DIVISION OF AGRICULTURAL SCIENCES UNIVERSITY OF CALIFORNIA 7/ 7 ^ '^^ RISK AND UNCERTAINTY UCD LIBRARY m MAY ^8 1973 'N LEnUCE PRODUCTION ^N2 5RQrD IN SALINAS VALLEY, CALIFORNIA C. V. Moore and J. Herbert Snyder JAN 31 1959 CALIFORNIA AGRICULTURAL EXPERIMENT STATION GIANNINI FOUNDATION OF AGRICULTURAL ECONOMICS Giannini Foundation Research Report No. 300 January 1969 ACKNOWLEDGMENTS The authors are indebted to a large number of individuals for their inter- est, time, and cooperation, without which this report could not have been com- pleted. In Monterey County, J. W. Huffman, Director, University of California Agricultural Extension Service; Peter Braschini; Dean Pryor; Karl Christierson; the late Ted Gotlieb; and the many growers, shippers, and managers of service firms supplied information and insight helpful in this study. Graduate stu- dents of the Department of Agricultural Economics, University of California, Davis; R. G. Bressler III; N. Pulchritudof f ; Arlo Biere; and C. D. Ditwiler worked on various phases of this project and contributed to it in many ways. Financial support for the research projects covering this study has been re- ceived from the Water Resources Center of the University of California, the Farm Production Economics Division, ERS , USDA, and Regional Research Project W-81. -i- HIGHLIGHTS OF THIS REPORT I. Selection of a cropping plan in the high-risk vegetables industry is analogous to selecting an investment portfolio. Growers must balance the pro- portion of crops to be grown and economic returns from different marketing ar- rangements against these alternatives—limited capital availability, individ- ual risk aversion or propensity for uncertain outcomes, and the desire for capital accumulation over time. II. Growers with limited capital supplies can be precluded from high- risk—high-gain marketing arrangements — but can gain entry to the vegetable industry through contractual arrangements with shipping firms who accept part of the risk of income variability for a share of the crop. III. Lettuce growing under a flat fee per acre contract may be less re- warding to a grower than other alternatives but it is an important source of operating capital and a low risk venture for farmers with little equity capi- tal of their own. IV. Growers with very limited capital supplies must use crop plans and marketing arrangements with high total income variance in order to maximize long-term expected income even if it requires foregoing some income in the current period. V. As growers accumulate additional operating capital, less current in- come need be sacrificed in order to achieve a satisfactory rate of financial growth and probability of failure is reduced. VI. Growers in the Salinas Valley appear to have a subjective measure of risk in different crops and diversify their crop programs and select contrac- tual arrangements accordingly. -ii- TABLE OF CONTENTS ^ Page INTRODUCTION . 1 The Region 1 The Vegetable Industry 3 Competition from Other Producing Areas 6 Credit Sources 6 The Problem 8 Theoretical Concepts 9 Decision Making and Uncertainty 12 Measurement of Variability 13 Transfer of Risk through Contractual Arrangements 15 Empirical Analysis of Alternative Contractual Arrangements for Lettuce 17 Open-Price Contract with a Guaranteed Minimum Return 18 Open-Price Contract with No Minimum Guarantee 25 Contract Selection under Price Uncertainty 27 Effect of Operating Capital Supply on Possible Contractual Arrangements 34 Planning under Conditions of Uncertainty 41 Estimating Transition Probabilities 43 ■ Results of the Djmamlc Programming Model 48 Implications for Growth 53 Limitations to the Dynamic Programming Model 53 Analysis of Actual Adjustments to Risk and Uncertainty 54 -111- LIST OF TABLES Table Page 1 Monthly and Annual Temperatures, Salinas Valley 2 2 Characteristics of Sample Farms by Total Crop Acres, Salinas Valley, 1962 5 3 Production Required for Growers to Break Even for Three Selling Prices, Varioxis Contractual Arrangements, Salinas Valley, 1962 19 4 Production Required for Shippers to Break Even for Three Selling Prices, Various Contractual Arrangements, Salinas Valley, 1962 20 5 Frequencies of Returns Per Acre After Packing Charges, 1961-1966 31 6 Net Income Per Acre to Growers for Various Lettuce Contracts 32 7 Net Income Per Acre to Shippers for Various Lettuce Contracts 33 8 Production Alternatives by State, 240-Acre Renter-Grower Farm 44 9 Income, Family Withdrawals, Loan Payments, Marginal Tax Rates, and Standard Deviation about Net Income, by State and Alternative, Salinas Valley ... 46 10 Transition Probabilities by State 47 11 Optimal Policy by State 51 Al Correlation Coefficients, Net Income 59 -iv- LIST OF FIGURES Figure Page 1 Hydrographlc Areas, Salinas Valley 4 2 Average Weekly Carload Shipments of Lettuce, 1949-1954 and 1963-1966 7 3 Variance of Net Income 11 4 Breakeven Chart, Contract No. 1 22 5 Breakeven Chart, Contract No. 2 23 6 Breakeven Chart, Contract No. 3 26 7 Distribution of Net Income Per Acre After Packing, 265 Blocks of Lettuce, Salinas Valley, 1961-1966 29 8 Optimal Cropping Program as Supply of Operating Capital is Varied, Owner-Grower Farm, Salinas Valley ... 38 9 Net Returns Above Variable Expenses as Operating Capital Is Varied, Renter-Grower Farm, Salinas Valley. . . 40 10 Schematic Diagram of Transition Matrix 41 11 Management Income and Standard Deviation of Income by Supply of Operating Capital (States), Salinas Valley ... 49 12 Gross Income and Standard Deviation Per Acre, Salinas Valley 56 -v- RISK AND UNCERTAINTY IN LETTUCE PRODUCTION IN SALINAS VALLEY, CALIFORNIA by C. V. Moore* and J. Herbert Snyder** INTRODUCTION The Salinas Valley of California has been described as the "Salad Bowl" of the United States. It is the largest producing area for summer head lettuce in the country. More than 20 other vegetable crops for fresh shipment or processing are also produced in the area. The average gross value^^ of production exceeds $100,000,000 per year — about half of which comes from lettuce. The Region The Salinas Valley lies within the central coast area of California ap- proximately 150 miles south of San Francisco. The Valley, with large areas of highly productive alluvial soil, is bounded by two mountain ranges — the Santa Lucia Range on the west and the Gabilan Range on the east. The northern end of the Valley is open to the sea at Monterey Bay. This unique geophysical setting provides the Valley with an almost ideal climate for summer vegetable production. During the summer, the prevailing winds are from the ocean pro- ducing cool weather and fog. From late fall through January, the prevailing winds reverse themselves and blow from the south providing the Valley with low mean annual temperatures with a small diurnal range especially in that half of the Valley nearest the ocean (Table 1) . The southern end of the Val- ley, represented by King City in Table 1, is too far removed from the ocean to benefit greatly from the temperature moderating influence of the ocean. *Agricultural Economist, Farm Production Economics Division, Economic Re- search Service, U.S. Department of Agriculture, stationed at University of California, Davis. **Professor of Agricultural Economics and Agricultural Economist in the Ex- periment Station and on the Giannini Foundation, University of California, Davis . 11 Monterey County, Agricultural Commissioner, Monterey County Annual Crop Report y Salinas: annual reports 1953-1966. TABLE 1 Monthly and Annual Temperatures, Salinas Valley Salinas Soledad-' King City Month Mean maximum Mean Mean minimum Mean Mean maximum Mean | Mean minimum January 60.7 49.7 38.6 49.2 60.2 47.5 34 .8 February 62.7 51.7 40.6 51.8 54.2 45.9 37 .6 March 65.4 53.6 41.8 55.3 68.8 54.0 39 .2 April 67.1 55.7 44.2 58.8 74.1 57.9 41 .7 May 69.1 58.5 47.9 62.8 78.5 62.0 45 .5 June 70.9 60.7 50.5 66.4 82.5 65.4 48 .3 July 71.5 62.0 52.4 66.6 85.2 68.0 50 .8 Augus t 71. o 62. 3 52.8 66.2 84.1 67.1 50 .1 September 75.7 63.6 51.5 66.1 85.8 67.1 48 .4 October 74.2 60.9 47.5 58.8 80.1 61.9 43 .7 November 70.0 56.1 42.2 54.5 71.8 54.6 37 .4 December 63.2 51.6 39.9 50.1 64.3 49.9 35 .5 Annual 68.5 57.2 45.9 58.9 74.2 58.5 42 .8 aj Mean maximum and mean minimum data not available. Source: U.S. Weather Bureau, Climate of Seen Benito County y San Francisco: Published by San Benito County Chamber of Commerce, 1963. -2- Warmer spring temperatures in the King City area allow an early spring crop of lettuce to be grown. However, hot dry winds during the summer preclude a mldseason lettuce crop and field crops predominate In this area. Rainfall, in the vegetable-producing portion of the Valley, decreases with distance away from the ocean. Average annual rainfall at Castrovllle, near the mouth of the Valley, Is 17.8 Inches; Salinas, 14.1; Gonzales, 12.3; Soledad, 10.2; and King City, 10.5 Inches (Figure 1). Over 90 percent of the rainfall occurs In the six-month period from November through April. Precipitation Is Inadequate for Intense crop production; therefore, all of the vegetable-producing lands have Irrigation available. Virtually all of the Irrigation water Is pumped from ground water, which Is recharged from the flow of the Salinas River. The Valley floor has been divided into five hydro- graphic areas based on the source and conditions of recharge to the under- ground basin (Figure 1) . Pumping costs vary directly with distance from the river, except for the pressure area which is characterized by a confined aqui- fer overlaid by a thick impervious clay layer. The Vegetable Industry The vegetable Industry in the Salinas Valley can be broadly characterized as a large number of small producers growing vegetables under contract to shippers for the fresh market and/or to processors for the frozen or canned market. These smaller producers grow an estimated 50 percent of the head let- tuce. The remaining 50 percent is grown, packed, and shipped by a small num- ber of large Integrated firms who contract for a nominal amount of additional lettuce acreage. Three cooperative marketing firms in the Valley deal primar- ily in lettuce, but also handle a limited amount of other vegetables to com- plement their sales and to utilize excess capacity in the off season. In recent years the membership in each of these cooperatives has remained at about ten growers. Not all farms in the Valley produce lettuce. Of the 91 farmers inter- viewed, 82 grew some vegetables and only 48 grew lettuce in 1962 (Table 2). Except for the largest size group, the intensity of cultivation decreases with farm size. Most of the farms in the largest size group operated their own packing sheds and coolers and the remaining growers either grew under contract or were members of one of the cooperative associations. -3- -4- TABLE 2 Characteristics of Sample Farms by Total Crop Acres, Salinas Valley, 1962 Total farms Lettuce farms Vegetable farms Total crop acres Number Percent of crop acres in vegetables Number Percent of crop acres in vegetables Number Percent of crop acres in vegetables 0 - 79 6 97.3 3 94.3 6 97.3 80 - 159 20 50.0 7 66.0 16 62.0 160 - 319 30 59.8 17 67.6 29 61.8 320 - 639 23 49.7 15 57.0 19 49.7 640 - 1,279 7 36.7 3 40.7 7 36.7 More than 1,279 5 58.0 3 79.3 5 70.0 TOTAL 91 48 82 Source: Survey data from interviews. Competition from Other Producing Areas Competition from other lettuce-producing areas has shortened the market- ing season for Salinas Valley growers and increased the degree of risk asso- ciated with vegetable-crop production. After World War II, a large increase in lettuce production in the Salt River Valley of Arizona, in the early spring and late fall, began to crowd Salinas producers out of this market. Prior to this period, the lettuce harvesting season ran from early March to well into December in Salinas. However, by the early 1950 's this season had been short- ened by about three weeks in the fall and two weeks in the spring. This trend has continued as shown by Figure 2 with about a one-week loss in the spring and an additional loss of three weeks in the fall market. The loss in the fall market since the early 1950's is primarily due to new areas of production opening up in the San Joaquin Valley of California. The San Francisco Bay area and the Santa Maria district of California compete directly with the Salinas Valley throughout the summer season. However, there have been no sig- nificant shifts of production between these three areas in recent years. Outside of the Pacific Southwest, further market competition originates from Texas, Colorado, and small areas around large metropolitan areas in the Midwest and the eastern seaboard. These areas market fresh vegetables during the summer months when Salinas Valley production is at its seasonal peak. Shortening of the harvest season in Salinas Valley has reduced the opportuni- ties for growers to "average out" price fluctuations over the season, reduced opportunities for double cropping, and forced growers to look to other vegeta- bles to diversify their cropping programs. The alternative vegetable crops are generally processed rather than shipped fresh. Thus, lower expected in- comes per acre from these crops tend to decrease expected total farm income. Credit Sources Lack of credit can be an important limiting factor to production planning particularly for vegetable production that may require from $250 to $600 per acre to bring a crop up to harvest. Commercial credit sources are reluctant to make large advances in the form of production loans without additional se- curity. A typical credit institution requirement is that at least half of a farmer's acreage be in field crops — characterized by relatively stable prices — as security against depressed vegetable prices. One bank calculates the loan -6- FIGURE 2 Average Weekly Carload Shipments of Lettuce, 1949-1954 and 1963-1966 1100 - / 1000 - / ■ -7- repajrment capacity on the basis of stable crops grown and does not include the risky crops of the grower. Another bank has a policy of not advancing direct loans on lettuce, celery, carrots, or cauliflower independently of the other crops grown unless the tangible assets of the grower or the balanced cropping program of the entire farm, including the stable crops, will meet the repay- ment capacity requirements. The commercial lending institutions prefer to make advances to a shipper based on physical assets. In tuim, he channels those funds to smaller growers based on their contractual agreements. This type of loan places the responsi- bility for repayment directly on the shipper in case a crop must be abandoned due to either economic reasons or disease problems. If the crop must be aban- doned due to economic reasons, i.e., f.o.b. price falls below the harvest (packing and shipping) cost, this decision is the responsibility of the ship- per or the management of the cooperative. The Problem Growers are faced with a large number of alternative crop enterprises each with different expected income levels, income variability, resource re- quirements, and alternative marketing institutions. Selection of a cropping program to maximize long-term expected income under these conditions is a formidable task. The objectives of this study were designed to develop guide lines for evaluating crop alternatives in this complex situation. They are: 1. To analyze possible and existing contractual arrangements between lettuce growers and shippers, under different market price levels, with respect to the incidence of risk. 2. To compare selected contracts for lettuce within a statistical decision theory framework to ascertain the expected net returns to growers and shippers. 3. To evaluate the grower adjustments with respect to efficiency criteria relating farm gross incomes to income variability. 4. To estimate the optimum cropping pattern for a vegetable farm for varying levels of operating capital supplies when market prices are assumed to be known and fixed. 5. To develop cropping programs that maximize long-term expected fam incomes under conditions of risk for various levels of operating capital availability. -8- Theoretical Concepts Individual fanner's adjustments to risk situations depend upon both his psychological makeup and the resources at hand. Selection of a cropping pat- tern for a farm under conditions of risk and uncertainty Is similar In many respects to the problem of selecting an Investment portfolio. Each crop enterprise can be considered as a marketable security such as a share of com- mon stock, a bond, or a deposit certificate in a savings institution. The proportion of a particular crop enterprise to the total crop acres is equiva- lent to the value of any one security to the total value of the portfolio. Two bases may be used in the analysis of either investment portfolios or cropping programs. The first is the expected Income or yield of the security or crop based on historical data. The second is the judgment of security ana- lysts or well-informed farmers as to the future prospects for income from the security or crop. Like most economic phenomena, prices and Incomes of securities or crops tend to move up and down together, i.e., they are highly correlated. However, they do not move together perfectly. Some prices and yields of crops, as well as of securities, have exhibited negative correlations. That is, when one crop is having a good income year, certain other crops in general have a bad year. Investors in securities and farmers in general desire a portfolio or crop program with the highest expected income return. However, this is usu- ally not the portfolio or cropping program with the lowest risk of return. The portfolio or cropping program with the highest likely return may be un- acceptable because of an extremely high degree of risk of income variability. Likewise, the portfolio or cropping program with a very low variability of outcome may have an unacceptably low expected income return. Portfolios and cropping programs can be diversified to take advantage of the correlation between returns. Heady^^ has explained the procedure in de- tail which can be siimmarlzed as follows: Two enterprises, A and B, with In- 2 2 come variance o. and a„ can be combined so that the total variance about the !_/ Heady, E. 0., Economioa of Agriaultural Produation and Resource Use^ New York: Prentice-Hall, Inc., 1952. -9- 2 2 2 mean income is 0^ = + + Zra^o^. This states that the total variance for the portfolio or cropping program is the variance of A plus the variance of B plus twice the correlation coefficient for the two enterprises. If there is a fixed amount of resources (in this case land or money) , then the propor- tion going to each enterprise must be taken into account. (1) a^^ - q^a^^ + (l-q)^ag2 + 2rq (l-q)a^ag Where q is the proportion of the resource used by A and 1-q is the proportion used by B. By taking the first derivative of Equation 1 and setting it equal to zero we can solve Equation 2 for the value of q, the proportion of the re- source used by A that minimizes income variance (Equation 3) . (2) = 2qa/ - 2 (l-q)ag^ + 2r (l-2q)a^ag ^B^ - "^Vb (3) q = ^a' -^b" -^'^Va The proportion of resources that minimizes the variance about the mean also minimizes the standard deviation (square root of the variance). Equations 1, 2, and 3 may be generalized to include any number of enterprises. A portfolio or cropping pattern I, which has the same expected income as portfolio or cropping pattern II and has a lower variability than II, is superior if the objective is to achieve the highest expected immediate gains. Similarly, if I had the same variance as II but a higher expected return, it would still be considered superior. These superior portfolios and cropping programs are termed "efficient," with respect to expected immediate returns. If the criteria for selection of portfolios was only that of maximizing immedi- ate returns, all inefficient portfolios and cropping plans could be eliminated from the analysis. Under certain other criteria this may not always be the case as we will show later. Calculation of an efficiency surface showing all of the "efficient" port- folios or cropping programs can be easily done with available digital com- puters, although the problem assumes quite large dimensions if small changes in the proportions of each security or crop enterprise to the total are con- sidered. Figure 3 shows a hypothetical efficiency surface relating expected returns and the variability of the returns. -10- FIGURE 3 Variance of Net Income Variance of Net Income Point A on the efficiency surface identifies a low risk combination with a small positive Income and a near-zero standard deviation.—^ This point could represent cash deposited in an insured savings account or in goverrunent bonds where the possibility of a loss would be extremely small. Most security advisors and farm management consultants strive for a bal- anced portfolio or cropping progrcim, selecting a combination of likely returns and uncertainty which best suits the needs of the client. This is generally a diversified portfolio or cropping pattern that contains portions of cash, bonds, "blue chip," and "growth stocks" for the security counselor and low- (Govemment price supported) and high-risk perishable crops for the farm manager. This allows the analyst to take advantage of the correlation between returns mentioned above in the discussion of diversification principles. Point C represents a high-risk high-return portfolio or cropping program. The surface is curvilinear and indicates that the small increase in net income obtained from shifting from point B to point C can be achieved only with a large increase in the variability (and therefore chance of a loss) in net in- come. In the analogy of the security counselor, point C would represent a portfolio made up entirely of a single speculative stock. For the farm mana- ger, this would represent a cropping program consisting of planting the entire farm to a high-risk crop such as lettuce. 2 1/ Much of the literature on portfolio analysis uses the variance (o ) or standard deviation squared as the measure of variability. The standard devi- ation is measured in the same units as that of the mean, in this case dollars, and is better understood by the nonprofessional investor. -11- A portfolio or cropping program represented by point D is an example of an inefficient investment in that moving to the left to a point on the effi- ciency surface would enable the investor or grower to obtain the same level of net income at a much lower variability. Similarly, moving vertically to the surface would allow a much higher income with the same degree of variability. There are several possible explanations for a grower or investor operating at point D. First, he may have only a subjective and limited estimate of the variability associated with the past performance of the security or crop. In- complete knowledge of the mean, variance, and correlation coefficients may cause a perfectly rational person to make a mistake in choosing his invest- ments. Second, since the hypothetical efficiency surface is based on the past record of the security or crop, he may have a judgment that the security or crop will not follow its previous pattern, but rather the future prospects are significantly different due to a change in market demand or a new technology becoming available. Third, the security or crop may be new and there would be insufficient information available from which to judge performance allowing for a wide error even to the rational investor. Fourth, the investor himself may be (economically) irrational in his decisions in that he is willing to gamble — and in fact derives a great deal of satisfaction from gambling. This individual may be found rolling for high stakes at the craps table even though he is fully aware that the gambling casino operates at a sizable profit. Decision making by the investor-grower is thus seen to be conditioned by his specific objectives. Even if we eliminate from consideration the irra- tional investor, there remains a wide range of behavioral patterns that would be considered rational based on the objectives of the individual concerned. Decision Making and Uncertainty " The literature on portfolio analysis makes the implicit assumption that any efficient portfolio is available to the potential investor.—^ However, unless purchase of fractional shares is allowed it is obvious that a small in- dividual Investor could not purchase certain portfolios if he only had a small amount of money to Invest. For example, if a diversified portfolio contained ten different securities, each with a value of $500, then an investor with 1/ Howard, R. A., Dynamic Programming and Markov Process, Cambridge: Technology Press of Massachusetts Institute of Technology, 1960; and Marko- witz, H. M. , Portfolio Selection , Efficient Diversification of Investments , New York: John Wiley & Sons, Cowles Foundation Monograph No. 16, 1959. -12- I I only $3,000 to invest would be precluded from this alternative. A farmer may also be limited in the alternative efficient cropping plans from which he is able to choose for two reasons. First, on technical considerations alone, the proportion of a single crop in the efficient cropping plan may be so small as to make it unfeasible to grow small acreages. For example, a grower with only 50 acres of crop land, and a cropping plan containing 50 percent lettuce, 25 percent carrots, 20 percent celery, and 5 percent potatoes would find planting only 2.5 acres of potatoes a technically inefficient operation. Second, a grower may be precluded from certain cropping patterns or contractual agree- ments to grow particular crops within a cropping pattern due to financial limitations. For example, growers selling their lettuce through a marketing co-op must furnish all of their own operating capital. Furthermore, member- ship in one of the marketing cooperatives requires sufficient financial re- sources to withstand several bad crop years in succession. Therefore, growers without large amounts of liquid assets available are excluded from selecting this method of marketing their production. This point will be dealt with in greater detail in a later section of the empirical analysis. Measurement of Variability Variability of income from crops stems from the fact that prices, produc- tion, and costs are affected by many variables both physical and economic in a random manner. Farmers recognize that present and future prices and yields are affected by such economic and physical trends as changes in the business cycle and the overall growth in technology. Looking backward in a historical iiense, deviations from these trends can be treated as the random elements in , a time series and are therefore the proper values to be used as the variances of these series.—^ Although there are several methods available of estimating these random 2/ elements, Tintner's variate difference method— is used in this study. This method assumes that each time series is made up of two additive parts, the 1/ Carter, H. 0., and G. W. Dean, "Income, Price, and Yield Variability for Principal California Crops and Cropping Systems," Hilgardia, Berkeley: Uni- versity of California, Agricultural Experiment Station, Vol. 30, No. 6, Octo- ber 1960. 2/ Tintner, G., The Vccriate Difference Method, Evanston: Principia Press, Cowles Foundation Monograph No. 5, 1940. -13- trend element and the random element. By taking a series of differences of the historical data, the systematic element can be eliminated leaving only the random element which becomes our estimate of the variance. The procedure consists of calculating the variance of the original time series and also of each succeeding series. The difference between the variances of two succes- sive series is then compared with the standard error. In developing this method, Tintner concludes that if this difference is smaller than three times the standard error then we can say, with a high degree of confidence, that a valid estimate of the random element has been made. Variability estimates of two sets of data were made for this study. First, the variability of gross income which includes both price and yield variability and second, variability of net income which includes cost vari- ability, as well as the elements contained in gross income variability. Mea- sures of gross income variability were derived from published Monterey County data.— ^ Net income variability was estimated from the gross income series less a series of cost data deflated by an index of prices paid for inputs in the 2/ fruit and vegetable industry.— lliese series were further adjusted to reflect major changes in vegetable harvesting technology implemented during the 14- 3/ year period under study, 1950-1963.— Some researchers argue that county data understate the variability faced by the individual grower because of data averaging during compilation. This problem is further accentuated by averaging data from different producing areas within the county. Also, average prices for the season mask some of the wide day-to-day fluctuations observed in the vegetable industry. However, since most growers harvest several blocks during a shipping season, the prob- lem of averaging is not as serious as may first be suspected, as will be shown from analysis of the detailed records of two growers over a six-year period. 1/ Monterey County, Agricultural Commissioner, loo. oit. 2J French, Ben C, and John A. Richter, Cost and Factor Price Changes in the Vegetable Producing and Processing Industry ^ 1947-61, Davis: University of California, Agricultural Experiment Station, Supplement to Giannini Re- search Report 241, 1963, 17 pp. processed. _3/ The statistical time series were adjusted to reflect the shift from ice packing in the shed to field packing in cardboard cartons and vacuum-cooling and the time trend in weight per carton. -14- The variate difference method was also extended to estimate the variance of gross and net Income under assmed contractual arrangements for marketing vegetables. The details of these contracts are discussed In the next section. Transfer of Risk through Contractual Arrangements Forward contracting — locally known as "deals" — provides a common method of providing growers an assured market for their production in the Salinas Valley. Contracts also benefit the shipper with a steady and predictable sup- ply of produce during the shipping season. Fresh vegetable "deals" can be broadly classified Into three categories. First, the flat-rate deal specifies the crop, approximate planting (and there- fore, the approximate heirvest date) , and the amount of money to be paid per acre for the crop. This is a closed-price contract and in essence the grower is paid a flat fee for growing a crop and at harvest time the crop belongs to the shipper. All of the risk of price variability is teiken by the packer. Under a flat-sum guarantee contract the grower can maximize profits only by minimizing preharvest production costs. The grower's major risk is due to production cost variability stemming from such unexpected events as insect or disease infestations. This type of contract is not too common. However, since the payments are made at predetermined times during the growing season, it does provide a source of operating capital, which may be more Important to smaller growers just starting in business. Shippers must provide a large amount of supervision on this type of contract in terms of seeing to it that pests are adequately controlled and sufficient irrigation water and fertilizer is applied since there is no incentive provided for the grower to do this on his own volition. The second type of "deal" is an open-price contract. The most important characteristic of this type of contract is that the price received by the grower is not specified — the price actually received by the shipper for the crop determines the proceeds which are divided between the grower and the shipper by formula. These contracts usually specify the planting date and the amount of production inputs each party will provide in the way of pesticides, fertilizer, hoeing, and thinning labor, etc. The shipper generally harvests and sells the crop, charging a flat rate per carton or other unit of harvest. -15- A portion of the grower's risk may be transferred to the shipper by a minimum guarantee provision in the open-price contract. For example, a let- tuce growing contract will specify an amount, say $150 per acre, that is guar- anteed to the grower. This amount is to be paid even if the crop goes unhar- vested. The payment is made in three equal installments during the growing period and thus usually serves as the grower's primary source of operating capital. Depending upon the individual contract, the guarantee advanced may or may not have to be returned to the shipper when the proceeds of the crop are divided. The guaranteed-retum provision minimizes the grower's losses in an extremely bad year but also reduces his opportunity to reap large gains during a season of strong prices. A second variation of the open-price contract does not contain a minimum guarantee provision. Under this variation the shipper purchases a specified share in the crop either by advances in the form of cash or by providing cer- tain inputs such as labor for hoeing and thinning plus paying the cost of a portion of other cash inputs. This type of contract reduces the magnitude of absolute income fluctuations to the grower. However, the relative variability remains unchanged as compared to selling on the open market, since the mean income over time and its standard deviation have been reduced in the same pro- portion. By sharing both the costs of production and also the proceeds from the sale of the crop, both the fluctuations in net returns and the average level of net returns to the grower are reduced. By forward contracting, even under an open-price agreement, it is possi- ble for a grower to buy a quasi-insurance policy against loss of assets and operating capital due to price fluctuations. However, as under any method of insurance (formal or informal), there is a cost associated or a premium that must be paid. The amount of this premium in a situation such as this is not explicitly stated; however, it is possible to impute some values by a careful examiaation of the expected returns over time to the grower under different contractual arrangements . The contractual agreement between the grower and the shipper, as has al- ready been observed, also provides a source of operating capital to the grower. Due to the large risks involved in vegetable production, regular production credit sources are reluctant to extend sizable lines of production credit, especially if the grower is farming rented land. Commercial lending agencies -16- prefer to channel credit through a well-established shipper, vho In turn ex- tends operating capital to growers In the form of advances on contracts. This tends to spread the risk of loan failure over a large number of growers and crops, with the assets of the shipper being used as security for the basic commercial loan. The small Individual grower who finds It difficult to gain access to normal sources of operating capital may find his only source to be through a contract with a shipper. The Interest rate paid for these funds Is not stated In the contract and cannot be separated from the "price-risk Insur- ance premium" mentioned earlier. The third type of contract, open to some growers, can be made only by membership with one of the vegetable marketing cooperatives operating in the Valley. Marketing cooperatives do not provide minimum guarantees of returns to growers nor do they advance operating capital to their members. As the name Implies , they perform only two functions : they maintain labor crews and equipment to hoe, thin, harvest, and pack the produce of their members and by maintaining a sizable sales force, they attempt to obtain the highest price possible for their members on any particular day. Members are charged only the actual cost of these functions since the cooperative is owned by the membership . These marketing cooperatives are limited membership organizations in that only those growers who can provide all of their own operating capital are ac- cepted. Since normal comnercial lending agencies are reluctant to provide a large proportion of the funds required to grow a crop under such high-risk conditions, only a relatively small number of growers have sufficient personal assets to allow them access to this alternative. There is no transfer of risk with marketing cooperative membership. Mem- bers face thfe same variability of Income situation as the shipper organiza- tions with which they compete. Therefore, a grower marketing through a coop- erative must have sufficient capital not only to finance his production during a growing season but also to be able to withstand several bad price years in succession with little short-term support from sources outside the firm. Empirical Analysis of Alternative Contractual Arrangements for Lettuce Lettuce is the primary crop grown under contract in the Salinas Valley. Therefore, major attention will be focused upon this crop in the following -17- analysis. As stated previously, there are three basic types of lettuce marketing contracts available to growers. The first sets a flat sum of money to be paid to the grower per acre for growing a crop. The grower assumes none of the risk and is paid regardless of whether or not the crop is harvested. The negotiated income received by the grower usually covers the total cost of production plus a small return to unpaid family labor and management. The open-price contract is the most common method used in the lettuce in- dustry. The grower and the shipper agree to provide certain inputs and to divide the proceeds from the crop in a set manner. The price used to calcu- late the gross returns is the price actually received from the crop when it is sold (Tables 3 and 4). There are many variations of this basic arrangement and because each one is negotiated individually it can be said that there are almost as many dif- ferent variations as there are growers of lettuce. However, representative contracts may be evaluated and particular contracts identified as being more or less favorable to either party in monetary terms alone . Presumably, con- summation of any contract between two parties is evidence of an equitable solu- tion based on consideration of nonmonetary as well as monetary terms. Open-Price Contract with a. Guaranteed Minimum Return The basic provisions of a representative open-price contract with a guar- anteed minimum return can be stmmiarized as follows: The grower will produce a crop of lettuce furnishing the land, irrigation water, machinery, and prehar- vest labor except for hoeing and thinning. The contractor provides hoeing and thinning labor and will harvest and sell the crop for a specified rate per carton. A minimum return is guaranteed to the grower even if the crop is not harvested. In 1964 when the Initial survey work for this study was completed, the typical charge for packing and selling lettuce was $0.90 per carton and the typical minimum guarantee was in the range of $125 to $150 per acre. It is possible in the short run for the price of lettuce per carton to fall below the cost of harvesting — but the shipper usually reserves the right to make the decision to harvest or leave the crop in the field. If the crop is harvested, the packing charges are deducted from the gross income.—^ 1/ It was observed that in some cases the minimum guarantee which had been advanced to the producer during the growing season was also deducted before (Continued on next page.) -18- TABLE 3 Production Required for Growers to Break Even for Three Selling Prices, Various Contractual Arrangements , Salinas Valley , 1962 Contract 1 1 2 1 3 cartons per acre Number of cartons required when price received per carton is : $1.25 1,125 1,097 1,022 1.68 553 394 458 2.11 225 254 295 Number of cartons required for owner-grower to re- cover varicible expenses when price received per carton is: $1.25 532 288 433 1.68 248 128 194 2.11 143 83 125 Note: The historical average production is about 445 cartons per acre planted which indicates that both shippers and growers would not expect to cover total costs when market price is $1.25 per carton. TABLE 4 Production Required for Shippers to Break Even for Three Selling Prices, Various Contractual Arrangements, Salinas Valley, 1962 Contract 112 13 cartons per acre Number of cartons required when price received per carton is : $1.25 633 878 732 1.68 300 394 329 2.11 183 279 212 Note: The historical average production is about 445 cartons per acre planted which indicates that both shippers and growers would not expect to cover total costs when market price is $1.25 per carton. Figure 4 shows the break-even production In cartons per acre (production required to cover average total costs) for three selling prices of lettuce. For Contract No. 1 the assumed guarantee Is a $135 advance to the grower which must be returned before the proceeds from the crop are divided. The assumed selling price for lettuce under Situation A In Figure 5 Is $1.68 per carton, which Is the average seasonal price received during the period 1950-1963. Situation A asstmes a packing cost of $0.90 per carton and an average total preharvest cost of $307.56 per acre.— ^ To cover packing and selling charges, as well as the $135 advance, would require about 175 cartons per acre yield. After recovering these costs, the net proceeds would be split equally between the grower and the shipper. The shipper would recover all remaining direct costs at a yield of about 300 cartons per acre. The grower would recover his average total costs If the yield per acre reached about 550 cartons. The grower could recover his average variable expenses at a yield of about 250 cartons per acre; any remaining Income would apply to recovering fixed costs Including unpaid family labor. Under Situation B, the same cost levels are assumed but a selling price of $1.25 per carton Is used. This situation reflects a below normal price or 2/ minus one standard deviation away from the mean historical price.— A yield of almost 400 cartons per acre Is required to cover the $135 advance to the grower and about 630 cartons per acre are necessary before the shipper recov- ers his Investment. The tenant grower will not recover his average total costs until the production per acre reaches 1,125 cartons, which Is extremely unlikely when the historical average production per acre for the Valley Is only about 445 cartons. An owner-operator grower, on the other hand, would be able to cover his variable production costs with a yield of about 530 cartons per acre. (Footnote 1 continued from page 18.) the net proceeds were divided. The net proceeds were then split on a 50-50 basis. Other contracts were observed where the advance payments were not deducted and proceeds after packing and selling charges were split on a 50-50 basis from the first carton harvested. Ij Based on field survey Information. 2/ The mean plus and minus one standard deviation for a normal shaped dis- tribution Includes about two-thirds of the expected observations. Growers can expect to receive a price of less than $1.25 per carton about one-sixth of the time or sibout 17 times out of 100. -21- FIGURE 4 Breakeven Chart, Confroct No 1 ^ Gross Income 0 100 200 300 400 500 600 700 800 900 1000 1100 800 y Gross Income 700 — Price- $2.11 per corton 600 Packing and Selling Charges 500 400 ^^''^Shipper Breakeven^*** 300 ^^183 cfn.) ^ 200 Grower Breokeven ■ (225 ctn.) 100 -^wner V.C. $135 Guarontee (143 ctn.) 0 1 1 1 II 0 50 100 150 200 250 300 350 400 450 500 550 Production ctn./oc. Various prices ($135 guarantee, plus hoeing and thinning, 50 percent split after guarantee returned, assumed packing charge $.90 per carton, average total cost $307 per acre). -22- FIGURE 5 Breakeven Chart, Contract No. 2 ^ ° Various prices ($135 guorontee plus half of fertilizer ond pesticide, 50 percent split after fertilizer and pesticide cost returned, assumed packing charge $.90 per carton, average total cost $307 per acre)- -23- Situation C in Figure 4 is based on the same cost rates as used in the other two examples; however, a selling price of $2.11 per carton is used, which is one standard deviation above the mean. The shipper recovers the $135 advance at a production level of about 110 cartons and recovers his entire investment after 180 cartons have been harvested. The owner-grower recovers all of his costs if production per acre reaches 225 cartons but can cover variable expenses only if about 140 cartons are harvested. An open-price contract with a minimum guarantee, which is not returned before the split is made, is illustrated in Figure 5. Under this contract, one-half of the fertilizer and pesticide cost is borne by the shipper and his investment in these two items is returned before the 50-50 split starts. In analyzing this contractual arrangement, the same packing and selling charges and average total preharvest costs as used previously are assumed. For a selling price of $1.68 per carton, the shipper's share of the fertilizer and pesticide expense is recovered at a yield of about 50 cartons per acre. Under this deal when the minimum guarantee advance is not returned to the shipper, both the shipper and the grower have a break-even point of about 395 cartons per acre. Alternatively, an owner-operator grower will be able to cover his variable costs with a yield of about 130 cartons per acre. If the selling price at harvest time is low, as represented by the $1.25 per carton in Situation B, then a production per acre of about 125 cartons is necessary for the shipper to recover his share of the fertilizer and pesti- cides used on the crop. The shipper will break even on his investment only if the production reaches about 880 cartons per acre; whereas, the grower will not break even unless about 1,090 cartons per acre are harvested. The owner- grower, however, can recover his variable costs if the yield is above about 290 cartons per acre. During a season of strong lettuce prices, as represented by Situation C with a selling price of $2.11 per carton, only 254 cartons per acre production are required for the owner-grower to break even and only 83 cartons per acre to cover his variable costs. The shipper, however, does not break even on his investment until about 280 cartons per acre are harvested. -24- Open-Price Contract with No Minimum Guarantee Under this type of contract, since there is no guarantee, funds advanced by the shipper essentially buy one-half of the crop. The shipper can advance the grower a sum equivalant to one-half of the preharvest costs or provide an equal amount in the form of actual cost sharing. In the contract analyzed below, the shipper shares one-half of the grower's cultural costs. In a typi- cal situation this would involve the shipper paying one-half of such produc- tion expenses as seed, fertilizer, dusts and sprays, and hoeing and thinning labor. It would not cover the grower's permanent hired labor costs, fuel and repairs on machinery, or irrigation costs. Under this agreement, returns — after deducting packing charges — are split 50-50 from the first carton harvested. Since the shipper's direct contribu- tion is slightly less than one-half of the average total preharvest costs, his break-even point is at about 330 cartons per acre for a selling price of ^il.b8 per carton (Figure 6, Situation A). The owner-grower recovers his in- vestment in the crop when about 460 cartons per acre are harvested and recov- ers his variable costs if the yield is about 195 cartons per acre. Any yield over the 195 carton level goes toward covering the fixed overhead costs of the grower. A poor lettuce price — $1.25 per carton (Situation B) — leaves only $0.35 per carton to be divided between the grower and the shipper after packing charges have been deducted. The shipper would not recover his direct invest- ment in the crop until the production reached about 730 cartons per acre and the grower would not recover his total costs until the yield reached about 1,020 cartons per acre. The grower would have recovered his variable expenses after about 435 cartons per acre had been harvested. A selling price of $2.11 per carton reflects a strong demand for lettuce. A market price equal to or greater than $2.11 per carton could only be ex- pected about 17 times out of 100 based on the period 1950-1963 experience. Under this favorable but relatively unexpected price situation, the shipper will break even after about 210 cartons have been harvested and the tenant grower will break even after 295 cartons are harvested. Owner-operator vari- able costs are recovered after a harvest of 125 cartons (Figure 6, Situation C) . -25- FIGURE 6 Breokeven Chort, Contract No 3 - 800 r- 700 600 — 500 - 400 - 300 200 100 0 700 600 500 400 300 200 100 0 Price; $1.68 per corton Gross Income Packing and Selling Chorges Shipper Breakeven (329 ctn.) Owner V.C. (194 ctn.) I Grower ^Breakeven ((458 ctn.) J \ L — Price: $1.25 per corton Gross Income Packing and Selling Charges Packer Breakeven (732 ctn.) jrower Breakeven( 1 022 ctn^ rbwner V C, (433 ctn.) I t I I 1 800 / Gross Income 700 — Price: $2,11 per carton / 600 / Packing and Selling Ch 500 400 300 200 X ^^^Shipper Breakeven ^^^^^^ / ^^1(212 ctn.) ^ 100 0 ^^00"""^ I Grower Breakeven r(295 cm.) ^ Owner V.C. (125 ctn.) 1 f 1 1 1 y 1 1 1 1 0 50 1 00 1 50 200 250 300 350 400 450 500 550 Production (ctn./oc.) Various prices (half of cultured costs plus 50 percent split from first carton. Assumed packing cost $.90 per carton, overage total cost $307 per acre). -2b- Lettuce growing contracts provide the grower with three necessary goods and services. First, they provide a market outlet for the small grower. Be- fore the advent of refrigerated railroad cars and trucks, fresh vegetables were grown close to the area of consumption, and marketing was a rather simple affair with most of the production being sold through local commission houses. The nationwide marketing institutions of today make it extremely difficult for small growers to gain direct access to the market. The elaborate sales organi- zations of present day shippers require vegetable harvest and delivery eveiry week of the season to make them feasible. A small independent grower is al- most precluded from the benefits of such sales outlets unless he grows under contract. A second major service of contracts, as discussed earlier, provides small growers with the necessary access to the capital market for production and in- termediate credit. Third, small growers without firm financial backing can transfer all or part of the risk of price variability to the shipper. All of these services come at a cost to the grower — the cost of providing a market outlet, the cost of interest on operating capital advanced, and the cost of an insurance "premium" for sharing the price risk. It is not possible to make more than a guess at the magnitude of each individual cost item be- cause they are fill lumped together. By comparing the contract where $135 is advanced but returned (Deal 1) with the straight cost-sharing contract (Deal 3), it would appear that under the low price of $1.25 per carton it required about 100 cartons more per acre under the first deal to break even as com- pared to Deal 3. If this 100 carton difference can be assumed to be the cost of the insurance premium, then growers choosing Deal 1 over Deal 3 have a rather strong aversion to risk becaxise the amount of capital advanced in each case is nearly the same. In other words, a grower under Deal 1 is paying a substantial premium in order to minimize his losses in a very bad year. Contract Selection under Price Uncertainty The prices used in the previous analysis were based on county-wide sea- sonal average prices for each growing season for the period 1950-1963. As ex- plained earlier, these data mask the variability faced by an individual grower. In an attempt to unmask this variability, we secured detailed information from two growers on each identifiable block of lettuce they grew for a six-year -27- period.— These data included net price received and total production per acre for a total of 265 blocks. By applying a variable cost of packing and selling, it was possible to develop estimates of returns to preharvest costs for these two growers. Figure 7 shows the frequency of occurrence of gross income per acre after packing to these two growers for the years 1961-1966. These data were grouped by $100 class intervals. The points in the figure in- dicate the midpoints. For example, 38 of the blocks fell into the $50 to $150 interval, two blocks fell into the $1,150 to $1,250 interval, and seven blocks fell into the $1,650 and over class. The frequency distribution curve in Figure 7 is not bell-shaped reflect- ing the normal distribution assumed for the analysis of the contracts dis- cussed earlier. Rather it is skewed to the right. The tail of the curve on the left-hand side is very short due to growers being unwilling to cut if the market price falls below the harvest cost. In the previous section, contracts were examined under conditions of three price levels assuming that the distri- bution of prices and incomes was from a normal distribution. The objective now becomes one of determining the best contractual arrangement to follow if the data on prices and yields from the two growers are assumed to be better estimators of the true situation with respect to prices and yields faced by growers. Statistical decision theory can be used to analyze these contracts and determine the most profitable contract based on past histories of the uncer- tain price and yield variables. The basic requisite for such a statistical decision analysis is the relative frequencies (probabilities) of the occur- 2/ rence of prices and yields.— Prices and quantities of lettuce harvested are positively correlated in the short run. This is contrary to the condition usually found for such crops as feed grains and other storable commodities. When the lettuce prices in the major consuming centers are high (generally assimied to be associated with hot weather) growers harvest every carton possible from the fixed acreage ready for cutting to take advantage of the situation. Even low quality lettuce that 1/ One of these farms fell into the 160-319 acre size group and the second was in the more than 1,279 size group. 2/ Luce, R. D., and Howard Raiffa, Games and Decisions; Introduction and Critical Survey, New York: John Wiley & Sons, 1957. -28- FIGURE 7 Distribution of Net Income Per Acre After Packing, 265 Blocks of Lettuce, Salinas Valley, 1961-1966 Net Income (dollars) would normally be left in the field is harvested at this time. When lettuce prices are low in the eastern markets, only the best quality heads are cut be- cause buyers become very choosy at this time. Frequently large quantities of low quality heads are dumped at the terminal market. Growers have found through experience that losses are minimized by not incurring the high vari- able harvest costs for low quality lettuce during these periods. Not only will growers lose the sunk production costs, but they also stand to lose the variable harvest costs. These latter costs account for more than half of the total cost of each carton sold. This correlation may be taken into account by combining the price and production data to obtain the gross income less packing charges as shown in Figure 7. The relative frequencies (probability of occurrence) are obtained by dividing the ninnber of blocks in each class interval by the total number of blocks observed (Table 5) . The type of contract that will give the highest expected return to grow- ers and shippers over time can be determined by applying the probabilities of occurrence of all possible returns to preharvest costs. The sum of the gains and losses under different contracts will then indicate the expected net in- come including production costs from each contract (Tables 6 and 7) .— Membership in a marketing cooperative brings the highest expected returns to the grower ($180.31) and also the highest absolute loss in an extremely poor price season (-$311.06). For the grower unable to market through a coop- erative, the highest expected net return per acre is provided by the contract guaranteeing $133.00 per acre plus one-half share of the cost of fertilizer and pesticides where the guarantee funds advanced are not returned before the proceeds are split. However, this contract does not have the lowest absolute loss under extremely poor price conditions. The smaller gains under this con- tract compared to marketing through the cooperative are due to the profit sharing associated with it. By the same token, the maximum possible losses are reduced by more than half under this contract as compared with marketing through the cooperative. 1/ These probabilities are referred to as a priori probabilities in deci- sion theory literature. See Markowitz, H. M., loa. oit. If additional in- formation could be obtained, such as growing conditions in other producing areas, then a "data problem" could be developed. Growers in the Salinas Val- ley indicated these additional data were not used in their decisions. -30- „ TABLE 5 Frequencies of Returns Per Acre After Packing Charges, 1961-1966 Class interval | Number of blocks | Percentage 1 Probability Less than 0 4 1.51 .015 0 - 200 69 26.04 .261 201 - 400 68 25.65 .256 401 - 600 50 18.87 .188 601 - 800 28 10.57 .105 801 - 1,000 15 5.66 .057 1,001 - 1,200 9 3.40 .034 1,201 - 1,400 11 4.15 .042 Greater than 1,401 11 4.15 .042 TOTAL 265 100.00 1.000 Note: Individual items may not exactly agree due to rounding. -31- TABLE 6 Net Income Per Acre to Growers for Various Lettuce Contracts Price condition Sell through cooperative Share cultural costs $135 guarantee plus 1/2 fertilizer and pesticide split after fertilizer and pesticide returned $135 guarantee plus hoeing and thinning split after $135 returned Probabilities dollars percent Extremely poor -311.06 -182.87 -135.06 -129.26 .015 Very poor -206 .53 -128 85 .261 Poor - 12.71 - 31.91 4.21 - 49.34 .256 Fair 202.54 75.68 101.24 58.29 .189 Above normal 379.94 164.38 189.94 146.99 .106 Good 654.10 301.46 327.02 284.07 .057 Very good 761.88 356.35 380.91 337.96 .034 High 976.53 462.67 488.23 445.28 .042 Very high 1,481.35 715.08 740.64 697.69 .042 Weighted average net income 180.31 64.56 93.12 77.90 1.000 TABLE 7 Net Income Per Acre to Shippers for Various Lettuce Contracts Price condition Market by self Share cultural costs $135 guarantee plus 1/2 fertilizer and pesticide split after fertilizer and pesticide returned $135 guarantee plus hoeing and thinning split after $135 returned Probabilities dollars percent Extreaely poor -311.06 - 67.60 -176.00 -181.80 .015 Very poor -206.53 - 13.50 -103.24 - 77.27 .261 Poor - 12.71 83.33 - 6.25 36.62 .256 Fair 202.54 190.95 101.60 144.25 .189 Above normal 379.94 289.65 190 .00 232.95 .106 Good 654.10 416.83 327.08 370.03 .057 Very good 761.88 470.62 380.97 423.92 .034 High 976.53 577.94 488.30 534.24 .042 Very high 1,481.35 830.35 740.70 783.65 .042 Weighted average net Incone 180.31 105.86 93.80 127.76 1.000 Note: The sum of the total net incomes is not equal for all contracts due to varying assumptions as to how losses are shared under each contract and rounding error. Growing and harvesting through their own operations brings the highest expected net Income to shippers, which Is equivalent to the grower selling through a marketing cooperative. As expected, It also has the highest abso- lute loss attached. For the shipper wishing to "contract In" additional acre- age of lettuce to utilize the excess capacity of his cooler and labor crews , the $135 guarantee plus hoeing and thinning where the guarantee Is returned offers the highest expected net Income per acre. However, this contract gives the lowest net Income to the grower and also has the possibility of the great- est absolute loss attached to It. It would be possible to analyze a much larger number of possible varia- tions In contractual arrangements. However, these additional variations would be relatively minor and should not alter the results substantially. A perti- nent question to ask at this point is, why, with the wide spread between ex- pected net returns from the different contractual arrangements, do so many growers choose those contracts with significantly lower returns? It would appear that the two major factors are risk aversion and operating capital availability provided by these particular contracts. The fear of a complete economic failure for a given block of lettuce appears to be overly emphasized in the mind of the small grower. For example, of the 263 blocks examined, only four were not harvested or sold for less than harvest cost. But for a grower operating on a very thin margin, the threat of such loss appears to be foremost in his mind. Lack of operating capital can also be an Important restraint to the grower and may exclude him from certain contractual arrangements. The Impact of the limited availability of operating capital on the contractual arrange- ments and possible cropping patterns will be analyzed in the following section. Effect of Operating Capital Supply on Possible Contractual Arrangements Linear programming is a sophisticated form of budget analysis which allows simultaneous consideration of the various physical constraints under which a farmer must operate. To calculate the optimum cropping program for a representative owner-grower and a renter-grower vegetable farm in the Salinas -34- Valley, a linear programming model was developed based on the resource endow- ment of a 240-acre farm in the Salinas area — the objective being to determine the cropping program which maximizes farm income depending on the level of operating capital available. It was assumed that lettuce, to be included as a crop in the program, would be grown in 24 blocks of equal size so that at least one block could be harvested every 15 days during the normal harvest season. By specifying 24 blocks to be harvested, it was possible to use historical average prices as the expected price. Five marketing arrangements for lettuce were considered and are described briefly as follows : 1. Marketing through a cooperative. All the operating capital is fur- nished by the grower and none of the proceeds from the crop are shared. 2. A joint venture with a shipper. The shipper advances one-half of the cultural costs in order to purchase a 50-percent share in the crop. Returns are split equally from the first carton harvested. The balance of the operating capital must be furnished by the grower from his own sources. 3. The lettuce is grown under a contract with a packer. A $135 per acre advance is made to the grower and the packer furnishes one- half of the cost of pesticides and fertilizer. Returns are split after the cost of the shipper's share of pesticides and fertilizer is returned. 4. A minimum income guarantee contract with a packer. All of the hoe- ing and thinning costs and $135 are advanced. Receipts are shared equally after the $135 is returned. 5. A flat fee of $300 per acre is paid to the grower to produce an acre of lettuce. There is no sharing of profits or losses by the grower. Two carrot growing contracts were also included in this analysis. The first is a $135 per acre guarantee paid by the shipper to the grower plus a one-half share of the pesticides and fertilizer being paid by the shipper. The receipts are split equally after packing charges from the first crate. The second is a flat-fee contract where the grower is paid $275 per acre to grow a crop to maturity. The latter contract does not provide for profit or loss sharing to the grower. -35- Small white beans and sugar beets were included in the program to repre- sent the low income, low variability field crop alternatives available to farm- ers in this area. These are typical rotation crops in the area. Double crop- ping is allowed up to a level of 290 crop acres for 240 acres of land. . A labor restriction, representing the maximum available from a permanent year-round crew including the operator, was defined for the peak months of March through August. Initially, machinery capacity both in terms of wheel tractors and crawlers for the peak period months was used as a restriction assuming a 10-hour operating day during the peak seasons. The number of trac- tors assumed available was based on the results of the basic survey.—^ Prices and production were based on historical averages used in the earlier sections of this report. Before discussing the results of this analysis the reader should be re- minded of one important qualification to the following solutions. This linear programming model assumes the historical average price will be the selling price the grower expects to receive. Linear programming does not take into account the variability of income or the grower's psychological reaction to this situation. In other words, this method of analysis takes into account the mean income but not the variance about this mean. Therefore, although the following results will not provide concrete recommendations for cropping pro- grams, they are useful in demonstrating the effect of limited operating capi- tal upon the alternative contracts available to a grower. The implications of planning under uncertainty will be developed in a subsequent section. In the solution to the linear programming problem, the operating capital supply was varied continuously from a starting point of zero to $75,000 per year. The program did not allow for internal generation of operating capital. In other words, profits from a crop grown in the spring could not be used later to finance a crop grown in the summer or fall. This lack of allowance for capital turnover within the year overstates the true quantity of capital needed, but it does not affect the final results if the reader keeps in mind that the figures used represent the gross capital employed during the year and not the net capital that was recirculated. 1/ It was foxmd, however, that the tractors available on a typical farm were far in excess of the number needed to do the work required in the time allowed for the task. -36- Figure 8 shows a map of the acreage of crops grown that maximize farm In- come as the gross capital supply Is varied for an owner-operator farm with 240 Irrigated acres of land. This figure shows that the highest return to the op- erator's capital comes from Investing In a carrot crop grown under a flat-fee contract. The shipper advances almost the entire amount of capital required to grow the crop Including sufficient funds to cover fixed overhead cost Items. The net returns per acre under this no- loss contract are quite small, but In relation to the amount of money Invested by the grower from his own funds, the return per dollar Invested Is high. Figure 8 shows, under severely limited capital supply conditions , the Initial dose of capital being used to produce 200 acres of carrots under this contract. Two hundred acres of car- rots exhausts the permanent labor supply during the critical cultivating and Irrigating months regardless of the contract voider which It Is grown. The next dose or Increment of operating capital Is most profitably In- vested In growing lettuce under a minimum return guarantee of $135 per acre contract (No. 3) where the guarantee Is advanced to the grower and the shipper shares half of the pesticide and fertilizer cost. Under this contract, 240 acres of lettuce are grown and carrot acreage under the flat-fee contract Is reduced to 50 acres. A cropping program of lettuce and carrots under these two contracts requires only $7,500 per year of gross operating capital and uses all of the land available for the 290 crop acres on 240 acres of land as Indicated by the vertical dashed line. Shippers are very reluctant to allow a single grower to produce lettuce under two different contracts simultaneously. Although Figure 8 shows that the next increment of available operating capital should be split first be- tween contracting lettuce under Deal No. 3 and Deal No. 4 and then between Deal No. 4 and Deal No. 1, the Institutional setting requires that the grower have sufficient funds to grow all of the crop under a single contract. In other words, in our example, if the grower has $7,500 of operating capital he can contract all of his lettuce acreage under Deal 4. He must wait until he has accumulated about $48,000 of gross operating capital in order to shift to the more remunerative Deal No. 1. He cannot split his acreage between Deal No. 4 and No. 1 if he only has, say, $30,000 of gross operating capital. -37- FIGURE 8 Optimal Cropping Program as Supply of Operating Capital Is Varied, Owner-Grower Farm, Salinas Valley Operating Capital (000 dollars) Once a grower h£is sufficient capital to join the marketing cooperative and assumes all of the Income variability associated with that method of mar- keting, additional operating capital can be profitably employed by shifting his carrot acreage to a higher risk contract (No. 1). This brings a lower return per dollar invested of operating capital but a higher total return to the farm. A similar solution was calculated reflecting the operating capital re- quirements of a renter-operator. The results were very similar — except the solution was shifted to the right as compared to the owner-operator due to the additional requirement of the renter paying a substantial cash rent per acre. Thus, the maximum gross operating capital required for a cropping program where lettuce w£is sold through the cooperative and carrots were grown under Contract No. 2 Increased from about $38,000 per year for an owner to about $73,000 per year for a renter. This increased operating capital requirement is reflected in Figure 9, which shows the net return over variable expenses as gross operating capital supplies are varied. The broken dashed horizontal line indicates the level of fixed costs and the point of Intersection with the returns curve indicates that total costs (Including both fixed and variable) are just met when a renter has Invested approximately $22,000 in operating capital. The dashed line that forms a step on the diagram reflects the Institutional requirement of a grower being unable to grow part of his acreage under one contract and also market a portion of the same crop through a cooperative. This shows the effect on income due to the Inability of a grower to split acreage of a crop between two different contractual arrangements. A reader familiar with the Salinas Valley will quickly note that the cropping patterns described In this section are not typical of those being followed by most growers. This brings us back to the original hypothesis — that analysis of Investment portfolios must Include both the problem of vari- ability of Income about the mean, as well as the capital position of the grower making the Investment decision. -39- FIGURE 9 Operating Capital (000 dollars) Planning under Conditions of Uncertainty Howard—^ has developed a computational method based on the earlier work 2/ of Bellman— whereby it is possible to analyze alternative cropping programs within a stochastic context. The basis of this method is the Markov process and its mathematical extension, dynamic programming. The important character- istics of dynamic programming are the "state" of a system and the "transi- tions" among the states of the system. This is best described by an example. Suppose a small machinery manufacturer has developed a machine which has found great favor among his customers in a given year. Let us call this State 1. The next year his machine has become obsolete and he is out of favor with his customers. Let us call this State 2. If we assume that these are the only possible states for the manufacturer, this will describe the first part of the system. Assume now that when he is in State 1, there is a 50-percent chance that by the end of the time period he will remain in the favor of his customers. This also means, therefore, a 50-percent chance of being out of favor with his customers. When he is in State 2, he experiments with a new machine design. If the design is successful, he returns to State 1; if not, he remains in State 2. Let us assume the probability of returning to State 1 in the next time period is two-fifths. If his new design fails the probabil- ity of remaining in State 2 is three-fifths. It is possible to attach rewards reflecting the gain or loss associated with each transition. Schematically, these transitions can be shown as in Figure 10. FIGURE 10 Schematic Diagram of Transition Matrix 2/5 1/ Howard, R. A., loo. cit. 1/ Bellman, Richard E. , Dynamia Progrcoming , Princeton, New Jersey: Princeton University Press, 1957. -41- In matrix form this can be stated as a transition matrix 1/2 1/2 2/5 3/5 We can define a state probability as iri (n) , which is the probability that the manufacturer will occupy the ith state after n transitions if its state at N = 0 is known. Since .Z, tt, (n) = 1 and i«l i N (N + 1) = Tr^(n) p^^ n - 0, 1, 2, . . . then Tr(l) - Tr(0) p"^. And since this is a recursive relation (that is, the results in one time period are dependent on what occurred in the previous time period) , it is pos- sible to find the probability that the manufacturer occupies each of the states in the system after n transitions by post-multiplying the initial state probability by the nth power of the transition matrix. Rewards for each transition may be included by defining a value v^ (n) as the expected total earnings in the next n transitions if the manufacturer is now in state i and that r^^ is the amount he will earn if he makes the transi- tion from state i to state j. Thus, the expected total earnings can be ex- pressed as N ^i " j=l ^ij ^""ij ''j ^"""^^^ i = 1, 2, . . . , N, n = 1, 2, 3, . . Howard^^ has developed an iterative procedure for solving problems stated in this form, including a criteria for terminating the process that works quite satisfactorily. In the application of dynamic programming to portfolio or cropping pro- gram analysis we define the states as the gross supply of operating capital. For each state a set of alternatives was defined in terms of proportion of each crop in the cropping program and the contractual arrangements under which \j Howard, R. A., Zoo, oit. -42- they were grown. In the preceding section It was shown that under different operating capital supplies only certain contracts were physically or economi- cally feasible. Therefore, we used the results of the variable capital prob- lem as a first approximation in defining the alternatives in the states for the dynamic programming problem with the exception that one state was defined as a proxy for bankruptcy. It is not a true representation of bankruptcy but rather that the operating capital supply is so low that production operations could not be carried on satisfactorily. Table 8 contains a description of the states and alternatives used in the analysis. The problem is defined for a 240-acre vegetable farm in the Salinas Val- ley with a typical line of machinery and equipment. A 55-percent equity in the machinery and equipment investment is assumed. Within each state a choice between several different cropping systems may be made. For example, in State 2 five alternative choices can be made, each with a different proportion of crops. An expected farm income is associated with each alternative. Whether or not this Income is realized depends strictly on chance. From this income must come the funds to pay for family living expenses, a payment on thena- chinery loan, and a tax on income, both State and Federal. The balance re- maining is a net addition to or reduction in the operating capital supply. Estimating Transition Probabilities Total variances of net income for each cropping program alternative were calculated, taking into account individual variances and covariances among all alternative crops. It was assumed that the net incomes were normally distrib- uted about the calculated incomes .— ^ The calculated net income from each al- ternative, plus or minus one standard deviation (square root of the total variance), would Include about two-thirds of the expected observations. In other words, the calculated income, plus or minus one standard deviation, could be expected about two-thirds of the time due to chance alone. Thus the grower can expect his Income to fall within this range two- thirds of the time. The transition probabilities were determined by calculating Income and operating capital requirements necessary to move from one state to the next. 1/ Although the distribution is skewed to the right, the assumption of normality is more conservative for planning purposes and is more convenient to work with. -43- TABLE 8 Production Alternatives by State, 240-Acre Renter-Grower Farm State Description and crops grown Alternatives 1 1 2 3 1 4 1 5 6 Proportions (percent) St ate Bankruptcy State 2 Very low capital 55-percent equity in machinery 37,500 operating capital Carrots (2) 30 25 20 15 15 Sugar beets 20 20 20 20 15 Small white beans 30 25 20 15 10 Lettuce $300 flat W 20 30 40 50 60 State 3 Very low capital 55-percent equity In machinery $11,250 operating capital Carrots (2) 30 25 20 15 15 Sugar beets 20 20 20 20 15 Small white beans 30 25 20 15 10 Lettuce $300 flat (4) 20 30 40 50 60 State 4 55-percent equity in machinery $15,000 operating capital Carrots (2) 20 15 15 30 15 20 Sugar beets 20 20 15 20 20 20 Small white beans 20 15 10 30 25 20 Lettuce $300 flat (4) 40 50 60 $135 guarantee (2) 20 30 40 State 5 55— percent equity in machinery $20,000 operating capital Carrots (2) 20 15 15 30 15 20 Sugar beets 20 20 15 20 JO 20 Small white beans 20 15 10 30 25 20 Lettuce $300 flat (4) 40 50 60 $135 guarantee (2) 20 JO 40 State 6 55~pGrccnt 3^ Cy ixx 013 cli Inc $27,500 operating capital Carrots (2) 20 15 15 30 15 20 Sugar beets 20 20 15 20 20 20 Small white beans 20 15 10 30 25 20 Lettuce S135 guarantee (2) 40 50 60 Share (3) 20 30 40 State 7 -jj percent ecjuxty in Tnacn xne ry 5 20 15 20 15 Sugar beets 15 20 20 15 20 20 Small white beans 10 20 15 10 20 15 Lettuce $135 guarantee (2) 60 Share (3) 40 50 60 Cooperative (1) 40 50 State 8 jj— percent equity in machinery ^)33,uuu operating capital Carrots (1) 15 Carrots (2) 15 15 20 15 15 Sugar beets 20 15 20 20 15 20 Small white beans 15 10 20 15 10 15 Lettuce Share (3) 50 60 Cooperative (1) 40 50 60 50 State 9 55-percent equity in machinery $75,000 operating capital Carrots (1) 20 20 20 20 Carrots (2) 20 20 Sugar beets 10 C 20 10 20 10 Lettuce Share (3J 70 8 Cooperative (1) 60 70 60 70 State 10 100-percent equity In machinery Unlimited operating capital Carrots (1) 20 20 20 Sugar beets 20 10 0 Lettuce Cooperat Ive 60 70 80 -44- The specific question asked was, how many standard deviations (converted to area under the normal curve) would be required to provide sufficient Income to overlap with the next state? Typical annual family living expenses were as- sumed to be $3,200. in addition, machinery loans and Income taxes had to be paid. The sum left over must then be large enough to Increase the operating capital supply sufficiently that It would be Included in the next state. For example, Alternative 1 In State 2 has a calculated net return to unpaid family labor and management of $9,556. Assuming a tax deduction of $1,000, a taxable Income of $8,556 remains. Taxes on this Income amount to $821. After with- drawal of family living expenses and a machinery loan payment of $1,845, leaves a net addition to the operating capital fund of $690. However, since the starting operating capital supply was $7,500, to reach the next state of $11,250 requires a net addition of $3,750 to the supply. To achieve the next higher state, the grower must receive an Income 1.121 standard deviations above the calculated Income after taxes.— ^ That Is, with a standard deviation of $3,497 and a marginal tax rate of 22 percent, each standard deviation above the mean contributes $2,728 to the operating capital fund; therefore, ($3,750 - $690) divided by ($3,497) (.78) = 1.121. The probability of a grower follow- ing Alternative 1 and achieving an Income sufficient to place him in State 3 or higher in the next time period (year) is 0.1314. The probability of reach- ing State 4 in the next time period using the same method of calculation is 0.0077. By subtraction, the probability of achieving State 3 but no higher is 0.1314 - 0.0077 = 0.1237.-^ The probability of a grower incurring a loss sufficiently large to elim- inate the $7,500 operating capital fund that he started the year with can be calculated in the same manner and is 0.0020. The sum of the probabilities for an alternative of either changing states or remaining in the same state must equal 1.0 (Zp.. = 1.0). By subtraction, the probability of remaining in State 2 is, 1.0 - (0.0020 + 0.1237 + 0.0077) = 0.8666. The data used to calculate the transition probabilities for each alternative in each state are shown in Table 9. Transition probabilities for all states and alternatives are shown in Table 10. Xj Based on the area under a normal curve. Ij It would be possible to relate the family living expenses to level of income in order to assess the Impact of present consumption on future growth of the firm. -45- TABLE 9 Income, Family Withdrawals, Loan Payments, Marginal Tax Rates, and Standard Deviation about Net Income by State and Alternative, Salinas Valley Return to Operating labor and Taxable Family Loan Marginal otanaara State capital Alternative management income withdrawal payment tax rate deviation dollars dollars percent dollars 1 0 1 0 0 0 0 0 0 2 7,500 9 , 556 8 ,556 5,200 1,845 3 ,497 2 9,561 8,561 5,200 1,845 0 0 11 3 ,227 3 9,566 8,566 5,200 1,845 zz 2 ,974 4 9,572 8,572 5 ,200 1,845 zz 2 , 740 5 9 ,582 8,582 5,200 1,845 zz 2 ,168 3 11,250 1 9 ,556 8,556 5,200 1,845 22 3.497 2 9 ,561 9,561 5 ,200 1,845 ZZ 3, 227 3 9,566 8,566 5,200 1,845 ZZ 2 ,974 9,572 8,572 5,200 1,845 ZZ 2 , 740 5 9 ,582 8,582 5,200 1,845 ZZ 2 , 168 A 15,000 1 9 ,566 8,566 5,200 1,845 22 2,974 2 9 ,572 8,572 5,200 1,845 22 2.740 J 9 ,582 8,582 5,200 1,845 22 2 ,168 A 10,417 9,417 5,200 1,845 ZZ 4 , 589 10,852 9,852 5,200 1,845 25 5 ,587 11 ,288 10 ,288 5,200 1,845 Z J 6 ,830 5 20,000 1 9 , 566 8,566 5,200 1,845 22 2.974 2 9,572 8,572 5,200 1,845 22 2,740 ■J 9 ,582 8,582 5,200 1,845 ZZ 2 , 168 A ** 10,417 9,417 5,200 1,845 22 4 , 589 K 10,852 9,852 5,200 1,845 25 5 .587 11 ,288 10,288 5,200 1,845 ZD D , o JU 6 27,500 1 11,288 10,288 5,200 1,845 25 6,830 2 11,723 10,723 5,200 1,845 25 8,210 J 12 ,164 11 ,164 5,200 1,845 25 99 ,675 A 11,438 10,438 5,200 1,845 25 5 , 153 11,840 10,840 5,200 1,845 Id 6 ,276 0 13,330 12,330 5,200 1,845 28 7,581 7 35,000 1 12,164 11,164 5,200 1,845 28 9,675 2 13,330 12,330 5,200 1,845 28 7,130 3 14,276 13,276 5,200 1,845 28 8,988 4 15,228 14,228 5,200 1,845 32 10,228 5 25,127 24,127 5,200 1,845 36 11,705 6 29,008 28,008 5,200 1,845 39 14,586 8 55,000 1 14,276 13,276 5,200 1,845 28 8,988 2 15,288 14 ,288 5,200 1,845 28 10,228 3 25,127 24,127 5,200 1,845 36 11,705 29,008 28,008 5,200 1,845 39 14,586 5 32,923 31,923 5,200 1,845 42 17,542 £ D 30,642 29,642 5,200 1,845 hZ 18 , 717 9 75,000 1 18,478 17,478 5,200 1,845 32 17,056 2 19,296 18,296 5,200 1,845 36 18,216 3 33,194 32,194 5,200 1,845 45 17,826 4 36,962 35,962 5,200 1,845 48 20,714 5 35,354 34,354 5,200 1,845 48 23,457 6 39,122 38,122 5,200 1,845 50 26,283 10 100,000 1 35.354 34,354 5,200 1,845 48 23,457 2 39,122 38,122 5,200 1,845 50 26,283 3 42,889 41,889 5,200 1,845 53 29,167 -46- TABLE 10 Transition Probabilities by State Alter- State native 1 2 3 A 5 6 7 8 9 1 .9999 .0001 1 .0020 .8666 .1237 .0077 2 .0010 .8611 .1330 .0049 3 .0006 .880A .116A .0026 4 .0001 .8998 .0989 .0012 5 .0001 .9A62 .0536 .0001 1 .038A .8054 .1539 .0023 2 .0287 .8334 .1368 .0011 3 .0217 .8593 .1186 .0004 4 .0116 .8881 .1002 .0001 5 .0037 .9426 .0536 .0001 1 .0006 .0211 .9773 .0010 2 .0001 .0115 .9570 .0314 3 .0001 ,0036 .9869 .0094 4 .0089 .0605 .7544 .1749 .0013 5 .0217 .0668 .6727 .2326 .0062 6 .0024 .0412 .0989 .5525 .2884 .0163 1 .0006 .0072 .9896 .0026 2 .0001 .0031 .9956 .0012 3 0 .0003 .9996 .0001 4 .0003 .0086 .0240 .9156 .0514 .0001 5 .0017 .0200 .0353 .8479 .0941 .0010 6 .0071 .0336 .0602 .7380 .1540 .0071 1 .0013 .0058 .0336 .7982 .1611 2 .0013 .0051 .0119 .0511 .7129 .2177 3 .0047 .0092 .0205 .0695 .6360 .2601 4 0 .0033 .0071 .8928 .0968 5 .0004 .0026 .0226 .8275 .1469 6 .0013 .0055 .0276 .9632 .0024 1 .0023 .0025 .0131 .0825 .8921 .0075 2 0 .0003 .0023 .0318 .9640 .0016 3 .0004 .0008 .0056 .0437 .9406 .0089 4 .0006 .0024 .0086 .0648 .9057 .0179 5 .0002 .0004 .0017 .0123 .7877 .1975 .0002 6 .0003 .0006 .0015 .0099 .6505 .3356 .0016 1 .0012 .9899 .0089 2 .0030 .9791 .0179 3 .0006 .8017 .1977 4 .0009 .6619 .3370 .0002 5 .0012 .0032 .6059 .3877 .0020 6 .0037 .0062 .6709 .3169 .0023 1 .0003 .0005 .0321 .8936 .0735 2 .0001 .0004 .0015 .0364 .8852 .0764 3 0 .0001 .0047 .7243 .2709 4 .0001 .0003 .0083 .6828 .3085 5 .0002 .0004 .0012 .0179 .6788 .3015 6 .0005 .0007 .0019 .0207 .6017 .3745 1 .0009 .0107 .9884 2 .0017 .0133 .9850 3 .0003 .0025 .0155 .9817 -47- Each time a grower makes a transition either to another state or remaining in the same state he earns a reward. If the transition is to a lower-numbered state, then the reward is negative or a cost reflecting the loss incurred to his operating capital supply. A transition to a higher-numbered state re- ceives a positive reward reflecting the net increase to the operating capital fund. For example, if a grower started in State 5 and the next move took, him to State 6, the reward attached would be $7,500. On the other hand, if the transition had caused him to go to State 4, a negative reward of $5,000 would be incurred. If a transition caused him to remain in the same state, the re- ward would be the expected net addition to operating capital based on the cal- culated net income. For example, in State 2, Alternative 1, we have shown this to be $690. Results of the Dynamic Programming Model Dynamic programming maximizes long-run expected income. This is in con- trast to the usual portfolio analysis or budgeting solution of a farm manage- ment problem that attempts to maximize the immediate expected income. That is, dynamic programming not only takes into account the immediate expected in- come for any starting state but also the income received if subsequent transi- tions cause the grower to land in a different state from where he started. For example, if a grower starts in, say. State 3, the program would calculate the rewards from each alternative as well as the rewards that would be forth- coming if in the next period he was in State 4. Therefore, the program must calculate the rewards from starting in State 3, as well as the rewards the grower would receive by following an optimal policy if he lands in State 4, allowing for the probability of making the transition to State 4. Although the expected income from a given alternative within a state may be lower than another alternative, the probability of making a transition to a higher state (with a higher expected income from the included alternatives) may be greater. Therefore, the alternative that maximizes expected immediate gains may not maximize the long-term expected gains if we consider a large number of time periods. Figure 11 shows the expected incomes from each alternative plotted against its standard deviation. The solid black lines in the figure indicate the restrictions Imposed by the supply of operating capital used in defining -48- FIGURE 11 Management Income and Standard Deviation of Income by Supply of a/ Operating Capital (States), Salinas Valley— 36 - 33 - 30 27 24 21 18 15 12 9 6 3 / / / / / / • / 1/ 2 & 3 4 & 5 7. J L 8 J L _L 10 _l_ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Standard Deviation (thousand dollars) a/ The points circled are the optimum alternatives in each State. 28 30 -49- the states. (The dashed curve represents an estimate of the efficiency fron- tier discussed earlier, p. 11.) Since some alternatives were repeated in more than one state, these lines were drawn to show the lowest state in which an alternative first appeared. For example, in State 10, three alternatives are allowed, two of which were also allowed in State 9. However, in Figure 11 only one alternative is shown, indicating that only one new alternative crop- ping program was defined for this state. All of the alternatives first used in States 2 and 4 were again repeated without new additions in States 3 and 5 causing the boundaries to be identical. These should be interpreted to mean that the capital supply available was sufficient to finance these cropping plans, but that it would take more than one year's accumulation of profit to increase the supply to make it adequate to finance a more risky program, i.e., move to a higher state. The table of transition probabilities indicates that although it is possible to make the transition from State 2 to State 4 in one time period, it is a rather low probability event. By defining an intermedi- ate state such as State 3, which accmulates the profits of State 2 following the same cropping plans, it is possible to make the transition to State 4 with a higher degree of probability. The policy iteration method as developed by Howard^^ defines an optimal policy as the set of alternatives that maximizes the present value of income in all states. That is, if the number of time periods is sufficiently long, this method will choose the alternative in each state that maximizes the pres- ent value of expected future rewards. The solutions will show for each level of gross operating capital (state) the cropping program and contractual ar- rangement a grower should follow if his objective is to maximize his long-term income. The optimal policy for this problem using a discount rate of 8 per- cent is shown in Table 11. The results of the dynamic programming analysis, as shown in Figure 10, indicate some results different from those achieved when only immediate income is maximized. The optimal strategy indicated for growers with a very low op- erating capital supply as represented by States 2 and 3 are to follow the al- ternatives which forego the highest immediate average income. Under low capi- tal conditions, the analysis indicates that in order to maximize long-term 11 Howard, R. A., loo. ait. -50- TABLE 11 Optimal Policy by State State Alternative Crop land nercGnt 1 1 a/ 2 1 Carrots (2) 30 Sugar beets 20 Small white beans 30 Lettuce $300 flat 20 3 1 Carrots (2) 30 Sugar beets 20 Small white beans 30 Lettuce (4) $300 flat 20 4 6 Carrots (2) 20 Sugar beets 20 Small white beans 20 Lettuce (2) $135 guarantee 40 5 6 Carrots (2) 20 Sugar beets 20 Small white beans 20 Lettuce (2) $135 guarantee 40 6 3 Carrots (2) 15 Sugar beets 15 Small white beans 10 Lettuce (2) $135 guarantee 60 7 6 Carrots (2) 15 Sugar beets 20 Small white beans 15 Lettuce Cooperative 50 8 5 Carrots (2) 15 Sugar beets 15 Small white beans 10 Lettuce Cooperative 60 q 5 Carrots (1) 20 Sugar beets 10 Lettuce Cooperative 70 10 3 Carrots (1) 20 Lettuce Cooperative 80 a/ Only one alternative was defined for State 1 which indicated a large penalty for being unable to grow any crops. -51- gains , a cropping program with a slightly lower immediate farm income but a higher variability should be followed. In terms of the table of transition probabilities, this means that in order to maximize long-term income the aL- temative with the highest variability of current Income gives a better chance of moving into a higher state after several transitions have occurred. That is, due to the higher variability of Income of Alternative 1 in State 2 the probability is 0.1237 of making a transition to State 3, whereas for Alterna- tive 5, for example, in the same state the probability is 0.0536. Therefore, the optimum alternative sacrifices a small amount of current farm income for a greater probability of shifting to a higher state in the future. This same situation of sacrificing a higher current income for a cropping program with a somewhat lower farm Income but a higher probability of making the transition to a higher numbered state in the future was found to be opti- mal in States 2, 3, 4, 5, and 6. The optimal policy in States 7, 8, and 10 indicates that long-term income can be maximized by choosing cropping programs on the income-variability efficiency frontier. The best alternative for State 9 was one very close in relative terms to the efficiency frontier. These results appear to indicate rather Important conclusions. First, decision making of vegetable growers with the objective of maximizing long- term expected gains must include inefficient alternatives — traditionally, analysis of cropping programs and investment portfolios in the context of income (variance space has precluded consideration of alternatives not located on the efficiency frontier) . The shift in objectives from current income maximization to one of maximizing long-term expected gains is consistent with reality. The frequently observed so-called "inefficient" alternatives may in fact indicate sound planning for long-term gains. Second, analysis of this type of problem that does not Include capital supply as an important variable will result in unrealistic solutions. For example, failure to Include oper- ating capital supplies in the definition of the states would have produced an optimal policy of always following Alternative 3 in State 10. Third, that as capital supplies incresise to a point near to where the productivity of addi- tional units of capital just equal the cost of that capital, solutions will show that less current income needs to be sacrificed in order to achieve a reasonable probability of making the transition to higher states. -52- Implications for Growth Although the dynamic programming model used in this analysis was not de- signed to explain the economic growth of an individual firm (in terms of in- creases in net worth) it does have certain Implications for this type of prob- lem. The model used in this analysis represented a fixed-sized farm in terms of acres of land and the line of machinery available. Only the variable re- sources such as labor £ind capital were allowed to change ailong with the con- tractual agreements. It is suggested that if a grower finds himself in such a resource situation as described by the lower number states, e.g., States 2-6, his best strategy would be to "plunge!" His best chance of success is found in terms of choosing the highest risk (income variability) cropping program permitted by his operating capital supply. Although the cloud of economic failure would be ever present, this would maximize his long-term income by providing a strategy of moving to higher states with their higher expected in- come alteimatives in the most rapid way possible. As operating capital is accumulated — as represented by the higher numbered states — growth can be achieved without a great sacrifice in current Income. Limitations to the Dynamic Programming Model Ideally many more alternatives could have been generated both along the efficiency curve and some distance away. The limited number of alternatives analyzed causes rather large jxmips in the final optimum policy. A more con- tinuous specification of the capital supply in a larger number of states as opposed to the large discrete jumps between states would have made the model more applicable to a larger number of growers. The most limiting factor in the model, however, was the assumption of the normal distribution of net Incomes by which the transition probabilities were determined. The use of a distribution more skewed to the right would, no doubt, have more nearly represented the true situation. The assumption of normality represents a conservative estimate. Inclusion of an analysis of possible changes in consumption as Income levels Increase would have offered a wider choice of possible alternatives to the grower and would also have added more realism. This would have been espe- cially true if the primary purpose of the model had been to explain strategies -53- for the growth of the farm firm where consumption in the present time period would affect the rate of growth in future time periods. Analysis of Actual Adjustments to Risk and Uncertainty In most of the previous discussion and analysis we have considered only the adjustments that a grower should make to risk and uncertainty. In this section the emphasis will be on the adjustments that a sample of growers actually have made . The questions asked are, what cropping programs and what contractual arrangements have vegetable growers actually made within the framework previously outlined, and how do they compare to the theoretical st£Uidards we have discussed? Growers previously contacted in the initial questionnaire survey were re- quested to provide information on the method of marketing their 1962 vegetable production in one of six possible categories. Growers were asked to report the crops and acreage sold by each of the following methods. 1. Sell through marketing cooperative. 2. Sell to highest bidder at harvest time, either in the field or after packing. 3. Contract with no minimum guarantee but split profits in some fixed proportion. 4. Contract with a guaranteed return per acre plus a split of profits. 5. Contract with guaranteed fixed return per acre. 6. Contract with guaranteed price per unit of production only. Sixty growers of all sizes of farms returned usable questionnaires. Using published yield and price information, the gross income received by growers was estimated using contract specifications typical of each of the categories listed above for the period 1950-1962. Mean gross incomes, variances, and co- variances were calculated for each contract category and for each crop reported. Total variance of each of the 60 growers' cropping programs was calculated using the method discussed earlier in this report. The results of these calculations were plotted using average gross income per acre on the -54- vertical axis and standard deviation of gross income per acre on the horizon- tal axis (Figure 12) . A curve was drawn connecting the most efficient farms to indicate an estimate of the efficiency frontier.—^ The data shown in Figure 12 indicates that growers may have both a sub- jective measurement of risk and the desire to diversify their cropping plans. They contract their production to meet these risks. Several conclusions may be warranted. First, the observations are fairly close to the efficiency curve throughout its entire range, as denoted by the close clustering below and to the right of the curve. This indicates that, to increase gross farm income, growers must and actually do accept a relatively large variability of income. In other words, to achieve a 10-percent increase in gross income a greater than proportionate Increase in variability must be accepted. Second, although no good statistical test is available to test the hypothesis, it would appear that there is more divergence from the curve at the lower end of the range than at the upper end. This might be interpreted by drawing some analogies from the dynamic programming results in that those growers in the lower range are short on capital and are accepting high-risk situations or "plunging" in order to accumulate additional capital. When the observations are classified £is to farm size (in acres), there is little correlation between size and location on the curve. This indicates that small farms may be able to accumulate sufficient operating capital per acre to take advantage of being able to assume all or most of the risk similar to the high-numbered states in the dynamic programming model. A suggestion for future research in this area would be to collect de- tailed information on the capital position and age of operator along with these variance-income data in an attempt to correlate these variables in an explanatory model of adjustments to risk and uncertainty. What may appear as rather irrational adjustments (as denoted by the divergence away from the efficiency boundary) may be quite rational vrtien these other variables are taken into account. An additional explanatory variable may be the psychologi- cal makeup of the individual grower and his affirmity or aversion to gambling. 1/ This is not the "true" efficiency frontier which could have only been determined by analyzing every possible proportion, crop, and contract. -55- FIGURE 12 Gross Income and Standard Deviation Per Acre, Salinas Valley Standard Deviation of Gross Income A grower with adequate capital resources may be willing to operate at the lower end of the curve, due to a high preference for insurance against mone- tary loss . -57- APPENDIX f 1 TABLE Al Correlation Coefficients , Net Income Spring lettuce Summer lettuce Fall .ettuce Sugar beets Small white beans Carrots Variance Standard deviation 1 1 2 1 3 1 4 1 1 2 1 3 4 1 1 2 1 3 1 ^ 1 1 2 Spring lettuce 1 2 3 4 1.00 1.00 1.00 1.00 - .62 - .64 - .61 .49 - , 12 - .01 .01 .50 d4 ,635 .Oo 13,784.76 12,982.15 25.28 on "7/. lAO 117.408516 113.939255 5 .027716 Summer lettuce 1 2 3 4 - .62 - .64 - .61 - .49 1.00 1.00 1.00 1.00 72,336.06 18,457.99 17,379.57 24.67 268.953636 135.860188 131.831587 4.967735 Fall lettuce 1 2 3 4 - .12 .01 .01 .50 .34 .41 .40 1.00 1.00 1.00 1.00 1.00 39,925.29 10,523.31 9,839.66 34.14 199.813137 102.583179 99.195050 5.843065 Sugar beets - .57 - .42 .23 .21 .41 .22 .41 .21 .05 .06 .06 .01 1.00 1,271.78 35.662105 Small white beans .34 .45 .42 .64 - .44 - .44 - .44 .64 - .24 - .10 - .11 .64 .10 1.00 1,203.49 34.691434 Carrots 1 2 .01 .02 .18 .20 .08 .10 - .19 .50 .50 .49 .51 .51 .52 .51 - .19 - .18 .77 .71 .75 .76 .74 .75 - .19 - .18 .17 .01 - .38 - .40 1.00 1.00 15,181.75 26.50 123.214225 5.148225 03 O