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Satiation (Bliss Point)

\[U(x, y) = -a(x - x^*)^2 - b(y - y^*)^2\]

Utility rises toward the bliss point \((x^*, y^*)\) and falls away in all directions. Indifference curves are closed ellipses centred on the bliss point. This violates the standard monotonicity axiom — a consumer can have "too much" of a good.

Parameters

Parameter Type Default Description
bliss_x float 5.0 \(x\)-coordinate of the bliss point \(x^*\)
bliss_y float 5.0 \(y\)-coordinate of the bliss point \(y^*\)
a float 1.0 Curvature along the \(x\)-axis (must be positive)
b float 1.0 Curvature along the \(y\)-axis (must be positive)

Optimisation

The consumer solves

\[\max_{x,\,y}\; -a(x-x^*)^2 - b(y-y^*)^2 \quad \text{subject to}\quad p_x x + p_y y = I\]

The Lagrangian is

\[\mathcal{L}(x, y, \lambda) = -a(x-x^*)^2 - b(y-y^*)^2 - \lambda\,(p_x x + p_y y - I)\]

First-order conditions:

\[\begin{aligned} \frac{\partial \mathcal{L}}{\partial x} &= -2a(x - x^*) - \lambda p_x = 0 \\[6pt] \frac{\partial \mathcal{L}}{\partial y} &= -2b(y - y^*) - \lambda p_y = 0 \\[6pt] \frac{\partial \mathcal{L}}{\partial \lambda} &= p_x x + p_y y - I = 0 \end{aligned}\]

Dividing the first two conditions gives the tangency condition:

\[\frac{a(x - x^*)}{b(y - y^*)} = \frac{p_x}{p_y}\]

If the bliss point \((x^*, y^*)\) is interior to the budget set (i.e.\ \(p_x x^* + p_y y^* \le I\)), the unconstrained maximum is attained at the bliss point itself and the consumer does not spend all income.

Note

Satiation is not compatible with solve() because the standard budget-tangency optimum may lie outside the bliss ellipse. Use --no-budget --no-equilibrium in the CLI, or omit add_budget / add_equilibrium in Python.

Usage

import numpy as np
from econ_viz import Canvas, levels
from econ_viz.models import Satiation

model = Satiation(bliss_x=6.0, bliss_y=4.0)

x_pts = np.linspace(0.1, 12, 300)
y_pts = np.linspace(0.1, 10, 300)
X, Y  = np.meshgrid(x_pts, y_pts)
lvls  = levels.percentile(model(X, Y), n=5)

Canvas(x_max=12, y_max=10, title="Satiation — bliss at $(6, 4)$") \
    .add_utility(model, levels=lvls) \
    .save("satiation.png")

econ-viz plot --model satiation --bliss-x 6 --bliss-y 4 \
              --x-max 12 --y-max 10 \
              --no-budget --no-equilibrium \
              --output satiation.png