The motivation of our study is how to redistribute income earning skills in a heterogeneous society to reach the social optimum. Leung and Yazıcı (2010) write the first paper that analyzes this issue analytically. In light of their study, we analyze the optimum skill distribution with utilitarian and egalitarian social welfare functions and conduct two analyses. Firstly, we provide numerical simulations to measure the welfare effects of skill distribution choice under different social welfare functions. Secondly, we characterize the optimum skill distribution for different objective welfare functions with different assumptions. Our first result indicates that, it is always optimal to distribute all skills to one type in a society regardless of whether we use egalitarian or utilitarian objective social welfare functions. Secondly, an increase in welfare from Mirrleesian taxation without skill distribution to Mirrleesian taxation with skill distribution is always much more than an increase from laissez faire market to Mirrleesian taxation without skill distribution in both utilitarian and egalitarian problems. Our final result is that the economy with perfectly unequal skill distribution provides a more egalitarian society in terms of how utilities are distributed across agents, in both utilitarian and egalitarian problems
