The high pay packages of U.S. CEOs have raised serious concerns about what would constitute a fair pay. Since the present economic models do not adequately address this fundamental question, we propose a new theory based on statistical mechanics and information theory. We use the principle of maximum entropy to show that the maximally fair pay distribution is lognormal under ideal conditions. This prediction is in agreement with observed data for the bottom 90%-95% of the working population. The theory estimates that the top 35 U.S. CEOs were overpaid by about 129 times their ideal salaries in 2008. We also provide an insight of entropy as a measure of fairness, which is maximized at equilibrium, in an economic system. Our analysis shows that a certain amount of seeming inequality of pay is inevitable in organizations. Given this reality, the lognormal distribution is the fairest inequality of pay. One may view our result as an ‘economic law’ in the statistical thermodynamics sense. The free market will ‘discover’ and obey this economic law if allowed to function freely and efficiently without collusion like practices or other such unfair interferences. This result is the economic equivalent to the Boltzmann distribution of the energy landscape for ideal gases.