Mathematical formulation

Scope

This document describes the mathematical structure of the Future Technology Transformations (FTT) model family used in energy–economy–environment modelling. The focus is on the core diffusion equations and their derivation.

FTT formalises technology transitions as an evolutionary competition process under bounded rationality and increasing returns. Its mathematical structure is designed to study transition dynamics, lock-in, and policy-driven tipping behaviour rather than optimal equilibrium outcomes.

Technology Shares

Consider a sector with \(N\) competing technologies. Each technology \(i\) has a market share

\[S_i(t) \in [0,1], \qquad \sum_{i=1}^N S_i(t) = 1.\]

Technology diffusion is modelled as substitution between technologies, not as optimisation of total system cost.

Generalised Costs and Heterogeneity

Each technology has a generalised cost \(C_i\), including:

• investment,

• fuel, operation and maintenance costs,

• non-pecuniary preferences,

• policy effects (taxes, subsidies).

Agents are heterogeneous. Costs are therefore represented by probability distributions rather than point values. Let \(F_i(C)\) denote the cumulative distribution of perceived costs for technology \(i\).

Pairwise Discrete Choice

Agents compare technologies pairwise. The probability that technology \(i\) is preferred to \(j\) is

\[P_{i>j} = \int F_i(C) \, dF_j(C).\]

Under standard assumptions on the cost distributions, this probability can be expressed as a smooth function of cost differences

\[P_{i>j} = \Phi(C_j - C_i),\]

where \(\Phi\) is typically a logistic or normal cumulative distribution function.

Substitution Dynamics

Market share changes arise from bilateral substitution flows. The evolution of \(S_i\) is

\[\frac{dS_i}{dt} = \sum_{j \neq i} S_i S_j \left( A_{ij} F_{ij} - A_{ji} F_{ji} \right).\]

Two conceptually distinct matrices appear in this formulation.

Substitution matrix (A) The matrix \(A_{ij}\) governs the rate at which substitutions are possible. It depends on:

• technology lifetimes and scrappage rates,

• relative speed of capacity expansion between technologies,

The matrix \(A\) therefore controls how fast technologies can replace one another, independently of their relative attractiveness. \(A\) is defined as \(A_{ij} = \frac{K}{\tau_i t_j}\), where

• \(K\) is a normalisation constant setting the overall time scale of diffusion,

• \(\tau_i\) is the lifetime of technology \(i\),

• \(t_j\) is the build time of technology \(j\).

Preference matrix (F) The matrix \(F_{ij}\) represents agent preferences derived from cost comparisons. It is defined as

\[F_{ij} = P_{i>j} = \Phi(C_j - C_i),\]

where \(C_i\) and \(C_j\) are generalised costs and \(\Phi\) is a cumulative distribution function reflecting cost heterogeneity across agents.

The matrix \(F\) therefore controls which direction substitution occurs, based on relative costs and preferences.

Dynamic Properties

The FTT system is:

• non-linear due to the \(S_i S_j\) interaction terms;

• path-dependent, with outcomes contingent on initial conditions;

• not derivable from a global welfare or cost minimisation problem.

Learning-by-Doing

Technology costs decline endogenously with cumulative deployment. For technology \(i\)

\[C_i(t) = C_i(0) \left( \frac{W_i(t)}{W_i(0)} \right)^{-b_i},\]

where \(W_i\) is cumulative installed capacity and \(b_i\) is a technology-specific learning exponent, based on literature values.

Learning feeds back into the choice probabilities \(P_{i>j}\), creating increasing returns and reinforcing early adoption advantages.

Constraints

FTT includes physical and institutional constraints such as:

• limits on resource availability (e.g. scrap for steel),

• infrastructure constraints (e.g. grid capacity for power),

• technology constraints (e.g. restrictions on heating water with air-to-air heat pumps)

Mathematically, these enter as bounds on \(S_i\) or as modifiers of the substitution and preference matrices \(A_{ij}\) or \(F_{ij}\).

Model Coupling

FTT technology modules are coupled to macro-econometric models. Energy demand affects the markets, while technology diffusion (including prices and investment) feeds back into the macroeconomic system.