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

author:

Michel Bierlaire, EPFL

date:

Wed Apr 12 21:00:10 2023

import sys
import biogeme.biogeme as bio
import biogeme.exceptions as excep
from biogeme import models
import biogeme.results as res
from optima_data import database, normalized_weight
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 = models.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, _, _, MarginalCostPT_after = scenario(factor=1.2)
prob_PT_after = models.nested(V_after, None, nests, 0)

Disaggregate elasticities

direct_elas_pt = (
    (prob_PT_after - prob_PT)
    * MarginalCostPT
    / (prob_PT * (MarginalCostPT_after - MarginalCostPT))
)

Formulas to simulate.

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

Create the Biogeme object.

the_biogeme = bio.BIOGEME(database, simulate)

Read the estimation results from the file

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

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.getBetaValues())
simulated_values

	weight	Prob. PT	direct_elas_pt
0	0.886023	0.476807	0.000000
2	0.861136	0.238681	-0.212447
3	0.861136	0.119136	-1.362762
4	0.957386	0.051136	-0.941739
5	0.861136	0.257192	-0.508224
...	...	...	...
2259	2.036009	0.285872	-0.514608
2261	0.861136	0.149688	-0.819720
2262	0.861136	0.205901	-0.407408
2263	0.957386	0.219850	-0.667577
2264	0.957386	0.209149	-0.727399

1906 rows × 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: -0.299