For both passenger and freight transport we opt for an aggregate representation of behavior. Disaggregate choice behavior is clearly superior if a representative sample is available for simulation. Our experience is that this sample is often not available if one wants to go for a quick check of an investment project, so we rely on more aggregate data. We distinguish between passenger transport and freight transport. Passenger transport
Passenger transport is modeled in an aggregate way by making use of a nested CES (Constant Elasticity of Substitution) utility function. For every user class and every
4.2. MODEL 69
OD pair we calibrate a nested CES utility function with four levels as is illustrated in Figure 4.2.
Figure 4.2: Nested Utility tree for passenger transport
We allow consumers on each OD pair to choose the aggregate transport level, the period in which they travel and the path. The path can contain combinations of modes. This means we rule out substitution between destinations. The elasticity of substitution chosen for every branching of the tree will determine the ease of substitution and the cross price elasticity between di¤erent paths and, implicitly, also modes. For perfect substitution possibilities between paths we end up in the Wardrop equilibrium but substitution can also be imperfect. This means that, in general, we do not use the Wardrop principle for the choice of paths. Instead, we rely on the concept of Stochastic User Equilibrium, which is well known in transportation, when the user choice on a network is described by a discrete choice model (see She¢ [59]). Here we use the same concept, but the discrete choice approach is replaced by a CES model. The Logit and the CES model are both demand models for di¤erentiated goods . The Logit model is a disaggregate demand model, while the CES model is a representative consumer model. The Logit model has a representative consumer formulation, where the direct utility has an entropic form. The CES model can be derived as a two step disaggregate model, where the …rst stage is which good to buy, while the second stage is concerned with how much to buy. In this case, the indirect utility function is logarithmic in income, i.e. there are income e¤ects. The two models are related, and can be given a common denominator, either at the aggregate level or at the disaggregate level (see Anderson et al. [6]) . The main advantage of the CES is that it can be easily calibrated , while the Logit approach (the discrete choice approach), is more amenable to the individual data. As such data are harder to …nd for several transportation studies,
we have preferred to use the aggregate CES form. The calibration of a CES requires the elasticities of substitution at each branch plus the total quantities and prices at the lowest level of the utility tree. Finally note that the Logit has been criticized in the literature, since it requires that the demands satisfy the IIA (independence of irrelevant alternative) property.
We de…ne the nested utility function by specifying the di¤erent utility components. Following Keller ([42], equations (4), (6), (8) and (10)), we assume that all the utility components are linear homogeneous CES functions of the associated components at the next lower level3. Formally, at the nth level, the utility component q
n;i (8n > 1) is given by qn;i = " X j2i ( n 1;j)1 n;i(qn 1;j) n;i # 1 1 n;j ; n;i= n;i 1 n;i ; (4.3)
where n 1;j 0; 0 n;i 1 and
X
j2i
n 1;j = 1: Note that we suppress here the
superscripts d and k corresponding to the OD pair and user type in order to lighten the notation. The parameter n;i in eq(4.3) is the elasticity of substitution at level
n of the tree and the parameter n 1;i is a share parameter at the next lower level.
The notation “j 2 i ”indicates those j’s for which qn 1;j 2 qn;i.
For each utility component, an aggregate quantity index and corresponding aggre- gate price index can be computed. It is a property of CES functions that the utility component, qn;i, is itself a consistent quantity index and that the corresponding
price index, pn;i, takes similar functional forms i.e.:
pn;i = " X j2i n 1;j(pn 1;j) 0 n;i # 1 0 n;j ; 0n;i= 0 n;i 1 0 n;i ; 0n;i= 1 n;i ; 8n; i . (4.4) At each level the sum of the expenditures of the lower level equals the total income computed with the price and quantity indexes at that level,
yn;i =
X
j2i
yn 1;j = pn;iqn;i; 8n; i . (4.5)
It can be shown that the demand functions are
q0;i = y p3 3 Y n=1 n 1;i pn;i pn 1;i n;i ; 8i . (4.6)
The demand for transport services q0;i will correspond to the aggregate number of
trips on a path r during a period m for a category of users k on OD pair d; also
3The utility components at di¤erent levels are said to be associated if the higher-level component
is a function of the component at the lower level. If two utility components qn;i and qm;j ( with
m < n) are associated we write qm;j 2 qn;i. Notice that for utility components at level n 1 the
4.2. MODEL 71
denoted by Xdk
mr . The lowest price index p0;i corresponds with the generalized price
of making a trip using path r during period m for a category of users k; previously denoted by pk
mr given in eq(4.1).
The main advantage of the nested CES formulation is its ease of calibration. The drawbacks of a limited number of parameters are the implied restrictions. First there are unitary income elasticities for all transport goods - this can be mitigated by recalibrating the utility function when large income variations are foreseen in future years. Second the compensated cross price elasticities between goods of the same nest are only a function of the initial budget share and the elasticity of substitution for that nest.
Freight transport
For freight transport we use a similar approach. We assume that the production function of each …rm that needs transport services is a nested CES function of the di¤erent production inputs: labor, capital and di¤erent transport services. For each …rm, keeping production levels constant, the minimization of the …rm’s production cost, generates demand functions for inputs including the demand for di¤erent trans- port services. In practice we aggregate the non-transport goods into one nest and transport into another nest. The transport nest is then disaggregated into peak/o¤ peak and into paths. The structure is then similar to the one used for passenger transport (see Figure 4.3).
Figure 4.3: Nested Cost tree for freight transport..
The demand functions for inputs are conditional on the production level of the …rm and the prices of all the inputs, including the prices of non transport inputs. MOLINO-II is a partial equilibrium model that concentrates on the transport market
and takes the prices of all other inputs, as well as all product prices other than transport services, as given.
A major assumption is that we take production locations and production levels as given. This allows us to derive and calibrate demand functions easily. If we want to integrate endogenous location we need a spatial general equilibrium model. This is beyond the model scope we have in mind: a ‡exible model that can be calibrated to an existing study. This leaves us with the question: how important are the changes in production and location compared to changes in transport and distribution over paths and modes in response to changes in the prices generated by a particular infrastructure project? As transport costs are for most …rms maximum 5 to 10 % of total costs , a new project may strongly a¤ect the modal choice but in a new equilibrium total production costs may change by 1 to 2% . This is in most cases a second order e¤ect compared to the transport changes for given production levels