We derive matrices of price elasticities progressively for each branching level. At each level the equation combines a price elasticity derived for the level below with elasticities taken from the estimated coefficients of the AIDS share equations at the branching level to obtain an element of the matrix of price elasticities.
The progression begins with the own-price elasticity,
ε, for all domestic mail with respect to its average revenue per piece,
P. This elasticity is combined with three demand elasticities drawn from the estimates of the coefficients of the AIDS share equations for the major branches that divide the mail by classes. These are:
-
The elasticity of demand for product i with respect to total postal expenditures, Y: ε i Y = (1 + β i /s i ),
-
The elasticity of the AIDS price index, P, with respect to the price, p j of product j: ε P j = α j + ∑ i = 1 N γ ji ln(p i ), 12 and
-
The elasticity of demand for product i with respect to the price, p j of product j: ε ij M = − 1(i = j) + (γ ij − β i ε P j )/s i .
The latter elasticity, ε ij M , is a Marshallian elasticity because it is derived under the assumption that the expenditure, Y, is fixed.
The Marshallian elasticity does not capture the entire effect of a change in the price of product
j on demand for product
i. The complete elasticity of demand for product
i with respect to the price of product
j may be derived under the assumption that revenue per piece in the trunk equation and the AIDS price index that emerges from the estimates of the share equations are the same.
13 Under this assumption we obtain the complete elasticity by adding to the Marshallian elasticity a term to capture the effect on demand for product
i,
Q i , of changes in the price
p j transmitted first to the price index,
P, and then on through the effect of
P on total postal expenditures,
Y :
$$ {\varepsilon}_{ij}=-1\left(i=j\right)+\left({\gamma}_{ij}-{\beta}_i{\varepsilon}_j^P\right)/{s}_i+\frac{\partial \ln \left({Q}_i\right)}{\partial \ln (Y)}\frac{\partial \ln (Y)}{\partial \ln (P)}\frac{\partial \ln (P)}{\partial \ln \left({p}_j\right)}, $$
where
\( \frac{\partial \ln \left({Q}_i\right)}{\partial \ln (Y)}={\varepsilon}_i^Y, \) \( \frac{\partial \ln (Y)}{\partial \ln (P)}=1+\varepsilon, \) and
\( \frac{\partial \ln (P)}{\partial \ln \left({p}_j\right)}={\varepsilon}_P^j. \) We calculate the elements of the matrix of price elasticities for postal volumes disaggregated to the class level by evaluating this formula using the elasticities obtained from fitting the trunk equation and the AIDS share equations for the classes. Substituting elasticities in the formula above we have:
$$ {\varepsilon}_{ij}={\varepsilon}_{ij}^M+{\varepsilon}_i^Y\left(1+\varepsilon \right){\varepsilon}_P^j. $$
This is the general version of an equation found in Hausman and Leonard (
2005). Technically, it is the formula for a Marshallian demand elasticity because long-term real GDP per household is fixed when we calculate
ϵ. However, postal expenditures are a very small part of an average household’s total income. So,
ε ij , differs little from the Hicksian (compensated) elasticity. Consequently, we can inspect the estimated elasticities, as we normally would, for compatibility with neoclassical demand theory (
ε ii ≤ 0), and to identify substitutes (
ε ij > 0) and complements (
ε ij < 0) when
i ≠
j.
The formula above is sufficient to compute all of the elements of the matrix of price elasticities by major class, i.e. for the major branches of the tree in Fig. 1. The information needed to apply the formula is of the same form and origin for every element of the matrix. Specifically, the trunk equation elasticity, ϵ, is the same for every element. Other elasticities in the right-hand side of the formula are all derived from the fit of a single combined share equation.
However, the structure of the matrix of price elasticities becomes more complex as we move up the tree. At the next branch level domestic mail for the major classes is subdivided among work-sharing categories or among customer categories. Each class has its own set of share equations combined and fit to postal data disaggregated into work-sharing and customer categories. The matrix of price elasticities by these categories has rectangular blocks corresponding to the elements of the matrix of elasticities by major class:
$$ \left[\begin{array}{ccc}\left[{\varepsilon}_{11}\right]& \cdots & \left[{\varepsilon}_{1N}\right]\\ {}\vdots & \ddots & \vdots \\ {}\left[{\varepsilon}_{N1}\right]& \cdots & \left[{\varepsilon}_{NN}\right]\end{array}\right] $$
The diagonal blocks are square matrices with elements that apply to mail categories within the same class. The off-diagonal blocks hold cross elasticities between the categories of two different classes. The formulas above are adapted somewhat differently to calculate the elasticities in the diagonal and off-diagonal blocks.
The block [
ε kk ] contains all of the own-price and cross-price elasticities for the work-sharing and customer categories within the major mail class
k. The information required to estimate the elements of the block is similar to that used to calculate the elements of the class-level matrix of elasticities. The own-price elasticity for all of the mail in class
k is just the
k-th diagonal element of the class-level matrix
ε kk . This elasticity now assumes the previous role of the aggregate own-price elasticity from the trunk equation,
ε. Additionally, we need the three elasticities derived from the coefficients of the AIDS share equations for the major branching point that divide the postal revenues of class
k. If
Y k denotes total revenue and
P k is the AIDS price index for class
k, the three elasticities that we calculate from the fits of the share equations are:
-
The elasticity of demand for product
i with respect to the postal expenditures,
Y k :
$$ {\varepsilon}_i^{Y_k}=\left(1+{\beta}_i/{s}_i\right), $$
-
The elasticity of the AIDS price index
P k with respect to the price of product
j:
$$ {\varepsilon}_{P_k}^j={\alpha}_j+{\displaystyle \sum}_{i=1}^N{\gamma}_{ji} \ln \left({p}_i\right),\;\mathrm{and}, $$
-
The Marshallian elasticity of demand for product
i with respect to the price of product
j:
$$ {\varepsilon}_{ij}^{M_k}=-1\left(i=j\right)+\left({\gamma}_{ij}-{\beta}_i{\varepsilon}_{P_k}^j\right)/{s}_i. $$
Both products
i and
j are members of class
k. The formula for an element,
ϵ ij , of the diagonal block [
ϵ kk ] is:
$$ {\varepsilon}_{ij}={\varepsilon}_{ij}^{M_k}+{\varepsilon}_i^{Y_k}\left(1+{\varepsilon}_{kk}\right){\varepsilon}_{P_k}^j. $$
An off-diagonal block [
ε kl ] holds all of the cross-price elasticities between the products in two different major classes,
k ≠
l. The cross-price elasticity of demand for postal services in class
k with respect to the class-level price index for class
l is the element
ε kl in the
k-th row and
l-th column of the class-level matrix. The division of the total revenue
Y k among the work-sharing and customer categories of class
k does not require and makes no direct use of the prices that apply to the categories of another class such as class
l. The elasticities taken from the fits of share equations are:
-
The elasticity of demand for product
i with respect to the postal expenditures,
Y k and using the coefficients of the share equations for class
k:
$$ {\varepsilon}_i^{Y_k}=\left(1+{\beta}_i/{s}_i\right). $$
-
The elasticity of the AIDS price index
P l with respect to the price of product
j:
$$ {\varepsilon}_{P_l}^j={\alpha}_j+{\displaystyle \sum}_{i=1}^N{\gamma}_{ji} \ln \left({p}_i\right) $$
-
The elasticity
ε ij consists entirely of the effect on demand for product
i, Q i , of changes in the price
p j transmitted indirectly through the effect on the postal expenditures for class
k,
Y k :
$$ {\varepsilon}_{ij}=\frac{\partial \ln \left({Q}_i\right)}{\partial \ln \left({Y}_k\right)}\frac{\partial \ln \left({Y}_k\right)}{\partial \ln \left({P}_l\right)}\frac{\partial \ln \left({P}_l\right)}{\partial \ln \left({p}_j\right)}. $$
-
After making the substitutions:
\( \frac{\partial \ln \left({Q}_i\right)}{\partial \ln \left({Y}_k\right)}={\varepsilon}_i^{Y_k}, \) \( \frac{\partial \ln \left({Y}_k\right)}{\partial \ln \left({P}_l\right)}={\varepsilon}_{kl}, \) and
\( \frac{\partial \ln \left({P}_l\right)}{\partial \ln \left({p}_j\right)}={\varepsilon}_{P_l}^j. \), we have:
$$ {\varepsilon}_{ij}={\varepsilon}_i^{Y_k}{\varepsilon}_{kl}{\varepsilon}_{P_l}^j. $$
At the highest level of the tree many of the expenditures for work-sharing and customer categories are further subdivided by shape. The mathematics simply repeats with the matrix of price elasticities composed of rectangular blocks that correspond to the elements of the matrix of price elasticities by category.
