Cobb-Douglas
\[U(x, y) = x^\alpha \cdot y^\beta\]
The most common textbook utility function. Indifference curves are smooth, strictly convex, and asymptotic to both axes.
Parameters
| Parameter | Type | Default | Description |
|---|---|---|---|
alpha |
float | 0.5 | Output elasticity of good \(x\) |
beta |
float | 0.5 | Output elasticity of good \(y\) |
Optimisation
The consumer solves
\[\max_{x,\,y}\; x^\alpha y^\beta \quad \text{subject to}\quad p_x x + p_y y = I\]
The Lagrangian is
\[\mathcal{L}(x, y, \lambda) = x^\alpha y^\beta - \lambda\,(p_x x + p_y y - I)\]
First-order conditions:
\[\begin{aligned}
\frac{\partial \mathcal{L}}{\partial x} &= \alpha x^{\alpha-1} y^\beta - \lambda p_x = 0 \\[6pt]
\frac{\partial \mathcal{L}}{\partial y} &= \beta x^\alpha y^{\beta-1} - \lambda p_y = 0 \\[6pt]
\frac{\partial \mathcal{L}}{\partial \lambda} &= p_x x + p_y y - I = 0
\end{aligned}\]
Dividing the first two conditions gives the tangency condition \(\mathrm{MRS} = p_x/p_y\):
\[\frac{\alpha y}{\beta x} = \frac{p_x}{p_y}\]
Substituting into the budget constraint yields the Marshallian demands:
\[x^* = \frac{\alpha}{\alpha + \beta}\cdot\frac{I}{p_x}, \qquad y^* = \frac{\beta}{\alpha + \beta}\cdot\frac{I}{p_y}\]
Usage
from econ_viz import Canvas, levels, solve
from econ_viz.models import CobbDouglas
model = CobbDouglas(alpha=0.5, beta=0.5)
eq = solve(model, px=2.0, py=3.0, income=30.0)
lvls = levels.around(eq.utility, n=5)
Canvas(x_max=20, y_max=15, title=r"Cobb-Douglas $x^{0.5} y^{0.5}$") \
.add_utility(model, levels=lvls) \
.add_budget(2.0, 3.0, 30.0, fill=True) \
.add_equilibrium(eq, show_ray=True) \
.save("cobb_douglas.png")