Modelos de cinética

The hypotheses which were summarised in section 3.1 related the formation of non-major parties to (1) voter dissatisfaction with major parties, (2) the salience to voters of the issues responsible for this dissatisfaction, and (3) voter turnout at elections. Equations (3.1) and (3.2) use the terms set out in section 3.1.1 in order to express these three hypotheses formally:18

A y(k) = |(xj - Gj)2 - (xf - Vi)2Kk) + 4>i00 + ay(k-l) - r(k)

k = 0, 1 , 2 . . . (3.1)

r(k) = yy(k-l) + Xr(k-l) k = 0, 1, 2 . . . (3.2)

This linear, first-order system of difference equations expresses an output (the change in the number of constituencies with non-major party candidates/non-major party support as a percentage of total votes cast between time k and time k-1) as a function of two exogenous inputs (voter dissatisfaction with major parties and an economic issue whose salience attains crisis proportions), a past output, an endogenous variable (voter turnout at election k) and a series of constants (a, y, Ä.).19 In order to make these relationships clearer, equation (3.1) may be re-written:

A y(k) = Uj(k) + u2(k) + ay(k-l) - r(k)

k = 0, 1,2, ...( 3 .3 ) where the u terms represent the system’s inputs:

Uj(k) = |(xi - G^2 - -Vjq)2! u2(k) = ({r.

Substituting y(k) - y(k-l) for A y(k) and re-arranging the terms of equation (3.3) gives >'(k) - y(k-l) = Uj(k) + u2(k) + ay(k-l) - r(k)

y(k) = Uj(k) + u2(k) + y(k-l) + ay(k-l) - r(k) y(k) = Uj(k) + u2(k) + (1 + a)y(k-l) - r(k)

38

k = 0, 1 , 2 . . . (3.4) The terms of equation (3.4) can be re-arranged in order to obtain an expression for r(k-l) in terms of y and u:

-y(k-l) + Uj(k-1) + u2(k-l) + (1 + a)y(k-2) = r(k-l)

k = 0, 1 , 2 . . . (3.5) The terms of equation (3.5) can be substituted into equation (3.2):

r(k) = yy(k-l) + X[-y(k-l) + u^k-1) + u2(k-l) + (1 + a)y(k-2)]

r(k) = yy(k-1) - Xy(k-1) - Xuj(k-1) - Xu2(k-1) + >.(1 + a)y(k-2)

r(k) = (y - X)y(k-1) + X(1 + a)y(k-2) + Xuj(k-1) + Xu2(k-1)

Equation (3.6) can be substituted into equation (3.4): y(k) = Uj(k) + u2(k) + (1 + oc)y(k-l) - (y-X)y(k-l) -

X(1 + a)y(k-2) - Xuj(k-1) - Xu2(k-1)

Equation (3.7) can be re-written:

y(k) = Uj(k) + u2(k) - Xuj(k-1) - Xu2(k-1) + (1 + a - y+ X)y(k-1) - X(1 + a)y(k-2)

k = 0, 1 , 2. . . ( 3 . 6)

k = 0, 1 , 2 . . . (3.7)

k = 0, 1, 2. . . (3. 8) Equation (3.8) is a second-order linear difference equation written in terms of the system’s inputs, Uj and u2, and its output, y. It is in all respects equivalent to equations (3.1) and (3.2). Equation (3.8) can be solved by means of a z-transform and an inverse z-transform (Boynton, 1980pp.59-90). As a first step, the system’s outputs and inputs must be segregated:

y(k) - (l+oc-yfX)y(k-l) + X(l+ot)y(k-2) = ut + u2 - X u ^ - l ) - Xu2(k-1)

k = 0 , 1 , 2 . . (3.9) The z-transform of the output sequence of equation (3.9) yields

Z[y(k) - (l-t-a-Yt-X)y(k-l) + X(l+a)y(k-2)]

= Y (z)- (l+ a-yfX )z'1Y(z) + X (l+ a )z 2Y(z) (3.10)

= [1 - (1+ a-Y + ^ z1 + X (l+ a )z 2]Y(z) (3.11)

(where Y(z) represents the z-transform of y(k)).

The z-transform of the input sequence of equation (3.9) yields Z[Uj(k) + u2(k) - Xu^k-1) - Xu2(k-1)]

= Uj(z) + U2(z) - XujZ^Ujfc) - Xu2z_1U2(z)

= (1 - X z^U ^z) + (1 - \ z l)\J2(z)

= 1 -Xz-1[U1(z) + U2(z)] (3.12)

(where Uj(z) and U2(z) represent the z-transform of Uj(z) and u2(z)).

Recombining equations (3.11) and (3.12) gives

[1 - (l+cc-y+XK1 + X(l+cx)z'2]Y(z) = (1 - X z ^ tU ^ z) + U2(z)] Dividing both sides by 1 - (l+ a-y+ X ^'1 + X (l+a)z'2 gives

Y(z) 1 -X z 1

l-(l+ a-y+ X )z_1

+ X ( l+ a ) z - 2

[Ul (z)+U1{z)).

Multiplying (3.14) through by z2 gives the solution to equations (3.1) and (3.2): Y(z) z2-(\+a-y+X)z+ X ( \ + a ) [Ux(z)+U2(z)l (3.14) (3.15) 3.1.3. Analysis

Equation (3.15) describes the manner in which the system’s inputs (dissatisfaction with major parties and issue salience) are transformed into an output (non-major party formation and non-major party electoral support). The system’s response characteristics can be obtained when numeric values are as­ signed to its constants and the characteristics of its input terms are specifies. Accordingly, let

a= .l Y=.2

h=.2

£/ (£ )= / 0 if*=0 1 \ 1 for all other k £/(*)= J O if£=0

2 \ 1 for all other k

(3.16) Equation (3.16) indicates that partisan attachment to major parties is great (ninety percent of voters identify with one of the major parties) and that the strength and durability of any change in vote choice from the major parties to a non-major party is relatively low (.2, where 0 < y ^ l and 0<X < 1). Further, equation (3.16) indicates that Uj(k) and U2(k) are step inputs - in other words, that voter dissatisfaction with major parties and the salience of a particular issue i is initially absent (k=0) and subsequently present (k > 0) and constant at the value of Uj = U2 = 1.

Secondly, let a=.2 y=A h=A

^ (*)=/? p=o

£/(*)= J O i(k=0 2 \ 1 for all other k

(3.17) Equation (3.17) indicates that partisan attachment to major parties is low relative to equation (3.16) (with only eighty percent of voters identifying with one of the major parties) and that the strength and durability of any change in vote choice from the major parties to a non-major party is high relative to equation (3.16). Again, issue salience and voter dissatisfaction with the major parties is absent at k=0 and present (at the constant level of Uj = U2= l) at k>0.

Equation (3.18) gives the response characteristics of equation (3.15) under the conditions set out in equation (3.16). Equation (3.19) gives the response characteristics of equation (3.15) under the conditions set out in equation (3.17). (Appendix D sets out these mathematical derivations).

Y(k) = -18.42(.84)k + 1.29(.27)k + 16.70(l)k

Y(k) = 15.0(.6)k -40.00(.8)k + 25.00(l)k k = 0 , 1,2, ...( 3 .1 9 )

Figure 3-1: Non-Major Parly Formation and Electoral Support in Response

to Two Unit Step Inputs

--- Equation (3.18) --- Equation (3.19)

Time

Figure 3-1 plots equations (3.18) and (3.19). In each case, non-major party formation and electoral support is nil at k=0, increases rapidly at 0<k<10, and stabilises at k>10. Given the conditions set out in equations (3.16) and (3.17), the non-major party exists indefinitely. Figure 3-1 also indicates that the lower the level of partisan identification with major parties, and the stronger and the more durable the response to the non-major party, the greater is non-major party formation and support: for all k, equation (3.19) >(3.18).

The indefinite presence of voter dissatisfaction with major parties and issue salience thus cause the indefinite presence of non-major party formation and support. However, equation (3.15) indicates that non-major party longevity does not require sustained levels of both inputs. Let

a = .l

y

h=.2

U (k)= S 0 if£= 0 1 \ 1 for all other k

U (k)—S 1 if£ = 0

2{ ) \ 0 for all other k

(3.20) With respect to the values of the constants, equations (3.16) and (3.20) are identical. With respect to the second input term, however, equations (3.16) and (3.20) differ. Uj(k) (voter dissatisfaction with major parties) remains a step input - absent at k=0 and present at a constant level at loO. U2(k) (issue salience), however, is a Kronecker Delta input - present at k=0 and absent at k>0. Equation (3.20) thus describes a

Situation in which the partisan attachment to the major parties is relatively high, the response to non­ major parties is relatively low, voter dissatisfaction with major parties is continuously present at k=0 but issue salience is absent after k=0.

Secondly, let a=.2 y=A X=.4

£/ (*)=J o if*=0 1 \ 1 for all other k

U (lc )= f 1 if£=0

2

\

(3.21) With respect to the values of constant terms, equations (3.21) and (3.17) are identical. With respect to input terms, equations (3.21) and (3.20) are identical. Equation (3.21) thus describes a situation in which partisan attachment to the major parties is relatively low, the response to major parties is relatively high, voter dissatisfaction with the major parties is present at k>0 and issue salience is absent at k>0.

Equation (3.22) gives the response characteristics of equation (3.15) to the conditions described in equation (3.20). Equation (3.23) gives the response characteristics of equation (3.15) to the conditions described in equation (3.21).

Y(k) = -7.46(.84)k -1.09(.27)k + 8.56(l)k k = 0 , 1 , 2 , . . . (3.22)

Y(k) = 2.50(.6)k - 15.00(.8)k + 12.50(l)k k = 0 , 1 , 2 , . . . (3.23)

Figure 3-2: Non-Major Party Formation and Electoral Support in Response to One Unit Step and One Kronecker Delta Input

20

Figure 3-2 plots equations (3.22) and (3.23). As in Figure 3-1, non-major party formation and support is 0 at k=0, increases rapidly at 0<k<10, and stabilises at k>10. As in Figure 3-1, a relatively low level of partisan attachment to the major parties, together with a relatively strong and durable response to the non-major party, is associated with higher levels of non-major party formation support: for all k, (3.23) > (3.22). At the same time, however, Figure 3-2 conveys new information. Most importantly, as long as voter dissatisfaction with the major parties remains unabated - even if the salience of the issue which engenders this dissatisfaction quickly dissipates - the non-major party continues to exist indefinitely. However, the number of constituencies in which non-major party candidates are present, as well as the non-major party’s percentage share of the total vote in these constituencies, is lower when the non-major party is sustained only by voter dissatisfaction with major parties than when it is is sustained by both voter dissatisfaction and issue salience: for all k, (3.22) < (3.18) and (3.23) < (3.19). Studies of non-major parties [i.e., (Rosenstone, Behr and Lazarus, 1984); (Pinard, 1975); (Mazmanian, 1974) anticipate neither of these findings.

Nor do these studies do not explicitly address the process of non-major party dissolution. Equation (3.15), however, provides insight into this process. Let

a= .l

Y=.2 k=.2

U (lc)=S 1 if£=0 ^ \ 0 for all other k

U (k)= S 1 if£=0 \ 0 for all other k

(3.24) and let a=.2 y=A h=A U (lc)= f 1 if&=0 ^ ’ \ 0 for all other k

II (k)= S 1 if£=0 \ 0 for all other k

(3.25) In terms of the value of its constants, equation (3.24) is equivalent to equations (3.16) and (3.20) -- the partisan attachment to the major parties is relatively strong and the strength and durability of any change in vote choice towards a non-major party is relatively low. In terms of the value of its constants, equation (3.25) is equivalent to equations (3.17) and (3.21) - the partisan attachment to major parties is relatively weak and the strength and durability of any change in vote choice towards a non-major party is relatively great. Input terms, however, distinguish equations (3.24) and (3.25) - both describe situations in which issue salience and voter dissatisfaction with major parties is initially (at k=0) present and subsequently (at k>0) absent.

Equation (3.26) gives the response characteristics of equation (3.15) to the conditions specified in equation (3.24); equation (3.27) gives the response characteristics of equation (3.15) to the conditions specified in equation (3.25).

Y(k) = 3.51(.84)k -3.5l(.27)k Y(k) = -10.00(.6)k +10.00(.8)k

k=0,1,2,. ..(3.26) k=0,1,2,. ..(3.27)

Figure 3-3: Non-Major Party Formation, Electoral Support and Dissolution in Response to Two Kronecker

Delta Inputs

Equation (3.26) Equation (3.27)

Figure 3-3 plots equations (3.26) and (3.27). Each equation indicates that non-major party formation and support is nil at k=0, increases rapidly at 0 < £ < 4 , and declines (somewhat less rapidly) at k>4. Figure 3-3, in short, indicates that if issue salience and voter dissatisfaction with major parties disappears, so too does the non-major party and its electoral support. Once again, however, the lower the level of attachment to the major parties, and the stronger and more durable the response to the non-major parties, the greater is non-major party formation and support: for all k, equation (3.21) > (3.24). Non-major party dissolution is therefore contingent upon the disappearance, not merely the attenuation, of voter dissatis­ faction with major parties. Again, studies of non-major parties (Rosenstone, Behr and Lazarus, 1984); (Pinard, 1975); (Mazmanian, 1974) do not anticipate this result.