Commuting choice model

The terms highlighted in orange relate to functional assumptions needed to solve the model. For each mode \(m\), residential location \(x\), job center \(c\), income group \(i\), and worker \(j\), expected commuting cost is:

\[t_{mj}(x,c,w_{ic}) = \chi_i(\tau_m(x,c) + \delta_m(x,c)w_{ic}) + \textcolor{orange}{\epsilon_{mxcij}}\]

• \(w_{ic}\) is the (calibrated) wage earned.

• \(\chi_i\) is the employment rate of the household, composed of two agents (household_size parameter).

• \(\tau_m(x,c)\) is the monetary transport cost.

• \(\delta_m(x,c)\) is a time opportunity cost parameter (the fraction of working time spent commuting).

• \(\textcolor{orange}{\epsilon_{mxcij}}\) follows a Gumbel minimum distribution of mean \(0\) and (estimated) parameter \(\frac{1}{\lambda}\).

Then, commuters choose the mode that minimizes their transport cost (according to the properties of the Gumbel distribution):

\[min_m [t_{mj}(x,c,w_{ic})] = -\frac{1}{\lambda}log(\sum_{m=1}^{M}exp[-\lambda\chi_i(\tau_m(x,c) + \delta_m(x,c)w_{ic})]) + \textcolor{orange}{\eta_{xcij}}\]

• \(\textcolor{orange}{\eta_{xcij}}\) also follows a Gumbel minimum distribution of mean \(0\) and scale parameter \(\frac{1}{\lambda}\).

In the import_transport_data function (inputs.data module), for a given income group, this is imported with:

        (transportCostModes, transportCost, _, valueMax, minIncome
         ) = calcmp.compute_ODflows(
            householdSize, monetaryCost, costTime, incomeCentersGroup,
            whichCenters, param_lambda, options)

Looking into the body of the compute_ODflows function (calibration.sub.compute_income module), we see that the definitions of the transportCostModes and transportCost variables correspond to the above definitions of \(t_{mj}(x,c,w_{ic})\) and \(min_m [t_{mj}(x,c,w_{ic})]\)

    # We first compute expected (hourly) total commuting cost per mode
    # Note that incomeCentersFull is already defined at the household level,
    # whereas monetaryCost is defined at the individual level: hence, we need
    # to multiply monetaryCost by householdsSize to get the value of the
    # monetary costs for all members of the households
    transportCostModes = (
        householdSize * monetaryCost[whichCenters, :, :]
        + (costTime[whichCenters, :, :] * incomeCentersFull[:, None, None])
    )

    transportCost = (
        - 1 / param_lambda
        * np.log(
            np.nansum(np.exp(- param_lambda * transportCostModes), 2))
    )

Given their residential location \(x\), workers choose the workplace location \(c\) that maximizes their income net of commuting costs:

\[max_c [y_{ic} - min_m [t_{mj}(x,c,w_{ic})]]\]

• \(y_{ic} = \chi_i w_{ic}\) (income_centers_init parameter)

This yields the probability to choose to work in location \(c\) given residential location \(x\) (logit discrete choice model):

\[\pi_{c|ix} = \frac{exp[\lambda y_{ic} + log(\sum_{m=1}^{M}exp[-\lambda\chi_i(\tau_m(x,c) + \delta_m(x,c)w_{ic})])]}{\sum_{k=1}^{C} exp[\lambda y_{ik} + log(\sum_{m=1}^{M}exp[-\lambda\chi_i(\tau_m(x,k) + \delta_m(x,k)w_{ik})])]}\]

In the import_transport_data function, this corresponds to:

            ODflows[whichCenters, :, j] = (
                np.exp(
                    param_lambda
                    * (incomeCentersGroup[:, None] - transportCost))
                / np.nansum(
                    np.exp(param_lambda
                           * (incomeCentersGroup[:, None] - transportCost)),
                    0)[None, :]
                )

From there, we calculate the expected income net of commuting costs for residents of group \(i\) living in location \(x\):

\[\tilde{y}_i(x) = E[y_{ic} - min_m(t_m(x,c,w_{ic}))|x]\]

In the import_transport_data function, this corresponds to:

        incomeNetOfCommuting[j, :] = np.nansum(
            ODflows[whichCenters, :, j]
            * (incomeCentersGroup[:, None] - transportCost), 0)