Globalization and automation might imply deep changes on the labour market. An important policy issue is whether and how the tax-transfer rules should be reformed to cope with those changes. While the prevailing response has consisted of more sophisticated designs of mean-testing and targeting, we also witness an increasing interest in policies inspired by simplicity and universality. In this paper we take the latter route. Using a combination of behavioural microsimulation and numerical optimization, we look for a social welfare optimal tax-transfer rule within a flexible class where total household disposable income is a 4th polynomial in total household taxable income. We use a model of household labour supply that makes it possible to account for equilibrium constraints and to evaluate the effects of exogenous labour demand shocks. We consider two stylized scenarios: the Jobless Economy (the robots take over 10% of jobs at every skill-level) and the Polarized Economy (the robots take over 10% of the unskilled jobs while skilled jobs increase by 10%). We compare the social welfare performance of the polynomial optimal rules and of the current rules under the Current Economy scenario and under the alternative Jobless Economy and the Polarized Economy scenarios. We present results using the 2015 EU-SILC data sets for France, Germany, Italy and Luxembourg. The polynomial optimal rules feature a universal basic income and an almost flat marginal tax rate profile and are social welfare-superior under the Current Economy scenario in all the countries and also under the alternative scenarios in France, Germany and Italy.