Measures of Tier-2 Commonality

We introduce three metrics of tier-2 commonality : 1) diamond ratio and 2) cosine com- monality score. The first metric is based on the binary customer-supplier relationship, while the second metric is based on customer-supplier relationship weighted by the percentage of purchase costs that a supplier represents.

3.4.1.1 Diamond Ratio:

In the previous chapter, we use the degree of commonality to characterize the degree to tier-2 commonality. While degree of commonality is an intuitive measure indicating the number of

tier-1 suppliers that share a tier-2 supplier, the measure is influenced by the number of tier-1 suppliers that a firm has. That is, a tier-0 firm with more tier-1 suppliers is more likely to have a higher degree of commonality. This may introduce a bias between the degree of com- monality and firm size, because a large firm with a higher number of identified immediate suppliers will tend to have a higher degree of commonality. To address this potential issue we propose another metric, the diamond ratio.

Let matrix A denote the binary customer-supplier relationship where Aji indicates whether firm j supplies to firm i. The diamond ratio of firm i is defined as DM Di DCi{°_j1Aji¡0 ° jrA 2_s ji{p ° j1Aji¡0 ° j1rA2_s

jiq, which normalizes the degree of com-

monality with the size of the tier-1 supply base. Specifically, the diamond ratio of each tier-0 firm is obtained by dividing the degree of commonality by the number of tier-1 suppliers.

Figure 3.1: Illustration of common tier-2 supplier with degree of commonalityk 3.

0 A B C X Y Z

Tier-0 Tier-1 Tier-2

Note.You may refer to matrices2.1for the first-order and second-order adjacency matrices for this illustration example.

This metric also has an alternative intuitive interpretation. One can view the diamond ratio as the number of observed tier-0 to tier-2 paths over the number of all possible paths in a firm’s supply network. Note that the number of all possible paths is precisely the product of the number of tier-1 suppliers and the number of tier-2 suppliers. For example, the diamond ratio of the tier-0 firm in the supply network depicted in Figure3.1equals5{p33q 0.56. By definition, the diamond ratio can only take a value between 0 and 1, and a higher value indicates the presence of more common tier-2 suppliers.

3.4.1.2 Cosine Commonality Score:

Our second measure, the cosine commonality score, considers cost-weighted supplier- customer relationships. First, we define the cost percentage matrix C where Cji denotes the percentage of firmi’s purchase cost attributed to supplierj. Note that the binary matrix Awhich indicates whether firm j supplies to firmi satisfiesAji 1Cji¡0. The rows of C

supplier-customer relationship data are complete, the column sum ofC should be1. Using matricesCandA, we define the Cosine Commonality Score (CCS) of firmias

CCSi median jm,AjiAmi1 cospC,j, C,mq median jm,AjiAmi1 xC,j , C,my }C,j}2 }C,m}2

where C,j is the jth column of C, representing firm j’s spending on its own suppliers.

cospC,j, C,mq represents the pair-wise cosine similarity between the cost distributions of tier-1 suppliersj andm. Cosine similarity is a common metric used in the social network literature describing how similar two vectors are. The value of the cosine similarity between the spending vectors of any two firms ranges from 0 to 1, where 0 indicates that the two suppliers have no shared supplier (the spending vectors of the two firms are orthogonal), and 1 indicates that the two suppliers have the exact same supply base: same suppliers and same spending (the spending vectors of the two firms are identical). For example, if tier-1 supplier jsingle sources from firm A while tier-1 suppliermequally sources from firm B and C, the spending vectors of the two tier-1 suppliers on the union of their supply base{A, B, C}are [1, 0, 0] and [0, 1/2, 1/2]. In this case, the cosine similarity between the cost distributions of tier-1 supplierj andmequals zero. If tier-1 supplierj instead sources equally from firm A and B, the spending vector ofj becomes [1/2, 1/2, 0]. Now the cosine similarity between tier-1 supplier j and m equals to a1{2, suggesting the existence and the scale of tier-2 sharing.

After obtaining the cosine similarities, we aggregate the pair-wise measure for a focal tier-0 firm over all pairs of its tier-1 suppliers. We choose median (rather than mean) among all the pair-wise cosine similarities because of the high skewness of the distribution of cosine similarities.5 _{Similar to the diamond ratio, a higher value of the cosine commonality score}

suggests the presence of more tier-2 sharing in the focal tier-0 firm’s supply network.