Quasi-Linear

\[U(x, y) = f(x) + y \quad \text{or} \quad U(x, y) = x + f(y)\]

One good enters linearly; the other enters through a strictly concave, strictly increasing transformation \(f\). Indifference curves have the same shape at every income level — there is no income effect on the non-linear good.

Parameters

Parameter	Type	Default	Description
v_func	Callable	numpy.log	Strictly increasing, strictly concave scalar function \(f(z)\)
linear_in	'x' or 'y'	'y'	Which good enters linearly

v_func is validated numerically on \(z \in [0.1, 10]\) at construction time — non-monotone or convex functions raise ValueError.

Optimisation

Taking \(U = f(x) + y\) as the canonical form, the consumer solves

\[\max_{x,\,y}\; f(x) + y \quad \text{subject to}\quad p_x x + p_y y = I\]

The Lagrangian is

\[\mathcal{L}(x, y, \lambda) = f(x) + y - \lambda\,(p_x x + p_y y - I)\]

First-order conditions:

\[\begin{aligned} \frac{\partial \mathcal{L}}{\partial x} &= f'(x) - \lambda p_x = 0 \\[6pt] \frac{\partial \mathcal{L}}{\partial y} &= 1 - \lambda p_y = 0 \\[6pt] \frac{\partial \mathcal{L}}{\partial \lambda} &= p_x x + p_y y - I = 0 \end{aligned}\]

From the second condition \(\lambda = 1/p_y\), so the optimal \(x^*\) satisfies

\[f'(x^*) = \frac{p_x}{p_y}\]

This pins \(x^*\) independently of income \(I\) — the hallmark of the quasi-linear form. Income is then fully absorbed by \(y^*\):

\[y^* = \frac{I - p_x\,x^*}{p_y}\]

Usage

Python

import numpy as np
from econ_viz import Canvas, levels, solve
from econ_viz.models import QuasiLinear

model = QuasiLinear(v_func=np.log, linear_in="y")   # U = log(x) + y
eq    = solve(model, px=2.0, py=1.0, income=20.0)
lvls  = levels.around(eq.utility, n=5)

Canvas(x_max=15, y_max=15, title=r"Quasi-Linear $\ln(x) + y$") \
    .add_utility(model, levels=lvls) \
    .add_budget(2.0, 1.0, 20.0) \
    .add_equilibrium(eq) \
    .save("quasi_linear.png")

Note

QuasiLinear is not available via the CLI. Use the Python API directly.