Solved problem · Beginner

Deriving Demand from Cobb–Douglas Preferences

A complete derivation of Marshallian demand for Cobb–Douglas utility.

Problem

A consumer has utility $u(x, y) = x^{\alpha} y^{1-\alpha}$ with $\alpha \in (0,1)$ , prices $p_x, p_y > 0$ , and income $M \geq 0$ . Derive the Marshallian demand functions.

Solution

The consumer solves

\max_{x,\,y \ \geq\, 0} \; x^{\alpha} y^{1-\alpha} \quad \text{subject to} \quad p_x x + p_y y = M.

Step 1 — The budget binds. Utility is strictly increasing in both goods, so the constraint holds with equality at any optimum.

Step 2 — Tangency. Preferences are strictly convex on the interior, so the optimum is characterised by the tangency condition $MRS = p_x / p_y$ :

\frac{\partial u/\partial x}{\partial u/\partial y} = \frac{\alpha\, x^{\alpha-1} y^{1-\alpha}}{(1-\alpha)\, x^{\alpha} y^{-\alpha}} = \frac{\alpha}{1-\alpha}\cdot\frac{y}{x} = \frac{p_x}{p_y}.

Step 3 — Solve the system. From the tangency condition, $p_y y = \frac{1-\alpha}{\alpha}\, p_x x$ . Substituting into the budget constraint:

p_x x + \frac{1-\alpha}{\alpha}\, p_x x = M \;\;\Longrightarrow\;\; p_x x \cdot \frac{1}{\alpha} = M.

So the demand functions are

x^d(p_x, p_y, M) = \frac{\alpha M}{p_x}, \qquad y^d(p_x, p_y, M) = \frac{(1-\alpha)\, M}{p_y}.

Interpretation. The consumer spends the fixed share $\alpha$ of income on $x$ and $1-\alpha$ on $y$ , regardless of prices.

Check your understanding

Verify that each demand function is homogeneous of degree zero in $(p_x, p_y, M)$ .
Compute the indirect utility function $v(p_x, p_y, M)$ and verify Roy’s identity.