Firm behavior under quantity controls: The theory of virtual quantities

•Multiproduct firm responses to quantity controls on inputs and outputs.•Conversion from command-and-control quotas to cap-and-trade property rights.•Virtual quantities are dual to virtual prices.•Elasticity of intensity under multiple quantity controls.•Command-and-control quota, transferable effort right in Malaysian fishery.

The theory of virtual quantities, the dual to virtual prices, provides a framework to analyze competitive multiproduct firm behavior under multiple quantity controls on inputs and outputs, including command-and-control quotas and transferable property rights. The framework addresses the firm׳s reactions to regulatory controls, impacts of adding or dropping quantity controls, inferring unrationed from rationed production, and conversion from command-and-control quotas to cap-and-trade systems with transferable property rights and secondary market behavior. The paper develops reasons for failure of quasi-concavity of technology, extends the elasticity of intensity׳s properties, and integrates the virtual price and virtual quantity frameworks. Virtual quantities are applied to assess potential firm responses to quantity controls and a potential transferable property right in a Malaysian fishery.