The Representative Consumer

Two identical-taste Cobb-Douglas consumers with $a = 1/2$ , total wealth $W = 120$ , $p_2 = 1$ .

Compute aggregate demand $X_1$ at $p_1 = 1$ for the wealth splits $s = 0.2$ and $s = 0.8$ , and verify they coincide regardless of the distribution.

X_1 = \frac{a w_1}{p_1} + \frac{a w_2}{p_1} = \frac{aW}{p_1}

s = 0.2:\; 12 + 48 = 60, \qquad s = 0.8:\; 48 + 12 = 60

a = 1/2

W = 120

s = 0.2

— exactly on top of the ghost curve (

s = 0.5

Show proof

With $s = 0.2$ , $w_1 = 24$ and $w_2 = 96$ , so $X_1 = 0.5 \cdot 24 + 0.5 \cdot 96 = 12 + 48 = 60$ .
With $s = 0.8$ , $w_1 = 96$ and $w_2 = 24$ , so $X_1 = 48 + 12 = 60$ — the individual demands swap, but the sum is unchanged.
In general $s$ cancels in $X_1 = a sW/p_1 + a(1-s)W/p_1 = aW/p_1$ . Aggregate demand is pinned down by total wealth $W$ alone, so a single consumer with wealth $W$ (Cobb-Douglas, parameter $a$ ) is this economy's "representative consumer".

Verify that Cobb-Douglas indirect utility $v(p, w) = b(p)\,w$ , with $b(p) = a^a (1-a)^{1-a} / (p_1^a p_2^{1-a})$ , is of the Gorman form with a common $b(p)$ ,

and use Roy's identity to find $X_1$ for $a = 0.6$ , $W = 150$ , $s = 0.3$ , $p_1 = 2$ .

v_i(p, w_i) = 0 \cdot \alpha_i(p) + b(p)\,w_i \quad \text{(Gorman, common } b\text{)}

X_1 = \frac{0.6 \cdot 150}{2} = 45

a = 0.6

W = 150

s = 0.3

p_1 = 2

—

X_1 = 45

, distribution-free.

Show proof

Plugging Cobb-Douglas demand $x(p, w) = (aw/p_1, (1-a)w/p_2)$ into utility gives $v(p, w) = w \cdot a^a (1-a)^{1-a} / (p_1^a p_2^{1-a}) = b(p)\,w$ . Both consumers share the same $a$ , hence the same $b(p)$ : the Gorman form with $\alpha_i(p) = 0$ .
Roy's identity: $x_1^i = -\frac{\partial v_i / \partial p_1}{\partial v_i / \partial w} = -\frac{b'(p) w_i}{b(p)} = \frac{a w_i}{p_1}$ (the last equality follows by differentiating the formula for $b(p)$ ).
Summing, the common $b(p)$ in the denominator gives $X_1 = -b'(p)W / b(p) = aW/p_1 = 0.6 \cdot 150 / 2 = 45$ — the individual wealths $w_1 = 45$ , $w_2 = 105$ contribute only through $13.5 + 31.5 = 45$ .

The Representative Consumer

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