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Question 1. (22 marks) Suppose that the following utility function represents the preferences of a household over units of clothing (x1) and expenditure on all other goods (x2), U(x1,x2) = 100Jx1 + 3:2 The household has an exogenous income of I. The second 'good' is referred to as a 'composite' good and is an amount of money. We assume throughout that p2 = 1. i) (4 marks) Derive the household's ordinary demand functions, x1(p1, 1,1) and x2 (p1, 1,1). [Note: you can work from the tangency condition if you choose] ii) (2 marks) For what prices of the first good will the household choose to consume other goods (x2 2 0)? Assume the household's exogenous income is I = $1000 for the rest of the question. iii) (2 marks) Find the household's optimal choices of clothing and composite when p1 = $20/unit, and when p1 = $16/um't. iv) (4 marks) Illustrate the price effect arising from a fall in the price of clothing from p1 = $20/unit to p1 = $16/unit, in a sketch. Be sure to include two budget lines, and two indifference curves illustrating the household's optimal choices and constraints. v) (4 marks) In the sketch in iv), illustrate the decomposition of the price effect using the Hicksian method of holding the household's purchasing power constant (ie. their utility) and isolate the substitution effect. (No calculations are necessary.) vi) (6 marks) Repeat the calculations and graphical exercise of illustrating the decomposition of the price effect in parts iii)-v) using the following demand functions (a rising from a Cobb-Douglas utility function with o: = i), I 2: 361031: 1,1) = i and x2031, 1,1) = % Remember to use the homothetic property in your sketch