Question 3 On the Iyplane. a my from the origin is a straight line passing through the origin. (Mathematically, it is given by the equation y = 7'3: for some real number 1 r.) A preference is said to be homothetc if, the MRS does not change as we move along each ray from the origin. In other words, along each ra}r from the origin: the indifference curves are parallel. See Figure 1 for an illustration. lV'lathematically, a preference is homothemtie if, for any number 0: > 0 and any bundle (.T. y) l\\-'IRS(a';c, Org) : MRSUU, y). FIGURE 1. Homothetic Preference: Indifference curves are parallel along each ray from the origin (a) (Level B) Consider a twogood world and suppose a consumer's preference is homothetic. Can either one of the two goods be an inferior good for this con sumer? (Hint: Sketch an indifference curve diagram with an increase in income.) (b) (Level B) What is the income elasticity of Good X when preferences are homothetic? (Hint: How does the ratio between the two goods consumed change when there is an increase in income?) (This question is of Level C from this point onward. They are OPTIONAL. You won't be tested on any of the material below. Skip it if it make you feel too anxious.) A commonly used family of homothetic preferences is represented by constant elasticity of substitution (CBS) utility function, which is of the form Us: y) : [cu-P + at (1) where (1,5 > 0 and p is between 700 and 1. 2 (c) Calculate the MRS at bundle (:c,y) given the utility function in Equation (1) Verify that the preference is indeed homothetic by showing that for any number a > 0, l\\-'IR.S((w, cry) = MRS(m,y). (d) When preference is homothetic. the ratio of :r: and y at any optimally chosen bundle depends only on the relative price but not the income. Hence we can think of the the ratio .1:*/y* as a function of Pin/Pg. Using your answer to part (c) and the tangency condition (for the optimal bundle), express 1"\" /y\" as a function of PI/Py. (e) The elasticity of substitution is dened as % change in iii _ 05111 (ft) '75 change in 7'2: d 111 () I Using your answer to part (d), nd the elasticity of substitution for the utility function in Equation (1) Can you see Why this utility function is known as a constant elasticity of substitution utility function? (Hint: Take natural log on both sides of your expression in part (d). Then treat ln(PI/Py) as a single variable: and differentiate ln(9:* /y*) with respect to it.) (f) There are a few special cases of CES utility functions: (i) Suppose p 2 1. What kind of utility function will Equation (1) become? (ii) What will the expression for the MRS be when p : 0? Can you recall which utility function yields such an expression for the MRS? (iii) Consider the limit of the MRS as p 4} 700 (Hint: You will need to consider two cases: 37 > y and a: