Optimal Commuting and Migration Decisions under Commuting Cost Uncertainty

Abstract

The aim of this paper is to assess the implications of economic uncertainty on individual mobility behaviour at the local and regional levels. Focusing especially on uncertainty with respect to the evolution of commuting costs, a model is developed that describes the interrelated decisions of residential choice and the appropriate type of mobility connecting workplace and residence. Results indicate that higher uncertainty about commuting costs increases the number of residentially immobile individuals in that the decision regarding a relocation of residence is accordingly deferred to a later point in time. This also means that individuals initially commuting between residence and workplace remain in this mobility mode despite an apparently favourable net present value of moving to the workplace region.

Keywords

commuting migration decision-making mobility costs uncertainty real options

1. Introduction

Mobility decisions of individuals and households in conjunction with their choice of workplace and residence location have been of growing interest among scholars in recent years. Migration and commuting as the two modes of mobility are the second major factor for changes in population size and composition, especially of the workforce, behind natural growth. Thus, they have an important effect on the demographics and the competitiveness of a region (Poot, 2008). Understanding the complex interactions between the labour market, the housing market and different forms of mobility is necessary for dealing with changing transport needs (for example, shorter travelling times for commuters, more efficient interaction of different modes of transport) and the evaluation of appropriate policy measures affecting these fields of action (for example, regional development, transport infrastructure, housing and commuting subsidies).

The consideration of waiting in individual or household mobility decision behaviour in the presence of uncertainty is important in that it more realistically reproduces the underlying circumstances leading to a decision. In this respect, work has been done primarily in the context of labour mobility at the national and international level. However, with the appearance of new settlement structures in more remote areas, it seems reasonable to extend the analysis to the regional levels. Uncertainty present in commuting and migration interrelations has received only limited attention in theoretical studies so far.

To this end, this paper introduces a theoretical model that shall shed some light on the influence of commuting cost uncertainty—in particular, on the optimal individual decision to locate one’s residence in close proximity to the workplace or in a more remote area.

In building the model, this paper goes beyond Burda’s (1995) idea of considering income uncertainty in the individual migration decision based on options valuation. One important distinction of this paper is the application of the options concept to migration and commuting decisions in a single model. Moreover, the focus is on reproducing the effect of cost uncertainty on mobility decisions in contrast to income variation typically analysed in the context of migration studies. The presented model, therefore, provides for a different and augmented representation of the optimal decision problem originally discussed by Burda.

The paper is structured as follows. The next section describes the background and motivation for the subsequent analysis. Section 3 introduces the theoretical model specifications describing an individual’s simultaneous decision problem for residence and workplace choice. It represents the basis on which closed-form solutions for an optimal commuting and migration decision are derived in section 4. For reasons of high analytical complexity, a numerical simulation further substantiates the obtained results. Section 5 concludes the paper.

2. Background

Starting with the monocentric city model developed by Alonso (1964), urban economics early distinguished between the city centre, also known as central business district (CBD), and the surrounding residential area in the study of urban spatial structure and the location of households and firms. Later, the appearance and consideration of new settlement types, in particular the so-called exurbia, led to an enhancement of the initial dichotomy of the model (for example, Boskoff, 1970; Nelson, 1992; Daniels, 1998). These remote communities are typically characterised by a high share of out-commuters connecting their residence with their workplace in the city centre.

What is common to these settlement types is the increasing willingness of residents to commute long distances, which places them in a position of growing dependence on individual motor car traffic and, thus, greater gasoline demand. However, gasoline prices are only one of many different types of user costs that the individual incurs for urban automobile transport. Kanemoto (2008) stresses the diverse cost structure of urban transport and distinguishes between four main types of user cost: user fees, such as tolls and fares; vehicle running costs, which for example include the mentioned costs of fuel, maintenance and the value of vehicle wear-and-tear; the value of travel time, which corresponds to the involved opportunity cost of travel time; and traffic accident costs. While costs of the first type usually do not vary significantly over time, vehicle running costs as well as the value of travel time can exhibit considerable fluctuations. Particularly, changes in gasoline prices and other monetary cost components exert influence on the budget and, hence, on the consumption possibilities of the individual. Besides these direct costs, opportunity costs arising from the time needed to travel the distance between residence and workplace represent another relevant share in the individual’s or household’s overall mobility costs. An important source of opportunity costs of travel time is traffic congestion, inherent to modern urban transport infrastructure. To this end, in his simulation of traffic congestion reduction policies, Noland (1997) finds that travel time variance is a driving force for the expected costs of commuting, allocating a higher weight to it than to just the travel time.¹ Against this background, the model introduced in section 3 of this paper also accounts for travel time costs in the aggregation of overall commuting costs.

The mobility of labour has been studied in a variety of ways. For example, neo-classical theory recognises the spatially equilibrating effects and forces of labour migration. In contrast, the micro-economic human capital theory advanced by Sjaastad (1962) views migration as an investment in an individual’s human capital in that it enables him to select the region that provides the highest present value of all future net benefits on his acquired skills (see also Simpson, 1980; Vickerman, 1984).

A significant limitation of these ‘pure’ migration studies lies in their non-observance of the implications of spatially separated workplace and residence locations. Instead, they implicitly assume a congruent move of residence with any labour (supply) mobility. However, due to improved infrastructure and transport facilities, workplace—hence labour supply—and residence often tend to diverge across regions. This raises the demand for a more thorough appraisal of the interrelated choice between workplace, residence, and the type of mobility connecting the two (for example, Evers and van der Veen, 1985; van Ommeren et al., 1999; Reitsma and Vergoossen, 1988; van der Veen and Evers, 1983; Zax, 1994).

According to search theory, the task in individual or household decision-making is to find the optimal time at which to stop searching and accept a given job offer (for example, David, 1974; Eliasson et al., 2003; van Ommeren, et al., 1997), thereby describing a step-wise (sequential) or simultaneous choice of workplace and residence.² That is, given uncertainty about potential job opportunities (i.e. imperfect information about a remote labour market), in a first step the individual changes only his workplace and may choose to relocate his residence in a second step later in the process. According to Vickerman (1984), another possibility for the individual is to engage in an anticipated or speculative move of residence in hope of a beneficial workplace change later on. If, however, in both examples labour market expectations are not fulfilled, the individual may decide not to move his workplace or residence, but rather stay with the more flexible commuting mode. This can result in an overall increase of commuting intensity and partly increase the effective rate of unemployment in that area. Regarding the decision structure, Evers and van der Veen (1985) empirically analysed the determinants and interdependencies between different types of labour mobility and workplace and residence choice. Their findings suggest that there is no sequentialness but rather simultaneity in the decision whether to supply one’s labour force outside the initial region, the choice to make a short or long distance move, and the choice to commute or migrate. This speaks in favour of a simultaneous treatment of these aspects in theoretical considerations.

Contrary to these search-theoretic approaches in the explanation of individual decision behaviour, a different strand of labour mobility literature tackles the presence of uncertainty within a framework closely related to investment decisions in the theory of the firm. The well-known real options theory from financial economics, hearkening back to work by Dixit (1989) and Dixit and Pindyck (1994), is based on optimal control theory and describes optimal decision behaviour in the presence of uncertainty. Generally, it analyses the decision of a risk-neutral individual between two possible states with the objective being to maximise his expected (discounted) future payoff from the particular strategy net of any costs. Switching between both states involves (partly) irreversible costs. Uncertainty in models based on this approach arises from some underlying price or cost dispersion peculiar to the respective state. Thus, the possibility to change the state induces an option whose value is determined by the value attached to waiting one further period in the current state. Accordingly, waiting is optimal as long as the corresponding option’s value exceeds zero. Higher uncertainty about the evolution of prices or costs increases the value of waiting and, therefore, leads to a postponement of the decision whether to switch states. Burda (1995) first applied this approach to individual migration decisions in an attempt to explain restrained migration from East to West Germany despite significant wage differentials between the two regions. A growing number of studies have advanced on this reasoning, for example explaining family risk diversification through transnational movement or the migration decision in a co-ordinated mass of individuals (for example, Anam et al., 2008; Khwaja, 2002; Moretto and Vergalli, 2008).

Little is known, however, about the effects of uncertainty on mobility decisions at the regional level. The model presented in this paper builds on the aforementioned real options approach to analyse optimal individual decision behaviour under cost uncertainty within a commuting–migration framework. In the subsequent analysis, I presume a given workplace located in the city centre and disregard job search and wage differential aspects in this study.³ This, in particular, allows one to focus on the minimisation of total costs as the sole objective and represents a theoretical differentiation from existing studies whose primary interest lies in the maximisation of utility or net benefits by the individual or household.

3. The Model

In order to study the effect of cost uncertainty on commuting and migration behaviour, I consider a risk-neutral individual who chooses simultaneously between his residence location and the mode of mobility—commuting and migration—that optimally connects his workplace and residence. Without loss of generality, the individual’s workplace is assumed fixed and located in region 1, denoted as the city or metropolitan centre. Hence, he can only change his place of residence.⁴ Region 2 comprises all other locations and is referred to as the suburban or exurban peripheral area, for which lower rental costs compared with the city centre are presumed. In the continuous time setting of the model, the mobility decision of the individual depends on his current mobility–location condition, for which four scenarios can be distinguished: the individual remains residing near his workplace in region 1; the individual becomes an in-commuter into region 1 as a consequence of out-migration to region 2, which requires the individual to engage in commuting but allows him to benefit from lower rental costs; the individual remains residing in region 2 and commutes to his workplace located in region 1; the individual migrates to region 1 in order to avoid commuting expenses for travelling between region 2 and 1.

Subsequently, based on this elaboration of available mobility choices, the model explicitly incorporates uncertainty about the evolution of commuting costs into the decision problem of the individual, who is assumed to be risk-neutral and infinitely lived. Both costs of housing and commuting costs introduced in the following represent instantaneous rates. The dispersion process of commuting costs, denoted $C$ , is captured through the application of the geometric Brownian motion (GBM)

\frac{d C}{C} = μ d t + σ d z

(1)

where, $μ$ is a drift parameter; $σ$ is the standard deviation of the GBM per unit time; and $z$ is the increment of the standard Wiener process $z (t) = ε (t) \sqrt{dt}$ , with the random parameter $ε (t) \to N (0, 1)$ . The instantaneous drift rate of this stochastic process $dC$ is $E [dC] = μ Cdt$ and its instantaneous variance rate is $V [C] = σ^{2} C^{2} t$ .⁵ The latter shows clearly that for this particular definition of uncertainty by means of a GBM the variance of the proportional change in $C$ increases linearly in the time-span t. In general, the proposed definition of commuting costs is to be understood as a general, all-encompassing approach, which comprises both monetary vehicle running costs (such as fuel) as well as non-monetary costs (such as opportunity costs of travel time) involved in the commuting activity of the individual, provided these costs share similar properties regarding the aforementioned instantaneous drift rate, $μ Cdt$ , and variance rate, $σ^{2} C^{2} dt$ . For instance, if opportunity costs of the time travelled depend on the individual’s personal wage and if one presumes that the variation in his wage is connected to comparable economic fluctuations that provide the basis for the development of gasoline prices, then both types of commuting costs could be combined to a general cost measure.

It is assumed that future costs are discounted at the positive rate $ρ > 0$ and living in the city centre incurs costs of housing $R_{1}$ and for living in the suburb $R_{2}$ . The difference between both, denoted by $R = R_{1} - R_{2}$ , represents the mark-up or premium of city centre housing costs over suburb housing costs and is assumed to be positive, so $R > 0$ . Under risk-neutrality, the aim of the individual is to minimise the expected present value of future costs peculiar to living in both locations. The present value of the costs of living in the city centre is simply $R_{1} / ρ$ , while the cost situation in the suburb is described by the sum of housing costs and commuting costs, hence its present value is $R_{2} / ρ + C / (ρ - μ)$ .⁶ According to Marshallian theory,⁷ the individual would migrate from one location to the other as soon as the expected cost savings from migration exceed the costs of moving to the new location. Moving from the city centre to the suburb involves one-time costs $ℓ$ and for migration from the suburb to the city centre $κ$ , which both represent sunk costs that are not recoverable should the individual revise his decision in the future. In case the individual lives initially in the city centre, this necessitates that $R_{1} / ρ > R_{2} / ρ + C / (ρ - μ) + ℓ$ and yields the Marshallian migration threshold with regard to commuting costs as

T_{L} = \frac{ρ - μ}{ρ} (R - ρ ℓ)

(2)

Similarly, the optimal condition for changing the individual’s residence from the suburban location to the city centre requires that $R_{2} / ρ + C / (ρ - μ) > R_{1} / ρ + κ$ . This translates into the commuting cost threshold

T_{H} = \frac{ρ - μ}{ρ} (R + ρ κ)

(3)

These Marshallian trigger values $T_{L}$ and $T_{H}$ , however, assume immediate migration even in the presence of uncertainty without considering a postponement of migration For this reason, they are inappropriate for a comprehensive analysis of optimal migration decisions according to the objective of this paper.

Real options theory, on the other hand, applied in this context helps to model the waiting option by utilising the underlying commuting cost uncertainty, which is included in the derived optimality conditions. Following Dixit (1989), a separate valuation is established for each locational situation. The total costs of living in the city centre (no commuting costs) and following optimal strategies in the future are denoted by $V_{1} (C)$ , while those of living at a suburban residence are denoted by $V_{2} (C)$ . As before, the risk-neutral individual minimises the expected total costs and chooses his residence and mode of mobility accordingly. The Bellman equations representing the cost-minimising problem can be written in the form

ρ V_{1} = R_{1} + \frac{1}{dt} E (d V_{1})

(4)

ρ V_{2} = R_{2} + C + \frac{1}{dt} E (d V_{2})

(5)

which, at the same time, provide the necessary no-arbitrage conditions. The obligation to pay periodically can be seen as a debenture bond, with $V_{i} (C) \equiv V (C)$ being its respective value (i = 1, 2). The left-hand side (LHS) in both expressions is the payment per unit time a holder of the ‘asset’ (debtor) would be willing to afford given the discount rate $ρ$ . On the right-hand side (RHS) the terms $R_{1}$ and $R_{2}$ again denote the location-specific rental costs and $C$ the required immediate payment of commuting costs arising from holding the particular asset. The last term in both equations is the expected instantaneous rate of change of the asset’s value. A solution to the problem at hand requires the knowledge of the yet unknown functional form of $V_{1}$ and $V_{2}$ . Fortunately, this can be approximated from changes in $V$ over small units of time. Since both value functions are dependent on commuting costs $C$ , the information contained in the stochastic process (1) can be utilised. Deriving the second-order Taylor expansion for $dV$ (using Ito’s Lemma), substituting $dC$ for (1), taking expectations, and dividing by $dt$ yields the second-order linear partial differential equations

\frac{1}{2} σ^{2} C^{2} V_{1} ″ (C) + μ C V_{1}^{'} (C) - ρ V_{1} + R_{1} = 0

(6)

\frac{1}{2} σ^{2} C^{2} V_{2} ″ (C) + μ C V_{2}^{'} (C) - ρ V_{2} + R_{2} + C = 0

(7)

The solution to (6) and (7) is given by the sum of the general solution for the homogeneous part and a particular solution for the respective non-homogeneous parts.⁸ Applying the expression $A C^{ξ}$ for $V (C)$ in (6) and (7) yields a fundamental quadratic equation with two roots, whose detailed description can be found in the Appendix. Based on its two solutions, the general solution for the homogeneous part of the partial differential equation can be derived as the following linear combination

V (C) = A C^{- α} + B C^{β}

(8)

where, $A$ and $B$ are arbitrary constants and $- α$ and $β$ are the two roots of the fundamental quadratic. Substituting this into (4) and (5) for $\frac{1}{dt} E (d V_{1})$ and $\frac{1}{dt} E (d V_{2})$ and including the particular solutions for the non-homogeneous parts yields the value functions

V_{1} (C) = A C^{- α} + \frac{R_{1}}{ρ}

(9)

V_{2} (C) = B C^{β} + \frac{R_{2}}{ρ} + \frac{C}{ρ - μ}

(10)

Expressions (9) and (10) give the valuation of optimal future strategies when starting in the central city or suburb respectively. In (9) the last term on the RHS represents the present value of all future rental costs when living in the city centre forever. Thus, the first term must be the value of the option to move to the suburb, i.e. out-migration. This cost value function only includes that term from the general solution (8) which involves the negative root, $- α$ . Constant $B$ is set to zero in this case since the term $B C^{β}$ would otherwise violate the assumption that the option to move to the suburb vanishes for high levels of commuting costs. Analogously, the last two terms on the RHS in (10) denote the present value of all future rental and commuting costs when living in the suburb forever. Hence, the first term must again be the value of the option to move to the city centre. As before, the property of the option to move to the city centre requires that constant $A = 0$ in this cost value function. The first term on the RHS of both equations can alternatively be interpreted as the potential cost savings arising from relocation.

The optimal commuting cost levels relevant for triggering the relocation of one’s residence to the metropolitan centre or the suburb are denoted by $C_{H}$ and $C_{L}$ respectively. At the optimal points, switching from one location to the other yields a payoff that must equal the incurred up-front fixed costs of moving there. These moving costs are differentiated between the costs for in-migration to the city centre (to the workplace), $κ$ , and the costs for out-migration to the remote residence, $ℓ$ . Both include, but are not limited to, direct costs of relocating as well as psychological costs from leaving familiar surroundings and forging new ties at the new location. In addition, costs for out-migration, $ℓ$ , can further include necessary costs for purchasing a car, while costs $κ$ capture opportunity costs of owning a car that is no longer used to the same extent as before. The so-called value-matching condition for the lower trigger value, $C_{L}$ , is given by

V_{1} (C_{L}) = V_{2} (C_{L}) + ℓ

(11)

and similarly for $C_{H}$

V_{2} (C_{H}) = V_{1} (C_{H}) + κ

(12)

Moreover, each value function satisfies the so-called smooth-pasting condition, which ensures that the two valuations that are replaced in the event of migration meet tangentially in the optimal points $C_{L}$ and $C_{H}$

V_{1}^{'} (C_{L}) = V_{2}^{'} (C_{L})

(13)

V_{2}^{'} (C_{H}) = V_{1}^{'} (C_{H})

(14)

An important aspect of both value functions can be shown with their explicit formulations (9) and (10): for an economically feasible specification, the signs of the constants $A$ and $B$ must in fact be negative. This conforms to the interpretation of the terms $A C^{- α}$ and $B C^{β}$ given earlier, in that both represent the potential cost savings from moving to the suburb and to the city centre respectively, in comparison with the current situation. The stated value-matching conditions permit the specification of the general optimality conditions for remaining in the waiting state. If the individual lives initially close to his place of work in the city centre, waiting is optimal for him as long as

\frac{R}{ρ} - \frac{C}{ρ - μ} - ℓ < A C^{- α} - B C^{β}

(15)

where, $R = R_{1} - R_{2} > 0$ represents the rental cost premium of the city centre over the suburb.

Evaluation of the RHS in the waiting region but close to $C_{L}$ yields a value of the first term that tends to be larger than the absolute value of the second term due to the impact of the particular exponents. Thus, the former is dominating the latter term. As a result, the individual remains residing in the city centre despite a positive net present value from out-migration on the LHS, for in sum $A C^{- α} - B C^{β}$ is positive. Commuting cost uncertainty, in this respect, acts as a deterrent to the relocation decision, in which the individual moves his residence as soon as the cost benefit from relocation exceeds the sum of the opportunity costs of migration. In comparison with conventional NPV analyses, the individual demands a cost benefit that is not only larger than zero but larger than some positive threshold value to be determined in the process. With respect to commuting costs, this means that the current level must exceed the Marshallian threshold value, $T_{L} < C_{L} < C$ , in order to trigger out-migration to the suburb. In the case without the possibility of future return migration back to the city centre, constant $B$ is zero and the condition for the waiting state reduces to the requirement that the sum of expected cost savings from migration is less than the opportunity costs of immediate migration (stemming from possible unfavourable future developments) given by the positive value $A C^{- α}$ . Furthermore, this examination reveals the consequence of simultaneously considering migration and return migration in the problem formulation. It becomes obvious that the flexibility of the future return migration option, captured by $B C^{β}$ , reduces the opportunity costs that the individual takes into account for his initial migration decision. In that event, the individual is inclined to move to the suburb already at a presumably higher commuting cost threshold as compared with the case without the return option. Likewise, it can be seen that, if the signs of both constants were not negative, the condition would allow for a negative net present value in the case of migration, which is in contradiction to the cost-minimisation objective of the individual.

Analogous, when living in the suburb, waiting is perceived optimal for

\frac{C}{ρ - μ} - \frac{R}{ρ} - κ < B C^{β} - A C^{- α}

(16)

Accordingly, cityward migration is deferred as long as the cost savings from migration, given by the difference between ceasing commuting costs and the sum of rental cost premia in the city plus moving costs, are less than the corresponding opportunity costs of relocation on present value terms. Evaluating the RHS at high levels of commuting costs but still below $C_{H}$ shows that the first term outweighs the second term. Hence, as before, the condition requires the cost benefit from moving to exceed not only zero but the positive value denoted by the sum $B C^{β} - A C^{- α}$ . If return migration were to be excluded from the consideration again, the requirement states that the cost savings from migration must exceed the positive term $B C^{β}$ . Thus, the additional return possibility in future periods reduces the opportunity costs to be overrated by the potential cost savings. More flexibility in the face of commuting cost uncertainty is, therefore, shown to encourage migration in either way.

Up to this point, the analysis has focused only on the particular solutions (9) and (10) to derive the general optimality conditions (15) and (16) describing both the waiting and the migration behaviour of the individual. Nonetheless, an explicit expression of both commuting cost thresholds, $C_{L}$ and $C_{H}$ , has not yet been derived. Substituting (9) and (10) into conditions (11)–(14) gives the necessary set of equations from which parameters $A$ , $B$ , $C_{L}$ and $C_{H}$ are to be determined simultaneously. A full analytical solution of this system of equations, however, yields complex expressions and is therefore omitted in this paper. To still allow for the discussion of the effect of commuting cost uncertainty on the residential mobility decision of individuals, the following section derives closed-form analytical solutions for $C_{L}$ and $C_{H}$ and delivers a first insight into the comparative static aspects of the involved parameters, which is further extended in section 5. These solutions provide in a next step the base for the comprehensive numerical analysis in section 4 regarding the properties of the thresholds $C_{L}$ and $C_{H}$ with respect to changes in the underlying parameter constellations.

4. Results

4.1 The Closed-form Solution

As previously mentioned, the full solution for $C_{L}$ and $C_{H}$ from the system of equations (11)–(14) involves highly non-linear expressions. For this reason and in order to obtain a practicable solution for analysis, some limiting conditions have to be applied, which allow the derivation of both trigger values in closed form. This can be achieved with a more limited solution in each case by neglecting return migration possibilities in the future. In effect, this narrows down the general solution and resembles a ‘once and for all’ migration decision problem. Despite this limitation, the underlying feature of the option value of waiting remains valid even in the absence of return migration. In fact, as was indicated in the previous section, allowing for return migration would further reduce the migration thresholds since it introduces additional flexibility into the decision problem.

In the case of the upper threshold, $C_{H}$ , consider a substantial increase in the costs for leaving the city centre, $ℓ$ , so that $ℓ > R / ρ$ . In this case, the individual would never choose to move to the suburb, since the excessively high costs of moving are not expected to amortize through the potential rental cost savings in the suburb. Recursively, this enters into the initial decision problem of the individual by cancelling out this option and leaving him with a one-way migration possibility in his decision. Formally, the intrinsic option to relocate one’s residence away from the workplace again in future periods becomes valueless, which implies that constant $A$ goes to zero in (11)–(14). Given these simplifying assumptions about $ℓ$ and $A$ , the application of the value-matching (12) and smooth-pasting (14) condition respectively, permits the derivation of a closed-form solution for the upper threshold of commuting costs, $C_{H}$ , as the optimality condition for relocating to the city centre/near the workplace

C_{H} = \frac{β}{β - 1} \cdot \frac{ρ - μ}{ρ} \cdot (R + ρ κ)

(17)

The property $β > 1$ for the positive root of the fundamental quadratic function ensures an economically feasible solution, for commuting costs are presumed to be strictly non-negative. Further, the convergence condition $ρ > μ$ is understood to hold. Since parameter $β$ depends positively on the underlying degree of commuting cost uncertainty, the first fraction on the RHS is increasing in $σ$ but with diminishing intensity. Hence, higher uncertainty amplifies the upper threshold, $C_{H}$ . Furthermore, a rise in the rental cost mark-up of the city centre over the suburb ( $R$ ) as well as higher costs for moving to the city centre ( $κ$ ) lead to a magnified threshold and, therefore, affect the migration decision negatively.

Along the same lines, a closed-form solution can be obtained for the lower threshold, $C_{L}$ . Similar to before, increasing the costs of (return) migration to the city centre, $κ$ , renders the option to do so worthless and constant $B$ goes to zero. In this case, the optimal trigger costs $C_{L}$ can be derived from the value-matching (11) and smooth-pasting (13) conditions stated before

C_{L} = \frac{α}{α + 1} \cdot \frac{ρ - μ}{ρ} \cdot (R - ρ ℓ)

(18)

Since $α$ has a positive sign and decreases with enlarged commuting cost uncertainty, the first fraction, per definition, is less than one and tends towards zero with increasing $σ$ . In comparison with (17), the lower threshold is further reduced for higher levels of uncertainty. This generates an extended range for commuting costs to fluctuate within before actual migration is triggered. The effect of the rental cost mark-up is identical for both thresholds and thus simply lifts the region of inertia without changing its size. Obversely, higher costs of relocating to the suburb ( $κ$ ) have a reducing impact on $C_{L}$ , requiring yet lower commuting costs for compensation.

The Marshallian migration triggers, $T_{L}$ and $T_{H}$ , introduced in section 3, can now provide the benchmark case with which the solutions obtained on the basis of real options theory are contrasted. Both approaches diverge in essence by the additional $α$ - and $β$ -fractions found in $C_{L}$ and $C_{H}$ , that in turn include the uncertainty parameter $σ$ as well as the drift rate $μ$ —both describing the stochastic process of $C$ . According to conditions (18) and (17), the threshold value $C_{L}$ is exactly $α / (α + 1)$ times the NPV trigger value $T_{L}$ , while $C_{H}$ is $β / (β - 1)$ times the upper trigger value $T_{H}$ . Consequently, it can easily be seen that the upper threshold $C_{H}$ , obtained from the real options approach, overrates the NPV trigger value, whereas the lower threshold $C_{L}$ underrates its NPV analogue. The reason is found in the underlying uncertainty about commuting costs $C$ . The higher the degree of uncertainty, the larger is the additional wedge that is driven between $T_{H}$ and $C_{H}$ and $T_{L}$ and $C_{L}$ respectively. Thereby, the overall distance between the thresholds $C_{H}$ and $C_{L}$ increases further. In the literature, this distance is usually referred to as the ‘region of inertia’ or ‘range of inaction’ and constitutes the co-domain of $C$ for which the status quo is maintained in both locations. Concerning the problem at hand, this means that the decision to migrate to the suburb when commuting costs are low is not undone if costs of commuting, up to the critical level $C_{H}$ , increase again. Likewise, the decision to move to the city centre because of excessive commuting costs is not reversed if costs fall back to a previously lower level. This relates directly to the concept of ‘hysteresis’ which describes the incidence of taking permanent decisions based on short-term fluctuations or ‘the failure of an effect to reverse itself as its underlying cause is reversed’ (Dixit, 1989, p. 622). This observable procrastination effect is more pronounced than the optimal migration triggers $T_{L}$ and $T_{H}$ would predict, for their distance is only determined by the (sunk) moving costs $κ$ and $ℓ$ as opposed to the additional terms $α$ and $β$ (with their arguments $σ$ and $μ$ ) that cdrive the solution for conditions (18) and (17).

4.2 Numerical Simulation

The analysis of the analytical solution provided thus far is limited, for it does not allow for derivations of the commuting cost thresholds with regard to changes in commuting cost uncertainty, moving costs and rental cost premium in a practicable form. This section, therefore, aims at contributing to a more complete characterisation of the defined lower and upper thresholds by means of numerical simulation. For this it is assumed that the discount rate is $ρ = 0.1$ and the rental cost premium in the city centre, $R$ , takes the normalised value of 1. Furthermore, moving costs $κ$ and $ℓ$ are set to zero in the base case, to focus on the resulting implications of changes in commuting cost uncertainty, $σ$ , and commuting cost growth, $μ$ , on the threshold levels $C_{L}$ and $C_{H}$ given by the closed-form solutions (17) and (18).⁹Figure 1 highlights the effect of various levels of commuting cost uncertainty, $σ$ , on both thresholds, with $μ$ assumed zero for simplicity.

Figure 1.

Numerical simulation of the influence of uncertainty, $σ$ , on the commuting cost thresholds $C_{L}$ and $C_{H}$ , with parameters $ρ = 0.1$ , $R = 1$ , and no growth rate ( $μ = 0$ ).

In the face of higher uncertainty with respect to the future evolution of commuting costs ( $σ$ ), the lower threshold, $C_{L}$ , that must be underrun by actual commuting costs for a positive decision to move to the suburb, is further reduced – tending towards its limit at zero. On the other hand, the upper threshold $C_{H}$ , which must be exceeded to trigger migration to the city centre, is increased substantially. Both these effects describe a widening of the region of inertia—the distance between $C_{L}$ and $C_{H}$ —in which individuals with similar cost structures postpone their decision to move from the city centre to the suburb and vice versa. Overall, permanent mobility in the region is therefore expected to be restrained.¹⁰ Starting with no uncertainty ( $σ = 0$ ) and costless relocation between both residence choices ( $κ, ℓ = 0$ ), the upper and lower threshold meet in their common limit at $R = 1$ , where the region of inertia eventually vanishes. Allowing for positive moving costs alters the lower limit of both thresholds and results in greater distance, hence a widening of the region of inertia. In the numerical simulation, $κ$ and $ℓ$ are arbitrarily set to a level of two for comparison. This is likely to resemble a rather low approximation of the real cost relation between rental cost premia and moving costs in the suburb–central city context, since it relates to a meagre five-to-one ratio on present value terms. Yet, the illustrative purpose is in the focus of this exercise. The moving costs augment $C_{H}$ by the size of $ρ κ$ , such that its limit at $σ = 0$ becomes $R + ρ κ = 1.2$ for the given parameter settings. Likewise, $C_{L}$ is scaled down by an amount equal to $ρ ℓ$ and is now given by $R - ρ ℓ = 0.8$ . Accordingly, relocation costs of one-fifth the rental cost premium correspond to 20 per cent higher and lower, respectively, Marshallian migration triggers in this example. This symmetry, however, disappears in the thresholds $C_{L}$ and $C_{H}$ as commuting cost uncertainty is controlled for. Moving costs lower the threshold $C_{L}$ increasingly less with higher uncertainty, $σ$ . In contrast, the influence of moving costs on the upper threshold $C_{H}$ , relevant for a decision to move from the suburb to the city centre, further accentuates at higher levels of uncertainty. This clearly demonstrates the strong effect of commuting cost uncertainty on the evaluation of the involved costs $κ$ , $ℓ$ and $R$ in the commuting–migration problem at hand.

The reason for this asymmetrical effect of commuting cost uncertainty on the influence of moving costs in determining both thresholds is found in its alternative interpretation as opportunity costs from immediate relocation. The individual’s calculus demands that the cost savings realised from a change in residence (due to reduced rental costs or ceasing commuting costs) need to balance the peculiar costs of residential relocation. Since moving costs are one-time costs accrued in the instance of migration, the individual expects that the capitalised potential future periodical cost savings outweigh these costs. When taking the decision the individual considers not only direct costs involved in the decision but also possible opportunity costs arising from unfavourable developments in future periods after the decision has been made. Uncertainty about the outcome of these future cost savings that are linked to the evolution of commuting costs, therefore, affects expected opportunity costs from the immediate decision. The higher the uncertainty about future commuting cost levels the stronger is the resulting influence on the calculated opportunity costs.

In the case of the upper threshold, $C_{H}$ , the relevant term causing the exaggerating effect of moving costs ( $κ$ ) with increasing uncertainty is $β / (β - 1)$ from condition (17). This term is increasing in the uncertainty measure $σ$ $(\frac{\partial β}{\partial σ} > 0)$ and captures the opportunity costs inherent in the immediate decision to move to the city centre without waiting. Higher uncertainty—i.e. variability—not only pays regard to the upside potential in $C$ but also means a higher possibility of seeing lower commuting cost levels in the future which would then indicate that the taken decision of cityward migration is effectively incorrect. To avoid this regret and to insure against bad outcomes, the individual only moves if commuting costs are sufficiently high.¹¹

Conversely, it is true that the lower threshold ( $C_{L}$ ) is reduced with increasing commuting cost uncertainty. The responsible term is $α / (α + 1)$ in (18) which is decreasing in $σ$ , tending towards its limit at zero. Since it is linked multiplicatively to the moving costs for outward migration to the suburban residence, the influence of these costs increasingly diminishes too with higher commuting cost uncertainty. As a consequence, the consideration of the trade-off between moving costs on the one side and the sum of rental cost reduction plus additional commuting costs in the case of migration on the other side fades into the background in the personal calculation of the individual. In fact, the opportunity costs of immediate relocation to the suburb due to the existing commuting cost uncertainty dominates the further decrease of the required commuting cost threshold $C_{L}$ . For low levels of uncertainty, therefore, additional moving costs lead to a stronger reduction in $C_{L}$ than the same moving costs do at higher levels of uncertainty, where opportunity costs are increasingly important.

Turning to the implication of expected commuting cost growth ( $μ$ ) on the respective thresholds $C_{L}$ and $C_{H}$ , Figure 2 draws a clear negative relationship between a higher trend rate and both threshold levels. However, the strength of this effect is different for $C_{L}$ and $C_{H}$ . Threshold $C_{L}$ is increasingly lowered as the growth rate, $μ$ , approaches its economically feasible level defined by $ρ$ , while to the other end with a negative drift $C_{L}$ tends towards $R = 1$ . Additional migration costs, $ℓ$ , reduce the lower threshold further, although not below zero, since commuting costs can only take positive values. Regarding the upper threshold, $C_{H}$ , the graph shows that for positive values $μ \leq ρ$ the threshold tends towards $R$ as well, now as the lower limit. Smaller or negative growth rates lead to a significant increase in the threshold $C_{H}$ that has to be underrun by actual commuting costs to trigger migration from the suburb to the city centre. Consequently, the expectation of strongly declining commuting costs for living at the suburban residence alleviates the likeliness of relocation considerably. The individual is better off by staying and waiting for lower future commuting costs, which make the suburban location more favourable than the city centre location. In summary, faster expected growth of commuting costs—i.e. a reduction of both thresholds—represents a preference for the city centre: in-migration is carried out sooner and out-migration is deferred.

Figure 2.

Influence of commuting cost growth rate, $μ$ , on thresholds $C_{L}$ and $C_{H}$ , with parameters $ρ = 0.1$ , $R = 1$ and $σ = 0.1$ .

5. Conclusion

Results of the theoretical evaluation of commuting cost uncertainty on the optimal decision of residence and mobility mode choice highlight the importance of considering uncertainty in the mobility decision process. It was shown that higher commuting cost uncertainty increases the number of residentially immobile individuals, both in the city centre and the surrounding area. The decision about the relocation of one’s residence is deferred to a later point in time due to significant opportunity costs of migration. This means that individuals initially living close to their workplaces in the city centre are inclined to stay there, while those initially residing in a remote community in the suburban or exurban area continue to bear the necessary commuting costs. Both are possible even under positive net benefits resulting from a potential relocation to the alternative region. Although rather specific in its analysis and therefore limited in scope, the model explains some important interrelations of residence and mobility mode choice in the presence of uncertainty.

There do exist many more aspects that could be included in the presented analysis. For example, costs for commuting between residence and workplace are the main determinants of the relocation decisions of the individual. In this regard, commuting cost subsidies in the form of tax relief could serve as an insurance against cost uncertainty and, therefore, be taken into account in this examination. Particularly due to the progressive income taxation in most countries, high-income earners would benefit relatively more from this ‘insurance’ and be inclined to move to the suburban area, thereby potentially supporting gentrification processes and influencing the social structure in a region.

Another point of concern could be the incidence of uncertainty in the housing market that would account for costs of housing in the individual’s migration decision. For instance, strong declines in the price of houses could restrain people’s residential mobility by increasing the costs of relocation. Uncertainty about the further evolution of house prices could additionally delay individual or household migration.

The analysis presented here was conducted under the assumption of risk neutrality—i.e. in the absence of peculiar risk loving or avoiding attitudes—in order to stress the role of uncertainty in mobility decision-making. As an example, risk-averse behaviour incorporated through a utility maximisation framework would be expected to confirm and reinforce the derived findings in this analysis.

Footnotes

Appendix. A Fundamental Quadratic

Using the general form $C^{ξ}$ to replace $V_{i}$ in the homogeneous part of (6) and (7) yields the following quadratic equation

(7.1)

Q \equiv \frac{1}{2} σ^{2} ξ (ξ - 1) + μ ξ - ρ = 0

with the roots $β$ and $- α$

(7.2)

β = \frac{σ^{2} - 2 μ + \sqrt{{(σ^{2} - 2 μ)}^{2} + 8 σ^{2} ρ}}{2 σ^{2}} > 1

(7.3)

- α = \frac{σ^{2} - 2 μ - \sqrt{{(σ^{2} - 2 μ)}^{2} + 8 σ^{2} ρ}}{2 σ^{2}} < 0

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Notes

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