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.
(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.
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