Aggregating Demand

Three Cobb-Douglas consumers with $a = (0.3, 0.5, 0.7)$ , $w_i = 60$ , $p_2 = 1$ .

At $p_1 = 1$ , compute aggregate demand $X(p)$ directly and verify that total spending $p \cdot X(p)$ exactly equals total wealth $W$ (Walras' law).

x_1^i = \frac{a_i w_i}{p_1}, \quad x_2^i = \frac{(1-a_i) w_i}{p_2}

X_1 = 18 + 30 + 42 = 90, \quad X_2 = 42 + 30 + 18 = 90

a = (0.3, 0.5, 0.7)

w_i = 60

p_1 = 1

— Walras' law, numerically.

Show proof

Individual demands are $x_1^i = a_i \cdot 60 / 1 = 18, 30, 42$ and $x_2^i = (1 - a_i) \cdot 60 = 42, 30, 18$ .
Adding up, $X_1 = 90$ and $X_2 = 90$ . Total spending is $p \cdot X = 1 \cdot 90 + 1 \cdot 90 = 180$ .
Total wealth is $W = 60 + 60 + 60 = 180$ , so $p \cdot X = W$ holds exactly. When every consumer spends her whole budget (individual Walras' law), the summed aggregate demand spends the whole aggregate wealth too — summation is linear, so the property survives.
$p \cdot X(p) = \sum_i p \cdot x^i(p, w_i) = \sum_i w_i = W$

Now consumer 1 develops a strong taste for good 1, so $a = (0.9, 0.2, 0.2)$ (wealths unchanged at $w_i = 60$ ).

Show by computation that the aggregate demand curve shifts up at $p_1 = 1$ , yet the Walras check still holds exactly.

X_1 = 54 + 12 + 12 = 78, \quad X_2 = 6 + 48 + 48 = 102

p \cdot X = 78 + 102 = 180 = W

a = (0.9, 0.2, 0.2)

w_i = 60

p_1 = 1

— Walras' law after a taste shift.

Show proof

New individual demands: $x_1 = 54, 12, 12$ , so $X_1 = 78$ — moved from $90$ in the previous example. The aggregate curve is sensitive to the distribution of tastes.
Meanwhile $x_2 = 6, 48, 48$ , so $X_2 = 102$ , and total spending is $78 + 102 = 180 = W$ . What consumer 1 added to good 1 spending ( $+36$ ) she subtracted from good 2 spending ( $-36$ ); total spending stays pinned to total wealth.
The lesson: the "shape" of aggregate demand moves with the distribution of tastes, but the "constraint" $p \cdot X = W$ never breaks, whatever the distribution.

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