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# Consider an individual whose preferences are defined over bundles of non-negative amounts of each of two commodities. Suppose that this individual's preferences can be

## Consider an individual whose preferences are defined over bundles of non-negative amounts of each of two commodities. Suppose that this individual's preferences can be represented by a utility function U: R2 R of the form U (x, x) = ln (x + 1) + 2x2, where x denotes the individual's consumption of commodity one, and x2 denotes the individual's consumption of commodity two. This individual is a price taker in both commodity mar- kets. The price of commodity one is p > 0, and the price of commodity two is p2 > 0. This individual is endowed with an income of y > 0. 1. Does this individual have quasi-linear preferences? Justify your answer. (3 marks.) 2. Are this individual's preferences locally non-satiated? Justify your answer. marks.) 3. What is this individual's budget-constrained utility maximisation problem? (2 marks.) 4. Suppose that the individual will optimally consume strictly positive amounts of both commodities. What is the individual's optimal consumption bundle in this case? Under what circumstances, if any, will this case occur? (7 marks.) 5. Can it ever be optimal for this individual to choose to consume zero units of com- modity one? If so, what would be his or her optimal consumption of commodity two? Under what circumstances, if any, will this case occur? (3 marks.) 6. Can it ever be optimal for this individual to choose to consume zero units of com- modity two? If so, what would be his or her optimal consumption of commodity one? Under what circumstances, if any, will this case occur? (3 marks.) 7. What are the ordinary demand functions (or possibly correspondences) for com- modity one and commodity two for this individual? (2 marks.) 8. What is this individual's indirect utility function? (2 marks.)

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