These definitions are not equivalent: a market or other economic system may be allocatively but not productively efficient, or productively but not allocatively efficient. There are also other definitions and measures. All characterizations of economic efficiency are encompassed by the more general engineering concept that a system is efficient or optimal when it maximizes desired outputs (such as utility) given available inputs.
There are two main strains of thought on economic efficiency, which respectively emphasize the distortions created by governments (and reduced by decreasing government involvement) and the distortions created by markets (and reduced by increasing government involvement). These are at times competing, at times complementary—either debating the overall level of government involvement, or the effects of specific government involvement. Broadly speaking, this dialog takes place in the context of economic liberalism or neoliberalism, though these terms are also used more narrowly to refer to particular views, especially advocating laissez faire.
Further, there are differences in views on microeconomic versus macroeconomic efficiency, some advocating a greater role for government in one sphere or the other.
A market can be said to have allocative efficiency if the price of a product that the market is supplying is equal to the marginal value consumers place on it, and equals marginal cost. In other words, when every good or service is produced up to the point where one more unit provides a marginal benefit to consumers less than the marginal cost of producing it.
Because productive resources are scarce, the resources must be allocated to various Industries in just the right amounts, otherwise too much or too little output gets produced. When drawing diagrams for firms, allocative efficiency is satisfied if output is produced at the point where marginal cost is equal to average revenue. This is the case for the long-run equilibrium of perfect competition.