I want to extend the classical definition of consumer surplus to multi-product case. For a single product, consumer surplus is given as \begin{equation} CS=\int_ {p_{market}} ^ {p _{\max}}D(p)dp. \end{equation}
Suppose I have two products $X$ and $Y$ with interdependent demand, that is the demand for product $X$ is given as a function of price of $X$ and price of $Y$: \begin{equation} D_X(p_X,p_Y). \end{equation} Similarly, demand of product $Y$ is given as $D_Y(p_Y,p_X)$.
In this case, how can one define the consumer surplus? Will it involve two double integrals? Thanks!