Utility Measurement for ThoseWho Need to Know

A. N. Halter and Robert Mason

A practical technique for estimating decision-makers' utility functions by survey orgroup methods is explained and illustrated. Results from a survey of 44 Oregon farmersare reported. Risk attitudes of respondents are related to farm and decision-makercharacteristics. Regression analysis found age, education, and percentage of land owner-ship, either separately or jointly, to be statistically significant variables related to riskattitude. Risk attitudes measured from the estimated utility functions were found to beuniformly distributed across risk aversion, neutral and preference. Even though furtherempirical work is needed, it appears that the distribution of risk attitude among thehuman population cannot be predicted from a single variable.

There appears to be sufficient need formeasuring decision-makers' utility functionsin empirical and practical situations to justifyshowing how to do it in an efficient mannerthat anyone can apply. Most of the literatureon eliciting utility functions has dealt withconceptual developments, perfection of pro-cedures, and has remained at a fairly theoret-ical and abstract level [Anderson, et al.].Empirical tests of behavioral hypotheseshave been done with extremely small sam-ples and lend little credence to the conclu-sions [Conklin, et al; Halter and Dean; Lin, etal.; Officer and Halter]. What is lacking aregood applied prescriptions from experiencedusers on how to implement these proceduresin an efficient, systematic fashion for a widerange of situations. Users of such procedures,besides empirical researchers who needlarger samples, are extension teachers, class-room teachers, consultants, and those withbusiness-customer relationships [Holt andAnderson; Robison and Barry].

The technique for obtaining the numbersrequired to estimate a utility function from a

A. N. Halter is with the Electric Power Research Insti-tute, Palo Alto, California and Robert Mason is with theSurvey Research Center, Oregon State University.

Oregon Agricultural Experiment StationTechnical Paper No. 4894

decision-maker has evolved considerablysince our attempts in the Interstate Manage-rial Study [Halter]. Our latest study designwas based on our ensuing research and teach-ing experience and that of others, [Anderson,et al.; Conklin, et al.; Lin, et al.; Officer andHalter] and provides a means for derivingutility functions for large numbers of indi-viduals at low cost. This means we can nowbegin to formulate empirical hypothesesabout the distribution of risk attitudes, theirdeterminants and/or consequences in thehuman populations. Also, it means that prac-titioners can begin to apply these proceduresin situations where knowing something abouta client's risk attitude may make a differencein the outcome.

Finally, even those who suggest indirectmethods for measuring risk attitude willsome day need to know, if they are to makemeaningful comparisons or to show theirmethods superior [Meyer, 1977a; 1977b;Moscardi and de Janvry; Porter et al.; Robi-son; Robison and Barry].

Utility Measurement Technique

The technique to obtain numbers for util-ity function estimation is to construct ques-tions presenting a choice between two uncer-tain alternatives, and to ask the respondent tomodify his answer to the questions until he

99

Utility Measurement

indicates that he is indifferent between thealternatives. The questions are asked in con-junction with the following format which canbe presented on a small card or blackboard.

Game 1

Alternative Actions

Probability '/2

ofOccurrence 1/2

A,d c

a b

The four positions are filled in the followingway: a is the respondent's income level(rounded);' b is 3/4 of the respondent's incomelevel or 3/4 a (rounded); c is varied in askingthe question until indifference is obtainedand is set initially between b and d; and d isset equal to zero. The numerical relationshipbetween the four positions will be in the al-phabetical order indicated:a> b > c> d.

The question asked is: Which alternativeaction do you prefer if you know that withprobability /2 your monetary outcome wouldbe either a or d if you take action A, andeither b or c if you take action A,? This ques-tion can be motivated with a brief discussionof the riskiness of farming and can be de-tailed to include the outcomes of the two al-ternatives. After a very few warm-up ques-tions, most respondents understand the situ-ation and proceed without further motiva-tion.2 Students in a classroom situation usual-ly respond quickly after the group dynamicstake over.

If the respondent picks A,, this means c istoo small and should be increased. If the re-spondent picks A2, this means c is too large

1We either ask the respondent directly for his approxi-mate gross income or we present him with a card thatshows monetary intervals and ask him to indicate whichinterval contains his previous year's gross income.

2Instructions to interviewers, layout of items on the in-terview schedule and card format used are availableupon request from the Survey Research Center, Ore-gon State University.

100

and should be decreased. Thus, c is varieduntil the respondent finds it difficult to de-cide between A, and A2 and you concludethat he is indifferent, or he tells you.

The second "game" is constructed from thefirst game by replacing the zero or d positionwith the monetary value of the indifferencepoint from Game 1.

Game 2

Alternative Actions

A1 A2Probability 1/2 Indifference 1 c'ofOccurrence 1/2 a b

The question is phrased again and thenumber in position c' varied until indiffer-ence is indicated. The prime on c indicatesthat the number will be different now. The aand b positions are as in Game 1 and themonetary size relationship among the fourpositions is still a > b > c' > Indifference 1.

With two games you have sufficient infor-mation to calculate one point on the respon-dent's utility function after anchoring thefunction to an origin and one other point.

The formula for making the calculation is:(1) Utility value of Indifference point 1 =

1/2 (utility of zero and utility value ofIndifference point 2) or in shorthand:

u (Ind. 1) is 1/2 [u (0) + u (Ind. 2)].Now let one anchor point be [0, u(0)], theorigin, and the other be [Ind. 2, u (Ind. 2)] =[Ind. 2, 200] and hence:

u (Ind. 1) = 1/2 (0 + 200) = 100.The calculated point on the utility function

is: (Ind. 1, u (Ind. 1) = 100).We will give several illustrative examples

from our survey and the reasons why theformula works after setting up more games toobtain additional points. The additionalgames (points) are necessary to extend theutility function so that it will include thegross income point and beyond if desired.

Game 3 has the same relationship amongpositions, but is tied to the first two games byusing the monetary parts of the three points,i.e., 0, Ind. 1, and Ind. 2.

Halter and Mason

Western Journal of Agricultural Economics

Probabilityof

Game 3

Alternative Actions

A1 A2

1/2 0 Indifference 1

Occurrence 1/2 a' Indifference 2

In this game the number in the a' positionis varied until Indifference point 3 is reached.The prime on a indicates that the numberwill be different now. However, we expectthe preference ordering a' > Indifference 2> Indifference 1 > 0 as before.

This basic structure can then be extended

to cover the desired number of points. This isillustrated by Game 4 where we rotate,counterclockwise, the three indifference val-ues, leaving the a" position variable for find-ing Indifference point 4.

Game 4

Alternative Actions

A1 A2

Probability /2 Indifference 1 Indifference 2ofOccurrence 1/2 a" Indifference 3

Since we know the utility values for threeof the monetary positions in Games 3 and 4,we calculate the utility of the new indiffer-ence points by addition and subtraction.

Using the letters to stand for the four posi-tions we have:

(2) u (a')= u (b) + u(c)- u(d)

Using: u (b) = u (Ind. 2) = 200

u (c) = u (Ind. 1) = 100

u (0) = (0),

then: u (Ind. 3) = 200 + 100 - 0 = 300from Game 3.

Using from Game 4:

u (b) = u (Ind. 3) = 300

u (c) = u (Ind. 2) = 200

u (d)= u (Ind. 1)= 100,

then: u (Ind. 4) = 300 + 200 - 100 = 400.

We now have 4 calculated points and twoanchor points to use in estimating the utilityfunction. We will illustrate the calculationsusing the answers from three respondents inour survey that are shown in Table 1.

We note that the three respondents werepicked because they have the same gross in-come of $150,000 in position a of Game 1 and2. From the utility functions estimated, andshown in Figure 1, you will note that thethree cases illustrate three different types offunctions; this is another reason for pickingthese three cases. Position b is 3/4 of$150,000, which rounds to $115,000. In posi-

TABLE 1. Numerical Indifference Values Obtained from Three Respondents(a)

Individual 1

Game 1Game 2Game 3Game 4

Individual 2Game 1Game 2Game 3Game 4

Individual 3Game 1Game 2Game 3Game 4

a

$150,000150,000180,000210,000

a$150,000150,000225,000275,000

a$150,000

150,00077,000

127,500

b$115,000

115,000100,000180,000

b$115,000115,000170,000225,000

b$115,000

115,00042,50077,500

Positionc

$ 65,000100,00065,000

100,000

c

$107,000170,000107,000170,000

c

$ 25,00042,00025,00042,500

d$ 0

65,0000

65,000

d$ 0

107,0000

107,000

d$ 0

25,0000

25,000

(a)Numerical values corresponding to indifference points are underlined.

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Western Journal of Agricultural Economics

tions c and d of Game 1 and 2 for Individual 1we have two indifference points, $65,000 and$100,000, and the anchor value zero. Fromformula (1) we calculate:

u ($65,000) = /2 [u ($0) + u($100,000)]. After assigning

u (0) = 0 and u (100,000) = 200,we obtain

u (65,000) = 100.

From formula (2) we obtain from Game 3:

u (180,000) = u (100,000) + u(65,000) - u (0)

u (180,000) = 200 + 100 - 0 = 300.

From Game 4:

u (210,000) = u (180,000) + u(100,000) - u (65,000)

u (210,000) = 300 + 200 - 100 = 400.

In positions c and d of Game 1 and 2 forIndividual 2 we have the Indifference points$107,000 and $170,000. From Formula (1) wehave

u (107,000) = 1/2 [u (0) + u (170,000)].After assigning

u (0) = 0 and u (170,000) = 200,we obtain:

u (107,000) = 100.

From Formula (2) and Game 3:

u (225,000) = u (170,000) + u(107,000) - u (0)

u (225,000) = 200 + 100 = 300.

From Game 4:

u (275,000) = u (225,000) + u(170,000) - u (107,000)

u (275,000) = 300 + 200 - 100 = 400.

In position c and d of Game 1 and 2 for Indi-vidual 3 we have the indifference points$25,000 and $42,500. From Formula (1) wehave:

u (25,000) = /2 [u (0) + u (42,500)]and

u (25,000) = 100, after assigning:

102

u (0) = 0 and u (42,500) = 200.

From Formula (2) and Game 3:

u (77,500) = u (42,500) + u (25,000)- u (0)

u (77,500) = 300

From Game 4:

u (127,500) = u (77,500) + u (42,500)- u (25,000)

u (127,500) = 300 + 200 - 100 = 400

The five points for each individual are plot-ted in Figure 1. Polynomial equations (up to3rd degree) were fitted to the points by ordi-nary least squares and illustrate three differ-ent types of utility functions, i.e., from leftto right, quadratic, linear and cubic. We willhave more to say about this when we discussthe results from our sample survey.

For the moment, notice that the measuredutility numbers always occur at the same val-ues, i.e., 0, 100, 200, 300, and 400. The dif-ferences between the curves occur becauseof the differences in dollar amounts whichcorrespond to the indifference points ob-tained from the games. Also notice that thedifferences between the utility numbers arealways the same, i.e., 100. Hence, the gamestructure establishes equal intervals of utilitybetween indifference points. In fact, thelength of this interval was established whenthe anchor points for the utility function weredesignated in Formula (1). Formula (1) comesfrom equating two equal intervals fromGames 1 and 2.

Look again at Game 1. When the respondentindicates indifference between actions 1 and2, he is telling us that his expected utilityfrom each action is the same. We can writethe expected utility for each action as follows:

½/2u (d)+ ½/2u (a)= ½/2u (c)+ /2u (b).

Rearranging, we have:

u (a) - u (b) = u (c) - u (d)

From Game 2, at the point of indifferencethe expected utility of each action is:

1/2 u (Indifference 1) + 1/2 u (a) =1/2 U (c') + 1/2 u (b)

December 1978

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Rearranging, we have:

u (a) - u (b) = u (c') - u (Indif-ference 1).

The common interval between Game 1 and 2is u (a) - u (b) and therefore:

u (c) - u (d) = u (c') - u (Indif-ference 1).

Now since c is Indifference point 1:

2 u (Indifference 1) = u (d) + u (c'),

and since c' is Indifference point 2 and d isequal to zero we have:

u (Indifference 1) = 1/2 [u (0) + u

(Indifference 2)].

Making the assignment of u (0) = 0 and u(Indifference 2) = 200 establishes the lengthof the utility interval from zero to Indiffer-ence point 1 of 100 utils. The other games arethen structured to maintain this interval be-tween subsequent indifference points. Thatis what makes the procedure work and whythe games are structured in the way thatthey are.

Utility Functions andEmpirical Applications

With an efficient method of obtaining util-ity numbers and an ordinary least squares re-gression routine one can now estimate utilityfunctions for larger samples of decision mak-ers than was previously possible [Conklin, etal.; Halter and Dean; Lin, et al.]. With largersamples more interesting empirical hypoth-eses can be formulated and tested. Fromthese tests and refutations new theoreticaldevelopments could be made. We reporthere on a sample of 44 decision makers who

3Other functional forms could be fitted. Theoretical de-velopments indicate that polynomial forms may not bethe best choice for some purposes. There is little empir-ical evidence to indicate that such a choice makes anydifference; however, the choice of function has beendiscussed recently in nonagricultural settings [Cohn, etal., Friend and Blume].

4These included: acres of grain, acres of grass seed, acres

of other crops, dollar investment in livestock, percent ofacres of grass seed of total farm acreage, percent of

were farming in Oregon at the time of inter-viewing in 1974 and are known in the Willa-mette Valley as "grass seed farmers." Threeof the utility functions estimated from thissample were shown in Figure 1. The entiresample showed linear, quadratic and cubicequations in almost equal proportions whenR2 and visual inspection were used as criteriaof goodness-of-fit. 3 .

In addition to the utility games our surveyobtained data on 11 farm and decision makercharacteristics. 4 These are the usual kinds ofvariables that one suspects might be relatedto risk attitudes [Halter; Halter and Dean;Officer and Halter]. Since there has not beenmuch theoretical work completed on whichto base the hypothesized relationships andsamples of too small a size to make empiricalestimates of parameters in the past, we had torely on regression methods to sort out mean-ingful relationships. The results from thisanalysis, while subject to further investiga-tion, indicate that the numbers obtained bythe utility measurement technique describedabove are capable of distinguishing amongindividuals who have other characteristicswhich are more directly measureable.

We took as the dependent variable a mea-sure of risk attitude that is defined as thenegative ratio of the second to the first de-rivative of the utility function evaluated atthe respondent's gross income level. This iscalled the Pratt coefficient after its founderand has the splendid property that it can becompared among individuals whereas the in-dividual utility functions cannot be so com-pared [Henderson and Quandt; Pratt; Robi-son]. The risk attitude so derived is only de-fined at a point and is, thus, relative to thepoint chosen. The utility function itself can-not be characterized as to risk attitude in its

acres in annual grasses to total grass seed acreage, re-spondent's age, educational level, percent ownership ofland operated, percent of total dollar investment whichwas money owed as a loan, mortgage or unpaid bills,

and the number of years lived on present farm. Notethat gross farm income is not included in this list. Theobserved correlation between the Pratt coefficientevaluated at the gross income level and gross incomewas -0.03.

104

December 1978

Utility Measurement

entirety unless it is linear throughout, i.e.,risk neutrality [Johnson]. When the 44 utilityfunctions were evaluated for the Pratt coeffi-cient at the decision maker's 1973 gross in-come level and classified by the sign of thecoefficient into risk averse, risk neutral, andrisk preference we found that the numberfalling into each classification was aboutequal.

The eleven farm and decision makercharacteristics were entered into a stepwiseregression routine with risk attitude (Prattcoefficient) as the dependent variable. Threevariables were selected for further considera-tion. They were (in order of their statisticalinfluence) percent of land owned, educa-tional level and age. All 11 variables werealso subjected to a backstep analysis whichselected the least important variable first anddropped it from the model. This proceduredropped (in order of their least statistical in-fluence) years on the farm and percent in-debtedness. The variables of percent of landowned, educational level and age remainedin the model. Visual inspection of the plotteddata showed nonlinear trend lines and hencea second stepwise analysis was performedwhich included the linear and quadraticterms of the three variables as well as theirlinear interaction terms.

Results

Results of the regression analysis areshown in Table 2.

Linear terms for the three variables aresignificant as is the quadratic term for educa-tional level and for ownership by educationand education by age interaction terms. Signsfor the coefficients for the linear terms of thevariables remained unchanged during thestepwise-backstep analysis. Signs of the par-tial regression coefficients for the linearterms of these variables also remained un-changed throughout the regression analysis.

Evaluation of effects by inspection of re-sults in Table 2 is difficult without plottinggraphs for the variables in the model. Thesegraphs are presented in Figures 2-6. Mea-sures of risk attitude are plotted on the verti-cal axis with positive values signifying riskaversion and negative values signifying riskpreferring.

Regression of risk attitude on percentownership for five levels of education isshown in Figure 2. The strong interaction ef-fect implies greater risk preference amonghigher educated farmers as percent owner-ship increases. Contrariwise, grade-schooleducated farmers demonstrated greater riskaversion with increasing levels of ownership.

TABLE 2. Results of Statistical Tests for Three Variables Remaining

Regression Stand. error of Student'sVariable Coefficient regress. coef. "t" value

Constant ........ 11.547 7.152 1.64Percent owner ... 0.192 0.041 4.70Education ....... -6.793 2.854 -2.38Age............. -2.088 0.933 -2.24Education squared 1.088 0.360 3.02Percent owned

x education.... -0.060 0.015 -4.08Education x age.. 0.587 0.284 2.07

(Variables excluded from model)Percent ownership

squared .......- 1.188Age squared .....- 0.071Percent owned x

age ........... -1.637

105

Halter and Mason

Western Journal of Agricultural Economics

Figure 2. Regression of risk attitude on per-cent ownership for five levels of education.

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106

Figure 5. Regression of risk attitude on edu-cation for two levels of age.

December 1978

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Figure 6. Regression of risk attitude on agefor three levels of percent ownership.

Age effects were held constant at the meanvalue for this figure by inclusion of the agemean in the constant term. Figure 3 shows asecond view of ownership and education in-teraction effects and the nonlinearity of therelationship. Farmers who owned high per-centages of their land tend to show a riskpreference attitude with increasing educa-tional levels. Moreover, farmers who owneda relatively small percentage of land tend tobecome risk preferers, then risk averse withincreasing educational levels. Data in thesetwo figures suggest that a major shift occursamong the regression lines in which educa-tion level plays a role. The slope for the edu-cation levels in Figure 2 changes from posi-tive to negative for post-high school farmers.And, all percent ownership lines intersect atthe same post-high school point in Figure 3.

Education effects change dramaticallywhen regressed in conjunction with age, asshown in Figures 4 and 5. (Again, effect ofpercent ownership was held constant at themean value by inclusion of the percent own-

ership mean in the constant term). Collegegraduates, according to Figure 4, becomemore risk averse with increasing age whilegrade school-educated farmers become morerisk preferring. Viewing the age-educationrelationship from a different perspective,Figure 5 shows that grade school-educatedfarmers over 50 are more risk preferring thanthose under 50 but become more risk aversewith increasing educational levels. Farmerswho completed college and who are over 50are more risk averse than are those who areunder 50.

Finally, the relationship between risk atti-tude, age and percent ownership is examinedin Figure 6. Risk preferences increase withage for all levels of percent ownership butolder farmers with relatively low levels ofownership exhibited greater risk preferencethan did those with high levels of ownership.(Educational level was held constant by in-clusion of the education mean in the constantterm.)

In summarizing the data we suggest thatone cannot account for the observed trendsbetween any one variable and risk attitudewithout considering effects of the other vari-ables jointly or conditionally. Educationlevel, for example, interacts with both ageand percent ownership and any evaluation ofthese two effects should consider level ofeducation jointly.

Viewing the relationships from a temporalcause-effect perspective, one could interpretthe results to mean that higher levels of edu-cational attainment lead to a greater likeli-hood of attaining ownership over resources,which results in a person being less risk-averse. However, one could also view it froma psychological motivation perspective andsay that the risk index is a measure ofachievement motivation which results inhigher educational attainment and greaterskill in acquiring ownership of economic re-sources and of wealth itself. This wouldsuggest that the risk attitude is acquiredrather early in life and is relatively stablethroughout one's lifetime.

The effect of chronological age is held con-

107

Utility Measurement

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Western Journal of Agricultural Economics

stant in this interpretation. The experience ofdecision making, possibly a function of age,suggests a slightly different interpretation.This interpretation implies that a high levelof achievement need is associated with a setof behaviors that is commensurate with anefficient entrepreneurial role. Among therole behaviors identified by McClelland andelaborated by Brewster is that of risk taking.Here, subjects with high need for achieve-ment more often choose to select tasks withmoderate risks than do subjects with lowneed achievement in games of skill as well asin games of chance. They tend to work insituations where their skill is most likely topay off in terms of success, or, in gamblingsituations, prefer the safest bet they can get.They do not gamble on "long shots" wherelarge amounts are rarely won, nor do theyselect tasks where only a small reward is as-sured. These latter strategies are associatedwith those of low achievement motivation.Thus, we find that older, less educated farm-ers are more risk preferring and that older,well educated farmers are risk averse. Thisinterpretation assumes that the effect of per-cent ownership is held constant and impliesthat the achievement motive is subject tochange through time as one ages.

Strictly speaking, one needs longitudinaldata - data gathered in more than one timeperiod - to draw conclusions about time ef-fects and trends. Age effects are confoundedwith cohort characteristics, historical periodand the biological effects of aging itself. Ide-ally, to disentangle such effects we wouldneed to study several cohorts at the same ageand measure the changes that occur throughtheir lifetimes. Even this strategy would notpermit us to remove all the confounding ef-fects mentioned above, but it would permitus to make finer discriminations than are pos-sible through cross-sectional analysis.

Summary and Conclusions

We have shown how to empirically esti-mate utility functions for monetary gains.The procedures are easy to understand, lowcost to apply, and efficient in terms of time

108

required to obtain answers when one recog-nizes the questions to ask.

We were not surprised that differentshaped utility functions fitted the data pointsbecause we do not believe that all decisionmakers have diminishing marginal utility formonetary gains for all sizes of gains [Johnson;Roumasset]. Furthermore, we were not sur-prised when we evaluated the Pratt coeffi-cient at the respondents' gross income pointthat the coefficients were not all positive.This was not surprising because we do notbelieve that all decision makers can becharacterized by positive Pratt coefficients,i.e., risk aversion. However, we do believethat the equal proportions result should besubjected to further empirical tests. Oneneed not expect equal proportions, but thatdoes not mean that there are no risk prefer-ences among farmers for any size of gain.

The relationships that we obtained be-tween age, education, and resource owner-ship and risk attitude are most interesting butare in need of further theoretical elaborationand empirical testing. Implications for educa-tional programs and policy formulation areclear: one cannot depend upon a single vari-able to predict the distribution of risk at-titude among the human population.

Finally, further empirical work needs to bedone with respect to monetary losses andhow to obtain the utility function and its im-plications across both gains and losses.

References

Anderson, J. R., J. L. Dillon, and J. B. Hardaker. Ag-ricultural Decision Analysis, Iowa State UniversityPress, Ames, 1977.

Brewster, John M. The Achievement Motive and Entre-preneurial Behavior. Washington, D.C., EconomicResearch Service, U.S. Department of Agriculture,1962.

Cohn, Richard A., Wilbur G. Lewellen, Ronald C.Lease, and Gary S. Schlarbaum. "Individual InvestorRisk Aversion and Investment Portfolio Composition"Journal of Finance 30 (1975): 605-620.

Conklin, F. S., A. E. Baquet, and A. N. Halter. ABayesian Simulation Approach for Estimating Value ofInformation: An Application to Frost Forecasting.

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Utility Measurement

Tech. Bul. 136, Oregon Agricultural Experiment Sta-tion, Oregon State University, Corvallis, Oregon.

Friend, Irwin and Marshall E. Blume. "The Demand forRisky Assets." American Economic Review 65 (1975):900-922.

Halter, Albert N. "Measuring Utility of Wealth AmongFarm Managers." Unpublished Ph.D. thesis, Michi-gan State University, 1956.

Halter, Albert N. and G. W. Dean. Decision Under Un-certainty with Research Applications. Cincinnati,South-Western Publishing Co., 1971.

Henderson, J. M. and Richard E. Quandt. Microeco-nomic Theory: A Mathematical Approach. New York,McGraw-Hill Book Co., 2nd Edition, 1971.

Holt, John and Kim B. Anderson. "Teaching DecisionMaking Under Risk and Uncertainty to Farmers."American Journal of Agricultural Economics 60 (1978):249-253.

Johnson, Glenn L. "Contributions of Economists to aRational-Decision-Making Process in the Field of Ag-ricultural Policy." Proceedings Issue of the Nairobi(1976) Conference of the International Association ofAgricultural Economists. Oxford, England.

Lin, William, G. W. Dean, and C. V. Moore. "An Em-pirical Test of Utility Versus Profit Maximization inAgricultural Production." American Journal of Ag-ricultural Economics. 56 (1974): 497-508.

McClelland, David C. The Achieving Society. New York:D. Van Nostrand Co.. 1961.

Meyer, J. "Choice Among Distributions." Journal ofEconomic Theory. 14 (1977a): 326-336.

."Second Degree Stochastic Dominance WithRespect to a Function." International Economic Re-view. 18 (1977b): 477-487.

Moscardi, Edgardo, and Alain de Janvry. "Attitudes to-ward Risk Among Peasants." American Journal of Ag-ricultural Economics. 59 (1977): 710-716.

Officer, R. R., and A. N. Halter. "Utility Analysis in aPractical Setting." American Journal of AgriculturalEconomics. 50 (1968): 257-277.

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