Papel de los aminoácidos del motivo “CAAX” en la

For notational convenience we omit dependency on time and write x(t) simply as x.

tional to adopt Technology i, given the current adoption levels, x. To determine the

fraction of adopters of each technology, we introduce the notion ofindifference points,

which identify thresholds in users technology valuation (θ)corresponding to changes

in technology preference. Specifically, θ0_i(x),i∈ {1,2}identify theθvalue separating

users with a negative utility for Technologyifrom those with a positive utility. In other words, for technology penetration levelsx,θ0_i(x)is such thatUi(θ0_i,x) =0, andUi(θ,x)

is positive (negative) forθvalues larger (smaller) thanθ0_i.

From eqs. (3.1) and (3.2),Ui(θ0_i,x) =0 gives

θ0₁(x) = p1−(x1+α1βx2)

q1

(3.5)

θ0₂(x) = p2−(βx2+α2x1)

q₂ (3.6)

Similarly, θ1₂(x) corresponds to the θ value separating users preferring Technology 1

from those preferring Technology 2,i.e., U₁(θ1₂,x) =U2(θ1₂,x)and users withθ>θ1₂(x)

derive greater utility from Technology 2 than Technology 1 (recall that q₂>q₁). Set- ting,U₁(θ1₂,x) =U2(θ12,x)gives

θ1₂(x) = (1−α2)x1−β(1−α1)x2+p2−p1

q₂₋q₁ (3.7)

Combining eqs. (3.5)-(3.7) gives

θ1₂(x)−θ0₁(x) = q2 q2−q1 (θ0₂(x)−θ0₁(x)), (3.8) θ1₂(x)−θ0₂(x) = q1 q₂₋q₁(θ 0 2(x)−θ01(x)), (3.9)

Proposition 4

Ifθ0₁(x)<θ0₂(x), thenθ1₂(x)>θ0₂(x)>θ0₁(x).

Ifθ0₁(x)≥θ0₂(x), thenθ1₂(x)≤θ0₂(x)≤θ0₁(x).

Proposition 4 constrains the possible orderings of the indifference points given by eqs. (3.5)-(3.7), so thatHi(x),i∈ {1,2}can be characterized in a compact manner.

H1(x) =        [θ1₂(x)]_[0,1]−[θ01(x)][0,1] ifθ01(x)<θ 0 2(x) 0 otherwise (3.10) H2(x) =        1₋[θ1₂(x)]_[0,1] ifθ0₁(x)<θ0₂(x) 1₋[θ0₂(x)][0,1] otherwise

wherey_[a,b]is the ‘projection15’ ofyinto[a,b].

The expressions forH₁(x)andH₂(x)determine the trajectory as well as the equilib- rium outcome of the adoption process as per eqs. (3.3) and (3.4) respectively. Eq. (3.10) characterizes H_i(x), i={1,2_} through multiple possible expressions that depend on the relative ordering ofθ0₁(x),θ0₂(x)andθ1₂(x), and the outcome of their projections on

[0,1]. Identifying the different combinations that eq. (3.10) gives rise to calls for par- titioning the (x1,x2)-plane intoregions, each corresponding to unique expressions for

the(H₁(x),H₂(x))pair. A method for constructing such a partition is described next. First, consider all values of x which satisfy θ0₁(x)_≥θ0₂(x). In the corresponding half-plane of the(x₁,x₂)-plane, H₁(x) is always 0, but the value ofH₂(x)depends on the projection of θ0₂(x)on[0,1](i.e.,whetherθ0₂(x)≤0,0<θ0₂(x)<1,or 1≤θ0₂(x)).

This creates three regions in the(x₁,x₂)-plane, each with a different expression for the 15_{i.e.,its value isyfory∈[a,b],afory<a, andbfory>b.}

(H₁(x),H₂(x))pair. These three regions, labeledR₁,R₂andR₃, and the corresponding conditions onθ0₂(x)appear in the left side of Table 3.1. The expressions forHi(x), i=

{1,2_}in each region are provided in Table 3.2.

Conversely, for values of x which satisfy θ0₁(x) < θ0₂(x) (the other half-plane),

eq. (3.10) indicates that expressions for H₁(x) and H₂(x) depend on the positions of both θ1₂(x) and θ0₁(x) relative to 0 and 1 (i.e., whether θ1₂(x) ≤0, 0<θ1₂(x) <1 or

1_≤θ1₂(x), and similarly forθ0₁(x)). This yields nine possible combinations. The num-

ber of feasible combinations can, however, be reduced to six using Proposition 4, which constrains θ0₁(x)<θ0₂(x)<θ1₂(x). For example, when θ0₁(x)<θ0₂(x), θ1₂(x)_≤0 and 1_≤θ0₁(x) is infeasible per Proposition 4. Thus in the half-plane θ0₁(x)<θ0₂(x), there

are six possible expressions for H_i(x), i={1,2_}. These expressions are reported on the right side of Table 3.1, with the corresponding regions labeledR₄toR₉. Combining

the two half-planes gives a total of nine regions, R1 to R9, where in each region the

(H1(x),H2(x))pair has a unique expression as specified in Table 3.2.

This partitioning in nine regions has a graphical representation, as shown in Fig- ure 3.1. The lineθ0₁(x) =θ0₂(x)splits the(x1,x2)-plane in the two previously mentioned

half-planes. The two lines corresponding to θ0₂(x) =0 andθ0₂(x) =1 are parallel, and

define the three regions R₁, R₂ and R₃ in the half-planeθ₁0(x)≥θ0₂(x). Similarly, the

linesθ0₁(x) =0,θ0₁(x) =1,θ1₂(x) =0 andθ1₂=1, divide the second half-plane into the

six regions,R₄toR₉. Figure 3.1 also illustrates that the lines θ0₂(x) =0,θ0₁(x) =0 and θ1₂(x) =0 always intersect at a point denoted asP, and the linesθ0₂(x) =1,θ0₁(x) =1

and θ1₂(x) =1 always intersect at a point denoted asQ, with bothP andQ16 lying on

the line θ0₁ =θ0₂. The points P and Q can be shown to act as “anchors” that ensure

that the(x1,x2)-plane is always partitioned into exactly nine regions with fixed relative

positions.

It should also be noted that all nine regions need not always be feasible. Whether or not they are feasible depends on their relative position in the solution space,

S=_{(x1,x2)s.t.0≤x1≤1,0≤x2≤1,x1+x2 ≤1}. The number of regions that lie

insideS(and hence are relevant to the analysis) is a function of the system parameters

(q_i,p_i,β,αi;i={1,2}). Last but not least, as shown in Appendix B.1, the partitioning

of the solution space into nine regions actually holds for more generic (monotonic) ex- ternality functions,i.e.,it is not an artifact of the simplified linear externality function.

Finally, we pause to briefly interpret the conditions that define each region, and their

implications for solutions. We do so by way of an example, focusing on Region R₈.

Region R8 is defined as the set of adoption levels, x= (x1,x2), for which 1≤θ1₂(x)

and 0_≤θ0₁(x)<1. The condition 1≤θ1₂(x) implies that in Region 8 all users prefer

Technology 1 over 2. Hence any existing Technology 2 adopter will disadopt. Thus, in

R₈, users can either be non-adopters of both technologies (0<θ<θ0₁(x)) or adopters

of Technology 1 (ifθ>θ0₁(x)). Similar interpretations are possible for other regions. 16_{The coordinates of the points PandQcan be readily found by solving simple systems of linear} equations given by eqs.(3.5)-(3.7).

Figure 3.1: Region Partitions

Table 3.1: Partitions characterizingHi(x) θ0₁(x)≥θ0₂(x) θ0₁(x)<θ0₂(x)

Region condition Region condition

R1 θ0₂(x)_≤0 R4 θ1₂(x)_≤0, 0_≤θ0₁(x) R2 0<θ0₂(x)<1 R5 0<θ1₂(x)<1, θ0₁(x)≤0 R3 1≤θ0₂(x) R6 0<θ₂1(x)<1, 0<θ0₁(x)<1 R7 1_≤θ1₂(x), θ0₁(x)_≤0 R8 1_≤θ1₂(x), 0<θ0₁(x)<1 R9 1_≤θ1₂(x), 1_≤θ0₁(x)

Table 3.2: Expressions forHi(x) R1 H1(x) =0 H2(x) =1 R2 H2(x) =1−p2−(βxq22+α2x1) R3 H2(x) =0 R4 H1(x) =0 H2(x) =1 R5 H1(x) =(1−α2)x1−β(1−α1)x2+p2−p1 q2−q1 H2(x) =1− (1−α2)x1−β(1−α1)x2+p2−p1 q2−q1 R6 H1(x) =(1−α2)x1−βq(12−−qα11)x2+p2−p1 H2(x) =1− (1−α2)x1−β(1−α1)x2+p2−p1 q2−q1 −p1−(x1+βα1x2) q1 R7 H1(x) =1 H2(x) =0 R8 H1(x) =1−p1−(x1+βα1x2) q1 R9 H1(x) =0