Question: Price change to maximize profit. A business sells \(n\) products, and is considering changing the price of one of the products to increase its total profits. A business analyst develops a regression model that (reasonably accurately) predicts the total profit when the product prices are changed, given by \(\hat{P}=\beta^{T} x+P\), where the \(n\) -vector \(x\) denotes the fractional change in the product prices, \(x_{i}=\left(p_{i}^{\text {new }}-p_{i}\right) / p_{i}\). Here \(P\) is the profit with the current prices, \(\hat{P}\) is the predicted profit with the changed prices, \(p_{i}\) is the current (positive) price of product \(i\), and \(p_{i}^{\text {new }}\) is the new price of product \(i\).

1. What does it mean if \(\beta_{3}<0\) ? (And yes, this can occur.)
2. Suppose that you are given permission to change the price of one product, by up to \(1 \%\), to increase total profit. Which product would you choose, and would you increase or decrease the price? By how much?
3. Repeat part (b) assuming you are allowed to change the price of two products, each by up to \(1 \%\).

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