Travel mode choice — the value of travel time (WTP)

When a commuter chooses between modes, they trade money against time. The rate at which they are willing to substitute one for the other — how many dollars an hour of travel time is worth to them — is the value of travel time savings (VTTS), the single most-used number in transport appraisal. It is a willingness to pay: the marginal rate of substitution between a time attribute and the cost (price) attribute in a discrete-choice model.

pymargins exposes this directly. For a model with a continuous time regressor and a continuous cost regressor, m.wtp(attribute, price) returns

\[ \text{WTP} \;=\; -\,\frac{\partial U/\partial\,\text{attribute}} {\partial U/\partial\,\text{price}}, \]

with the standard error propagated jointly through both slopes — a ratio, so the uncertainty is not just the two marginal SEs bolted together. This demo estimates VTTS on the classic Greene–Hensher travel-mode data and shows why the ratio’s interval rewards a simulation cross-check.

import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf

from pymargins import Margins

# The TravelMode data ships in long form: one row per (traveller, mode),
# 210 travellers × 4 modes (air / train / bus / car).
long = pd.read_csv("data/travelmode.csv")
print(long.head(8).to_string(index=False))

 individual  mode choice  wait  vcost  travel  gcost  income  size
          1   air     no    69     59     100     70      35     1
          1 train     no    34     31     372     71      35     1
          1   bus     no    35     25     417     70      35     1
          1   car    yes     0     10     180     30      35     1
          2   air     no    64     58      68     68      30     2
          2 train     no    44     31     354     84      30     2
          2   bus     no    53     25     399     85      30     2
          2   car    yes     0     11     255     50      30     2

1. A binary choice with attribute differences

statsmodels multinomial logit takes traveller-level regressors, but cost and time are alternative-specific — each mode has its own. The textbook device that turns alternative-specific attributes into a chooser-level regression is the difference specification: restrict to two alternatives and regress the choice on the difference in each attribute between them. The coefficient on a difference is the utility weight on that attribute, which is all VTTS needs.

We take the two ground modes, car vs train, and difference their in-vehicle time (travel, minutes) and out-of-pocket cost (vcost, dollars). A full appraisal would fit a conditional logit over all four modes; the two-alternative difference model is the version that fits a chooser-level GLM, and it is plenty to exercise wtp() on real data:

wide = long.pivot(index="individual", columns="mode",
                  values=["vcost", "travel"])
wide.columns = [f"{attr}_{mode}" for attr, mode in wide.columns]

chosen = long[long["choice"] == "yes"].set_index("individual")[["mode", "income"]]
df = wide.join(chosen)

# Restrict to travellers who chose car or train, and build the binary outcome.
df = df[df["mode"].isin(["car", "train"])].copy()
df["car"] = (df["mode"] == "car").astype(int)
df["cost_diff"] = df["vcost_car"] - df["vcost_train"]   # dollars
df["time_diff"] = df["travel_car"] - df["travel_train"]  # minutes

print(f"{len(df)} travellers; car share = {df['car'].mean():.2f}")
print(df[["car", "cost_diff", "time_diff", "income"]].describe().round(1))

122 travellers; car share = 0.48
         car  cost_diff  time_diff  income
count  122.0      122.0      122.0   122.0
mean     0.5      -26.8      -28.6    32.3
std      0.5       24.4      146.7    19.9
min      0.0      -87.0     -327.0     4.0
25%      0.0      -48.8     -130.0    15.0
50%      0.0      -23.5       -7.5    30.0
75%      1.0      -12.2       52.5    45.0
max      1.0       26.0      522.0    72.0

2. Fit the choice model

fit = smf.glm(
    "car ~ cost_diff + time_diff + income",
    data=df,
    family=sm.families.Binomial(),
).fit()
print(fit.summary().tables[1])

==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -2.7097      0.617     -4.392      0.000      -3.919      -1.501
cost_diff     -0.0419      0.014     -2.913      0.004      -0.070      -0.014
time_diff     -0.0117      0.003     -4.353      0.000      -0.017      -0.006
income         0.0379      0.015      2.505      0.012       0.008       0.067
==============================================================================

Both attribute coefficients are negative and significant: a car that costs more, or takes longer, relative to the train is chosen less often — exactly the sign theory demands. Cost and time are individually well identified; the question is what their ratio says, and how sure we are of it.

3. Average marginal effects

On the probability scale, each extra dollar of relative car cost and each extra minute of relative car time both lower P(choose car):

m = Margins.linear_scale(fit, vcov="HC3", at="overall")

print(m.dydx("cost_diff").summary())
print(m.dydx("time_diff").summary())

==============================================================
              Margins Result (delta, level=0.95)              
==============================================================
           estimate  std err        z  P>|z|  [95% Conf. Int.]
--------------------------------------------------------------
cost_diff   -0.0053   0.0020  -2.7081  0.007  -0.0092, -0.0015
==============================================================

n = 122
κ: 0.145
Delta-vs-sim disagreement: 11.463%
==============================================================
              Margins Result (delta, level=0.95)              
==============================================================
           estimate  std err        z  P>|z|  [95% Conf. Int.]
--------------------------------------------------------------
time_diff   -0.0014   0.0008  -1.7687  0.077   -0.0029, 0.0002
==============================================================

n = 122
κ: 0.277
Delta-vs-sim disagreement: 62.608%

4. Willingness to pay — the value of travel time

wtp forms the ratio with joint inference. Because time_diff is a nuisance attribute (more time lowers utility), the WTP for one more minute is negative — travellers would need to be compensated to accept it. The interpretable headline number is its negation: the value of travel-time savings, in dollars per hour.

wtp_minute = m.wtp("time_diff", "cost_diff")
print(wtp_minute.summary())

vtts_per_hour = -float(wtp_minute.estimate) * 60
print(f"\nValue of travel time savings ≈ ${vtts_per_hour:.2f} per hour")

===================================================================
                 Margins Result (delta, level=0.95)                
===================================================================
                estimate  std err        z  P>|z|  [95% Conf. Int.]
-------------------------------------------------------------------
WTP(time_diff)   -0.2622   0.2184  -1.2002  0.230   -0.6903, 0.1659
===================================================================

n = 122
κ: 0.277

Value of travel time savings ≈ $15.73 per hour

A value in the mid-teens of dollars per hour is squarely in the range transport economists report for this dataset — a sanity check that the difference specification recovers a sensible number, not just a significant coefficient.

5. Why the ratio wants a simulation cross-check

The two slopes are each tightly estimated, but their ratio is a nonlinear function of β, and the denominator (cost_diff) is the noisier of the two. That curvature is exactly what the κ diagnostic watches. Re-running the same WTP under simulation shows how much the delta-method interval understates the asymmetry:

m_sim = Margins.linear_scale(
    fit, vcov="HC3", at="overall",
    method="simulation", n_sim=4000, rng_seed=0,
)
wtp_sim = m_sim.wtp("time_diff", "cost_diff")

def ci_str(res):
    lo, hi = (float(x) for x in res.conf_int())
    return f"[{lo:+.3f}, {hi:+.3f}]  width {hi - lo:.3f}"

print(f"delta      WTP/min = {float(wtp_minute.estimate):+.3f}  95% CI {ci_str(wtp_minute)}")
print(f"simulation WTP/min = {float(wtp_sim.estimate):+.3f}  95% CI {ci_str(wtp_sim)}")
print(f"\nκ on the ratio: {float(np.max(wtp_minute.kappa)):.3f}")

delta      WTP/min = -0.262  95% CI [-0.690, +0.166]  width 0.856
simulation WTP/min = -0.262  95% CI [-0.920, -0.008]  width 0.912

κ on the ratio: 0.277

The simulation interval is wider and skewed — the right behaviour for a ratio whose denominator is uncertain. When a WTP, elasticity, or any other ratio is the deliverable, report the simulation interval (or let the κ guard trip the fallback automatically); the delta interval is a linearization of something visibly curved.

Note

wtp() builds the ratio from two dydx calls and composes them. If subgroup κ values straddle the fallback threshold (so one slice falls back to simulation while another stays on the delta method), composition refuses to mix inference methods. Pin the method explicitly — method="simulation" on the session — whenever you compute WTP across subgroups.

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