Will privatization reduce costs?

I develop a model of public sector contracting based on the multitask framework by Holmström and Milgrom (1991). In this model, an agent can put effort into increasing the quality of a service or reducing costs. Being residual claimants, private owners have stronger incentives to cut costs than public employees. However, if quality cannot be perfectly measured, providing a private firm with incentives to improve quality forces the owner of the firm to bear risk. As a result, private firms will always be cheaper for low levels of quality but might be more expensive for high levels of quality. Extending the model to allow for differences in task attractiveness, I find that public firms shun unattractive tasks, whereas private firms undertake them if incentives are strong enough.