An age structured demographic theory of technological change

Abstract

At the heart of technology transitions lie complex processes of social and industrial dynamics. The quantitative study of sustainability transitions requires modelling work, which necessitates a theory of technology substitution. Many, if not most, contemporary modelling approaches for future technology pathways overlook most aspects of transitions theory, for instance dimensions of heterogenous investor choices, dynamic rates of diffusion and the profile of transitions. A significant body of literature however exists that demonstrates how transitions follow S-shaped diffusion curves or Lotka-Volterra systems of equations. This framework is used ex-post since timescales can only be reliably obtained in cases where the transitions have already occurred, precluding its use for studying cases of interest where nascent innovations in protective niches await favourable conditions for their diffusion. In principle, scaling parameters of transitions can, however, be derived from knowledge of industrial dynamics, technology turnover rates and technology characteristics. In this context, this paper presents a theory framework for evaluating the parameterisation of S-shaped diffusion curves for use in simulation models of technology transitions without the involvement of historical data fitting, making use of standard demography theory applied to technology at the unit level. The classic Lotka-Volterra competition system emerges from first principles from demography theory, its timescales explained in terms of technology lifetimes and industrial dynamics. The theory is placed in the context of the multi-level perspective on technology transitions, where innovation and the diffusion of new socio-technical regimes take a prominent place, as well as discrete choice theory, the primary theoretical framework for introducing agent diversity.

Appendices

Appendix A: Growth under full employment

In a situation where an innovation was to follow a path free of competitors and grow as fast as its industry can produce it, it would follow the fastest possible rate $t_{i}^{-1}$ of exponential growth determined by:

$$ 1 ={\int}_{0}^{\infty} R_{i} e^{-b/t_{i}} {\ell_{i}^{K}}(b) {P_{i}^{K}}(b) db. $$

(44)

This is a transcendental equation that can only be solved numerically. As demonstrated by Lotka (1911) (see Kot 2001 for a clearer derivation), it has only one real solution, all others being complex of the form u±i v which give rise to oscillatory behaviour in the real part. The non-oscillatory real solution, an exponential, can be approximated if simple forms are taken for ℓ(b) and P(b) (see Fig. 3, right)

$${\ell_{i}^{K}}(b) \simeq e^{-b/{\tau_{i}^{K}}}, \quad P(b) = \left\{ \begin{array}{rl} 0, &\quad b < b_{0}\\ P_{i}, &\quad b \geq b_{0} \end{array}\right. $$

$$\Rightarrow {\lambda b_{0} \over (R_{i} P_{i} b_{0})} = e^{-\lambda b_{0}} \quad \text{with} \, \lambda = {{1\over t_{i}} + {1 \over {\tau_{i}^{K}}}},\\ $$

where b ₀ is the time between investment and construction, P _i is a production rate and therefore (R _i P _i )⁻¹ is the rate of expansion of production capacity (in inverse years), and ${\tau _{i}^{K}}$ is the timescale of capital depreciation, its life expectancy (and therefore we always have ${\tau _{i}^{K}} >> b_{0}$). To first order, one can find which is the dominant of these timescales in particular situations, using limits for the dimensionless parameter R _i P _i b ₀ (which determines the slope of the linear left hand side of the equation, see graph).

Case 1

(R _i P _i b ₀)is small ⇒λ b ₀ is small

Fig. 6

ᅟ

We perform a Taylor expansion around λ b ₀ = 0,

$$ \lambda = R_{i}P_{i} \left( 1 - b_{0} \lambda - {{b_{0}^{2}}\lambda^{2} \over 2} + ... \right)\quad \Rightarrow \quad t_{i} \simeq \left[ {1 \over (R_{i}P_{i})^{-1} + b_{0}} - {1 \over {\tau_{i}^{K}}}\right]^{-1} $$

(45)

Since R _i P _i cannot be a very small quantity, it is most likely that b ₀<<(R _i P _i )⁻¹, for instance with small technologies that are ready to use as they come out of the factory (e.g. vehicles). And since $R_{i}P_{i} >> 1/{\tau _{i}^{K}}$, then t _i ≃(R _i P _i )⁻¹. In this case the rate of production is constrained by the rate of re-investment into production capacity, which depends on the rate of production of technology units but not on the time of construction of production capacity. For large systems (e.g. power plants, wind turbines, infrastructure), the time of construction may be long (i.e. several years), constraining money flows used for firm expansions. For small modular technologies (e.g. electronics), the time of production is short and other timescales dominate the time ‘bottleneck’.

Case 2

(R _i P _i b ₀) is of order 1 ⇒λ b ₀≃1

We perform a Taylor expansion around λ b ₀=1,

$$ \lambda = R_{i}P_{i} e^{-1} \left( 1 - (b_{0} \lambda-1) - ... \right)\quad \Rightarrow \quad t_{i} \simeq \left[ {1 \over (eR_{i}P_{i})^{-1} + b_{0}} - {1 \over {\tau_{i}^{K}}}\right]^{-1} $$

(46)

If $b_{0} << R_{i}^{-1}P_{i}^{-1}$ then t=e R _i P _i and the limiting timescale is again the re-investment rate. However, if $b_{0} >> R_{i}^{-1}P_{i}^{-1}$ then t=b ₀ and the rate of growth is limited by the rate of completion of capital installation. For example, for technologies with complex production capital structures with a long time of for installation with no income constrains the rate of return, the dominant bottleneck timescale is b ₀, a situation where a firm must wait for expansion projects to come to completion and income to be brought in before launching itself into further expansions.

Case 3

(R _i P _i b ₀) is large ⇒λ b ₀ is large

No Taylor expansion is possible. However if $R_{i}P_{i} e^{-\lambda b_{0}} \rightarrow 1/{\tau _{i}^{K}}$, then

$$ {1 \over t_{i}} = R_{i} P_{i} e^{-\lambda b_{0}} - {1 \over {\tau_{i}^{K}}} \simeq \rightarrow 0 $$

(47)

and the timescale of expansion diverges. This corresponds to a case where a firm struggles with its cash flows to maintain its production capacity. Beyond this the timescale can also become negative, where a firm scales down its activities.

Thus in many cases one of the three timescales dominates, the bottleneck timescale. In other cases, if two timescales are similar, t _i must be calculated numerically using Eq. 44.

These three cases however only occur if consumers are ready to buy all that this particular industry is able to produce, and consequently its growth is limited by its ability to expand. If, however, the demand grows more slowly than these maximal rates, then the demand constrains the rate of growth, a demand-led case. Furthermore, if consumers have a choice of products and competition occurs, then the rate of growth is further constrained and a model of competition must be derived, as done in Section 4.

Appendix B: A binary logit choice model

A model of choice is constructed here using a pairwise comparison, which will be performed for all possible pairs in order to rank exhaustively consumer preferences, the latter being distributed. I use for this generalised cost distribution of sales obtained from recent sales data. This calculation is the basis of discrete choice theory adapted to the purposes of this work, and more information can be obtained in Ben-Akiva and Lerman (1985) and Domencich and McFadden (1975).

I assume two distributions for the relative numbers of situations where agents, stating their individual preference between technologies i and j, face different situations and state different choices. By counting how many agents prefer which technology in each pair, one can determine what the probabilities of preferences between these two technologies are for future situations where choices are to be made (e.g. 70 % of agents choose i and 30 % j). It does not mean however that when the time comes to invest or purchase, these are the choices that would be made, since depending on the state of diffusion of these technologies, they might not necessarily be available to every agent. By going through an exhaustive list of pairwise preferences, final choices can be determined.

I denote these (normalised) distributions f(C,C _i ,σ _i )d C=f _i (C−C _i )d C and f(C,C _j ,σ _j )d C=f _j (C−C _j )d C, where C _i ,C _j are the means and σ _i ,σ _j are the standard deviations for technologies i and j. These distributions can be of any kind, but they require to have a single well defined maximum and variance (e.g. they cannot have two maxima²⁹). I can then evaluate the probability of choosing i over j using the following. First, I calculate the probability of choosing i in all cases where j has an arbitrary cost C. The central assumption here is that the fraction of agents for whom the generalised cost of j is C and for whom the cost for i is lower than C will choose technology i over j if given a choice, and this fraction is equal to the cumulative probability distribution F _i (C−C _i ). But this situation occurs a fraction f _j (C−C _j ) of the time, giving a total probability

$$ P(C_{i} < C | C_{j} = C) = F_{i}(C-C_{i}) f_{j}(C-C_{j})dC, $$

(48)

while the converse is

$$ P(C_{j} < C | C_{i} = C) = F_{j}(C-C_{j}) f_{i}(C-C_{i})dC. $$

(49)

In order to evaluate how often the cost of technology i is lower than that of technology j, and the converse, a sum over all possible values of C must be taken. For simplicity, I use as variables C ^′=C−C _j and C ^″=C−C _i , with the mean cost difference ΔC=C _i −C _j :

$$ F_{ij}({\Delta} C) = P(C_{i} < C_{j}) = {\int}_{-\infty}^{+\infty} F_{i}(C^{\prime}-{\Delta} C) f_{j}(C^{\prime})dC^{\prime}, $$

$$ F_{ji}({\Delta} C) = 1- F_{ij} = P(C_{j} < C_{i}) = {\int}_{-\infty}^{+\infty} F_{j}(C^{\prime\prime}+{\Delta} C) f_{i}(C^{\prime\prime})dC^{\prime\prime}. $$

(50)

This appears difficult without further knowledge of the distribution type, however it is possible to take a derivative with respect to ΔC, which makes the integral a convolution of the two distributions

$$ {d F_{ij} \over d{\Delta} C} = -{\int}_{-\infty}^{+\infty} f_{i}(C^{\prime}-{\Delta} C) f_{j}(C^{\prime})dC^{\prime} = - f_{ij}({\Delta} C) $$

(51)

$$ = {\int}_{\infty}^{-\infty} f_{i}(C^{\prime\prime}) f_{j}(C^{\prime\prime}+{\Delta} C)dC^{\prime\prime} = - f_{ji}(-{\Delta} C) = {d F_{ji} \over d{\Delta} C}. $$

(52)

This convolution yields a new distribution f _{i j} (ΔC)dΔC of which the standard deviation is $\sigma _{ij} = \sqrt {{\sigma _{i}^{2}} + {\sigma _{j}^{2}}}$. This is the probability distribution of technology switching in terms of ΔC. The convolution having been computed, this distribution can be integrated again as a function of ΔC to yield a cumulative probability distribution that the cost of technology i is less than that of j (and conversely):

$$ F_{ij}({\Delta} C) = {\int}_{-\infty}^{+\infty} f_{ij}({\Delta} C)d{\Delta} C = 1 - {\int}_{-\infty}^{+\infty} f_{ji}({\Delta} C)d{\Delta} C = 1-F_{ji}({\Delta} C). $$

(53)

Thus given a choice between technologies i and j, the fraction F _{i j} of agents tends to choose technology i and the fraction F _{j i} chooses j, these fractions being functions of the generalised cost difference, and this cumulative choice function has a width that follows the sum of the squares $\sigma _{ij} = \sqrt {{\sigma _{i}^{2}} + {\sigma _{j}^{2}}}$. Note that this calculation is independent of probability distribution type; however F _{i j} (ΔC) should have roughly the shape of a ‘smooth’ step function, its ‘smoothness’ determined roughly by the widths of both cost distributions.

In discrete choice theory, the Gumbel distribution is often used, $f_{i} = e^{-e^{-(C-C_{i})/\sigma _{i}}}$, and the result of the convolution of two Gumbel distributions is a logistic distribution of the average cost difference ΔC _{i j} relative to the root mean square width σ _{i j} :

$$ f_{i} = e^{-e^{-(C-C_{i})/\sigma_{i}}}, \quad F_{ij} = {\int}_{-\infty}^{\infty} \left( f_{i} \ast f_{j} \right) d{\Delta} C = {1 \over 1 + e^{\Delta C_{ij}/\sigma_{ij}}}. $$

(54)

Appendix C: From the binary to the multinomial logit in the replicator equation

The derivation of the binary logit in Appendix B gives a relationship between the cost probability distributions and the cumulative distribution of choice between two options f _i and f _j (with parameters C _i ,σ _i and C _j ,σ _j ), as a convolution, consistent with Domencich and McFadden (1975). The probability of cost of option i being less than the cost of option j is

$$ P(C_{i} < C_{j}) = {d F_{ij} \over d{\Delta} C_{ij}} = -{\int}_{-\infty}^{+\infty} f_{i}(C^{\prime}-{\Delta} C_{ij}) f_{j}(C^{\prime})dC^{\prime} = f_{i} \ast f_{j}, $$

(55)

the star denoting a convolution. The binary form of the replicator dynamics equation requires summing the result of binary choices, however distributions can be first grouped and afterwards convolved:

$$ \sum\limits_{j} A_{ij}F_{ij} S_{j} = {\int}_{-\infty}^{\infty} \left( f_{i} \ast \sum\limits_{j} A_{ij}S_{j} f_{j} \right) d{\Delta} C_{ij}. $$

(56)

In this picture, each cost distribution of possible alternatives f _j is weighted by the factor A _{i j} S _j which involves shares and changeover timescales. This weighted sum of distributions results in a composite distribution with new mean and standard deviation parameters $\overline {C}$ and $\overline {\sigma }$.³⁰, which cannot be expressed analytically in any simpler form. The convolution corresponds roughly to the probability that the cost of option i is less than the cost of the ‘average’ alternative (with average cost $\overline {C}$) weighted by the frequency of occurrence of these choices, $P(C < \overline {C} | C_{i} = C)$.

As an approximation, I replace this ‘average’ probability distribution with that of an arbitrary cost value C being lower than the minimum of all available alternatives simultaneously. This corresponds to the product of the individual distributions,

$$ P\left( C < \min[ C_{1}, C_{2}, C_{3} ... C_{n}] \right) = P(C < C_{1})P(C < C_{2}) ... P(C < C_{n}). $$

When the weighting of these choices is equal, if each of these probability functions are Gumbel, then the result of this is also a Gumbel distribution, of cost parameter proportional to $\ln {\sum }_{j} e^{-C_{j} / \sigma _{j}}$, a classic result of discrete choice theory when deriving the multinomial logit (Ben-Akiva and Lerman 1985, p. 105). Equal weighting in the multinomial logit corresponds to perfect access to information and technology options by all agents. In the theory presented here, each agent has access to a different set of choices, which, when correctly weighted, is

$$ P(C < C_{1})^{A_{i1}S_{1}}P(C < C_{2})^{A_{i2}S_{2}} ... P(C < C_{n})^{A_{in}S_{n}}, $$

$$ P\left( C < \min[ C_{1}, C_{2}, C_{3} ... C_{n}] | C_{i} = C \right) = F_{i}(C_{i}-C) \prod\limits_{j} P(C < C_{j})^{A_{ij}S_{j}}. $$

(57)

The weighted (representative alternative) cost parameter is instead

$$ \hat{C} = \hat{\sigma} \ln \sum\limits_{j} A_{ij}S_{j} e^{-C_{j} \over \sigma_{j}} . $$

(58)

This unequal weighting of alternatives is generally overlooked in discrete choice theory, but crucial when exploring the diffusion of technology since part of the dynamics stem from restricted access to options in early states of diffusion. Following Appendix B, the convolution becomes

$$ {\int}_{-\infty}^{\infty} \left( f_{i} \ast \hat{f} \right) d{\Delta} C_{i} = {1 \over 1 + e^{A_{ii}{C_{i} \over \sigma_{i}} - \ln \left( {\sum}_{j} A_{ij}S_{j} e^{-C_{j} / \sigma_{j}} \right)}} = {A_{ii} e^{-C_{i} /\sigma_{i}} \over {\sum}_{j} A_{ij}S_{j} e^{-C_{j} / \sigma_{j}}} . $$

(59)

This is the multinomial logit weighted by A _{i j} S _j , i.e. adjusted for restricted access to alternatives. I now define the fitness in the evolutionary theory sense,

$$ \mathcal{F}_{i} = {{\overline{t} \over t_{i}} e^{-C_{i} /\sigma_{i}} \over {\sum}_{j} {\overline{t} \over t_{j}} S_{j} e^{-C_{j} / \sigma_{j}}} - {{\overline{\tau} \over \tau_{i}} e^{-C_{i} /\sigma_{i}} \over {\sum}_{j} {\overline{\tau} \over \tau_{j}} S_{j} e^{-C_{j} / \sigma_{j}}} $$

(60)

This is the fitness of a technology to capture the market, a growth minus a survival term.³¹ The average fitness is then

$$ \overline{\mathcal{F}} = \sum\limits_{j} S_{j} \mathcal{F}_{j} = 1 - 1 = 0. $$

(61)

Thus the replicator dynamics (37) can in fact be written as

$$ {\Delta} S_{i} = S_{i} \left( \mathcal{F}_{i} - \overline{\mathcal{F}} \right) {\Delta} t. $$

(62)

This is the classical replicator dynamics equation in general evolutionary theory (e.g. Hofbauer and Sigmund 1998), where the ability of a proponent option or biological specie to capture market or space is proportional to the difference of its fitness to the average fitness.

This transformation however has required an approximation which is a simplification of the distribution of alternatives. This is a useful simplification for the sake of exposition, but leads to a less accurate form of the replicator equation and associated market response. This is due to leaving out, at Eq. 57, some of the details of the restricted access to technology and information, as well as the complexity emerging from interactions.³² The binary form essentially maintains the information as to which options are seen by which agent in aggregate. It is however heavier computationally since it involves pairwise comparisons, which scales as n ²/2−n for the binary form, compared to n for the multinomial form (n the number of options).