groups: those with incomes less than 1/16 of US income; those between 1/16 and 1/8 of US income; those between 1/8 and 1/4 of US income; those between 1/4 and 1/2 of US income; and those with income higher than 1/2 of US income.

Each cell of Panel I of Table 4 and Table 5 indicates the number of transitions from one relative income group to another one. Similarly, each cell of Panel II shows the probability of making such transition. The sum of each row of Panel I amounts to the number of states observed between 1950 and 2008.<sup>21</sup> The sum of each row of Panel II should equal unity. Hence, a cell _aij_ in Panel I illustrates the total number of transitions from group _i_ to group _j_ over ten-year intervals for the period in consideration. _pij_ , on the other hand, in Panel II, denotes the probability of transition between relative income groups _i_ and _j_ . These transition matrices can be used to describe the evolution of world income over time. For instance, after _n_ periods, the relative distribution of income at time _t + n_ will be given by , where is a N x 1 vector describing the relative income distribution at time _t_ .

**Table 4: Ten-year transition matrix 1950-2008**

||**_Panel I:_**|**_Total numb_**|**_er of transi_**|**_tions_**|||**_Panel II_**|**_I: Mean firs_**|**_t passage t_**|**_ime_**||
|---|---|---|---|---|---|---|---|---|---|---|---|
||[0-0.15)|[0.15-0.3)|[0.3-0.45)|[0.45-0.6)|≥0.6||[0-0.15)|[0.15-0.3)|[0.3-0.45)|[0.45-0.6)|≥0.6|
|[0-0.15)|356|20|0|0|0|[0-0.15)|20.9|146.0|63.1|75.0|69.6|
|[0.15-0.3)|25|88|23|2|0|[0.15-0.3)|374.5|52.3|44.3|56.2|50.8|
|[0.3-0.45)|1|11|19|14|3|[0.3-0.45)|723.4|127.2|106.5|34.7|28.0|
|[0.45-0.6)|0|0|4|17|11|[0.45-0.6)|973.4|250.0|250.0|35.6|9.6|
|≥0.6|0|0|0|1|87|≥0.6|1061.4|338.0|338.0|88.0|1.1|
||**_Pa_**|**_nel II: Transi_**|**_tion matrix_**|||||||||
|p(i,j)|[0-0.15)|[0.15-0.3)|[0.3-0.45)|[0.45-0.6)|≥0.6|**_Ergodic distri_**|**_bution for tr_**|**_ansition m_**|**_atrix_**|||
|[0-0.15)|0.95|0.05|0.00|0.00|0.00||0.05|0.01|0.01|0.03|0.90|
|[0.15-0.3)|0.18|0.64|0.17|0.01|0.00|**_Eigenvalues fo_**|**_r transition_**|**_matrix_**||||
|[0.3-0.45)|0.02|0.23|0.40|0.29|0.06||0.20|0.56|0.76|0.98|1.00|
|[0.45-0.6)|0.00|0.00|0.13|0.53|0.34|**_Half-life for tr_**|**_ansition ma_**|**_trix_**||||
|≥0.6|0.00|0.00|0.00|0.01|0.99||40.0|||||

The total number of states observed is as follows: 376, 138, 48, 32 and 88. Not surprisingly, Table 4 shows that transitions between adjacent cells are not uncommon in a tenyear span. On the other hand, transitions between non-adjacent groups are less frequent. Moreover, results in Table 4 suggest that countries are more likely to stay in their relative income group over a ten-year period: the probability of being in group _i_ and remaining in group _i_ is greater than the probability of being in group _i_ and moving to group _j_ after a decade –i.e. _pii_ > _pij_ for _i ≠ j_ for each single row. In other words, diagonal elements are larger than non-diagonal elements.

Table 4 indicates that low income countries relative to the US are much more likely to remain in that group ( _p11_ equals 0.95). This may suggest the low income countries may be subject to a poverty trap. On the other hand, the probability of being a high income country and remaining as such is also high ( _p55_ equals 0.99), indicating that once you join the high income

> this can lead to a distribution in which most countries have more than twice the world average income. Countries can be classified relative to a leading country or a group of leading countries to avoid this problem. 21 We also include in the tables the transition for the final years of our sample, i.e. the transitions from 2000 to 2008.

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