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Suppose we have two goods, price changes in the two are independent; having seen this question I am considering why the change in total Consumer Surplus is the sum of the change in Consumer Surplus for the two goods separately (if this is true)?

In my reference I see that when we hold $p_y$ fixed at $p_y^0$ and only $p_x$ changes:

\begin{equation} CS = \int_{p^0}^{p^1} x^m (p_x, p_y^0, I^0) dp_x \end{equation}

Where $x^m$ is the Marshallian demand curve for $x$.

If we are going to calculate the consumer surplus when both $p_x$ and $p_y$ change, how should I write the Consumer Surplus equation? Guessing it is one of the following:

$$ total \; CS = \int_{p^0}^{p^1} x^m (p_x, p_y^0, I^0) dp_x + \int_{p^0}^{p^1} y^m (p_x^0, p_y, I^0) dp_y $$

or

$$ total \; CS = \int_{p^0}^{p^1} \int_{p^0}^{p^1} x^m (p_x, p_y^0, I^0) y^m (p_x^0, p_y, I^0) dp_x dp_y $$

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