A General Equilibrium Framework for Integrated Assessment of Transport and Economic Impacts

Abstract

The use of separate transport and economic models in urban planning provides a limited view of economic impacts, restricts the testing of network design options and lengthens the planning process. Furthermore, the standard methodology for economic appraisal assumes partial economic equilibrium and cannot determine the distribution of impacts from the transport sector to households. Computable general equilibrium (CGE) models can capture general equilibrium effects and measure welfare at the household level, but mostly lack integration with transport models and do not represent all trip generators. This paper develops an integrated traffic assignment and spatial CGE model in nonlinear complementarity form, casted as a framework for economic appraisal of urban transport projects. The CGE submodel generates commuting, shopping and leisure trips as inputs into the transport submodel, which then assigns trips to the network according to user equilibrium. The resulting travel times then feed back into household prices and freight margins. Households and firms fully account for travel times in decisions on where to shop, how much labour to supply and where to source production inputs. Calibration and applications of the model are demonstrated for 14 regions and 2 industries across Sydney using GAMS/PATH on the NEOS server. The welfare of various network improvements is measured using equivalent variations. The model can be calibrated to external strategic transport models, and be extended to simulate additional trip generators and land-use.

Appendices

Appendix A

1.1 Household Equations

Households are assumed to spend a constant share of their income on each type of good. Composite consumption c _h is then a Cobb–Douglas combination of consumption from each industry c _h , i :

$$ {c}_h=\prod_{i\in I}{c_{h, i}}^{\beta_{h, i}} $$

(27a)

$$ \sum_{i\in I}{\beta}_{h, i}=1 $$

(27b)

where β _h , i is the Cobb–Douglas share parameter for c _h , i . Total expenditure to obtain c _h in terms of prices and travel time is:

$$ \sum_{i\in I}\left({p}_{c, h, i}+{p}_{t, h}\cdot {v}_{c, h, i}\right)\cdot {c}_{h, i} $$

(28)

where p _{c , h , i} is the price index and v _{c , h , i} is the travel time index to obtain c _h , i . Since c _h is already specified from the upper nest, minimising expenditure in equation (28) subject to the composite consumption function in equation (27a) yields:

$$ {c}_{h, i}={B}_{h, i}\cdot {c}_h $$

(29a)

$$ {B}_{h, i}={\prod_{i^{\prime}\in I}\left(\frac{\beta_{h, i}\cdot \left({p}_{c, h, i}+{p}_{t, h}\cdot {v}_{c, h,{i}^{\prime }}\right)}{\beta_{h,{i}^{\prime }}\cdot \left({p}_{c, h, i}+{p}_{t, h}\cdot {v}_{c, h, i}\right)}\right)}^{\beta_{h,{i}^{\prime }}} $$

(29b)

where Β _h , i is the input–output coefficient for c _h , i . Due to the CRS specification, upper nest prices p _c , h and travel times v _c , h are:

$$ {p}_{c, h}=\sum_{i\in I}{B}_{h, i}\cdot {p}_{c, h, i} $$

(30a)

$$ {v}_{c, h}=\sum_{i\in I}{B}_{h, i}\cdot {v}_{c, h, i} $$

(30b)

Regional sources of goods from each industry are assumed to be imperfect substitutes. Composite consumption from each industry c _h , i is then a CES combination of consumption from each region c _{h , (i, r)}:

$$ {c}_{h, i}={\left(\sum_{r\in R}{\gamma}_{h,\left( i, r\right)}\cdot {c}_{h,\left( i, r\right)}\right)}^{\frac{\sigma_{h, i}}{\sigma_{h, i}-1}} $$

(31a)

$$ \sum_{r\in R}{\gamma}_{h,\left( i, r\right)}=1 $$

(31b)

where γ _{h , (i, r)} is the CES share parameter for c _{h , (i, r)} and σ _h , i is the elasticity of substitution for regional inputs of c _h , i . Total expenditure to obtain c _h , i is:

$$ \sum_{r\in R}\left({p}_{g,\left( i, r\right)}+{p}_{t, h}\cdot {v}_{c, h,\left( i, r\right)}\right)\cdot {c}_{h,\left( i, r\right)} $$

(32)

where p _{g , (i, r)} is the price of good (i, r) and v _{c , h , (i, r)} is the travel time required to obtain a unit of good (i, r). Minimising expenditure in equation (32) subject to the composite consumption function in equation (31a) yields:

$$ {c}_{h,\left( i, r\right)}={\Gamma}_{h,\left( i, r\right)}\cdot {c}_{h, i} $$

(33a)

$$ {\Gamma}_{h,\left( i, r\right)}={\left(\frac{\gamma_{h,\left( i, r\right)}}{p_{g,\left( i, r\right)}+{p}_{t, h}\cdot {v}_{c, h,\left( i, r\right)}}\right)}^{\sigma_{h, i}}\cdot {\left(\sum_{r\in R}{\gamma}_{h,{\left( i, r\right)}^{\sigma_{h, i}}}\cdot {\left({p}_{g\left( i, r\right)}+{p}_{t, h}\cdot {v}_{c, h,\left( i, r\right)}\right)}^{1-{\sigma}_{h, i}}\right)}^{\frac{\sigma_{h, i}}{\sigma_{h, i}-1}} $$

(33b)

where Γ _{h , (i, r)} is the input–output coefficient for c _{h , (i, r)}. Similar to earlier, prices p _{c , h , i} and travel times v _{c , h , i} are:

$$ {p}_{c, h, i}=\sum_{r\in R}{\Gamma}_{h,\left( i, r\right)}\cdot {p}_{g,\left( i, r\right)} $$

(34a)

$$ {v}_{c, h, i}=\sum_{r\in R}{\Gamma}_{h,\left( i, r\right)}\cdot {v}_{c, h,\left( i, r\right)} $$

(34b)

Appendix B

1.1 Firm Equations

In accordance with the Armington assumption, composite intermediate input x _{g , f , j} is a CES combination of sources from each region after transportation x _{g , f , (j, r)}:

$$ {x}_{g, f, j}={\left(\sum_{r\in R}{\theta}_{f,\left( j, r\right)}\cdot {x_{g, f,\left( j, r\right)}}^{\frac{\tau_{f, j}-1}{\tau_{f, j}}}\right)}^{\frac{\tau_{f, j}}{\tau_{f, j}-1}} $$

(35a)

$$ \sum_{r\in R}{\theta}_{f,\left( j, r\right)}=1 $$

(35b)

where θ _{f , (j, r)} is the CES share parameter for x _{g , f , (j, r)} and τ _f , j is the elasticity of substitution for regional inputs of x _{g , f , j} . Minimising expenditure to obtain x _{g , f , j} yields:

$$ {x}_{g, f, j\left( j, r\right)}={\Theta}_{f,\left( j, r\right)}\cdot {x}_{g, f, j} $$

(36a)

$$ {\Theta}_{f,\left( j, r\right)}={\left(\frac{\theta_{f,\left( j, r\right)}}{p_{g, f,\left( j, r\right)}}\right)}^{\tau_{f, j}}\cdot {\left(\sum_{r\in R}{\theta_{f,\left( j, r\right)}}^{\tau_{f, j}}\cdot {p}_{g, f,\left( j, r\right)}\right)}^{\frac{\tau_{f, j}}{\tau_{f, j}-1}} $$

(36b)

where Θ _{f , (j, r)} is the input–output coefficient and p _{g , f , (j, r)} is the price index for x _{g , f , (j, r)}. Price index p _{g , f , j} represents the cost of a unit of x _{g , f , j} :

$$ {p}_{g, h, i}=\sum_{r\in R}{\theta}_{f,\left( j, r\right)}\cdot {p}_{g, f,\left( j, r\right)} $$

(37)

Quantity x _{g , f , (j, r)} represents the input of good (j, r) after transportation to firm f’s region. Transportation from the good’s source to firm f’s region is accomplished by a Leontief combination of the good at its source x _{s , f , (j, r)} and a freight margin x _{m , f , (j, r)}:

$$ {x}_{g, f,\left( j, r\right)}= \min \left({x}_{s, f,\left( j, r\right)},\frac{x_{m, f,\left( j, r\right)}}{\rho_{f,\left( j, r\right)}}\right) $$

(38)

where ρ _{f , (j, r)} represents the units of transport required to relocate a unit of good (j, r) from its source region to firm f’s region. Minimising expenditure to obtain x _{g , f , (j, r)} yields:

$$ {x}_{s, f,\left( j, r\right)}={x}_{g, f,\left( j, r\right)} $$

(39a)

$$ {x}_{m, f,\left( j, r\right)}={\rho}_{f,\left( j, r\right)}\cdot {x}_{g, f,\left( j, r\right)} $$

(39b)

The freight margin x _{m , f , (j, r)} is supplied by the transport industry in the good’s source region x _{s , f , (1, r)}, such that:

$$ {x}_{s, f,\left(1, r\right)}=\sum_{j\in J}{x}_{m, f,\left( j, r\right)} $$

(40)

Price index p _{g , f , (j, r)} is then:

$$ {p}_{g, f,\left( j, r\right)}={\rho}_{g,\left( j, r\right)}+{\rho}_{f,\left( j, r\right)}\cdot {p}_{g,\left(1, r\right)} $$

(41)

In the primary factor composite nest, x _p , f is a CES combination of labour x _l , f and capital x _k , f :

$$ {x}_{p, f}={\left({\zeta}_{l, f}\cdot {x}_{l, f}\frac{\varphi_f-1}{\varphi_f}+{\zeta}_{k, f}\cdot {x_{k, f}}^{\frac{\varphi_f-1}{\varphi_f}}\right)}^{\frac{\varphi_f}{\varphi_f-1}} $$

(42a)

$$ {\zeta}_{l, f}+{\zeta}_{k, f}=1 $$

(42b)

where ζ _l , f and ζ _k , f are the CES share parameters for x _l , f and x _k , f respectively, and φ _f is the elasticity of substitution between labour and capital. Minimising expenditure to obtain x _p , f yields:

$$ {x}_{l, f}={\mathrm{Z}}_{l, f}\cdot {x}_{p, f} $$

(43a)

$$ {x}_{k, f}={\mathrm{Z}}_{k, f}\cdot {x}_{p, f} $$

(43b)

$$ {Z}_{l, f}={\left(\frac{\zeta_{l, f}}{p_{l, f}}\right)}^{\varphi_f}\cdot {\left({\zeta_{l, f}}^{\varphi_f}\cdot {p_{l, f}}^{1-{\varphi}_f}+{\zeta_{k, f}}^{\varphi_f}\cdot {p_k}^{1-{\varphi}_f}\right)}^{\frac{\varphi_f}{1-{\varphi}_f}} $$

(43c)

$$ {Z}_{k, f}={\left(\frac{\zeta_{k, f}}{p_{l, f}}\right)}^{\varphi_f}\cdot {\left({\zeta_{l, f}}^{\varphi_f}\cdot {p_{l, f}}^{1-{\varphi}_f}+{\zeta_{k, f}}^{\varphi_f}\cdot {p_k}^{1-{\varphi}_f}\right)}^{\frac{\varphi_f}{1-{\varphi}_f}} $$

(43d)

where Ζ _l , f and Ζ _k , f are the input–output coefficients for x _l , f and x _k , f respectively, p _l , f is the labour wage and p _k is the rental rate of capital. Price index p _p , f is then:

$$ {p}_{p, f}={Z}_{l, f}\cdot {p}_{l, f}+{Z}_{k, f}\cdot {p}_k $$

(44)