MODELING THE DIFFUSION OF PUBLlC POLICY INNOVATIONSAMONGTHE

U.S. STATES:

VIJAY MAHAJAN School of Management, Staic University of New York, Buffalo. NY 14314, C.S.A.

KItiGSLEY E. HAYNES

Lyndon R. Johnson School of Public .Affxirh and Deparrment of Geography. The University of Texas, Austin. TX

7871’. U.S.A.

and

K. C. HAL KCMAR Department of Geograph!. The University of rexas, /\ustin. TX 78712. U.S.A.

(Received !i Irow 1976; recked 18 February 1977)

Abstract--This paper describes and validates a general model of slate diffusion of public policy innovations. The model can be used IO identify whether new policies are likely to become fully adopted into the system of states and the time horizon over which this is likeI) IO occur. Limitations and extensions of the model are also concluded.

I?ITRODUCTION

Comparative rescarch on state policies has focused on the characteristics of states that arc useful in predicting their behavior. These characteristics range from socio- economic conditions(C~ttendcn[ I]: Hofferbert[?]) to political organization (Jacob and Vines 131: Sullivan [4]) while the state behavior being analyzed has ranged from welfare expenditures (Dau\on and Robinxm[5]: Dye[h)) to policy outputs (Crew[7]; Hoffcrbert[8]). This paper represents a continuation of that tradition. In this case the behavior we are analyzing represents the state adop- tion of a new policy. From the perspective of the adop- ter, the policy is an innovation. This does not mean that the adopting state originated the policy or program but that the policy or program i\ ne\v for that state.

This distinction between origination of innovation and adoption of an innovation is of critical importance in diffusion theory (Rose[9]). Originators are considered leaders or opinion makers whereas all other adopters are referred to as followers or more specifically as early adopters, early majority, late majority and laggards (Rogers [ IO]). More generally WC can refer to all but the originators as imitators. imitators are influenced in the timing of adoption by the decision of other states in t he same system m,hcreas originators are not influenced by prc- vious adopters. Rogers[lOl defines these originators quite arbitrarily as the first 2$/c of the adopters. Furthermore, he defines originators as vcnlurzsome and daring a> well as most likely to interact with other potential Ltdopterh. Our purpose here is to effectively model the entire adoptive process and hence adopter groups are not differentiated. However, by implication an innovation is not considered unless at least one potential adopter does in facl adopt the new policy or program. Hence it is not the process of origination but rather the spreading process by which

+Thi, paper is part of ongoing wearch on the diffusion of innovation supported by the tiational Science Foundation (G- 36829). This support is apprcciared.

something (a policy) that has already been originated (is new) spreads (diffuses) among potential adopters (U.S. States) over time. In summary, the concern of this paper is the development of a model of the diffusion of public policy innovations and an empirical validation of that model.

Such a development and validation is a response to an interchange between Gray[ll] and Walker1 121. In an earlier paper, Gray[ 13, 141 outlined a simple model of interaction diffusion in the context of comparative state policy research. This was an attempt “to extend in a more rigorous fashion the investigation of the innovation by states” along the lines of Walker’s[ 151 exploratory study (Gray[l3/). In her commentary Gray[ I II notes that her model is simplified and that her primary interest was “to work in the direction pointed out by failure of the empirical model to yield accurate predictions”. This interest, howecer. focused on the empirical limitation of the model and not the refinement of the model itself. Both Gray and Walker have emphasized the need to develop a general model of state diffusion of public policy innovations and to that end we have described and validated such a model.

MODEL

It is interesting to note that the diffusion process models used in the social sciences have often been modeled after physical or bioplogical processes such as heat transfer (Haynes, Mahajan and White[l6]), the spread of epidemics (Brown[l71) and the growth of population (Peilou[l8]). A review of such efforts is given in Mahajan and Schocman[19]. To model a diffusion process the analyst works with a few macro-parameters which locate a curve describing the spread of an in- novation over time.

In applying diffusion theory to the timing of the adoption of a new policy we specify the following relationship. Let f(f) = Proportion of adopters at time t and F(t) =Cum- illative proportion of adopters at time 1. then at time T

2’9

V. .\i,!HAJAS. K. E. HAYSFS and K. C. RAI. KWAK

‘,

(1) 0

and at any time f

f(i) = y. f2)

The assumption is that f(r) and F(t) are continuous functions with derivatives that exist at all points. Fur- thermore, if it can be assumed that f(t) is a unimodal function, whi<h may or may not be true in the case of public policy diffusion, then f(t) has a maxima when

dJ’(i”) -=O at 1=I*.

df

Therefore, using eqn (Z), if we have the cumulative adopters djstributjon function F(r) the adopters dis- tribution function f(t) can be obtained. Equation (2) ako gives the rate of diffusion at time t. i.e. the differential change in cumulative adoption in the time period (t, t tdl). If F is the ceiling on proportion of adopters, Mahajan and Schoeman[l91 have su_Mested the following representation for the rate eqn (2).

f(t) = d!! = g(t) (F - F(r))

and

Fft = tO) = F. (41

In eqn (4) (8 -F(t)) contains nonadopters at time t and g(t) is a growth coeficient. One initial v&e condition is required to so&e eqn (4) for F(r), As can be observed, the rate of diffusion or proportion of adoptions at time f is controlled by the growth coefficient g(t), which will vary from innovation to innovation, the social system in which it is diffused, the channel, and the change agents used to diffuse it

g(f) = h(innovation. social system, channels, change agents). (5)

In our case, the social system being considered is the U.S.A. Mahajan and Aga~~[20] have analyzed the state adoption of public policies in terms of eight factors ~Gubernatoriai Power-Legal Professiona~is~Ur- bank&ion, Citizen Political rnvo~vement and Inter-party competition, welfare expenditures, natural resources, Le- gislature Professionalism, tndustraiization, Interest Group Pressure and Party Cohesiveness) derived from political and socio-economic characteristics of the states. They have suggested that these factors may be con- sidered in the development of g(f). However, no attempt has been made in the diffusion literature to find an empirical or mathematical equation for g(t).

In the absence of functions for g(t), Mahajan and Schoemantl9J have proposed the use of the previous number of adopters as a surrogate measure of the diffusion elements. Expressing g(f) as a linear function of F(t), i.e.

g(t) = a + bF(tj

and substituting this in eqn (4), i.e.

f(r)=!y= (u+hF(t))(F-F(l))

= a(F - F(l)) + bF(t)fF- F(t)) (6)

they derive

(F (u + OFy exp ( - (a + bF) (f - to)) f(q = (0 + hF0)

(, + m- m (8)

(a exp (-(a + hR (t - id)’ ’ 0

and fft) has a maxima when

fQ*) = (0 + m* 4b

Since for new policies F(t = to) = 0, i.e. there are no previous adopters

Maximum values will be the same as given in eqns (9) and (10). The diffusion model. eqn (7). can also be expressed in terms of the absolute number of adopters by defining the following:

n(t) = number of adopters at time t Nft_) = cumulative number of adopters at time I

N = ceiling on the number of adopters M = total potential number of adopters

N(t = t,,) = No

Therefore.

Substituting these expressions in eqns (1 I) and (12) we

Modeling the diiusion of public policy innovations among the U.S. states

where constants

261

obtain

N(t) = fi - Tiy i,:i exp (-(a t b’N) (t - 6,))

1 + b’(fi - N:) (a + b,N ) exp (-(a + b’N) (t - to))

0

(13)

0 (a + b’I@ exp (-(a + b’N) (t - to)) n(t) = (a + b’Nn)

1 t $ifj exp (-(a + b’N) (t - t$i)’

(14) where

b’=$

t*=&)-L a

(a tb’&lnb’N (15)

n(t*) = (a + b’@)* 4b’

and

N(t*) = (b’# - a) 2b’ (17)

We will now discuss how the parameters a, bj and N, in these models can be evaluated from time series data by using the discrete analogue of these models (Mahajan and Schoeman [ 191).

Estimation of parameters The proposed model is (recall eqn (6)).

f(&!E& (a t bF(t)) (F-F(t))

or

f(t)=aF+(bP-a)F(t)-bF’(t)

which may also be approximated by

F(t t 1) - F(t) = aP t (bF - a)F(t) - bP(t)

or

F(t t l)= aF t (bF- a +l)F(t)- bF’(t) (18)

or in terms of absolute nnmbers of adopters

N(t + 1) = aN t (b’?? - a t 1) N(t) - b’N’(t)

= aI t a2N(t) + a&‘(t) (19)

‘TSince the examination of data suggested that the adopters distribution (f(t) or n(t)) was not a unimodal function, other special features of the model were not calculated (i.e. eons (14H16).

(Y,=aN,a,=b’~-atl,cu,=-b. (20)

We can use regression analysis to estimate aI, a2 and a3 and once the a’s have been estimated, a, b’ and N can be calculated. Since

b’ = -a3

a=$L N'

so

a*=-a#-%t 1 N

or

Therefore

fi= -(az-1)+~(at-1)2-4a1a3 2%

Behavioral considerations Although the proposed model is derived without any a

priori behavioral assumptions, Mahajan and Schoeman[191 have suggested that in equation (6) the constant a may be interpreted as an index of un- influenced adoption with the result that the term contain- ing a reflects adoptions by innovators. On the other hand, b is an index of influenced adoption as it takes into account the interaction between adopters and nonadop- ters. The second term, then, reflects adoptions by non- innovators. Depending upon the behavioral emphasis given a and b the resulting diffusion model may take several forms. In fact it has been shown (Mahajan and Schoeman[l9]) that behaviorally based models used in economics, geography, marketing and sociology are spe- cial cases of the proposed model.

RESULTS Ten public policies were selected to test the model.

The diffusion time span of these policies range from a minimum of eleven years to a maximum of fifty-six years. The data was restricted to the 48 contiguous states and most policies had completely diffused although three examples of incomplete diffusion were utilized. The policies represent a wide range in histerical period, time span and program characteristics.

Table 1 displays the regression results. The data ap- pear to be in close agreement with the model. The RZ values indicate that the discrete analogue of the model (eqn (19)) describes the diffusion behavior of the public policy innovations rather well (average value of R* is 0.9724). The regression constants seem reasonable. The values of (Ye are negative as required and the values of N are plausible (note that one of the advantages of this model is that we can determine the ceiling on the number of adopters).

The regression estimates of a,b’ and fl are used in the cumulative adopters function (eqn (13)) to predict the adoption. Examination of rz values, between the actual and predicted adoptions, in Table 2 indicate that over the diffusion span the model performs quite well (average r* value is 0.9366)t

SEPS Vol. 11, No. 5-D

262 V. MAHAJAN, K. E. HAYNES and K. C. BAL KUMAR

Table 1. Prediction of adopters by regression analogne of diffusion equation

Time Span Number of Regression Constants Diffusion Model Parameters Public Policies* m-s) Adopters R*

oil az 013 a b N

ACCOUIltant

Licensing

Architect Licensing

Defense & Disaster Supervision

Gasoline Tax

Mental Health Standards

Nurse Licensing

Planning & Development Agencies

Police & Highway Patrol

Uniform Commercial Code

56 48 .1601 1.2678 -.0058 .0034 .0058 48 .9963

15

38 3.3789 1.0543 -.0036 .0866 .0036

48 2.2102 1.1487 -.0042 .0470 .0042

48 1.1755 1.2988 -.0066

-.0043

-.0122

.0240 .0066

48 .0506 1.2112

48

.4477 1.0713 -.0016 .0090 .OOlb

.0886 1.6065 -.0147 .0022 .0147

4.2362

3.4316

2.9809

1.5757

-.0454

-.0141

.0921

.0730

.0454

.0141

.2816 1.6048

.OOlO

.0057

.0043

.0122

50

41

47

48

39

48

49

49

50

.9947

.9951

.8990

.9507

.9562

.9896

.9772

.9933

.9714

Source: *Public Policy, State Adoption information derived from Mahajan and Agarwal (1977).

Table 2. Forecasting accuracy of the model for the public poli- cies

Public Policies Period r=

Accountant Licensing 1896-1951 .9972

Architect Licensing 1897-1951 .9903

Community Affairs Licensing 1959-1970 .9776

Defense & Disaster Supervision 1949-1959 .7933

Gasoline Tax 1919-1929 .9565

Mental Health Standards 1955-1969 .9763

Nurse Licensing 1903-1933 .9935

Planning 6 Development Agencies 1935-1954 .9730

Police & Highway Patrol 1903-1947 .9531

Uniform Colmnercial Code 1954-1968 .7547

CONCLUSIONS

Now only is it possible to effectively model the diffusion of public policy innovations, but it has been demonstrated that this can be done with no a prior’ behavioral assumptions. Furthermore such a model can be used to identify whether new policies are likely to become fully adopted into the system of states and the time horizon over which this is likely to occur. However, the proposed model possesses certain limitations. For example, it assumes that:

(1) The number of potential adopters at any time t is constant. i.e. N or F or Mjs constant. For example, in our case we have implicitly assumed that during the entire diffusion process, the number of potential adopter-s in the U.S.A. remained constant, i.e. M= 48 and iV, although unknown, remains constant over time. Mahajan and Peterson[21] have proposed an extension of the model that relaxes this assumption and studies diffusion of an innovation in a dynamic population.

(2) State adoption of a policy does not complement, enhance or inhibit the adoption of some other policies, i.e. policies are adopted independent of each other.

Cases where innovations complement or substitute each other are considered by Mahajan and Koutrovellis[22] and Peterson and Mahajan [23]. Furthermore, rejection of a policy after adoption or readoption (amendments) are not incorporated in the model.

Finally the direction of further modelling research should concentrate on the full interpretation and un- derstanding of g(t) which encompasses a series of in- termediate functions in the diffusion process. The au- thors are presently empirically examining g(t) to deter- mine the role of socio-economic structure and political organization characteristics on the spatial diffusion of public policy innovations among U.S. states. Analytically these advances at the macro level of modelling the diffusion process of public policy innovation would argue for a more detailed analysis of the individual behavioral characteristics of the decision process in adoptions. Only with a balanced development in our understanding of both the macro and micro aspects of the diffusion process can a complete untangling of the innovation process be accomplished.

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