The head of the division of the budget shall be the director of the budget who shall be appointed by the governor and hold office during his pleasure. He shall receive a salary to be fixed by the governor within the amount appropriated therefor. It shall be the duty of the director of the budget to assist the governor in his duties under the constitution and laws of the state respecting the formulation of the budget and the correlating and revising of estimates and requests for appropriations of the civil departments, and also to assist the governor in his duties respecting the investigation, supervision and coordination of the expenditures and other fiscal operations of such department. The director, subject to rules prescribed by the governor, may appoint and fix the compensation of subordinates and employees of the division within the amounts appropriated therefor.