Stone-Geary

\[U(x, y) = (x - \bar{x})^{\alpha}(y - \bar{y})^{\beta}\]

A generalisation of Cobb-Douglas that introduces subsistence quantities \(\bar{x}\) and \(\bar{y}\) — the minimum amounts of each good required before any utility is derived. The consumer first secures subsistence, then allocates the remaining supernumerary income like a Cobb-Douglas consumer.

Utility is defined only in the supernumerary region \(x > \bar{x}\), \(y > \bar{y}\). Indifference curves have the same shape as Cobb-Douglas but are shifted away from the origin by \((\bar{x}, \bar{y})\). Dashed reference lines at \(x = \bar{x}\) and \(y = \bar{y}\) are drawn automatically on the canvas.

Parameters

Parameter	Type	Default	Description
alpha	float	0.5	Expenditure share on supernumerary \(x\) (must be positive)
beta	float	0.5	Expenditure share on supernumerary \(y\) (must be positive)
bar_x	float	1.0	Subsistence quantity of good \(x\) (must be non-negative)
bar_y	float	1.0	Subsistence quantity of good \(y\) (must be non-negative)

Optimisation

Let \(m = I - p_x \bar{x} - p_y \bar{y}\) be the supernumerary income — the budget remaining after securing subsistence. The consumer solves

\[\max_{x,\,y}\; (x - \bar{x})^{\alpha}(y - \bar{y})^{\beta} \quad \text{subject to}\quad p_x x + p_y y = I\]

Substituting \(\tilde{x} = x - \bar{x}\) and \(\tilde{y} = y - \bar{y}\), the problem reduces to

\[\max_{\tilde{x},\,\tilde{y}}\; \tilde{x}^{\alpha}\tilde{y}^{\beta} \quad \text{subject to}\quad p_x \tilde{x} + p_y \tilde{y} = m\]

which is a standard Cobb-Douglas problem in the supernumerary quantities. The Marshallian demands are therefore

\[x^* = \bar{x} + \frac{\alpha}{\alpha + \beta}\cdot\frac{m}{p_x}, \qquad y^* = \bar{y} + \frac{\beta}{\alpha + \beta}\cdot\frac{m}{p_y}\]

A necessary condition for an interior solution is \(m > 0\), i.e.

\[I > p_x \bar{x} + p_y \bar{y}\]

If this fails, solve() raises InvalidParameterError.

Relationship to Cobb-Douglas

Setting \(\bar{x} = \bar{y} = 0\) reduces Stone-Geary to Cobb-Douglas exactly. The expansion path for Stone-Geary does not pass through the origin — it is a ray from \((\bar{x}, \bar{y})\) — so no expansion-path ray is drawn on the canvas.

Usage

Python

from econ_viz import Canvas, levels, solve
from econ_viz.models import StoneGeary

model = StoneGeary(alpha=0.5, beta=0.5, bar_x=2.0, bar_y=2.0)
eq    = solve(model, px=2.0, py=3.0, income=30.0)
lvls  = levels.around(eq.utility, n=5)

# Subsistence lines (dashed) are drawn automatically
Canvas(x_max=20, y_max=15, title=r"Stone-Geary  $\bar{x}=2,\ \bar{y}=2$") \
    .add_utility(model, levels=lvls) \
    .add_budget(2.0, 3.0, 30.0, fill=True) \
    .add_equilibrium(eq) \
    .save("stone_geary.png")

Note

StoneGeary is not yet available via the CLI. Use the Python API directly.

LaTeX parsing

Stone-Geary does not have a standard compact LaTeX form, so parse_latex() does not support it. Construct the model directly via the Python API.

Comments