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

author:

Michel Bierlaire, EPFL

date:

Wed Apr 12 21:01:41 2023

import sys

from IPython.core.display_functions import display

import biogeme.biogeme as bio
from biogeme import models
import biogeme.results as res
from biogeme.exceptions import BiogemeError
from biogeme.expressions import Derive
from biogeme.data.optima import read_data, normalized_weight

from scenarios import (
    scenario,
    TimePT,
    TimeCar,
    MarginalCostPT,
    CostCarCHF,
    distance_km,
)

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 = models.nested(V, None, nests, 0)
prob_CAR = models.nested(V, None, nests, 1)
prob_SM = models.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 = bio.BIOGEME(database, simulate)

Read the estimation results from the file

try:
    results = res.bioResults(pickle_file='saved_results/b02estimation.pickle')
except BiogemeError:
    sys.exit(
        'Run first the script b02estimation.py in order to generate '
        'the file b02estimation.pickle.'
    )

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     0.886023   0.508061  ...             -0.179272            -6.416591
2     0.861136   0.574899  ...             -0.021838            -0.705481
3     0.861136   0.888093  ...             -0.030361            -6.195871
4     0.957386   0.790872  ...             -0.027865            -1.431063
5     0.861136   0.733898  ...             -0.057884            -4.774826
...        ...        ...  ...                   ...                  ...
2259  2.036009   0.710676  ...             -0.130517           -10.886598
2261  0.861136   0.849664  ...             -0.055027            -8.233209
2262  0.861136   0.688689  ...             -0.033734            -1.752372
2263  0.957386   0.747552  ...             -0.051470            -3.335287
2264  0.957386   0.767021  ...             -0.059843            -3.914194

[1906 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: -0.0353

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: -0.0993

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: -0.231

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: -0.366

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: -1.12