Extensive margin models use Rogerson (1988) style lotteries in order to simplify the savings problem under non-full-insurance.

However, one could still write down the model in that style but instead of assuming full insurance (that is, marginal utility of consumption equalized among all individuals in the household, no matter whether working or not), do it with partial insurance, in order to match the data better.

I however have not seen this approach so far. Did I miss something? Or what would be arguments against this approach? Besides the obvious one: If you don't have full insurance but forbid savings at the individual agent level, you are an evil person.

  • $\begingroup$ What do you mean by partial insurance? Would self-insurance by borrowing and saving in an incomplete markets economy count? Or do you mean literally partial insurance, e.g. insured against some shocks but not others? The latter is in, for instance, Heathcote, Storesletten, and Violante's latest AER. $\endgroup$ Dec 18 '14 at 17:19
  • $\begingroup$ I mean that the unemployed individuals in the household get, say, 40% of the consumption that the employed ones get, to reflect the fact that in reality, unemployment insurance does not perfectly insure against unemployment $\endgroup$
    – FooBar
    Dec 18 '14 at 17:42
  • $\begingroup$ ah. If there's no borrowing/saving, and people who don't work get an unemployment benefit of 40% of the wage of people who do work, then not clear how employment would be affected by real wages: since return from employment and unemployment would move proportionally, for (e.g.) KPR preferences real wages would be neutral for employment. If, on the other hand, unemployment benefits are constant, then there's an unambiguous sign of employment response w.r.t. real wages, but magnitude still in question. $\endgroup$ Dec 19 '14 at 0:49
  • $\begingroup$ in fact, in benchmark Rogerson case with no heterogeneity, the elasticity of labor supply wrt real wages would still be either zero or infinity, probably zero - depending on whether you happened to be at the point where everyone was indifferent between working and not working. Without agents being able to borrow and save, the model behaves in strange ways, which is why ivansml's focus below on self-insurance through borrowing/saving makes sense. $\endgroup$ Dec 19 '14 at 0:52

I'm not very well versed in macro-labor, but I'm aware of a few papers that might be close to what you mean. One could make unemployment uninsurable, and instead let agents partially self-insure by saving capital, like in Chang & Kim (2007) AER paper. Also Krusell & coauthors have some papers (like 2010 ReStud) where they combine uninsurable unemployment with search/matching framework.

The obvious problem is that dealing with heterogeneity is painful, because one must keep track of distribution of variables across agents (e.g. employment status and savings). If you're interested in computation and what can go wrong with it, check out comment on Chang & Kim paper by Takahashi (as well as reply by original authors in the same issue).


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