigation system per-mits it, the benefits of the best years can be passed on to later, less fortunate yearsby storing paddy. This possibility is not one which is likely to be automaticallyconsidered if a technical rule-of-thumb, such as the one-in-five rule is applied,and further weakens the case for using it.

School of Development Studies, University of East Anglia.Institute of Economics and Statistics, Oxford.

31 Rees, op. ci&, p. 147.