Spatial interaction models from Irish commuting data: variations in trip length by occupation and gender

Abstract

Core and peripheral contrasts in journey-to-work trip length can be interpreted as imputing the relative value of origin and destination accessibility (yielding theoretical proxies for rent and wages). Because the main variables are shown to be critically dependent on spatial structure, they may be interpreted as showing the shadow prices due to comparative location. There is also a unifying connection between these results and the existing literature on many dimensions: rent gradients, accessibility, and emissivity. In an empirical example, the advantages of a panoramic view of national commuting statistics are shown, using an Irish data set. Variations in the rates of participation in trip making by location, occupation, and gender are examined. Places that emit more trips than would be expected from their relative location are identified. Further, examining ways in which such emissivity is sensitive to a change in trip length highlights the regions where trips could possibly be adjusted to produce a shorter average trip length or which might be especially sensitive to reduction in employment. A careful reinterpretation of one of the key outputs from a calibrated spatial interaction model is shown to be consistent with the declining rent gradient expected from Alonso’s theory of land use.

Appendix: Sensitivity analysis

Connecting the model notation:

$$ \frac{1}{{A_{ik} }} = O_{ik} \exp ( - \lambda_{ik} ) = \sum\limits_{j} {\exp (\mu_{jk} - \beta_{k} C_{ij} )} $$

Differentiating the second and third expressions with respect to β _k

$$ - O_{ik} \exp ( - \lambda_{ik} )\frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} = \sum\limits_{j} {\exp (\mu_{jk} - \beta_{k} C_{ij} )} \left( {\frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} - C_{ij} } \right) $$

Recognizing that $T_{ijk} = \exp (\lambda_{ik} + \mu_{jk} - \beta_{k} C_{ij} )$ and after rearranging:

$$ - O_{ik} \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} = \sum\limits_{j} {\exp (\lambda_{ik} + \mu_{jk} - \beta_{k} C_{ij} )} \left( {\frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} - C_{ij} } \right) $$

$$ \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} = \frac{1}{{O_{ik} }}\sum\limits_{j} {T_{ijk} } \left( {C_{ij} - \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }}} \right),\quad O_{ik} > 0 $$

Given that $O_{ik} = \sum\nolimits_{j} {T_{ijk} } ,$ we can restate as:

$$ \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} = \bar{C}_{ik} - \frac{{\sum\nolimits_{j} {T_{ijk} } \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }}}}{{\sum\nolimits_{j} {T_{ijk} } }} = \bar{C}_{ik} - \bar{P}_{ik} ,\quad {\text{where}}\quad \bar{P}_{ik} = \left( {\frac{{\overline{{{\text{d}}\mu_{jk} }} }}{{{\text{d}}\beta_{k} }}} \right) $$

which gives the key result for the interpretation of the sign of the origin effects shown in the maps:

$$ \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} > 0\quad {\text{if}}\quad \bar{C}_{ik} > \bar{P}_{ik} ,\quad {\text{and}}\quad \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} < 0\;{\text{otherwise}}. $$

Similarly, in the case of destinations, connecting the model notation:

$$ \frac{1}{{B_{jk} }} = D_{jk} \exp ( - \mu_{jk} ) = \sum\limits_{i} {\exp (\lambda_{ik} - \beta_{k} C_{ij} )} $$

Differentiating the second and third expressions with respect to β _k

$$ - D_{jk} \exp ( - \mu_{jk} )\frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} = \sum\limits_{i} {\exp (\lambda_{ik} - \beta_{k} C_{ij} )} \left( {\frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} - C_{ij} } \right) $$

Rearranging as before:

$$ - D_{jk} \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} = \sum\limits_{i} {\exp (\lambda_{ik} + \mu_{jk} - \beta_{k} C_{ij} )} \left( {\frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }} - C_{ij} } \right) $$

$$ \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} = \frac{1}{{D_{jk} }}\sum\limits_{i} {T_{ijk} } \left( {C_{ij} - \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }}} \right),\quad D_{jk} > 0 $$

Given that $D_{jk} = \sum\nolimits_{i} {T_{ijk} } ,$ we can restate as:

$$ \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} = \bar{C}_{jk} - \frac{{\sum\nolimits_{i} {T_{ijk} } \frac{{{\text{d}}\lambda_{ik} }}{{{\text{d}}\beta_{k} }}}}{{\sum\nolimits_{i} {T_{ijk} } }} = \bar{C}_{jk} - \bar{Q}_{jk} ,\quad {\text{where}}\quad {\bar{Q}}_{jk} = \left( {\frac{{\overline{{{\text{d}}\lambda_{ik} }} }}{{{\text{d}}\beta_{k} }}} \right) $$

which gives the key result for the interpretation of the sign of the destination effects:

$$ \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} > 0\quad {\text{if}}\quad \bar{C}_{jk} > \bar{Q}_{jk} ,\quad {\text{and}}\quad \frac{{{\text{d}}\mu_{jk} }}{{{\text{d}}\beta_{k} }} < 0\;{\text{otherwise}}. $$