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:

$$t_{mj}(x,c,w_{ic})={\chi }_i({\tau }_m(x,c)+{\delta }_m(x,c)w_{ic})+{ϵ}_{mxcij}$$

• $w_{ic}$ 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).

• ${ϵ}_{mxcij}$ 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):

$$mi{n}_m[t_{mj}(x,c,w_{ic})]=-\frac{1}{\lambda }log(\sum _{m=1}^{M}exp[-\lambda {\chi }_i({\tau }_m(x,c)+{\delta }_m(x,c)w_{ic})])+{\eta }_{xcij}$$

• ${\eta }_{xcij}$ 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 $mi{n}_m[t_{mj}(x,c,w_{ic})]$

    # 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:

$$ma{x}_c[y_{ic}-mi{n}_m[t_{mj}(x,c,w_{ic})]]$$

• $y_{ic}={\chi }_i{w}_{ic}$ (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 $x$:

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

In the import_transport_data function, this corresponds to:

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