This paper provides a directed search model designed to explain the residual part of wage variation left over after the impact of education and other observable worker characteristics has been removed. Workers have private information about their characteristics at the time they apply for jobs. Firms value these characteristics differently and can observe these characteristics once workers apply. They and hire the worker they most prefer. However, the characteristics aren't contractible, so firms can't condition their wages on them. The paper shows how to extend directed search arguments to handle this, allowing for arbitrary distributions of worker and firm types. The model is used to provide a functional relationship that ties together the wage distribution and the wage duration. This relationship provides a testable implication of the model. This relationship is used to provide a common property of wage distributions that guarantees that workers who leave unemployment at the highest wages also have the shortest unemployment duration. This is in strict contrast to the usual (and somewhat implausible) directed search story in which high wages are always accompanied by higher probability of unemployment.
