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
import biogeme.biogeme as bio
from biogeme import models
import biogeme.results as res
import biogeme.exceptions as excep
from biogeme.expressions import Derive
from optima_data import database, 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,
}

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 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.getBetaValues())
simulated_values

	weight	Prob. car	Prob. public transportation	Prob. slow modes	direct_elas_pt_time	direct_elas_pt_cost	direct_elas_car_time	direct_elas_car_cost	direct_elas_sm_dist
0	0.886023	0.521791	0.476807	0.001402	-0.128761	0.000000	-0.044176	-0.155490	-6.282280
2	0.861136	0.563268	0.238681	0.198051	-0.236955	-0.215072	-0.007565	-0.020018	-0.690123
3	0.861136	0.875824	0.119136	0.005040	-0.278714	-1.550950	-0.008245	-0.030060	-6.242559
4	0.957386	0.814110	0.051136	0.134754	-2.020961	-1.030288	-0.020782	-0.022100	-1.734068
5	0.861136	0.732059	0.257192	0.010749	-0.255844	-0.525291	-0.015470	-0.052004	-4.727090
...	...	...	...	...	...	...	...	...	...
2259	2.036009	0.714085	0.285872	0.000043	-0.305144	-0.531971	-0.033015	-0.115081	-10.671587
2261	0.861136	0.849040	0.149688	0.001272	-0.948767	-0.881532	-0.051836	-0.049301	-8.100940
2262	0.861136	0.689083	0.205901	0.105016	-0.551441	-0.417948	-0.029794	-0.030061	-1.862736
2263	0.957386	0.744274	0.219850	0.035876	-0.291011	-0.697499	-0.017718	-0.046520	-3.407459
2264	0.957386	0.765468	0.209149	0.025383	-0.342744	-0.765284	-0.018957	-0.053750	-3.969568

1906 rows × 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: -0.0441

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: -0.0906

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: -0.274

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: -0.32

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