Some Consequences of Ignoring Relocations in the Cost–Benefit Analysis of Transportation Infrastructure Investments

Abstract

Traditional cost–benefit models of investments in road infrastructure are often based on demand curves assuming a given spatial distribution of jobs and households. We first use numerical experiments based on a spatial general equilibrium model to illustrate how this potentially introduces a serious prediction bias in the willingness-to-pay for the investments. Our experiments illustrate that it is not in general possible to say whether ignoring relocation effects leads to over- or underprediction of commuting flows. We identify cases of both kinds, and also cases where substantial changes in the road transportation network affect total commuting flows only marginally.

Appendices

Appendix

A. A Technical Presentation of the Model

This appendix provides a technical presentation of the mechanisms represented in Fig. 1. The spatial distribution of basic sector firms is considered to be exogenously given, which means that aspects of local innovativity and competitiveness are not explicitly accounted for. In this version of the model, we further ignore the possibility that migration decisions are affected by job diversity and local amenities, while housing prices and wages are assumed to be exogenously given.

1.1 A.1 Basic and Local Sector Firms; the Economic Base Multiplier

Total employment in zone i $(E_i$) is defined to be the sum of basic sector employment ($E^b_i)$ and local sector employment ($E^l_i)$ in the zone:

$$\begin{aligned} E_i \equiv E^b_i + E^l_i \end{aligned}$$

(1)

Let $\varvec{L}$ be a vector representing a given residential location pattern of workers, while $T_{\textit{ij}}$ is the probability that a worker lives in zone i and works in zone j. Hence, $\varvec{T}=[T_{\textit{ij}}]$ represents the commuting matrix in the geography, and by definition:

$$\begin{aligned} \varvec{TE}=\varvec{L} \end{aligned}$$

(2)

The spatial distribution of local sector activities reflects both the spatial residential pattern and the spatial shopping behaviour. Assume that the number of workers living in a zone is proportional to the number of residents/consumers in the zone, and let $C_{\textit{ij}}$ be the number of local sector jobs in zone i which are supported by shopping from worker living in zone j. Hence, $\varvec{C}=[C_{\textit{ij}}]$ is a shopping matrix, and the spatial distribution of employment in local sector activities is given by:

$$\begin{aligned} \varvec{E^l} = \varvec{CL} \end{aligned}$$

(3)

Given that the inverse of the matrix $\varvec{(I-TC)}$ exists, it follows from Eqs. (1), (2) and (3) that:

$$\begin{aligned} \varvec{L}= & {} (\varvec{I}-\varvec{TC})^{-1} \varvec{T}\varvec{E^b} \end{aligned}$$

(4)

$$\begin{aligned} \varvec{E^l}= & {} \varvec{C(I-TC)}^{-1} \varvec{T}\varvec{E^b} \end{aligned}$$

(5)

These solutions capture the economic base multiplier process: people attract local sector activities, while local sector employment opportunities attract workers (see Sect. 2). As mentioned above, the spatial distribution of basic sector activities ($\varvec{E^b}$) will be considered exogenous in the model, while the other variables ($\varvec{C}, \varvec{E^l}, \varvec{T}, \varvec{L}$) are represented by a set of equations representing shopping, commuting, location and migration decisions of households and firms. In the next two subsections of this appendix, we consider how residential location and migration decisions reflect spatial disparities in the labour market situation as well as characteristics of the spatial structure and the road transportation network.

1.2 A.2 Interzonal Migration Flows and Spatial Equilibrium

In modelling migration probabilities, Nævdal et al. (1996) introduced a nice trick to facilitate construction of Markov chains. The construction uses a symmetric matrix $\varvec{Q}=\{ Q \}_{i,j=1}^{N}$, where all the elements (${Q}_{\textit{ij}} \ge 0, i,j=1,2,\ldots ,n$) are dependent on the characteristics of the geography. The transition matrix $\varvec{M}=\{ P_{\textit{ij}} \}_{i,j=1}^{N}$ is given by:

$$\begin{aligned} P_{\textit{ij}} = \frac{Q_{\textit{ij}}}{\sum _{k,k\ne j}{Q_{kj}}} \quad i,j=1,2,\ldots ,N \end{aligned}$$

(6)

Nævdal et al. (1996) showed that any assumption about the coefficients $Q_{\textit{ij}}$ can be interpreted as an assumption about migration flows in the equilibrium state. As a next step, Nævdal et al. (1996) introduced some network characteristics which are symmetric between zones and which are relevant in explaining the relevant kind of spatial interaction. For a connected network with fixed $Q_{\textit{ij}}$-s, the construction produces regular Markov chains. Nævdal et al. (1996) showed that the equilibrium condition is given by the eigenvector:

$$\begin{aligned} \varvec{L} = \left[ \begin{array}{c} \frac{\sum _{i,i \ne 1}{Q_{i1}}}{1-P_{11}} \\ . \\ . \\ . \\ \frac{\sum _{i,i \ne 1}{Q_{iN}}}{1-P_{NN}} \end{array} \right] \end{aligned}$$

(7)

Let $\alpha _i=1-P_{ii}\ne 0, i=1,2,\ldots ,N$ be the probability that a person will not stay in zone i within the given time frame. In this subsection, we focus on internal migration flows, and the diagonal elements $\alpha _i$ of the migration probability matrix are assumed to be given. Assume next a strategy where a migrant evaluates destinations successively outwards over the network and moves to the first zone where the conditions are ‘satisfactory’. In addition, we introduce a simplifying assumption of constant absorption, defined by the absorption parameter s:

$$\begin{aligned} s = \frac{\text {Probability of moving to (}m+1\text {)-th neighbours}}{\text {Probability of moving to } m\text {-th neighbours}}, \hbox {constant in } m \end{aligned}$$

As an operational assumption accounting for the impact of both distance and absorption the migration flows between zone i and its m-th neighbour will be proportional to $\frac{s^m}{d_{\textit{ij}}^{\beta }}$, where $\beta$ is a distance deterrence parameter. The step parameter n defines the maximum transition length, i.e. the transition probability is zero between neighbours of order greater than n. The symmetric ${\varvec{Q}}$-matrix derived from this procedure defines the transition matrix $\varvec{M}$ by (6). As an illustration, the transition matrix for a simple linear three-node system is given by:

$$ \left[ \begin{array}{ccc} 1-\alpha _1 &{} \frac{s\alpha _2}{d_{21}^{\beta }} \left( \frac{s}{d_{12}^{\beta }} + \frac{s}{d_{32}^{\beta }} \right) &{} \frac{s^2 \alpha _3}{(d_{12}+d_{23})^\beta } \left( \frac{s}{d_{23}^{\beta }} + \frac{s^2}{(d_{12}+d_{23})^\beta } \right) \\ \frac{s\alpha _1}{d_{21}^\beta } \left( \frac{s}{d_{21}^{\beta }} + \frac{s^2}{(d_{21}+d_{32})^\beta } \right) &{} 1-\alpha _2 &{} \frac{s\alpha _3}{d_{23}} \left( \frac{s}{d_{23}^{\beta }} + \frac{s^2}{(d_{12}+d_{23})^\beta } \right) \\ \frac{s^2\alpha _1}{(d_{21}+d_{32})^\beta } \left( \frac{s}{d_{21}^\beta } + \frac{s^2}{(d_{21}+d_{32})^\beta } \right) &{} \frac{s\alpha _2}{\left( \frac{s}{d_{21}^\beta } + \frac{s}{d_{32}^\beta } \right) d_{32}^\beta } &{} 1-\alpha _3 \end{array} \right] $$

In our chapter, the coefficients will be state-dependent, i.e. the $Q_{\textit{ij}}$-s are functions of $\varvec{E}$ and $\varvec{L}$. In that case, the equilibria are no longer unique, but the interpretation in terms of the strength of migration flows in the equilibrium state remains valid; see Nævdal et al. (1996). A standard application of Brouwer’s fixed-point theorem gives that equilibria always exist in the state-dependent case.

1.3 A.3 The Decision to Stay or Move from a Zone

It is a central hypothesis in the model that the decision to stay or move from a zone depends on the labour market accessibility of the zone. Labour market accessibility is introduced by a measure of generalised distance, rather than, for example, by a gravity-based Hansen accessibility measure (Hansen 1959). The generalised distance from zone i is given by:

$$\begin{aligned} d_i = \sum _{j \ne i} \frac{W_j}{\sum _{k \ne i} W_k}d_{\textit{ij}} \end{aligned}$$

(8)

Labour market accessibility is of course not just a matter of distances, the weight $W_i$ represents the size of alternative job destinations. The size, and thickness, of a potential destination is assumed to be represented by the number of jobs; $W_j = E_{j},j=1,2,\ldots ,N$, defining $d_{i}$ as the average Euclidean distance to potential employment opportunities in the geography. In a spatial labour market context, however, it can be argued that potential destinations within a reasonable commuting distance should be put more weight on than more distant destination alternatives. This is done through the introduction of a distance deterrence function $D(d_{\textit{ij}})$, that places a relatively high weight on destinations which lie within a short distance from the residential location:

$$\begin{aligned} W_j = E_j (1-D(d_{\textit{ij}})) \end{aligned}$$

(9)

The distance deterrence, and, hence, the weights are parameterised by $d_{\infty }$, $d_0$ and $\mu$ in the following logistic expression:

$$\begin{aligned} D(x) = \frac{1}{1+e^{-k(x-x_0)}}, \quad x_0 = \frac{1}{2}(d_0+d_\infty ), \quad k=\frac{2\log (\frac{1}{\mu }-1)}{d_\infty -d_0} \end{aligned}$$

(10)

$d_\infty$ is the upper limit for how far workers, as a rule, are willing to commute on a daily basis, $d_0$ is the lower limit (internal distance) where people are insensitive to further decreases in distance, while $\mu$ captures friction effects in the system. The values of $x_0$ and k are given to satisfy the conditions $D(d_0)=\mu$ and $(1-D(d_\infty ))=\mu$. If , e.g. $\mu =0.05$, this means that the function will fall to 5% of its value outside the range where $d_0 \le x \le d_\infty$. Glenn et al. (2004) gave a microeconomic and geometric justification for the use of such a function.

Finally, the definition of generalised distance also accounts for the competition for jobs at alternative locations (Liu and Zhu 2004; Shen 1998), represented in the model by the proportion of the total number of job seekers in each potential destination, $\frac{E_j}{L_j}$:

$$\begin{aligned} W_j = E_j (1-D(d_{\textit{ij}})) \frac{E_j}{L_j} \end{aligned}$$

(11)

The definition of generalised distance is included in the diagonal elements of the migration matrix, reflecting workers spatial interaction response to an unfortunate local labour market situation ($L_i>E_i$). A high value of $d_i$ [and $D(d_i)]$ means that the migration decisions are very sensitive to the local labour market situation. On the other hand, a high local unemployment does not in itself bring about a significant out-migration from zones in highly accessible labour market location (low $d_i$), with an excellent commuting potential. This is captured by the following specification of $\alpha _i$:

$$\begin{aligned} \alpha _i = \alpha _i(L_i) + D(d_i)\max \left\{ \rho \left( \frac{L_i - E_i}{L_i} \right) ,0 \right\} \end{aligned}$$

(12)

Here, the parameter $\rho$ reflects the speed of adjustment to an unfortunate labour market situation, towards a situation with a balance in the local labour market, $L_i=E_i$.

1.4 A.4 The Spatial Distribution of Local Sector Employment

It is reasonable to assume that local sector activities in a whole region ($E^l_r$) are proportional to population in the region ($L_r$):

$$\begin{aligned} E^l_r=\sum _{i}^{n}E^l_i=b\sum _{i}^{n}L_i=bL_r \qquad b>0 \end{aligned}$$

(13)

where b is the proportion parameter. Let the spatial distribution of local sector employment be represented by $\frac{E^l_i}{L_i}$, that is the number of shop-employees per resident at location i. Assume, as a simplification, a monocentric region, offering agglomeration benefits for local sector firms and price savings for households in shopping. Shopping decisions then results from a trade-off between price savings and transport costs.

Transportation costs provide an incentive for local sector firms to decentralize in order to cater for local demand. The trade-off between transport costs and potential price savings plays a central role in Gjestland et al. (2006), providing a theoretical base in favour of the hypothesis that the frequency of shopping locally is a smooth, concave, function of the Euclidean distance from the CBD. In our chapter, we assume that there is only one CBD and define the local sector density by:

$$\begin{aligned} \hbox {Local sector density}&=\frac{E^l}{L}(\hbox {distance to CBD})\end{aligned}$$

(14)

$$\begin{aligned}&=R_\infty (1-\exp [-\beta _\mathrm{CBD}\cdot \hbox {distance to CBD}])\end{aligned}$$

(15)

$$\begin{aligned}&+C\cdot \exp [-(\gamma \cdot \hbox {distance to CBD}/d_\mathrm{dispersion})^2] \end{aligned}$$

(16)

The only free parameter is $\beta _\mathrm{CBD}$ which controls the decay in the local sector density curve. The other parameters are defined as follows:

$$\begin{aligned} R_\infty =\frac{\sum _{i=1}^NE^l_i}{\sum _{i=1}^NL_i}\hbox { (average local sector density in the system as a whole)} \end{aligned}$$

(17)

$d_\mathrm{dispersion}$ is the spatial extension of the CBD, $\gamma =\sqrt{-\ln [\kappa ]}$ forces the effect of the second term (16) down to $\kappa \%$ of its peak value at the boundary of the CBD. Given values for $\beta _\mathrm{CBD},R_\infty$ and $d_\mathrm{dispersion}$, C are chosen such that the integral

$$\begin{aligned} \int \limits _0^{d_\mathrm{dispersion}} \hbox {Local sector density}(r)\cdot {{2r}\over {d_\mathrm{dispersion}^2}}\cdot L_\mathrm{CBD}dr=E^l_\mathrm{CBD} \end{aligned}$$

(18)

The spatial distribution of local sector activities reflects the net effect of the price savings resulting from agglomeration forces and the transport costs of shopping in the CBD rather than locally.

1.5 A.5 Commuting Flows

In the model to be used in this chapter, the location of workers and jobs is assumed to be fixed, calling for a doubly-constrained version of the gravity model. It is well known that a doubly-constrained gravity model is equivalent to the multinomial logit model; see Anas (1983) for details. This means that the model can be derived from random utility theory. The doubly-constrained gravity model incorporates set of balancing constraints, representing an assumption of a given spatial distribution of jobs and households. The following model specification ensures that the column sums of the predicted commuting flow matrix equal the total number of jobs at the corresponding destinations, and that each row sum equals the number of workers residing in the corresponding zone:

$$\begin{aligned} T_{\textit{ij}}=A_iO_iB_jD_je^{(-\beta _\mathrm{gravity} d_{\textit{ij}})} \end{aligned}$$

(19)

$$\begin{aligned} A_i=\left[ \sum _{j}B_jD_je^{(-\beta _\mathrm{gravity} d_{\textit{ij}})} \right] ^{-1} \end{aligned}$$

(20)

$$\begin{aligned} B_j=\left[ \sum _{i}A_iO_ie^{(-\beta _\mathrm{gravity} d_{\textit{ij}})} \right] ^{-1} \end{aligned}$$

(21)

where

$T_{\textit{ij}}$:

is the number of commuters from origin i to destination j

$O_i$:

is the observed number of commuting trips originating from zone i

$D_j$:

is the observed number of commuting trips terminating in zone j

$d_{\textit{ij}}$:

is the travel time from origin i to destination j.

$A_i$ and $B_j$ are the balancing factors which ensure the fulfilment of the marginal total constraints: $\sum _jT_{\textit{ij}}=O_i$ and $\sum _iT_{\textit{ij}}=D_j$.

1.6 A.6 An Iterative Process Towards Spatial Equilibrium

To initiate the iterative process, we begin with more or less random random initial values for employment and population ($\varvec{E}_0 =\varvec{E^l}_0+\varvec{E^b}_0$ and $\varvec{L}_0$). These values are fed into a state-dependent migration matrix $\varvec{M}$ and adjusted to fit a local sector density curve, which is then iterated until we find a fixed point $\varvec{L}$, which represents the equilibrium solution for population (workers), i.e. that $\varvec{M L} = \varvec{L}$, and equilibrium values fitting the local sector (jobs) $\varvec{E^l}$ to the local sector density curve.

1.7 A.7 Parameter Values Chosen for the Numerical Experiments

Absorption effects are ignored in the very simple transportation network (Fig. 3), $s=1$. The distance deterrence in internal migration flows is represented by an elasticity of $\beta =-1.0$, unless else is stated. The logistic distance deterrence function involved in determining the decisions to stay or move from a zone is specified by $d_{\infty }=80, d_0=5$, and $\mu =0.05$, while the speed of adjustment to an unfortunate labour market situation is given by $\rho =1$. The form of the local sector density function is given by $\kappa =0.05$ and a spatial extent of the CBD of $d_{\mathrm{dispersion}}=4$. Estimated commuting flows reflect a distance deterrence parameter of $\beta _{\mathrm{gravity}}=0.07$.