Translog

Translog adds a flexible log-quadratic utility specification that can approximate a wide range of smooth preferences.

from econ_viz.models import Translog

model = Translog(
    alpha_x=0.5,
    alpha_y=0.5,
    beta_xx=0.1,
    beta_yy=-0.05,
    beta_xy=0.08,
)

Functional form

ln U(x, y) = alpha_0
           + alpha_x ln x + alpha_y ln y
           + 0.5 beta_xx (ln x)^2
           + 0.5 beta_yy (ln y)^2
           + beta_xy ln x ln y

The implementation returns:

U(x, y) = exp(ln U(x, y))

Setting beta_xx = beta_yy = beta_xy = 0 collapses the model to a Cobb-Douglas-style log-linear form.

Parameters

Parameter	Default	Meaning
alpha_x	0.5	First-order coefficient on ln x
alpha_y	0.5	First-order coefficient on ln y
beta_xx	0.0	Second-order own effect for ln x
beta_yy	0.0	Second-order own effect for ln y
beta_xy	0.0	Cross term ln x ln y
alpha_0	0.0	Log-utility intercept

alpha_x and alpha_y must be strictly positive.

Properties

• utility_type is UtilityType.SMOOTH
• works with Canvas.add_utility(...)
• works with solve(...) through the library's numerical optimisation path

Example

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

model = Translog(alpha_x=0.6, alpha_y=0.4, beta_xy=0.12)
eq = solve(model, px=2.0, py=3.0, income=30.0)
lvls = levels.around(eq.utility, n=4)

Canvas(x_max=18, y_max=12, title="Translog utility") \
    .add_utility(model, levels=lvls, label="$U$") \
    .add_budget(2.0, 3.0, 30.0, label="$B$") \
    .add_equilibrium(eq) \
    .show_legend()

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