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In this study, I analyze the optimal choice of a board of directors using a theoretical model. In my model, shareholders choose either an informed board that can bring additional private information to the firm or an uninformed board that merely considers the inside information already available within the firm. Then, the board randomly chooses a good chief executive officer (CEO) who has inside information or a bad CEO without such information. Next, the CEO decides whether to consult with the board when making a project decision. I show that shareholders generally choose an informed board in order to maximize the firm's value by utilizing the private information available to the board. However, shareholders optimally select an uninformed board if the CEO is reluctant to communicate with an informed board out of fear that it will reject his/her decisions. The uninformed board is also optimal when it derives a sufficiently large private benefit from monitoring the CEO, the cost of choosing the informed board is large, or the firm is involved in many unrelated businesses, especially when the inside information is valuable and many outsiders are needed to observe useful outside information. Some of these implications are used to discuss recent trends in the board structures of Japanese firms.