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Why does an increase in the Dixit-Stiglitz price index lead to an increase in demand?

If the price index $P_j$ rises with $p_{ij}$ held constant, then it must be because the prices of goods other than $i$ have risen. Since the goods are substitutes, demand for good $i$ will rise.
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Why does a strong dollar make imported goods cheaper?

Let's say we have a product, like a shoe, that is worth 100 USD and 100 EUR in America and Europe respectively. If the conversion rate between USD and EUR becomes 1:2, but the prices of the shoes ...
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