There are several ways to respond to claims of market failure: definitional, institutional, entrepreneurial, and comparative analysis. Economists can show that terms such as externality, public goods, monopoly, and macro-instability have conceptual difficulties and that by clarifying terms the critique of the market economy fades away. In The Economic Way of Thinking, for example, the ambiguity associated with the term monopoly is explored --- if you define the relevant market broadly enough no monopoly is evident, but if your definition is narrow then every good will exhibit monopolistic characteristics. In Rothbard's Man, Economy and State externalities and public goods are also called into question as well as monopoly. However conceptually correct this response is, it tends to dismiss critics of the market out of hand. And while it doesn't necessarily lead to this, it has the potential of cutting the discussion off too early so that the failure of government and the power of the market are not explored in their full details.
Another response is institutional. If property rights are clearly defined and strictly enforced than market failures also fade away. Externalities result, for example, due to conflicts over property rights, clarify the property rights and the so-called externality is internalized. Market failures are really legal failures. Again, however correct this intellectual exercise may be it results in missing out on explaining the reasons for government ineptness and market robustness.
An understanding of government ineptness results from an examination of government efforts to serve as a corrective to so-called market failures. Government decision-making is prone to certain systemic perversities that public choice theory has exposed. But public choice theory begins with recognition that markets may indeed fail, but that government 'cures' may be worse than the identified 'disease'.
On the other hand, the entrepreneurial perspectives sees the market as an on-going process of adjustment to changing circumstances. Today's inefficiency represents profit opportunities for those individuals who can act to eliminate them. Admitting the existence of 'market failure' sets up the analysis of entrepreneurial responses. By admitting the frictions in the market, the economists can see the way that market participants respond to ease those frictions and realize the gains from trade and technological innovation.
There is a new book out by Brian P. Simpson, Markets Don't Fail that pursues the definitional-conceptual path to great effect. Brian is a professor at the National University in CA, and a product of our PhD program at GMU. Brian wrote his thesis under Richard Wagner, but I was fortunate to be on his committee as well. Brian is an Objectivist and in particular follows closely the work of George Reisman.
In the pecking order of effective responses to claims of market failure I do believe that the entrepreneurial response is the most powerful followed by the public choice critique and institutional analysis. However, correct the conceptual points are they are not as persuasive to the trained economist as the others. Still it is important to insist on definitional and conceptual clarity. Simpson's book does an excellent job of pushing that line and insisting on consistent and persistent applications of terms such as voluntary. Congratulations Brian and wishing you continued success in your teaching and writing career.