The acquisition of information and the adoption of new technology

UJA)~J The Acquisition of Information and the Adoption of New Technology by Gershon Feder and Roger Slade World Bank September 1982 The views expressed in this paper are those of the author only and cannot be taken to represent the views of the World Bank, The Acquisition of Information and the Adoption of New Technology Abstract This paper presents a dynamic model of diffusion of a new technology involving a variable input. The model highlights the role of information accumulation, and distinguishes between active gathering of information, which entails costs, and passive information acquisition, which takes place in an autonomous manner. The model generates a number of hypotheses regarding the likely pattern of adoption and use of the variable input over time by farmers of differing holding sizes and different access to information. It provides a possible explanation to the often observed lag in adoption of innovations by smaller farmers. Data from a recent survey of farming practices in North-West India is presented in order to assess the implications of the model. Focusing on the use of different types of fertilizers, the empirical results are generally consistent with the hypotheses suggested by the theoretical framework. Key words: New Technology ... Information... Knowledge... Adoption The Acquisition of Information and the Adoption of New Technology I. Introduction Improved knowledge regarding new technologies through the accumulation of information over time is hypothesized to be one of the main dynamic elements of innovation adoption processes. Several studies have modeled innovation diffusion as a Bayesian learning process, where each period's experience is used to update initial beliefs about the character- istics of the new technology [e.g., O'Mara; Lindner, Fischer and Pardey; Feder and O'Mara (1982)]. Provided that the innovation is profitable, the accumulation of favorable experiences will eventually induce most farmers to adopt the new technology,. A related model, although not explicitly following a Bayesian approach, has been presented by Hiebert who argued that over time, the probability distributions, as perceived by farmers, of new (and unfamiliar) technological parameters, will shift due to learning and experience. Probabilities will be redistributed from lower to higher payoffs, inducing farmers to increase their use of the new technology. A more explicit treatment of learning has been provided by Kislev and Shchori-Bachrach, who argue that the production function associated with the new technology incorporates an efficiency factor which is positively related to the level of knowledge. The latter is represented by cumulative output, as is common in the learning-by-doing literature. Over time cumulative output increases, raising the level of knowledge and efficiency, thus rendering the new technology attractive to an increasing number of producers. Focusing on the role of uncertainty and risk-aversion, Feder and O'Mara (1981) introduced knowledge (measured as cumulative acreage allocated -2- to high-yielding varieties) as a factor which reduced uncertainty and thereby induced adoption by risk averse farmers. One common feature of the models mentioned above is that information accumulates in a passive way, without any explicit effort on the part of farmers to obtain information other than that which happens to come their way. But in reality, farmers actively gather information when they expect it to provide an economic return. As pointed out by Hornik (p. 3) "...individual farmers are willing to bear the cost of obtaining such information directly. They will invest time or money or both because the return is high." It follows that the amount of information acquired during any given period is, in part, a decision variable. As the level of knowledge is important in explaining adoption behavior, it is of much interest to explore the factors affecting decisions regarding information acquisition, and thn impact of these decisions on adoption. This paper present a formal decision model involving a new technology represented by a new input and explicitly incorporates the decision to acquire information. Unlike earlier works which have focussed on a discrete innovation, or which have dealt only with the level of use of new inputs, the present discussion addresses both the issue of non-adoption, and the intensity of use once adoption takes place. The outline of the paper is as follows: in the next two sections, the model is presented and its implications are discussed. These are followed by a review of some empirical evidence drawn from a recent farm sample survey in India. The last section summarizes the results. -3- II. The Model rarmers are assumed to grow a crop with a technology that is new to them and which involves a variable input N (e.g., fertilizer) with which they are not familiar. It is reasonable to expect that as a farmer accumulates knowledge of and experience with the new technology he will be able to produce more output with a given bundle of inputs. Hence, the specification of the production function should exlicitly incorporate the level of knowledge. Such a formulation has been suggested by Kislev and Shchori-Bachrach in the context of a model of an innovation cycle, and it is adopted here with some modifications; (1) = (K t)F(L,Nt where Yt = output at time t L = land N. = variable input at time t Kt = cumulative information at time t g and F are concave functions. It is assumed further thatF(*) is of constant returns to scale, and equation (1) can thus be rewritten in per acre terms: (2) y t tg(t )f(nt ) where y E Y /L; n t N /L and f = ?(l;n )/L t tt t t -4- The function f(*) is assumed to possess the characteristics of a well behaved production function, i.e., positive but decreasing marginal productivities. In addition, it is assumed that the marginal productivity at n=O is finite and that some output can be obtained even when n=O. The assumptions regarding the knowledge function g(*) follow those of Kislev and Shchori-Bachrach: the marginal contribution of knowledge is positive and declining. Furthermore, as knowledge increases the knowledge function converges to an upper limit 7. This implies that as knowledge accumulates, the marginal productivity of increments in information tends to zero.. It is also reasonable to assume that when knowledge is zero, the value of g(*) is very low or zero. The above assumptions are summarized below, and are illustrated in Figure 1. (i) f' > 0 ; f" < 0 (ii) f'(0) = f' > 0 (iii) f(0) > 0 (iv) g' > 0 (V) Jim g(K) = g (3)*C (vi) Am g' = 0 (vii) g(K) = 0 (viii) g" < 0 Figure 1 f !,fA f(n) f'(0) g9 g(K) 0 - n 0 -5- In defining accumulated information Kt and the way it is acquired, we modify the original formulation to allow for active (and costly) information acquisition. thus, the farmer gains information both passively (e.g., by listening to discussions among other farmers, through learning-by- doing or incidental by observation) or by purposefully seeking to expand his knowledge, with the expectation that more knowledge will increase his productivity. It is reasonable to assume that active information gathering (unlike passive learning) involves costs in tirms of time, cash, or both. At any given time, the marginal cost of acquiring information is assumed to be increasing, reflecting the tendency to approach first the more convenient (cheaper) sources of information. The state of knowledge (K) at time t is defined by (4) K = K + A + h t t-1 t t where ht describes passively acquired information gained during the period. The size of ht is probably increasing over time, as more individuals increase their knowledge and discuss their experiences with others. The variable At denotes the extent of purposeful knowledge acquisitionl/ and is therefore a decision variable. The cost of information (C) is given by (5) Ct = C(A ); C' > 0; C" > 0. 1/ In the Kislev-Shchori-Bachrach model ht is total output of all farmers produced in the current period and At is zero for all t. The left-hand- side of equation (4) becomes therefore cumulative output since the introduction of the new technology, which represents the experience gained over time. -6- Assuming (without loss of generality) that output price is unity, and denoting the price of N by p, the profit function Ut for a farmer with holding of size L is given by (6) IT = L[g(Kt )f(n) - pnJ - C(At The farmers myopicL/objective is (7) Max 11t, subject to equation (4) and nt > 0; At>O. nt,At First order conditions for an optimum require (8) (i) T = L*[g(K )*f' - p1 < 0 (ii) nt *IT = 0 n t n (9) (i) IT = L*g'*f(n ) - C' < 0 (ii) A *HT = 0 a t t a where I , IT denote the respective partial derivatives of the profit na function. It is shown in the appendix that essentially the same first order conditions will result from a model which incorporates maximization of expected utility when output is uncertain. Tn that case knowledge serves to reduce risk rather than increase productivity. 1/ Alternatively, the farmer's objective can be formulated as the discounted stream of operating profits minus the current cost of acquiring information, assuming that for a given level of n profits are constant over time. The resulting effect would be to add a constant multiplicative term to the short-term operating profit; but this will not change the results. -7- Second order conditions require that the Hessian matrix be negative definite. This implies (10) 11 *IT - 12 2 L*g*f"*[L*f*g" - C"1 - L2.(g. f)2 > 0 nn aa na The analysis assumes that these conditions hold throughout. In the absence of this assumption, the problem is characterized by uninteresting (and unrealistic) corner solutions. III. Implications of the Model Some characterization of the optimal solution can be deduced in a straightforward manner from conditions (8) and (9). Setting Hn = 0 and n differentiating, we obtain d = -g'*f'/g*f". Three observations can be made t on the basis of this result. First, at high values of K, where the marginal productivity of information is negligible (g'=0) the optimal level of n is fixed and is independent of the particular level of Kt. Furthermore, the optimal level of n at that range is independent of holding size L. Since over time all farmers acquire high levels of K (whether by active or passive information it follows that event-ally all farmers adopt n and utilize it at identical levels. Second, given that g(0)+0, there exists a unique value of cumulative information, say K*, such that farmers who have not yet acquired this level do not adopt n. The level of K* is defined by the condition g(K*) *f'(0) - p = 0, and is independent of farm size. But we show below that the speed with which farmers of different size classes reach level K* is different, and hence that different farmers will adopt at different times. Third, once n is adopted, as long as the marginal productivity of information (g') is positive, farmers with higher levels of cumulative information utilize higher levels of N per acre. If, however, the critical information level K* is close to the range of K values where the marginal productivity of information is low (K' of Figure 1), then, once adoption takes place, there will be no differences in the level of n used by adopters with different K levels. In order to characterize the likely pattern of adoption and use of n, we focus first on the early period (t=0) when Kt-1 = 0 and ht (the. autonomous increase in knowledge) is negligible. Consider the implicit functions T (n, AO) =0 and Ia(n , A) =0 defined by equations (8) and nO 0 a 00 (9). It can be shown that these describe positively sloping curves in the positive quadrant of (n0,A0) space, and that Rn = 0 intercepts on the A axis. Furthermore, second order conditions imply that the slope of the Il curve is steeper than that of the Ia curve [see appendix]. Figure 2 na illustrates two possible situations with distinct optimal solutions. Figure 2 case 1 case 2 A, A A A =0 n =0 n n 1U < 0 110 11A < =0 A 0 0 a 0l< pitimal A a ...... .......... 0 opt imal solution 0 n0 n n0 -9- In case 1 of Figure 2 there is no combination of (n,A) which will simultaneously set IT and I to zero. The optimal solution in this case is to n a forgo use of n, but to allocate some resources to the acquisition of information. Note, however, that the amount of information gained is positively related to farm size. This can be verified by differentiation of I (0,A) yielding n (11)dA f()g' > 0 if K < K' dLjI =0,n=O f-O).g, a aa =0 if K > K' That larger farmers allocate more resources to information gathering in the early stages of diffusion process accords with intuition. For such farmers, owing to the large area that they command, the benefit of marginal increments in knowledge is higher than for small farmers.l/ Equation (11) is also implies that where farm size is large enough, the level of A will reach the critical value K* at which adoption of n takes place. Similarly, where farm size is small enough the condition fa = 0 will not hold and no resources will be devoted to information gathering.2/ The initial period of the diffusion process may thus be characterized by differential behavior on the part of farmers with different holding sizes (and thus different wealth). Farmers above a certain critical holding size adopt n. Smaller farmers to not adopt n, but actively seek information. Very small farmers do not adopt n, neither do they actively seek to expand their knowledge. 1/ Lindner arrives at a similar conclusion on the basis of a Bayesian model focusing on a dichotomous innovation. 2/ In the context of Figure 2 (case 1), such farmers are characterized by a if curve with an intercept on the n axis. a - 10 - The dynamics of decisions made by farmers who fall under the regime of case 1 of Figure 2 are of much interest, as these dynamics define the diffusion path of the new technology. Over time, the cumulative information level increases due to both autonomous increments and the active acquisition of knowledge. It can, however, be shown that if the autonomous information increment ht affects all farmers equally, then prior to adoption the level of cumulative information is positively related to farm size./ This implies that larger farmers reach the critical level of information (K*) faster, i.e., they adopt n earlier than smaller farmers. Considering farmers of a given holding size, it is easy to show that those who have better access to information, e.g., farmers who are visited regularly by extension agents, will adopt faster because they attain the critical level K* earlier. In the context of the present model, such farmers can be characterized by either a higher ht trajectory, or a lower cost function C(A) (Hornik, p. 12), or both. It is more convenient to derive the* results for a formulation involving change in ht. as we do below, but results will be the same for the alternative formulations. Differentiating the first order condition H = 0 while maintaining n=O (i.e., non adoption) we obtain n 1/ The proof is obtained by negation. Suppose K t(L ) > K t(L 2) where L < L . Since it was has been shown that K(L1) < XO(L2), there must exist a time t' such that r,(L1 t,(L2). But at such a point A(11) < A(L?) (by equation (11)) so that in the subsequent period K(L ) < K(L 2). Thus at no period prior to adoption can an equality K t(L ) 1 t (L ) be established, and the conclusion is Kt(L2) > Kt(Li). - 11 - dAt -L*f (0)*g" (12) dA< 0 dtH =0, n=0 L-f()0f- n dKt + dAt L*f(0)*g" - -C"T (13) = 1 +-= 1-=>0 dh dh L*f(O) *g" - C" L*f(0)*g" > 0 tt Equation (12) implies that autonomous information is a substitute Zor information acquired as a result of the farmers' own initiative. But Equation (13) confirms that farmers with better access to information have higher levels of cumulative information, and will therefore adopt n earlier than other farmers, ceteris paribus. Considering adopters of n, the analysis covers two possible situations. The first is when the critical level of cumulative information required for adoption (K*) is close to the range of K values where the marginal productivity of information is negligible (K' of Figure 1). In this case, once n is adopted, there is no further purposeful acquisition of information and any subsequent accumulation of autonomouc information does not affect decisions. The quantity of n used is constant and farm size or ease of access to information have no impact on the per acre input of N. In the second, the critical information level K* is well within the range of K values where the marginal productivity of information is positive. In this case, for each farmer adopting n, the level of n increases over time until his Kt exceeds K'. This is verified by differentiating ITn = 0 with respect to Kt to obtain (14) dn=c gf t' - - 12 - Since Kt increases over time, equation (14) implies that dn > 0 as long as 1t < K'. This second situation also implies for that group of adopters who have not yet reached K', that at any period t input use per acre is positively related to farm size and to the ease of access to information. This is verified by differentiating the system (Tn = 0, Ia = 0) and obtaining (15) dn - f(g,)2f' > 0 (for K < K') dL H *H - nn aa na (16) dn L2*C"*g'*f' > 0 (for K < K') dht I *H - nn aa na Farmers who have reached K', however, utilize the same quantity of input per acre, irrespective of farm size or ease of access to information. As in the earlier discussion, it can be shown that at any period t, farmers with larger holdings (or with easier access to knowledge) will have higher levels of cumulative knowledge and will therefore reach the saturation level K' earlier. The impact of two policy variables can be established within the framework of the present model. First, consider a subsidy on p, the price of n. A reduction in p lowers the critical cumulative information level K* required for adoption (as can be verified by a straightforward differentiation of the term I (n=0). This implies that adoption of n will take place earlier n than otherwise for most farmers. Por adopters, in the case K* < K', a differentitation of the system (T = 0, T = 0) establishes n a - 13 - (17)dn = -(L*f*g" - c"t) (17) ='a un aa na (18) dA- L*g'f' F> 0 if K < K' -dp I . rF aa nn na =0 if K > K These results indicate that a subsidy on the price of n increases the per acre input of n, as one would expect. The investment in information gathering increases as well (for those farmers who have not yet reached the saturation level K'). This latter result implies that the pace at which farmers reach the saturation level K' (and the corresponding ceiling level of n use) becomes faster. Early adopters will also confer externalities on other farmers by generating information and experience. Hence, in the early phase of a diffusion process, there may be a justification for government to subsidize the input price. The second policy variable of interest is agricultural extension. Extension may be a source of information to many farmers, either directly through farmers' contact with extension agents, or indirectly, as farmers who have benefitted from direct extension exposure transmit information to other farmers. The preceding analysis regarding the impact of changes in ht reflects the effects of intensified extension efforts. Thus equations (13) and (16) imply that increased extension activities will speed t'e time of adoption for most farmers, and will induce higher levels of input use until the saturation level of information K' is obtained. Such intensified extension efforts entail increased costs to the economy, but a faster adoption - 14 - and a faster attainment of the ceiling level of N per acre provide additional and presumably greater benefits. Therefore, there is an a priori rationale for expanding extension activities and other public information services when new technology becomes available. Although the model presented in the preceding sections deals with a single variable input, the hypotheses can be generalized, and the data on adoption of several inputs (e.g., different fertilizers) can be analyzed within the general framework of the model. IV. Empirical Evidence The preceding analysis suggested a number of hypotheses regarding the extent of knowledge, adoption and the intensity of use of a variable input by farmers with different holding sizes and differing access to information. Some evidence follows which is consistent with these hypotheses. The analysis is based on data from a large sample survey of farmers conducted in fall 1981 in two districts of the state of 'Haryana (India). In the survey area the main crop grown at this time of year (the Kharif season) is rice, mainly under irrigated conditions, removing a-substantial portion of uncertainty regarding water availability. High yielding (dwarf) varieties of rice were introduced in this area in the early 70's and by the time of the survey 95% of rice growers were using such varieties. The yield of dwarf varieties is strongly affected by the use of inorganic fertilizers: Nitrogen, Phosphate and Potash. A trace element (Zinc Sulphate) is also recommended, although it can be applied in the preceding season by those following a wheat/rice cycle. Since fertilizers are divisible variable inputs, and since the use of fertilizers is still expanding quickly in Haryana, it seems that the diffusion process of fertilizer based technology (and the accumulation of - 15 - knowledge) has not yet reached the saturation stage. There is also evidence to show that adoption has not proceeded at an equal pace for all types of fertilizers. The data suggest that there were no serious problems or interruptions in the timely supply of fertilizers or other complementary inputs. Neither was there a significant shortage of short-term credit.l/ Hence, differences in fertilizer use may be related to differences in the level of information more clearly than is generally the case. Moreover, in the area covered by the study a radical change in the structure and intensity of agricultural extension took place in 1979 when the state reorganized its network of multi-purpose rural agents into a purely agricultural extension system following the Training and Visit system (Benor and Harrison). Inter alia the number of agents was increased and provision was made for the continuous updating of their knowledge. The nature of agents' contact with farmers also has changed: about ten percent of the farmers were selected as "contact farmers," who receive an extension visit from the same agent every two weeks on a pre-specified date. Other farmers can join meetings held by the agents with their contact farmers, but the initiative is left to them. Indeed, the survey established that within a reference period of one month, 80% of the contact farmers had at least one meeting with extension agents, while 31% of the other farmers had some direct exposure to extension in the same period. Contact farmers are expected to serve as propagators of extension messages, either by discussing extension 1/ Farmers were asked whether they obtained finance from money-lenders, who are the most expensive source of credit. Those who did, were asked whether they could have obtained more credit if needed. Only 6% of the farmers said they could get no more credit. - 16 - advice with others, or by adopting recommendations and thus prompting inquiries from other farmers. In the context of the model discussed in the preceding section, contact farmers have higher levels of autonomous information (ht), and the cost to them of acquiring accurate information is lower than for other farmers, [see also Hornik, p. 121, because they obtain information at first hand and do not need to seek out the extension agent.l/ Rather, the agents' duty is to visit them, preferably in their fields. Hence the hypothesis that ceteris paribus, contact farmers will have more knowledge and will adopt a given input, in this case fertilizers, earlier than other farmers. Table 1 describes the extent of knowledge regarding the most recently introduced fertilizer, Zinc Sulphate. To facilitate comparisons between sub-groups the data are broken down by farm size-class and by extension contact status:2/ It is apparent that in 1978, before the transformation and intensification of extension, differences between contact and non-contact farmers irrespective of farm size were small as there were no contact farmers in the pre-1979 period. However, the rate of knowledge among large farmers was significantly higher than among smaller farmers (for the pooled sample). In 1980, after the reorganization of the extension l/ Our survey data show that 71% of the contact farmers using fertilizers pointed to extension as the main source of information regarding timing and quantity of fertilizers. In contrast, only 39% of non-contact farmers stated that extension is their main source of information on this matter. More than half of all non-contact farmers (54%) indicated that they have learned from other farmers. 2/ Obviously, there are many other factors explaining differential behavior of individual farmers, but here their combined effect is assumed to be a random element with expected value zero. - 17 - Table 1: RATE OF KNOWLEDGE REGARDING USEFULNESS OF ZINC-SULPHATE AMONG RICE GROWERS (HARYANA, INDIA) Contact Farmers Non-Contact Farmers Small farmers/a Large farmers Small farmers Large farmers Year (N=96) (N=178) (N=145) (N=136) - - - ----------- - - - Percent---------- ----- 1978 17.7 31.4 23.4 30.1 1980 40.6 62.9 42.8 49.3 /a Small farmers are defined as those who own less than 10 acres. The rest are considered large farmers. Source: World Bank, RPO 672-29. system, we note that the rate of knowledge of larger contact farmers is significantly higher than that of both smaller contact farmers and large non- contact farmers, and that this is in accord with our earlier hypothesis. Similarly, the rate of knowledge is higher by 6.5 percentage points among larger non-contact farmers than among of small non-contact farmers. For the small farmers group, however, there is no significant difference between contact and non-contact farmers, and our hypothesis cannot be confirmed. A study of Filipino rice farmers cited by Hiebert (p. 767) also provides evidence that the level of "expertise" manifested by farmers with intensive extension contact is consistently higher than that of other farmers. The survey data do not contain information regarding knowledge of other fertilizers, but detailed information on the adoption and use is available. Table 2 describes the adoption of the three main fertilizers among farmers of different holding size and different exposure to extensionv./ 1/ The use of zinc sulphate in any given season is not a proper indication of its adoption owing to the possibility of using it in alternate seasons. - 18 - Table 2: THE RATE OF ADOPTION OF FERTILIZERS AMONG RICE GROWERS (HARYANA, INDIA) Contact Farmers Non-Contact Farmers Fertilizer Small farmers/a Large farmers Small farmers Large farmers type (N=100) (N=186) (N=163) (N=145) ---------- - - - - - Percent--------- ----- Nitrogen 98.0 98.4 95.7 97.2 Phosphate 32.0 47.3 20.2 30.0 Potash 14.0 19.9 9.8 16.6 /a Small farmers are defined as those who own less than 10 acres. The rest are considered large farmers. Source: World Bank, RPO 672-29. The data indicate that there is almost complete (more than 95% of the farmers in each sub-group) adoption of nitrogen. The uniformly high rate of adoption, and the fact that the average quantity applied is quite close to the recommended dose with negligible differences among sub-group (see Table 3), correspond to the situation where most farmers have attained the saturation point K' (see Figure 1). Given that nitrogen tended to be the most commonly available fertilizer in the early stages of T{YV diffusion,1/ it is reasonable to expect that a substantial stock of knowledge about this type of fertilizer has already been accumulated. 1/ In the year 1969/70, only 5,120 tonnes of Phosphates and 1,800 tonnes of Potash were used throughout the state of Haryana, while the corresponding figure for Nitrogen was 47,000 tonnes. - 19 - Table 3: AVERAGE USE OF FERTILIZERS BY ADOPTERS (HARYANA, INDIA) Average quantity applied (kg/acre) Contact farmers Non-contact farmers Fertilizer Recommended type dose (kg/acre) Small/a Large Small Large Nitrogen 48 45.6 43.3 39.2 43.3 (16.7)/b (14.9) (14.3) (15.7) Phosphate 24 15.1 14.2 14.1 10.6 (4.6 (6.4) (10.0) (5.1) Potash 24 7.0 10.4 7.1 8.3 (1.6) (10.4) (3.0) (8.7) /a Small farmers are defined as those who own less than 10 acres. Others are considered larger farmers. /b Figures in parentheses are standard deviations. The adoption of phosphates and potash is not so advanced, probably because knowledge is not as widespread. The model postulates that when diffusion is incomplete, there will be differences in adoption rates between larger and smaller farmers, and between farmers with different access to information. This is indeed borne out by the data on phosphate and potash adoption in Table 2. For each farm size group, the rate of adoption is higher among contact farmers, and regardless of exposure to extension, adoption is higher among larger farmers. Not all differences are statistially significant, but the pattern is consistent. Considering adopters, the data in Table 3 indicates that there are no significant differences between sub-groups with respect to the average quantity of phosphates and potash applied. This can be interpreted on the - 20 - basis of the theoretical model as a case where the critical level of knowledge necessary for adoption(K*)is close to the saturation level of knowledge K'. If this interpretation is correct, it also implies that the average quantity of phosphates and potash applied per acre by adopters is not likely to increase over time unless prices change, since it corresponds to the optimal dosage of the ceiling level of cumulative knowledge. The present data do not permit further analysis to substantiate this point. V. Summary and Conclusions This paper presented a dynamic model of diffusion of a new technology involving a variable input. The model highlighted the role of information accumulation, and distinguished between active gathering of information, which entails costs, and passive information acquisition, which takes place in an autonomous manner. The model generated a number of hypotheses regarding the likely pattern of adoption and use of the variable input over time by farmers of differing holding sizes and different access to information. It was shown that during the initial phases of diffusion, larger farmers are likely to allocate more resources to the acquisition of information, and will therefore possess higher levels of cumulative information at any given period, ceteris paribus. Farmers with more convenient access to information will also enjoy higher levels of knowledge. The analysis suggested that a certain critical level of cumulative information must be attained before adoption of the variable input takes place. It therefore follows that larger farmers, or farmers with better access to information, will adopt earlier than other farmers. Once adopted, the quantity applied may increase over time up to some ceiling level, or it may remain constant, depending on the relationship - 21 - between the critical level of knowledge for adoption and the level of information at which the marginal contribution of additional information becomes negligible. If the quantity used increases over time, then larger farmers and farmers with better access to information apply larger quantities than other farmers. But over time most farmers adopt, and differences in quantities used disappear as the marginal effect of information slackens, and the level of use converges to a ceiling. Data from a recent survey of farming practices in North-West India is presented in order to test the implications of the model. Focusing on the use of different types of fertilizers, the empirical results are generally consistent with the hypotheses suggested by the theoretical framework. - 22 - Appendix I. A Model of Adoption Under Uncertainty and Risk Aversion This section demonstrates that the full certainty model presented in the text is equivalent to a model involving risk-aversion and uncertainty. Suppose that the per acre production function embodying the new technology follows the Just-Pope stochastic specification: (A.1) y = f(n) + E(q(n)]I/2 Where f(n) is mean output, q(n) is a positive function and e is a random variable distributed normally with zero mean and variance equal to a. Further assume that the variance declines as knowledge accumulates, reflecting reduced uncertainty. (A.2) a = a(K) ; a < 0 Assume also that farmer maximizes an exponential utility function [Yassour, Zilberman and Rausser] of the form: (A.3) U = -e-8X Where n is profit and 6 the parameter of absolute risk aversion. As suggested by Just and Zilberman, it is appropriate to assume that the parameter of absolute risk aversion delcines with wealth, measured by land holding size. For convenience, we assume an inverse relationship: - 23 - (A.4) 6; P > 0 Using the notatios of the text and equations (4) and (5), the farmer's optimization problem is: (A.5) Max. V E {-exp(-6*L*f(n)+6*p*n*L+6*C(A) - 8*L*E*q(n2 n,A Taking the expectation and utilizing equation (A.4), the objective function becomes (see the procedure of Yassour, Zilberman and Rausser): (A.6) Max V E -exp{-P*f(n) + P*p*n + P*C(A)/L-P2*q(n) *c/2} n,A First order conditions require (with some rearrangement): (A.7) V = f -p +l1/g3..qI 0 ; neVn 0 (A.8) Va =1/3*q(n)*a' -C'/L < 0 ; A*Va = 0 It can be shown that with suitable assumptions regarding d", q", all the results derived in the text from the first order conditions (8) and (9) can be obtained from the system (A.7), (A.8). Purthermore, in the special case where f(n) = q(n), and denoting g(K) : (14/8*Q(X)1, the first order conditions for the objective function (A.6) are exactly those derived for the certainty model in the text. While the role of knowledge in the two models is interpreted differently, the hypotheses generated are identical. - 24 - II. Background for Figure 2 Focusing on the early period t=0 such that Kt.- = 0 and ht is negligible), consider the representation of 1 = 0 and H = 0 in the (n,A) n a space. First, it is easy to show dA- - > = Hna 0, i.e., dn Hn= Ena dn ITaa Un= 0 IIa= 0. the two implicit functions are described by upward sloping curves in the (n,a) space. Next, by second order conditions, dnT - dA 0 = -IThnIaa+I-Aa> 0. dn II =0 -dn IT =011*l n a aa na Thus, if the two curves intersect at all, the curve representing Fa =0 will cut a the curve I =0 from above. n The value of A which maintains the quality IT =0 when n=0, is clearly n positive because at A=0 the value of g(.) will be very low, or zero [see equation (3(vii))], thus implying FH <0 and requiring an increase in A so as to n raise IT n References (1) Benor, D. and J.Q. Harrison, Agricultural Extension: The Training and Visit System, (Washington, D.C,: The World Bank), 1977, (2) Feder, G. and G.T. O'Mara "Farm Size and the Adoption of Green Revolution Technology", Economic Development and Cultural Change 30 (1981): 59-76. (3) Feder, G. and G,T, O'Mara "On Information and Innovation Diffusion: A Bayesian Approach" American Journal of Agricultural Economics, 64 (1982): 141-145. (4) Just, R. and R. Pope, "Stochastic Specification of Production Functions and Economic Implications" Journal of Econometrics 7 (1978): 67-86. (5) just, R. and D. Zilberman, "Stochastic Structure, Farm Size, and Technology Adoption in Developing Agriculture" Oxford Economic Papers, (Forthcoming) (6) Hiebert, D., "Risk, Learning and the Adoption of Fertilizer Responsive Seed Varieties" Amrican Journal of Agricultural Economics, 56 (1974): 764-768. (7) Hornik, R.C., Communication and Agriculture, Stanford University and the University of Pennsylvania, July 1982. (8) Kislev , Y. and N. Schori-Bachrach, "The Process of an Innovation Cycle", American Journal of Agricultural Economics 55 (1973): 28-37. (9) Lindner, R.K., A.J. Fischer, and P. Pardey, "The Time to Adoption", Economic Letters 2 (1979): 187-190. (10) Lindner, R.K., "Farm Size and the Time Lag to Adoption of Scale Neutral Innovation", Adelaide, University of Adelaide, (mineo), 1980. (11) O'Mara, G. "A Decision-Theoretic View of Technique Diffusion in a Developing Country", (Unpublished Ph.D. Dissertation, Stanford University, 1971). (12) Yassour, J., D. Zilberman and G. Rausser, "Optimal Choices Among Alternative Technologies with Stochastic Risk", American Journal of Agricultural Economics, 63 (1981): 718-723