We examine the role of uniform and non-uniform social insurance to supplement a general income tax when neither public nor private insurers can observe individual risk, which is positively correlated with wages (e.g., for old age dependency). In the (private market) Rothschild and Stiglitz (1976) equilibrium low-wage/low-risk individuals are not fully insured. While social insurance provided to the poor has a negative incentive effect, it also increases their otherwise insufficient insurance coverage. Social insurance to the rich produces exactly the opposite effects. Whichever of these effects dominates, some social insurance is always desirable irrespective of the pattern of correlation. Finally, we introduce risk misperception which exacerbates the failure of private markets. Rather surprisingly, this does not necessarily strengthen the case for public insurance.

Introduction

A significant part of government intervention in the economy is, or can be, justified by redistributive considerations. In modern welfare states a large variety of instruments is used, including taxation, transfers, price subsidies, in-kind transfers, pension benefits, and more generally social insurance. From an economic perspective this raises many questions, for instance, which policies ought to be used, or how should they be designed and financed? A starting point for addressing these questions is the so-called Atkinson and Stiglitz (1976) theorem which states, roughly speaking, that when preferences are separable between labor supply and goods, any (incentive compatible) Pareto-efficient allocation can be implemented by using only a general income tax. When the income tax is designed in an optimal way given the information that is available, an extra instrument is valuable only if it provides “better” information. In the Mirrleesian world considered by Atkinson and Stiglitz (AS), where individuals differ only in productivity, the separability of preferences implies that all these extra instruments do not provide any additional, pertinent information. If we take the AS theorem at its face value, the welfare state should be downscaled dramatically and replaced by a “simple” well-designed tax and (cash) transfer policy. However, it is by now well known that one of the major limitations of this result is that it relies on the assumption that individuals differ only in a single (non observable) dimension, namely productivity. When there are other factors of individual heterogeneity, including preferences or “risk”, the result no longer applies. In this paper we focus on one of the both most prominent and most debated instrument, namely social insurance. Private insurance redistributes ex post between states of nature and premiums reflect individual risk. But only social insurance (or a suitable regulated private system) can effectively redistribute between ex ante heterogenous risk types. Put differently, social insurance can provide insurance against the “risk of being a bad risk”, which private insurance cannot. This can be achieved through uniform premiums or, more generally, by a rate schedule where premium differences do not fully reflect risk differences. Rochet (1991) and Cremer and Pestieau (1996) have shown that social insurance is desirable to supplement an optimal income tax, even when private insurance is actuarially fair, as long as productivity and risk are negatively correlated, that is when less productive individuals face the higher risk.