Exercises · 4. Classical Demand Theory

Easy

1Easy

With $u(x_1, x_2) = x_1^{1/3} x_2^{2/3}$ , $p_1 = 2$ , $p_2 = 3$ , $w = 120$ , find Walrasian demand $x(p, w)$ and indirect utility $v(p, w)$ .

x_l(p, w) = \frac{a_l w}{p_l}

The tangency at

a = 1/3

,

p = (2, 3)

,

w = 120

.

Show solution

Plug $a = 1/3$ into the Cobb-Douglas demand formula: $x_1 = \frac{(1/3) \cdot 120}{2} = 20$ , $x_2 = \frac{(2/3) \cdot 120}{3} = \frac{80}{3}$ .
Indirect utility is $v = 20^{1/3} (80/3)^{2/3}$ . Checking the budget: $2 \cdot 20 + 3 \cdot 80/3 = 120$ — Walras' law holds.

2Easy

Starting from the budget line with $p_1 = 1$ , $p_2 = 2$ , $w = 100$ , how does the budget line move when $w$ rises to $150$ ?

Find the new optimum of a Cobb-Douglas consumer with $a = 1/2$ .

Show solution

A pure income rise shifts the budget line outward in parallel: intercepts move from $(100, 50)$ to $(150, 75)$ .
Demand is proportional to wealth, so $x_1 = 0.5 \cdot 150/1 = 75$ , $x_2 = 0.5 \cdot 150/2 = 37.5$ — exactly $1.5$ times the original $(50, 25)$ .

3Medium

Let $u = x_1^{1/4} x_2^{3/4}$ , $p_2 = 1$ , $w = 80$ .

When $p_1$ rises from $1$ to $4$ , compute the substitution, income, and total effects of the Slutsky decomposition.

A = x(p, w), \quad w' = p' \cdot A, \quad B = x(p', w'), \quad C = x(p', w)

a = 1/4

,

p_1: 1 \to 4

,

w = 80

— substitution

-11.25

, income

-3.75

.

Show solution

With $a = 1/4$ , $A = (0.25 \cdot 80/1,\ 0.75 \cdot 80/1) = (20, 60)$ . Slutsky compensated income is $w' = 4 \cdot 20 + 60 = 140$ .
$B = (0.25 \cdot 140/4,\ 0.75 \cdot 140) = (8.75, 105)$ , $C = (0.25 \cdot 80/4,\ 0.75 \cdot 80) = (5, 60)$ .
Substitution effect $= 8.75 - 20 = -11.25$ , income effect $= 5 - 8.75 = -3.75$ , total $= 5 - 20 = -15$ . The substitution effect is negative, and since the good is normal the income effect is negative too.
$-15 = \underbrace{-11.25}_{\text{substitution}} + \underbrace{(-3.75)}_{\text{income}}$

4Medium

Prove directly from the definition of the EMP that the expenditure function $e(p, u)$ is homogeneous of degree one in prices.

e(p, u) = \min_{u(x) \ge u} p \cdot x

Show solution

For $\alpha > 0$ , $e(\alpha p, u) = \min_{u(x)\ge u} (\alpha p) \cdot x = \alpha \min_{u(x)\ge u} p \cdot x = \alpha e(p, u)$ . The scalar $\alpha$ factors out because the constraint set does not involve prices.
The economics: doubling every price exactly doubles the cheapest cost of a given utility — only the unit of money changed.

5Hard

Prove with the Slutsky equation that a Giffen good cannot arise under Cobb-Douglas utility.

\frac{\partial x_1}{\partial p_1} = \frac{\partial h_1}{\partial p_1} - \frac{\partial x_1}{\partial w} x_1

Show solution

In Cobb-Douglas, $\partial x_1/\partial w = a/p_1 > 0$ — good 1 is always normal. So the income effect $-(\partial x_1/\partial w) x_1$ is always negative.
The substitution effect $\partial h_1/\partial p_1 \le 0$ is also never positive (compensated law of demand), so the total effect is a sum of two negatives — always negative. A Giffen good ( $\partial x_1/\partial p_1 > 0$ ) is impossible.

6Hard

Prove the duality identity $e(p, v(p, w)) = w$ from the definitions of the UMP and EMP (assume interior solutions and local nonsatiation).

v(p, w) = \max_{p \cdot x \le w} u(x), \qquad e(p, u) = \min_{u(x) \ge u} p \cdot x

Show solution

Let $x^* = x(p, w)$ solve the UMP.
By local nonsatiation $p \cdot x^* = w$ and $u(x^*) = v(p, w)$ .
Since $x^*$ is feasible for the EMP at target utility $v(p,w)$ , $e(p, v(p, w)) \le p \cdot x^* = w$ .
Conversely, if $e(p, v(p,w)) < w$ , the EMP solution $y$ satisfies $p \cdot y < w$ and $u(y) \ge v(p, w)$ .
Local nonsatiation gives a nearby feasible bundle with even higher $u$ , contradicting the maximality of $v(p, w)$ .
Hence equality holds.