World Bank Reprint Series: Number 207 Gershon Feder and Gerald T. O'Mara Farm Size and the Diffusion of Green Revolution Technology On Information and Innovation D'ffusion: A Bayesian Approach Reprinted with permission from Economic Development and Cultural Change, vol. 30, no. 1 (October 1981), pp. 59-76; and American Journal of Agricultural Economics, vol. 64, no. 1 (February 1982), pp. 145-47. Farm Size and the Diffusion of Green Revolution Technology* Gershon Feder and Gerald T. O'Mara Development Research Center, World Bank I. Introduction The introduction of high-yield-crop varieties (HYV) is one of the more significant technological changes taking place in less developed countries' agricultural sectors. This so-called Green Revolution tech- nology consists of hybrid seeds, chemical inputs (fertilizers, pesti- cides), and special cultivation practices. The yield potential and the income-generating capacity of the modern variety are significantly su- perior to those of the traditional crops, when properly cu-ltivated. Fur- thermore, the technology is divisible and neutral to scale. Thus, one may expect that adoption patterns will not be affected by farm size. But the facts are, as the record of diffusion experience in a substantial number of regions throughout the world shows, that adoption rates and the time pattern of adoption are related to farm size. One obvious reason for differential adoption rates in many regions is the credit constraint. Working capital requirements associated with the new technology are substantially higher (fertilizers, pesticides, and hybrid seeds are cash inputs). Thus, where credit for smaller farmers is severely limited, they may not be able to adopt HYV at the same rate as larger farmers.' * The paper benefited from useful comments by W. C. Thiesenhusen and an anon- ymous referee. M. Parthasarathy provided valuable assistance with the computations. The views expressed in this paper are those of the authors and do not necessarily reflect the views of the World Bank or its affiliate organizations. I For instance, in a study of Indian agriculture, Bhalla (1979) reported that small and large farmers differed in the reasons offered for not using fertilizer in 1970-71. Lack of credit was a major constraint for 48% of small farmers and only 6% of large farmers. Bhalla concludes- that "access to credit may be responsible for the gain in income (and HYV area) made by the large farmers." Similarly, in a study of HYV adoption by Pakistani farmers, Lowdermilk (1972) found that a majority of small farmers reported shortage of funds as a major constraint on fertilizer use. Khan (1975) found that small farmers in Pakistan faced difficulties in obtaining fertilizer and seeds because they had no way to pay the premium prices during the sowing season for these inputs. Both Rochin © 1981 by The University of Chicago. 0013-0079/82/3001-0012$01.00 60 Economic Development and Cultural Change A number of other obstacles discriminating against smaller farmers are mentioned in the literature,2 but the most frequiently cited one is the interaction between the uncertainty associated with the innovation and the prevalence of risk aversion among farmers. Unlike the credit constraint, which is exogenous to the farmers, risk aversion is an en- dogenous factor and, thus, the implications of risk aversion in terms of farmers' decisions may change if farmers' perceptions change. Since the new technology is perceived as (and sometimes it indeed is) more risky, and provided that smaller farmers are more risk averse, it is argued that smaller farmers will be less inclined to adopt the innovation. However, when more rigorous analysis is applied to the relation be- tween risk aversion and farm size, the argument is not as straightfor- ward as it seeins at first glance. Assuming that (i) no fixed adoption costs are required; (ii) exogenous constraints (such as credit) are not effective; and (iii) that farmers are aware of the superiority, on the average, of the HYV, one can show that some adoption is beneficial for all risk-averse farmers, irrespective of the degree of uncertainty or of the size of the farm. Furthermore, it is quite- possible theoretically that even though smaller farmers are less willing to undertake risks in terms of the absolute size of income involved, they may demonstrate a higher rate of adoption (in terms of the relative share of land they allocate to HYV) compared to larger farmers.3 Thus, when the as- sumptions above are valid, the risk-aversion argument cannot explain (1973) and Bose (1974) found that small farmers in Bangladesh were at a disadvantage in obtaining scarce credit. Similarly, Frankel (1971), Rajpurohit (1972), and Wills (1972) found evidence of difficulty of access to credit by small farmers in India, with adverse consequences on their choice of new technology. See S. S. Bhalla, "Farm Size, Pro- ductivity, and Technical Change in Indian Agriculture," in Agrarian Structure and Pro- ductivity in Developing Countries, ed. R. Berry and W. Cline (Baltimore: Johns Hopkins University Press, 1979); Max Lowdermilk, "Diffusion of Dwarf Wheat Production Tech- nology in Pakistan's Punjab" (Ph.D. diss., Cornell University, 1972); M. H. Khan, The Economics of the Green Revoluition in Pakistan (New York: Praeger Publishers, 1975); S. W. Bose, "The Comilla Co-operative Approach and the Prospects for Broad-based Green Revolution in Bangladesh," World Development 2, no. 8 (August 1974): 21-28; R. 1. Rochin, "A Study of Bangladesh Farmers' Experiences with IR-20 Rice Variety and Complementary Production Inputs," Bangladesh Economic Reviewv 1, no. I (January 1973): 71-94; F. R. Frankel, India's Green Revolution-Economic Gains and Political Costs (Princeton, N.J.: Princeton University Press, 1971); A. R. Rajpurohit, "Study of the HYV Programme in Mysore (1967-68): Mexican Wheat in Bijapur District," mim- eographed series no. 17 (Poona: Gokhale Institute, 1972); and I. R. Wills, "Projection of Effects on Modern Inputs on Agricultural Income and Employment in a C.D. Block, U.P., India," American Journal of Agricultural Economics 54, no. 3 (1972): 452-60. 2 See Wayne Schutjer and Harlin G. Van Der Veen, Economic Constraints on Ag- ricultural Technology Adoption in Developing Nations, USAlD, Occasional Paper no. 5 (Washington, D.C.: USAID 1977); and Michael Lipton, "Inter-Farm, Inter-regional and Farm-Non-Farm Income Distribution: The Impact of the New Cereal Varieties," World Development 6 (March 1978): 319-37. 3 These results depend on the properties of the underlying utility function with respect to the notions of absolute and relative risk aversion. See Gershon Feder, "Farm Size, Risk Aversion and the Adoption of New Technology under Uncertainty," Oxford Economic Papers 32, no. 2 (1980): 263-83. Gershon Feder and Gerald T. O'Mara 61 the prevalence of nonadoption, at least in the early years, by smaller farmers; nor does it provide an explanation for the higher share of land planted to HYV observed on larger farms in many regions. Should it thus be concluded that small farmers' higher risk aversion per se is not a valid explanation (or a complementary explanatory factor) of observed adoption pattern? "Not so," is the conclusion pro- pounded in the present paper. The nature of new innovations, and the limitations of the rural environment are such that fixed adoption costs do exist, in which case higher risk aversion among smaller farmers is a factor which can explain, by itself, the differential, farm-size depen- dent pattern of technology adoption observed, from both static and dynamic perspectives. In order to isolate the role of risk and risk aversion in generating differential behavior by farm size, all other factors which may cause differentiation will be held identical: All farmers are assumed to possess identical utility functions and to share the same perceived notions of risks involved with the new technology. Similarly, all farmers face the same prices (for both outputs and inputs), have the same quality of land, and none are affected by exogenous constraints such as credit or labor constraints, etc. Thus, the only thing differing among farmers is, in the framework of the present paper, the size of land holding. These assumptions are just analytical devices which will allow a proper assessment of risk aversion and its role as a factor inhibiting diffusion of innovations among smaller farmers. Furthermore, the dynamic pat- tern of adoption as implied by the present analysis can be compared with the diffusion process as observed in reality, so as to reveal the extent to which the latter can (or cannot) be explained by risk aversion and land distribution alone. The program of the paper is as follows: The next section presents the theoretical decision model and its implications for optimal farmers' behaviors. These are followed by a specific example which serves as a basis for numerical situations tracing the pattern of adoption in a hypothetical rural region over time. The simulations provide insights regarding the individual and the aggregate adoption performance and demonstrate the income distribution effects of the innovation at any point in time and over time. II. The Model and Its Implications It is assumed that two distinct technologies are available to farmers: The traditional technology requires no specialized inputs and yields a given net return per acre with certainty. The new technology is char- acterized by superior average performance per acre if certain special- ized inputs (e.g., fertilizers, improved seeds) are applied, and appro- priate cultivation practices (e.g., proper timing of planting and weeding) are observed. However, the yield from an acre allocated to the modern 62 Economic Development and Cultur al Change crop4 is associated with some degree of uncertainty. This uncertainty stems from both objective and subjective factors: In the first instance, the modern, high-yielding variety may be more susceptible to diseases, pest damage, and climatic variation. The dependence on timely avail- ability of the specialized inputs introduces another elemnent of objective uncertainty. Subjective uncertainty is almost inevitable with any in- novation being introduced into a system where farmers have for many years known only the traditional .echnology. It is likely, however, that over time the subjective uncertainty will be reduced as information regarding the innovation spreads, information and marketing channels are established, and the experience gained by the early adopters affects the perceptions of other farmers.' Also, research aimed at reducing pest and weather susceptibility may eliminate some of the objective uncertainty. While per acre input costs are scale neutral, one can conceive of several factors which may introduce a fixed-cost (independent of farm size) element, even though production itself is highly divisible and scale neutral.6 One such factor is the monetary and time cost involved in acquiring the essential technical information related to the cultivation of the new, high-yielding varieties. Part of the cost is an investment, and it is pos- sible that the cost declines over time; but nonetheless this fixed cost, which is independent of the intended scale of operation, may be in- curred at least in the initial states. A similar view, based on the results of a case study in El Salvador, is expressed by Cutie: ". . . There seem to be fixed technological-transitional costs associated with adop- tion."7 In a study of small Mexican farmers, O'Mara notes that "the set-up cost nature of information acquisition should tend to create a greater responsiveness to technical change by larger farmers. This greater responsiveness for larger farmers seems to hold even for com- parisons between quite small farmers."8 Schutjer and Van Der Veen9 emphasize the fact that HYV practices continue to change, thus im- plying further adjustments (and fixed information-ac.quisition costs) even after the first year of adoption. '0 4 The terms "new technology," "modern crop," "innovation." and 'high-yielding variety-HYV" will be used interchangeably. 5 For an empirical study of the variation w ith experience in the subjective uncertainty of small fairmers with respect to an HYV innovation, see Gerald T. O'Mara, "The Microeconomics of Technique Adoption by Smallholding Mexican Farmers," in Pr'o- gratnining Stuidies for Mexican Agriicdlt'ttial Policy, ed. R. D. Norton and L. Solis (Wash- ington, D.C.: World Bank, in press). 6 Schutjer and Van Der Veen, p. 4. Jesus Cutie. Diffiusiont of Ilybrid Cornl Rechnology: Tlle Case of El Salhador (Mex- ico D.F.: Internat,onal Center for Improvement of Rice and Wheat, 1976). p. 9. 8 O'Mara, p. 24. 9 Schutjer and Van Der Veen, p. 6. 0 Other fixed costs may be related to the purchase of specialized tools (fertilizer applicators and pesticide sprayers). While these are investments, they may be viewed Gershon Feder and Gerald T. O'Mara 63 As it is very likely that working capital will be borrowed if adoption takes place (since the cash requirements of the new technology are considerably higher than those of traditional crops), fixed transaction costs are incurred in relation to loan application and the time required to handle the request within the official system which is characterized by "red tape." These costs are likely to be iower if the informal (mon- eylender) credit market is used, but then the cost of credit is much higher. Another element of fixed costs may be incurred due to the time needed to obtain the specific inputs associated with the new t3chnology (seeds, fertilizers, pesticides). These costs are independent of the amount required and are likely to be higher in the initial introduction period when distribution channels are not yet well organized. In the following it is assumed that a fixed cost totaling C is required if the new technology is adopted. While part of C is, in fact, a one- time investment, it is assumed that farmers are myopic (at least at the introductory state of the new technology), so that they assign most of the investment cost to the first period. The dynamics of the adoption piocess will be affected, however, by the fact that once adopted, the fixed cost related to the new technology is reduced. As the amount of land available to each farmer is fixed, maximi- zation of expected utility (assuming that farmers are risk averse) is obtained by selecting optimal levels of the proportion of land allocated to the modern crop and optimal intensity of variable input per acre. The following results are implied by the model (the derivations are presented in the Appendix). i) The optimal level of per acre variable input is independent of both farm size and the fixed cost, C. (ii) There exists a critical land- holding size, say L, such that farmers with holdings smaller than L will not adopt the modern variety, while larger farmers will plant at HYV on at least a portion of their land. (iii) On those farms larger than L the share of area allocated to HYV increases with farm size, if relative risk aversion is constant or decreasing with income, while absolute risk aversion is decreasing." Even if relative risk aversion is increasing, the share of HYV may increase with farm size if the fixed cost is sufficiently high. (iv) For a given farm larger than L, the share of HYV declines as a fixed cost if the farmer adopts a myopic planning horizon. This may well be the case in view of the high uncertainty about the innovation in the first few years since introduction. I Absolute risk aversion measures the insistence of a risk-averse individual for more-than-fair odds when faced with a bet whereby he can win or lose a given sum of money. Relative risk aversion measures the same insistence when the bet is such that a given proportion of wealth or income can be won or lost. It is generally accepted that absolute risk aversion declines as wealth increases. See Kenneth J. Arrow, "The Theory of Risk Aversion," in Essays in the Theory ofRisk Becar ing (Chicago: Markham Publishing Co., 1971). 64 Economic Development and Cultural Change when fixed costs are higher or when the degree of uncertainty regarding the nrw variety is higher. (v) The critical landholding size L becomes lower when the fixed costs C and/or the degree of uncertainty decline. Results (iv) and (v) have important dynamic implications: Over time uncertainty declines (e.g., due to learning and dissemination of information), and the fixed costs may be reduced. Thus, farms which may have been initially too small to adopt HYV even partially will find it attractive, and farmers who have previously been partial adopters will allocate increasing proportions of their land to HYV. Eventually, when the degree of uncertainty and the fixed costs are sufficiently low, the farmer will switch into full adoption, that is, all of his land will be planted to the modern crop. Indeed, the evidence cited by Schutjer and Van Der Veen'2 confirms a pattern of initial experimentation which is expanded in subsequent periods until full adoption is reached. The role of fixed adoption costs in the model is better understood if it is noted that, in the absence of such costs, nonadoption cannot be an optimal policy, given that the mean net return under the new tech- nology is higher than the return from the traditional technology (see Appendix). However, even if these fixed costs remain high over time, the decline in uncertainty reduces the range of holding sizes which will opt for no adoption. It is quite possible, given the discrete pattern of crop seasons, that smaller farms find it advantageous to start adoption when uncertainty is sufficiently low to induce a very high initial level of adoption or even full adoption. Thus, while larger farmers (who are also early adopters) start with experimentation and gradually shift to full adoption, the smaller farmers can skip the experimentation stage. Obviously, this is not done without a cost, as the period of waiting entails the loss of the average higher income associated with the modern technology. III. A Simulation of the Model rfo demonstrate the implications of the preceding analysis, and in order to gain further insights on the dynamic aspects of innovation diffusion, an example is presented below. The utility function is specified as a logarithmic transformation of income,'3 that is, U = In nl, where r1 denotes income. This specification implies that absolute risk aversion is decreasing, while relative risk aversion is constant at unity. Per acre yield is assumed to take the 12 Schutjer and Van Der Veen, p. 5. '13 The approximation of utility by a logarithmic function dates back to Bernoulli (see D. Bernoulli, "Specimen theoriae novae de mensura sortis," Coimmentarii acade- miae scientiarumn imperiales Pet-o politanae 5 [1738]: 175-92, translated by L. Sommer as "Exposition of a New Theory on the Measurement of Risk," Economneirica 12 [1954]: 23-36) and was more recently advocated by Arrow. Gershon Feder and Gerald T. O'Mara 65 values (1 + a) y or (1 - u) * y with equal probabilities, where y is average yield. It is easy to see that or reflects the degree of uncertainty, as u2 . y2 is the variance of yield. Using the results presented in the Appendix (eq. [A2]), one obtains [R -(CIL)] (y - m-R) er y2 -(y - m-R)2 where l* is the share of land allocated to HYV, mn is the level of variable input costs per acre, R is the pf i acre return from the tradi- tional crop, L is the holding size, and C is the fixed cost. Equation (1) holds only for farmers with a holding beyond the critical size. As is evident from (1), in the absence of fixed costs, the same share of land will be allocated to the modern crop, irrespective of farm size. However, with C > 0, larger farmers will have higher adoption rates. The signs of the derivatives al*Iaoc2 and al*IaC can easily be verified to conform with our previous results. Using the relevant results in the Appendix, one can calculate the critical landholding size L: L = (C/R)/(1 -i , (2) where 4, {1 - [(y - m- R)/(cr y)]2}1/2 > 0. It is noted that 4, < 1, thus the critical landholding size is larger than CIR. In view of this result, together with equation (1), it follows that at the switching point from nonadoption to adoption there is a discontinuity such that new adopters do not start with minuscule experimental plots. Simple deri- vation of (2) verifies that the signs of aL/aC and aL/aCr2 are positive, as predicted earlier. For a given land distribution and specific numerical values of the parameters R, m, cr, y, and C it is easy to calculate the adoption rates at a point of time, using equations (1) and (2). In addition, dynamic elements may be introduced in both C and C so that the process of adoption may be simulated over time. Accordingly, table 1 describes TABLE 1 LAND DISTRIBUTION FARM SIZE (Acres) 1 2 3 4 5 6 7 8 9 10 20 30 40 50 90 Families (N) ...... 26 24 22 20 18 16 14 12 10 8 5 4 3 2 1 NOTE.-Total land area: 1,300 acres; total N of families: 185; Gini coefficient of land distribution: .519. 66 Economic Development and Clultural Change a distribution of land which is quite typical for many rural regions in developing countries. The Gini coefficient of the distribution of land (which applies also to the distribution of income prior to the introduction of the innovation, given that yield per acre is R for all farms) is .52, reflecting significant inequality. T1,'L following values are assigned to the parameters: y = 1.0; R = .4; m = .2. Thus, on the average, an acre of the modern crop yields twice the income generated by traditional technology (exclusive of fixed costs). The fixed cost is assumed to be .7 for the first-time adopt- ers, dropping to .35 in the subsequent year, due to the fact that a portion of the fixed cost is incurred only once. Thus, if full certainty prevailed, the new technology would be profitable for all farms with 2 acres of land or nmore, justifying a complete and immediate shift. The specification of 02 needs to take into account the accumulation of experience and information dissemination over time. Essentially, two dynamic processes affect the degree of uncertainty. The first one is endogenous and cumulative in nature: As more acres are cultivated using the new technology, more experience is gained and uncertainty regarding the performance of the innovation is reduced. The speed of this communal "learning-by-doing" process depends on the efficiency of communication channels among the different farmers.'4 It is natural to specify the index of this experience factor by the time-cumulative acreage (since the time of initial introduction of the innovation) planted to the modern crop.'5 The other process causing a reduction in uncertainty is exogenous and is brought about by the improvement in extension services, ad- vances in research reducing crop susceptibility, the gradual elimination of input supply uncertainties through a more efficient distribution sys- tem, and impro' 'd information from outside sources (radio, written materials, etc.). The effect of this factor is best captured by assuming that the progress of time, by itself, reduces uncertainty. Following this discussion, the variance of yield is specified as ot2 = o2 * exp ( -a t - a*E Q) (3) 14 For instance, P. K. Voon found that in Malaysia innovations in rubber processing were adopted by Chinese growers much faster than the rate observed for other ethnic groups. This can be explained, in part, as the result of more effective information dis- semination in the Chinese community, due to the structure of their settlements. See Phin- Keong Voon, "The Adoption of Technological Innovations in Rubber Processing: The Case of Malaysian Smallholders," Malayan Economic Review1 22 (October 1977): 33-51. 15 This procedure was used in the learning-by-doing literature by several authors, e.g., Simone Clemhout and Henry Wan, "Learning by Doing and Infant Industry Pro- tection," Review of Economic Studies 37 (January 1970): 33-56. Gershon Feder and Gerald T. O'Mara 75 dent of farm size, uncertainty, or risk aversion. In addition, onle can show that the optimal x is independent of the fixed cost, C. Thus, treating x as given at its optimal levAl, and denoting c * x + w m, the optimality condition with respect to I is E{U' * N, * y - m - R} = O, (A2) where E is the expectations operator. Second-order conditions are satisfied by the concavity of U. In addition it must hold (if HYV is adopted) that EU{L * [1*(£y - m - R) + R] - C} a U(L * R), (A3) where l* is the value of I which satisfies equation (A2). The relation between l* and farm size is obtained by differentiation of (A3), using the notation B -y - m -R: al* = E{U'" B * (, + C)} aL -E{UJ' B2} * L2 . If absolute risk aversion is decreasing (a plausible assumption), then E(U' * B) > 0; however, E{U' * B fR,} is negative, zero, or positive, depending on whether relative risk aversion is increasing, constant, or dccreasing in 11.23 The denominator is positive, and the sign is thus determined by the numerator. It follows that the existence of fixed costs contributes toward a positive relation between L and 1*. If relative risk aversion is approximately constant, then (A4) implies a1*/aL > 0. It can be easily verified that aI*/aC < 0, and that a higher degree of uncertainty (represented by a mean preserving spread of the distribution of the random variable e) will induce lower levels of 1*.24 The circumstances underlying a decision of nonadoption ( = 0) can be identified from (A3). It is important first to note that, in the absence of fixed costs, nonadoption cannot be an optimal decision. To see that, suppose to the contrary l* = 0. Then (A2) becomes U' * (y - m - R) < 0, where U' is nonrandom. But this contradicts our prior assumption of HYV's superiority, on average, over the traditional technology (which implies y - m > R). Thus, it must be concluded (C = 0) - (1* > 0). With C > 0, one can define a function 4 such that 4(L, C, y)-EU{[l*.(ey - m -R) + R]L - C} - U(R *L) (A5) where y is a parameter reflecting the degree of uncertainty as will be explained below, and l* satisfies (A2). For given values of y and C, the function 4 is 23 Ibid. 24 Gershon Feder, "The Impact of Uncertainty in a Class of Objective Functions," Journal of Economic Theory 16 (December 1977): 504-12. 76 Economic Development and Cultural Change increasing in L (i.e., a1/8L > 0). This implies that there is a unique value of L, say , which maintains the equality W, C, y) = 0. In view of condition (A3), it thus follows that farmers with holdings smaller than L will opt for nonadoption, given the values of C and y, while larger farms will be at least partial adopters. A diffirentiation of the implicit function 0 = 0 yields aL __OOC -EU' ac =- a,/aK at,aL >° .(A6) -c a/aL 4IaL In order to investigate the relation between L and the degree of uncertaintyv a specific formulation of the random term is needed. Following Feder,25 define E = - + y ' (e - 1), where - is a random variable with mean equal to I and y is a positive parameter. An increase in y implies a mean preserving spread of the distribution of E and, thus, reflects an increase in the degree of uncer- tainty. Differentiating * yields dL _ad/ay l*.L.y.E{U'.(E - i)} a-y a-.aL . (A7) The term E[UV(i - 1)] has the same sign as the covariance between U' and E and, therefore, is negative. It is thus concluded that aLlay > 0. 25 Ibid. Reprinted from AMERICAN JOURNAL OF AGRICULTURAL ECONOMICS Vol. 64, No. 1, February 1982 On Information and Innovation Diffusion: A Bayesian Approach Gershon Feder and Gerald T. O'Mara Learning and information accumulation are hy- The Model pothesized to play a major role in innovation diffu- sion. For instance, Hiebert argues that the probabil- The traditional technology provides an average ity distributions of new (and unfamiliar) technolog- profit of R dollars per acre. If the technology is ical parameters, as perceived by farmers, will shift familiar to farmers and has been available for some over time due to learning and experience. Prob- time, it is reasonable to assume that farmers know abilities will be redistributed from lower to higher the true mean of R. payoffs. This induces farmers to increase their use Assume that the new technology is risky. Profits of the innovation, which was new seed varieties in per acre cultivated with the new technology, say nr, the Hiebert model. are normally distributed with mean , and variance The model presented by Kislev and Shchori- a'2, and ,u > R. Farmers are assumed to be risk-neu- Bachrach introduces in the new technology produc- tral and believe the mean profit from the new tech- tion function an efficiency factor which is positively nology is normally distributed with mean in and related to learning. It is approximated by the variance 62. Also assume that farmers know the cumulative (over time) output produced with the variarnce of profits, o2, with the new technology, innovative technology. As learning increases, the and that they modify their beliefs about ,u based on innovation becomes advantageous for more and new information generated by observed outcomes more producers who then adopt it. on areas cultivated with the new technology by In a similar vein, Feder and O'Mara construct a adopters in accord with Bayesian procedures. The diffusion process where uncertainty about an inno- assumption that farmers' beliefs about A are nor- vation (high-yielding varieties [HYV]) depends on mally distributed is not at all restrictive if 7r is cumulative area allocated to HYV. This represents normally distributed. It is easily shown that if farm- experience. With the accumulation of experience, ers' initial beliefs are completely diffuse (i.e., uni- uncertainty declines, and the innovation is adopted formly distributed), and the underlying process is by an increasing proportion of producers. normal, then an application of Bayes Theorem in These models produce sensible (and empirically computing a posterior distribution, using new in- valid) hypotheses about the dynamics of innovation formation, will produce beliefs about ,u that are diffusion. However, the accumulation of informa- normal. tion, which is the basic driving element in these Since cr2 is known, the initial beliefs about the models, is not treated explicitly. Thus, the use of variance of ,, (802), imply an "equivalent sample cumulative output, or a cumulative input as an size," no, associated with prior information, i.e., index of learning, while plausible, still requires formal justification. One possibility is a Bayesian (1) cr2 = a2/n0. learning process. Indeed, the work by O'Mara Using the properties of conjugate prior distributions (1971, 1981) and more recently by Lindner, (Raiffa and Schlaifer, chap. 3) it can be shown that Fischer, and Pardey have utilized Bayes theorem to the mean and variance of y at time t are given characterize an individual farmer's adoption behav- respectively by ior. The purpose of this note is to formulate an ag- (2) mt - _2mt + 82t1NAtirel I gregative innovation diffusion model based on the 0r2 + 82t_1NAt_1 assumption that individual farmers revise their be- liefs in a Bayesian fashion. (2282t (3) 82 - a2 + 6211NA1-1 Gershon Feder and Gerald T. O'Mara are economists at the De- where frt- is the sample mean of observations on velopment Research Center, World Bank. the new technology generated by the adopters and Views expressed in this paper are the authors' and do not neces- observed in period t - 1, N is the total number of sarily reflect those of the World Bank. The paper benefited from the valuable comments of two ref- farmers, and At-, is the proportion of farmers who erees. have adopted the new technology in period t - 1. Copyright 1982 American Agricultural Economics Association 146 Februiary 1982 Ainer. J. Agr. Econ. For simplicity, assume that all farmers are the same This condition may be written, after some manipu- size and that the information content of a farmer's lation, as own trial is the same as that of another farmer's. N Also, all farms in the population are observable by (13) mi, R - v ( R) | A(T)dT = In all farmers. Hence, each farmer acquires the same n, 0 information about the new technology in each time Thus, if a farmer is an adopter at time t, his initial period, whether or not he is an adopter. These beliefs about the expected value of A must be at strong assumptions limit the generality of our re- least as large as m*,(t). Assuming that initial beliefs sults. Definey, = 1/812 and substitution in (2) yields about the expected value of ju (i.e., mo) are ran- (4) Y Y ± NAt- domly distributed over all farmers with cumulative 4t =t-I a'2 density function F(*), the expected aggregate adop- which upon recursive summation. becomes tion rate at time t is N( t-1 (14) Et A(t) = Prob[m0 > m*%(t)] ()yt =Y" + 2 EAi, I - F[tn*.(t)]. 1==1 or equivalently, For simplicity, denote P(t) E,A(t). It can be 1 I N t-l shown by straightforward derivation (utilizing equa- (6) 52 = - - + 2-~ A,) tiorns [13] and [14]) that A 0 -o pt Similarly, (1) may be written as (i) d ( )> O, (7) Mi Mt1.1 N (15) (V = 2-,- 2 Xt-lAt(ii) -)< 0. Defining zt mt/8,2 and substituting in (6), one aR obtains after recursive summation, As one would expect, the expected rate of adop- (8) = z + N tion is higher at any time if the innovation is more =o + Eprofitable, ceteris paribuis, and lower if the tradi- 1=0 tional technology is more profitable. Combining (6), (8), and (1) yields The expected rate of adoption at time zero is nt-l + N 7-I I given by (9) Mt= t-I (16) P(O) = I - F(R). nO + N I A, I=0 It follows, therefore, that the expected rate of or in a continuous time formulation, adoption at the initial stage does not depend on the innovation's profitability (represented by ,) but [1 rather on the profitability of the old technology and n0m0 + N J r(T)A(7)dr the parameters of the distribution, F. Now, the (10) m(t) A(T)d distribution of initial beliefs is probably strongly l + N J affected by sources of information other than ob- served fields, such as the farmer's education and Given the assumption of risk neutrality, the condi- innate ability, sales promotion, word of mouth re- tion for adoption becomes ports, extension, etc. The distribution of initial (I1) Et(7T,) = Et(;x) = m(t) ;- beliefs will also affect the expected rate of adoption That is, the time of adoption Lror a given farmer later, but then there will be an additional factor- occurs when the mean of his beliefs about I, the the difference in average profitability between the expected value of profits from the new technology, new and the old technology (ut - R). is at least as great as the return from the old tech- Inspection of equations (14) and (13) verifies that nology. However, m(t) depends not only on initial the Bayesian learning process introduces the subjective beliefs (min0 n.) but also upon a stochastic . r' component from the realized observations on the c process 7T(t), which is distributed N(A, oa2) for all t. in the aggregate expected adoption function. Within Hence, the realization 7t(t) is also normally distrib- the model's context, this is equivalent to the inclu- uted with expected value Aufor all t. Accordingly, sion of cumulative acreage (or output) with the new the expected time of adoption for a given farmer is technology in the adoption function. Previous diffu- given by the condition, sion models have simply assumed such a relation- n0m, + Ntz A(T)dr ship. These results serve as a theoretical justifica- (21) E,m(t) = Jo Ž R. tion for such an assumption. By differentiating P(t) n,, + N r A(r)dr twice with respect to time, one obtains, after some Jo manipulation, Fed/er aniel 0' Mlora I/Iufl 1ttio- anld Innovation Dityi/sijon 147 (17) P = (t) Conclusions This note has shown that, given our assumptions, 12 the Bayesian learning process introduces the (p. - R)* A(t)[(F')- - F`A(t)], cumulative adoption rate as an element in the ," aggregate expected adoption function. This is where A(t) is the actual adoption rate at time t. equivalent to the inclusion of cumulative acreage Cleairly, (or cumulative output) in the adoption function. (18) PW Z O as F" - (F' )2'/A (tl. Previous models of diffusion have assumed such a (Felationship. These results serve as a justification Now. P provides significant information about the for such an assumption. In addition, it was shown curvature of the expected adoption rate function, that Eayesian learning can generate a characteristic P(t). If P > 0 initially, then P(t) will be strictly sigmoid-shaped adoption function for a dominant convex over that range. Similarly, if P < 0 after a innovation. Although strong assumptions were certain time i, then P(t) will be strictly concave made, this analysis constitutes a first step toward a after that time. Thus, P(t) will have the sigmoid more general theory of innovation diffusion. shape of a typical successful innovation diffusion .Received February 1981; revision accepted process if P is first positive and then becomes nega- JiiDti 1981.1 tive after some time i (Griliches). The key to the change in sign of P is the behavior of F", the latter being the slope of the probability density function, References F'(m,n) -f(m,,). Note that if F' has a smooth uni- modal shape, as in figure 1, then F" causes P to be at Feder, G., and G. O'Mara. "Farm Size and the Diffusion first positive and then negative. That is, m*,0(t) will of Green Revolution Technology," Ec-oni. Develop. move from the right to the left in figure 1. High Cultit,r. Change. 30(1981):59-760. values of tn,, will be the first to meet the condition Griliches, Z. "Hybrid Corn: An Exploration in the Eco- III, : "1,,(t), insuring that P is first positive and riomics of Technological Change." Econometrica then negative. Clearly, the behavior of F" implies 25(1957J 231-52 constraints on the distribution of initial beliefs if the Hiebert, D. "Risk, Learning and the Adoption of Fer- aidoption curve is to be sigmoid. For- example, if F is Heet ."ik erigadteAoto fFr tilizer Responsive Seed Varieties." Amer. J. Agr. a uniform distribution, then F" = O and the adoption Econi. 56(1974):764-68. ctirve is exponential. Kislev, Y., and N. Shchori-Bachrach. "The Process of an Innovation Cycle." Amer. J. Ag,r. Econ. 55 (1973):28-37. Lindner, R., Fischer, A., and P. Pardey. "The Time to F AF lAdoption." Econ. Letters 2(1979):187-90. O'Mara, G. T. 'A Decision-Theoretic View of Technique Diffusion in a Developing Country." Ph.D. thesis, Stanford University, 1971. ".The Microeconomics of Technique Adoption by / Smallholding Mexican Farmers." Tlhe Book of Plt,......j...... _ CHAC: Progr'amnming Stiidies fi)r Mexican Agricul- -__l_*_, tur'e, ed. R. D. Norton and L. Solis. Baltimore, Md.: F i F O Johns Hopkins University Press, 1981. Raiffa, H., and R. Schlaifer. Applied Statistical Decisiont Figure 1. Probability density function of mean Theorv. Boston: Graduate Schooi of BuLsiness Admin- profits istration, Harvard University, 1961. World Bank Headquarters: 1818 H Street, N.W. Washington, D.C. 20433, U.S.A. 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