In times of turbulent economic outlook and scarce financial resources, the margin for error in public spending and regulatory burden is vanishingly small. Policymakers need reliable instruments to assess the economic impact of environmental regulations before implementation—not after.

Computable General Equilibrium models have become the workhorse of climate policy evaluation. They simulate how millions of economic agents—firms, households, governments—react to policy shocks. But here's the critical point: the "behavior" of these agents is governed mathematically by elasticity parameters.

"If the elasticity of substitution is high, the economy can cheaply decarbonize by swapping dirty energy for clean capital. If it is low, decarbonization requires painful reductions in output. The difference between these two parameters is the difference between a policy that is economically viable and one that is politically ruinous."

Yet the current situation of elasticities is "rather unsatisfying"—a problem acknowledged since Mansur and Whalley's critique in 1984. Modellers frequently employ elasticities from unrelated sources, creating what we call a "Frankenstein" approach to parameterization. In extreme cases, when estimates are unavailable, researchers resort to what Dawkins et al. (2001) termed the "idiot's law of elasticities"—assuming unity (Cobb-Douglas) or using arbitrary "coffee table elasticities" chosen based on intuition rather than empirical rigor.

The question of how factors of production can be substituted originates in the fundamental work of Robert Solow (1956). He considered three cases: the "Harrod-Domar" case with zero elasticity (fixed proportions), the "Cobb-Douglas" case with unit elasticity, and a third, more general case with flexible elasticity.

Five years later, Arrow, Chenery, Minhas, and Solow (1961) formalized the general Constant Elasticity of Substitution (CES) production function—a generalization that nests both Leontief and Cobb-Douglas as special cases.

However, a basic CES framework assumes equal substitution elasticities between all inputs. To overcome this, Sato (1967) introduced nested CES functions—constructing separate CES functions for groups of inputs with similar substitutability, combined hierarchically.

We leverage the World Input-Output Database (WIOD)—covering 40 regions (85% of world GDP), 35 industries, and annual data from 1995 to 2009. For the first time, we can derive elasticities from the same data that researchers use to calibrate their simulations.

The Battle of Algorithms

CES functions are non-linear in parameters. Most researchers linearize them via the Kmenta approximation (1967), but Kmenta himself noted problems when elasticities deviate from unity. The approximation works best when the world is Cobb-Douglas—creating circular logic.

Instead, we estimate directly from the non-linear form using five optimization algorithms: Levenberg-Marquardt, PORT routines, Differential Evolution, Nelder-Mead, and Simulated Annealing. This ensures we find the global minimum rather than getting stuck in local optima.

Sectoral Heterogeneity

The estimated elasticities reveal significant heterogeneity. Energy-intensive sectors (Basic Metals, Chemicals) show low elasticity between Capital-Labour and Energy—reflecting engineering constraints. Service sectors exhibit higher flexibility. The top nest (Materials) is often highly inelastic—you cannot substitute steel with labor to build a car.

The elasticity of substitution acts as a shock absorber. When a carbon cap hits the economy, agents try to substitute away from the constrained factor. If elasticities are high, the carbon price stays low. If they're low, it spikes.

Our finding that most sectors exhibit elasticities below unity for energy nests suggests that models assuming Cobb-Douglas flexibility may systematically underestimate the costs of climate policy in the short to medium term.