I think the claim would be that it works better than capitalism at reducing inequality
It would be interesting to measure the actual inequality in various capitalist and socialist countries, in various decades. I am not sure how that would properly be done, though. A naive solution would be to compare how much money people have, but that would not reflect the reality correctly; in a command economy, money is relatively less useful. -- There is no point in having lots of money, if the shops are empty, and the things you want most are illegal anyway. On the other hand, if you are a Party boss, all your wishes get fulfilled immediately, often without any corresponding flow of money.
It would be interesting to measure the actual inequality in various capitalist and socialist countries, in various decades. I am not sure how that would properly be done, though. A naive solution would be to compare how much money people have, but that would not reflect the reality correctly; in a command economy, money is relatively less useful. -- There is no point in having lots of money, if the shops are empty, and the things you want most are illegal anyway. On the other hand, if you are a Party boss, all your wishes get fulfilled immediately, often without any corresponding flow of money.