Question

Consider a consumer with the following utility function:

U(X, Y ) = XY.
(a) Derive this consumer’s marginal rate of substitution, MUX/MUY

(b) Derive this consumer’s demand functions X∗ and Y∗.

(c) Suppose that the market for good X is composed of 3000 identical consumers, each with income of $100. Derive the market demand function for good X. Denote the market quantity demanded as QX.

(d) Use calculus to show that the market demand function satisfies the law-of-demand.

Answer 1

(a)

MUx = U/X = Y

MUy = U/Y = X

Marginal rate of substitution (MRS) = MUx / MUy = Y / X

(b)

Utility is maximized when MRS = Px / Py

Y / X = Px / Py

X.Px = Y.Py

Substituting in budget line,

M = X.Px + Y.Py

M = X.Px + X.Px

M = 2X.Px

X* = M / (2Px)

Similarly,

M = Y.Py + Y.Py

M = 2Y.Py

Y* = M / (2Py)

(c)

Market demand is the horizontal summation of individual demand functions. So,

QX = 3000.X* and M = 100.

X* = QX / 3000

QX / 3000 = 100 / (2.Px)

QX / 3000 = 50 / Px

QX = 150,000 / Px

(d)

As per law of demand, (dQX/dPx) < 0.

dQX/dPx = - 150,000 / (Px²)

Since Px > 0, [- 150,000 / (Px²)] < 0, which satisfies law of demand.