Question: Suppose a consumer's preferences can be represented by the utility function: \(U\left( X,Y \right)={{X}^{2}}Y\)

1. Derive the function for the marginal rate of substitution holding utility constant: \({{\left. \frac{\Delta Y}{\Delta X} \right|}_{{\bar{U}}}}\)
2. Derive the demand curves for the two goods, X and Y.
3. Confirm that both demand curves slope downward.
4. Are both goods normal goods? Explain.
5. Calculate the price elasticity for each of the goods.
6. Calculate the income elasticity for each of the goods.
7. Does the fact that the cross-price elasticity is zero imply that the two goods are not substitutes?

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