*6.1. Theil Inequality Index*

An inequality index that belongs to the entropy class has the following general form:

$$GE(a) = \frac{1}{n(a^2 - a)} \sum\_{i=1}^{n} [(\frac{y\_i}{\overline{y}})^a - 1] \tag{10}$$

where *n* is the size of the sample, *yi* is the *i*-th observation and *y¯* is the mean value of the sample. The parameter α represents the weight given to distances between values at different parts of the distribution. For smaller values of α, *GE(*α*)* is more sensitive to changes in the bottom tail of the distribution. For higher values of α, *GE(*α*)* is more sensitive to changes in the upper tail of the distribution. The most commonly used values of α are −1, 0, 1 and 2. The most well-known member of the entropy class of inequality indices is the *GE*(1) index, called Theil index (*T*), by the name of Henri Theil who introduced it in 1967. All the members of the entropy class of inequality indices have the advantage of being perfectly decomposable (i.e., with a zero residual part).

Following Fernandez de Guevara et al. (2007), the Theil index for bank market power can be calculated from Equation (11):

$$T = \sum\_{\mathcal{S}=1}^{G} s h\_{\mathcal{S}} T\_{w,\mathcal{S}} + T\_b \tag{11}$$

where *T* is the total market power inequality, *G* is the number of observation groups in the sample, *shg* is the share in the sample (in terms of total assets) of a group *g, Tw,g* is the within-group inequality in group *g* and *Tb* is the between-group inequality.

The within-group inequality in group *g* is defined by Equation (12):

$$T\_{w, \mathcal{g}} = -\sum\_{i=1}^{N\_{\mathcal{G}}} (\frac{sh\_i}{sh\_{\mathcal{G}}}) \ln(\frac{\mathbf{x}\_i}{\mu\_{\mathcal{G}}}) \tag{12}$$

where *Ng* is the number of entities (e.g., countries or banks) in group *g*, *shi* is the share in the sample (in terms of total assets) of an entity *i* belonging to group *g* and μ*g* is the weighted average of the Lerner index of entities belonging to group *g*.

The between-group inequality is defined by Equation (13):

$$T\_b = -\sum\_{\mathcal{S}=1}^G \text{sh}\_{\mathcal{S}} \ln(\frac{\mu\_{\mathcal{S}}}{\mu}) \tag{13}$$

where μ is the weighted average of the Lerner index (variable *xi*) in the sample.
