Alice always maximizes her utility subject to her budget constraint. Suppose we observe her choices at two different points in time, before a price change and after the price change. That is, we observed her old choice bundle when we saw old prices and her new choice bundle after seeing the new prices. Suppose Alice tells us that as a result of the price change she ended up better off (that is, the new bundle gave higher utility). Based on this, what can we infer about the cost of her old bundle compared with what the new bundle would have cost at the old prices? Give a mathematical argument. Note that the question does not specify how many goods are in a bundle.