Note
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Direct point elasticities¶
- We use a previously estimated nested logit model and calculate
disaggregate and aggregate direct point elasticities.
Details about this example are available in Section 3 of Bierlaire (2018) Calculating indicators with PandasBiogeme
Michel Bierlaire, EPFL Sat Jun 28 2025, 19:21:30
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.expressions import Derive
from biogeme.models import nested
from biogeme.results_processing import EstimationResults
from scenarios import CostCarCHF, MarginalCostPT, TimeCar, TimePT, distance_km, scenario
Obtain the specification for the default scenario The definition of the scenarios is available in Specification of a nested logit model.
v, nests, _, _ = scenario()
Obtain the expression for the choice probability of each alternative.
prob_pt = nested(v, None, nests, 0)
prob_car = nested(v, None, nests, 1)
prob_sm = nested(v, None, nests, 2)
Calculation of the direct elasticities. We use the ‘Derive’ operator to calculate the derivatives.
direct_elas_pt_time = Derive(prob_pt, 'TimePT') * TimePT / prob_pt
direct_elas_pt_cost = Derive(prob_pt, 'MarginalCostPT') * MarginalCostPT / prob_pt
direct_elas_car_time = Derive(prob_car, 'TimeCar') * TimeCar / prob_car
direct_elas_car_cost = Derive(prob_car, 'CostCarCHF') * CostCarCHF / prob_car
direct_elas_sm_dist = Derive(prob_sm, 'distance_km') * distance_km / prob_sm
Formulas to simulate.
simulate = {
'weight': normalized_weight,
'Prob. car': prob_car,
'Prob. public transportation': prob_pt,
'Prob. slow modes': prob_sm,
'direct_elas_pt_time': direct_elas_pt_time,
'direct_elas_pt_cost': direct_elas_pt_cost,
'direct_elas_car_time': direct_elas_car_time,
'direct_elas_car_cost': direct_elas_car_cost,
'direct_elas_sm_dist': direct_elas_sm_dist,
}
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 Pandas 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. car ... direct_elas_car_cost direct_elas_sm_dist
0 0.893779 0.519165 ... -0.156968 -6.275368
1 0.868674 0.560868 ... -0.020208 -0.690945
2 0.868674 0.875040 ... -0.030370 -6.230584
3 0.965766 0.813507 ... -0.022260 -1.734141
4 0.868674 0.729616 ... -0.052688 -4.718355
... ... ... ... ... ...
1894 2.053830 0.711097 ... -0.116747 -10.650385
1895 0.868674 0.849018 ... -0.049505 -8.084431
1896 0.868674 0.688197 ... -0.030267 -1.857448
1897 0.965766 0.742202 ... -0.047084 -3.406664
1898 0.965766 0.763156 ... -0.054497 -3.968513
[1899 rows x 9 columns]
We calculate the aggregate elasticities.
First, the weighted probabilities.
simulated_values['Weighted prob. car'] = (
simulated_values['weight'] * simulated_values['Prob. car']
)
simulated_values['Weighted prob. PT'] = (
simulated_values['weight'] * simulated_values['Prob. public transportation']
)
simulated_values['Weighted prob. SM'] = (
simulated_values['weight'] * simulated_values['Prob. slow modes']
)
Then the denominators of the aggregate elasticity expressions.
denominator_car = simulated_values['Weighted prob. car'].sum()
denominator_pt = simulated_values['Weighted prob. PT'].sum()
denominator_sm = simulated_values['Weighted prob. SM'].sum()
And finally the aggregate elasticities themselves.
Elasticity of car with respect to time.
direct_elas_term_car_time = (
simulated_values['Weighted prob. car']
* simulated_values['direct_elas_car_time']
/ denominator_car
).sum()
print(
f'Aggregate direct point elasticity of car wrt time: '
f'{direct_elas_term_car_time:.3g}'
)
Aggregate direct point elasticity of car wrt time: -0.0437
Elasticity of car with respect to cost.
direct_elas_term_car_cost = (
simulated_values['Weighted prob. car']
* simulated_values['direct_elas_car_cost']
/ denominator_car
).sum()
print(
f'Aggregate direct point elasticity of car wrt cost: '
f'{direct_elas_term_car_cost:.3g}'
)
Aggregate direct point elasticity of car wrt cost: -0.0917
Elasticity of public transportation with respect to time.
direct_elas_term_pt_time = (
simulated_values['Weighted prob. PT']
* simulated_values['direct_elas_pt_time']
/ denominator_pt
).sum()
print(
f'Aggregate direct point elasticity of PT wrt time: '
f'{direct_elas_term_pt_time:.3g}'
)
Aggregate direct point elasticity of PT wrt time: -0.268
Elasticity of public transportation with respect to cost.
direct_elas_term_pt_cost = (
simulated_values['Weighted prob. PT']
* simulated_values['direct_elas_pt_cost']
/ denominator_pt
).sum()
print(
f'Aggregate direct point elasticity of PT wrt cost: '
f'{direct_elas_term_pt_cost:.3g}'
)
Aggregate direct point elasticity of PT wrt cost: -0.324
Elasticity of slow modes with respect to distance.
direct_elas_term_sm_dist = (
simulated_values['Weighted prob. SM']
* simulated_values['direct_elas_sm_dist']
/ denominator_sm
).sum()
print(
f'Aggregate direct point elasticity of SM wrt distance: '
f'{direct_elas_term_sm_dist:.3g}'
)
Aggregate direct point elasticity of SM wrt distance: -1.09
Total running time of the script: (0 minutes 4.108 seconds)