Arc elasticities

We use a previously estimated nested logit model and calculate arc elasticities.

Details about this example are available in Section 3 of Bierlaire (2018) Calculating indicators with PandasBiogeme

Michel Bierlaire, EPFL Sat Jun 28 2025, 20:59:31

import sys

from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.data.optima import normalized_weight, read_data
from biogeme.models import nested
from biogeme.results_processing import EstimationResults

from scenarios import scenario

Obtain the specification for the default scenario The definition of the scenarios is available in Specification of a nested logit model.

v, nests, _, MarginalCostPT = scenario()

Obtain the expression for the choice probability of the public transportation.

prob_pt = nested(v, None, nests, 0)

We investigate a scenario where the price for public transportation increases by 20%. We extract the corresponding scenario.

v_after, _, _, marginal_cost_pt_after = scenario(factor=1.2)
prob_pt_after = nested(v_after, None, nests, 0)

Disaggregate elasticities

direct_elas_pt = (
    (prob_pt_after - prob_pt)
    * MarginalCostPT
    / (prob_pt * (marginal_cost_pt_after - MarginalCostPT))
)

Formulas to simulate.

simulate = {
    'weight': normalized_weight,
    'Prob. PT': prob_pt,
    'direct_elas_pt': direct_elas_pt,
}

Read the data

database = read_data()

Create the Biogeme object.

the_biogeme = BIOGEME(database, simulate)

Read the estimation results from the file

try:
    results = EstimationResults.from_yaml_file(
        filename='saved_results/b02estimation.yaml'
    )
except FileNotFoundError:
    sys.exit(
        'Run first the script b02estimation.py in order to generate '
        'the file b02estimation.yaml.'
    )

simulated_values is a Panda dataframe with the same number of rows as the database, and as many columns as formulas to simulate.

simulated_values = the_biogeme.simulate(results.get_beta_values())
display(simulated_values)

        weight  Prob. PT  direct_elas_pt
0     0.893779  0.479431             NaN
1     0.868674  0.241046       -0.212075
2     0.868674  0.119893       -1.366282
3     0.965766  0.051229       -0.942913
4     0.868674  0.259609       -0.508500
...        ...       ...             ...
1894  2.053830  0.288859       -0.514652
1895  0.868674  0.149688       -0.822786
1896  0.868674  0.205928       -0.408671
1897  0.965766  0.222025       -0.667457
1898  0.965766  0.211584       -0.727161

[1899 rows x 3 columns]

We calculate the aggregate elasticities.

First, the weighted probabilities.

simulated_values['Weighted prob. PT'] = (
    simulated_values['weight'] * simulated_values['Prob. PT']
)

Then the denominator of the aggregate elasticity expression.

denominator_pt = simulated_values['Weighted prob. PT'].sum()

And finally the aggregate elasticities themselves.

direct_elas_pt = (
    simulated_values['Weighted prob. PT']
    * simulated_values['direct_elas_pt']
    / denominator_pt
).sum()

print(
    f'Aggregate direct arc elasticity of public transportation wrt cost: '
    f'{direct_elas_pt:.3g}'
)

Aggregate direct arc elasticity of public transportation wrt cost: -0.303