Do More Open Economies Still Have Bigger Governments? Rodrik's Insurance Hypothesis in 169 Economies, 2000-2023, a Mean-Variance Derivation, an Exact Omitted-Variable Identity for Country Size, and India

Rodrik found that economies more exposed to trade have larger governments, and explained this as social insurance against external risk. I derive the hypothesis in a mean-variance model where government consumption is the safe sector: the optimal government share rises with openness whenever the private sector's expected return exceeds the government's, and the effect is larger where terms-of-trade risk is higher. Using World Bank data for 169 economies averaged over 2000 to 2023, the elasticity of the government consumption share with respect to the trade share is 0.217 (robust standard error 0.072). Controlling for income per head it is 0.138 (0.079), and adding population it is minus 0.035 (0.082). An exact omitted-variable identity shows that the whole change comes from country size: small economies trade more (the correlation of log openness with log population is minus 0.50) and have larger governments (elasticity minus 0.083 with respect to population). The interaction of openness with terms-of-trade volatility, the specific prediction of the insurance model, is positive but not significant (0.31, standard error 0.22). Within economies the evidence is different: between the 1990s and 2014 to 2023, the log change in the government share rose with the log change in openness (slope 0.239, standard error 0.080, 146 economies). India is the clearest exception. Its trade share more than doubled, from 20.7% to 43.5% of GDP, while its government consumption share was almost flat (11.1% to 10.8%).