A consumer\'s preferences over two goods are represented by: U(x1, x2) = (x1^2)(

Question

A consumer's preferences over two goods are represented by: U(x1, x2) = (x1^2)(x2^3)/100 The prices are P1 and P2, and an amount of money E can be spent on these goods. a) Show that the two uncompensated demand functions are: x1 = 2E/5p1 and x2 = 3E/5p2 b) How do you know that these demands maximise utility rather than minimising it? c) Suppose that p1 = $4, p2 = $5 and E = $100. Use the results in a) to find the optimal quantities demanded of the tw0 goods, and the resulting amount of 'utils' obtained. Assume that both goods are divisible, so fractions are possible

Explanation / Answer

U = X12X23 / 100

Budget line: E = X1P1 + X2P2

(a)

MU(X1) = dU / dX1 = (1/100) x 2 x X1 x X23

MU(X2) = dU / dX2 = (1/100) x 3 x X12 xX22

MRS = MU(X1) / MU(X2) = (2/3) x (X2 / X1)

Utility is maximized when MRS = price ratio

(2/3) x (X2 / X1) = P1 / P2

3 x P1X1 = 2 x P2X2

P1X1 = (2/3) x P2X2

Substituting in budget line,

E = (2/3) x P2X2 + P2X2

E = (5/3) x P2X2

X2 = 3E / 5P2 [Demand function, X2]

X1 = (2/3) x P2X2 / P1

X1 = (2/3) x (P2 / P1) x (3E / 5P2)

X1 = 2E / 5P1 [Demand function, X1]

(b) The optimization problem we set up aims to Maximize Utility, subject to Budget line. When the constraint is either constant or of a minimization type, the objective function is that of a maximization, as is in this case.

(c) Budget line:

100 = 4X1 + 5X2

From (a), we get

X1 = 2E / 5P1 = (2 x 100) / (5 x 4) = 200 / 20 = 10

X2 = 3E / 5P2 = (3 x 100) / (5 x 5) = 300 / 25 = 12

U = X12X23 / 100 = (10)2(12)3 / 100 = 100 x 1,728 / 100 = 1,728 Utils

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