Exercises · 5. Aggregate Demand

Easy

1Easy

Three Cobb-Douglas consumers have parameters $a = (0.4, 0.5, 0.6)$ , wealths $w_i = 50$ , and $p_2 = 1$ .

At $p_1 = 2$ , find each consumer's $x_1^i$ and aggregate demand $X_1$ .

x_1^i(p_1) = \frac{a_i w_i}{p_1}

Aggregating Demand

The sum at

a = (0.4, 0.5, 0.6)

,

w_i = 50

,

p_1 = 2

.

Show solution

$x_1^1 = 0.4 \cdot 50 / 2 = 10$ , $x_1^2 = 0.5 \cdot 50 / 2 = 12.5$ , $x_1^3 = 0.6 \cdot 50 / 2 = 15$ .
Adding up, $X_1 = 10 + 12.5 + 15 = 37.5$ . Aggregate demand is just the sum of individual demands — the definition of a "market demand curve" itself.

Medium

2Medium

Suppose each consumer $i$ satisfies Walras' law $p \cdot x^i(p, w_i) = w_i$ (no assumption on the functional form of demand).

Prove that aggregate demand $X(p) = \sum_i x^i(p, w_i)$ satisfies $p \cdot X(p) = \sum_i w_i$ .

X(p) = \sum_i x^i(p, w_i)

Aggregating Demand

Show solution

The inner product is linear in addition, so $p \cdot X(p) = p \cdot \sum_i x^i(p, w_i) = \sum_i p \cdot x^i(p, w_i)$ .
Applying individual Walras' law to each term gives $\sum_i p \cdot x^i(p, w_i) = \sum_i w_i$ . The shape of demand (Cobb-Douglas or not) was never used — because summation is a linear operation.
$p \cdot X(p) = \sum_i p \cdot x^i(p, w_i) = \sum_i w_i$

3Medium

Two identical-taste Cobb-Douglas consumers with $a = 1/2$ , total wealth $W = 100$ , $p_1 = 1$ , $p_2 = 1$ .

Compute $X_1$ for the splits $s = 0.25$ and $s = 0.75$ , and verify that individual demands change while aggregate demand does not.

The Representative Consumer

a = 1/2

,

W = 100

,

s = 0.25

— thin individual curves move, the thick aggregate curve stays.

Show solution

With $s = 0.25$ , $w_1 = 25$ and $w_2 = 75$ , so $x_1^1 = 12.5$ , $x_1^2 = 37.5$ , and the sum is $X_1 = 50$ .
With $s = 0.75$ , $w_1 = 75$ and $w_2 = 25$ , so $x_1^1 = 37.5$ , $x_1^2 = 12.5$ — the individual demands swap, but $X_1 = 50$ is unchanged. Both equal $X_1 = aW/p_1 = 0.5 \cdot 100 = 50$ .

Hard

4Hard

Suppose $I$ consumers have indirect utilities of the Gorman form $v_i(p, w_i) = \alpha_i(p) + b(p)\,w_i$ with a common $b(p) > 0$ .

Use Roy's identity to derive each demand, prove that aggregate demand $X(p)$ depends only on total wealth $W = \sum_i w_i$ , and exhibit a representative consumer's indirect utility reproducing it.

v_i(p, w_i) = \alpha_i(p) + b(p)\,w_i

The Representative Consumer

Show solution

Roy's identity $x^i = -\nabla_p v_i / \partial v_i / \partial w$ gives numerator $-(\nabla \alpha_i(p) + \nabla b(p)\,w_i)$ and denominator $b(p)$ , so $x^i(p, w_i) = -\frac{\nabla \alpha_i(p)}{b(p)} - \frac{\nabla b(p)}{b(p)}\,w_i$ .
Summing over $i$ , the common $b(p)$ gives $X(p) = -\frac{\sum_i \nabla \alpha_i(p)}{b(p)} - \frac{\nabla b(p)}{b(p)}\,W$ — individual wealths enter only through total wealth $W$ . The distribution is irrelevant to aggregate demand.
$X(p) = -\frac{\sum_i \nabla \alpha_i(p)}{b(p)} - \frac{\nabla b(p)}{b(p)}\,W$
Representative consumer: a single consumer with indirect utility $V(p, W) = \sum_i \alpha_i(p) + b(p)\,W$ reproduces exactly this $X(p)$ via Roy's identity. The Gorman form (common marginal utility of wealth $b(p)$ ) is thus sufficient for a representative consumer to exist.