Question

Eugenia can consume two goods, good 1 and good 2 where xi and Xz denote the quantity consumed of each good. These goods sell at prices P. and P2, respectively. Eugenia's income is I and her preferences are given by: U(x1, x2) = x2x2 a) Are goods 1 and 2 perfect complements, perfect substitutes or imperfect substitutes to Eugenia? Explain. b) Derive Eugenia's demand functions for the two goods. c) Assume that p1 = P2 = $5 and 7 = $100. Which are the optimal quantities of good 1, xi, and good 2, X2, demanded by Eugenia given her preferences and her income? d) What share of her income is spent on good 1?

Answer 1

a) Goods 1 and 2 are imperfect substitutes because the given utility function is a well behaved utility function i.e of the form U(x1,x2) = x1a x21-a ,which represents imperfect substitutes.

They are not perfect complements because perfect complement utility functions are of the form U(x1,x2) = min {ax1,bx2}

They are not perfect substitutes because perfect substitute utility functions are linear i.e of the form U(x1,x2) = ax1 + bx2

b) Demand functions are calculated by equating the marginal rate of substitution (MRS) with the price ratio, i.e at the point whare indifference curve and budget constraint are tangent.

U (2, 4) = Ja, na = x MRS = MUX au/2 MUN au da, * MUM = t. x. - U = 1 . x . Y MRS = 12.*? 12 ry ra Yo 7 px 7 MRS = Now, - MRS = po MRS = Budget constraint: + 2 y = 1 Pin + Piss = I -

- 2 P, х = І x = 1 putting in 10

c)   optimal quantities can be calculated using the demand functions derived above and the given prices and income.

100 2X5 E 100 2x5

d) share of income spent on good 1 = x1*/I = 10/100 = 1/10

hence, one-tenth of the income was spent on good 1.