Domestic Product and Income

Let x denote a vector of output and input quantity, and x* is a price. The profit is $$\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle$$. A price/quantity pair $$\left ( \boldsymbol {x}^{\ast },\boldsymbol {x}\right )$$ is a production equilibrium if the quantity x maximizes the profit $$\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle$$ for the price x*. Returns to scale are constant: the technology set K is a cone. The polarK∘ is all prices for which the maximum profit is zero. The Fenchel equality$$\delta _{K}\left ( \boldsymbol {x}\right ) + \delta _{K^{\circ }}\left ( \boldsymbol {x}^{\ast }\right ) -\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle =0$$ is necessary and sufficient for the price/quantity pair $$\left ( \boldsymbol {x}^{\ast },\boldsymbol {x}\right )$$ to be a production equilibrium. One can model production equilibrium via a domestic-product and income duality. Decompose the quantity and the price into separate components for output and input:$$\begin {aligned}\boldsymbol {x} & =\left ( \boldsymbol {y},-\boldsymbol {z}\right ) \\ \boldsymbol {x}^{\ast } & =\left ( \boldsymbol {y}^{\ast },\boldsymbol {z}^{\ast }\right ) \!. \end {aligned}$$ The primal maximizes the domestic product: choose output to maximize its value $$\left \langle \boldsymbol {y}^{\ast },\boldsymbol {y}\right \rangle$$, such that this output can be produced with input z.$$\sup _{\boldsymbol {y}}\left [ \left \langle \boldsymbol {y}^{\ast },\boldsymbol {y}\right \rangle -\delta _{K}\left ( \boldsymbol {y},-\boldsymbol {z}\right ) \right ] \!.$$ The dual minimizes the domestic income: choose the input price z* to minimize the input value $$\left \langle \boldsymbol {z}^{\ast },\boldsymbol {z}\right \rangle$$, among all prices for which the maximum profit is zero.$$\inf _{\boldsymbol {z}^{\ast }}\left [ \left \langle \boldsymbol {z}^{\ast },\boldsymbol {z} \right \rangle + \delta _{K^{\circ }}\left ( \boldsymbol {y}^{\ast },\boldsymbol {z}^{\ast }\right ) \right ] \!.$$ If $$\left ( \boldsymbol {y}^{\ast },\boldsymbol {z}^{\ast },\boldsymbol {y},\boldsymbol {z}\right )$$ is a production equilibrium, then it solves the domestic-product primal and the domestic-income dual, and there is no duality gap. If $$\left ( \boldsymbol {y}^{\ast },\boldsymbol {z}^{\ast },\boldsymbol {y},\boldsymbol {z}\right )$$ solves the primal and the dual and there is no duality gap, then it is a production equilibrium.