Walrasian Demand and the Demand Curve

With $u(x_1, x_2) = x_1^{1/2} x_2^{1/2}$ , $p_1 = 2$ , $p_2 = 1$ , $w = 60$ , find Walrasian demand $x_1(p, w)$ for good 1 and check that the budget is exhausted.

x_1(p, w) = \frac{aw}{p_1} = \frac{30}{2} = 15

x_2(p, w) = \frac{(1-a)w}{p_2} = 30, \quad p \cdot x = 2 \cdot 15 + 30 = 60 = w

a = 1/2

p_1 = 2

w = 60

— the point

x_1 = 15

on the demand curve.

Show proof

Plugging $a = 1/2$ , $w = 60$ , $p_1 = 2$ into the Cobb-Douglas demand formula $x_1 = aw/p_1$ gives $x_1 = 30/2 = 15$ .
Since $x_2 = (1-a)w/p_2 = 30$ as well, total spending is $2 \cdot 15 + 1 \cdot 30 = 60 = w$ — the budget is exhausted. The marker on the curve is exactly this quantity.

Let $u(x_1, x_2) = x_1^{1/2} x_2^{1/2}$ , $p = (1, 1)$ , $w = 60$ .

For $\lambda = 1.5$ , show by direct computation that $x_1(\lambda p, \lambda w) = x_1(p, w)$ .

x_1(p, w) = \frac{0.5 \cdot 60}{1} = 30

x_1(\lambda p, \lambda w) = \frac{0.5 \cdot 90}{1.5} = 30

x(\lambda p, \lambda w) = x(p, w) \quad \text{(homogeneous of degree 0)}

\lambda = 1.5

—

x_1(p, w) = x_1(\lambda p, \lambda w) = 30

, zero difference.

Show proof

Original demand is $x_1 = 0.5 \cdot 60 / 1 = 30$ . Scaling all prices and wealth by $1.5$ multiplies the numerator and the denominator by the same $\lambda$ : $x_1(\lambda p, \lambda w) = \frac{0.5 \cdot 90}{1.5} = 30$ .
The two values coincide, so $|x_1(p, w) - x_1(\lambda p, \lambda w)| = 0$ . This is no coincidence: the budget constraint $(\lambda p) \cdot x \le \lambda w$ describes the same set as $p \cdot x \le w$ — Walrasian demand is homogeneous of degree zero.

Walrasian Demand and the Demand Curve

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