Exercises · 3. Consumer Choice

Easy

1Easy

With $p_1 = 2$ , $p_2 = 1$ , $w = 60$ , find the two intercepts and the slope of the budget line.

B_{p,w} = \{x \ge 0 : p_1 x_1 + p_2 x_2 \le w\}

The Budget Set

Intercepts

(30, 60)

, slope

-2

— the probe

b = (15, 30)

sits on the budget line.

Show solution

The $x_1$ -intercept is $w/p_1 = 60/2 = 30$ and the $x_2$ -intercept is $w/p_2 = 60/1 = 60$ .
The slope is $-p_1/p_2 = -2$ — the opportunity cost of one unit of good 1 is two units of good 2.

Medium

2Medium

At $p = (1, 1)$ , $w = 60$ the bundle $(30, 30)$ lies on the budget line.

When $p_1$ rises to $2$ , is this bundle still affordable? Describe the geometric change of the budget line as well.

The Budget Set

Show solution

The original intercepts were $(60, 60)$ . At $p_1 = 2$ only the $x_1$ -intercept shrinks to $30$ ; the budget line rotates inward around the $x_2$ -intercept.
The bundle $(30, 30)$ now costs $2 \cdot 30 + 30 = 90 > 60$ , so it is no longer affordable. Bundles heavy in the now-expensive good are hit hardest — the geometry of the rotation made visible.

3Medium

With $u(x_1, x_2) = x_1^{1/4} x_2^{3/4}$ , $p_1 = 2$ , $p_2 = 1$ , $w = 80$ , find $x_1(p, w)$ and verify that doubling all prices and wealth leaves demand unchanged.

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

Walrasian Demand and the Demand Curve

a = 1/4

,

p_1 = 2

,

w = 80

,

\lambda = 2

—

x_1 = 10

unchanged.

Show solution

With $a = 1/4$ , $x_1 = \frac{0.25 \cdot 80}{2} = 10$ .
Doubling prices and wealth gives $x_1(2p, 2w) = \frac{0.25 \cdot 160}{4} = 10$ — the factor of 2 cancels between numerator and denominator. The point on the demand curve does not move.

Hard

4Hard

For $\lambda > 0$ , prove the equality below, and deduce directly from the definition of the UMP that Walrasian demand $x(p, w)$ is homogeneous of degree zero.

B(p, w) = \{x \ge 0 : p \cdot x \le w\}

x(p, w) = \arg\max_{x \in B(p, w)} u(x)

B(\lambda p, \lambda w) = B(p, w)

Show solution

$x \in B(\lambda p, \lambda w)$ means $(\lambda p) \cdot x \le \lambda w$ .
Since $\lambda > 0$ , dividing by $\lambda$ gives $p \cdot x \le w$ , i.e. $x \in B(p, w)$ .
The converse is identical, so the two sets are equal.
The UMP maximizes utility over the budget set, so equal sets give equal maximizers: $x(\lambda p, \lambda w) = \arg\max_{B(\lambda p, \lambda w)} u = \arg\max_{B(p, w)} u = x(p, w)$ .
$x(\lambda p, \lambda w) = x(p, w) \quad \forall \lambda > 0$
Scaling prices and wealth together leaves the budget set itself unchanged, so the choice cannot change either — the geometric reason behind the degree-zero homogeneity of demand.