A paper forthcoming in the QJE by Gabaix & Landier provides a straightforward answer:
"This paper develops a simple equilibrium model of CEO pay. CEOs have different talents and are matched to firms in a competitive assignment model. In market equilibrium, a CEO’s pay depends on both the size of his firm, and the aggregate firm size. The model determines the level of CEO pay across firms and over time, offering a benchmark for calibratable corporate finance.
We find a very small dispersion in CEO talent, which nonetheless justifies large pay differences. In recent decades at least, the size of large firms explains many of the patterns in CEO pay, across firms, over time, and between countries.
In particular, in the baseline specification of the model’s parameters, the six-fold increase of U.S. CEO pay between 1980 and 2003 can be fully attributed to the six-fold increase in market capitalization of large companies during that period."
The paper is a very nice application / extension of Rosen's Economics of Superstars paper in the AER.