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Why is it rational to pay out a dividend?
Paying out dividends and financing new projects with debt also lessens the agency problem. The consequences of a failed project are greater when debt is used, so the manager now has a greater incentive to see that the project is a success. This, in addition to the paid divided is a benefit to the shareholder. If equity wasn't paid out and instead used for the project then the manager may not be so interested in its success. And if it's a failure then the shareholders are worse off.