Office of the Auditor-General comments 
A healthy and productive relationship between the governance arm of a local authority and its chief executive is an important factor in an authority's effectiveness. This relationship is the vital link between governance and management, and between decision-making by elected representatives and operational activity. Problems in that relationship can have a significant effect at all levels of the organisation.
The governing body is responsible for appointing the chief executive. The chief executive is the only employee directly appointed by the governing body. All other employees are appointed by the chief executive 
The minimum legal requirements for employing chief executives are set out in schedule 7 of the Local Government Act 2002 (clauses 33-36). In summary:
The council has to appoint a chief executive for a term of no more than five years.
In the year before the contract expires, the council must review the chief executive's performance and skill mix, and consider how this mix fits with the local authority's expected future needs.
Based on that review, the local authority can then decide to either reappoint the incumbent for another two years, without advertising or any other process, or advertise the pending vacancy and start afresh.
Appointing a chief executive is an important decision for a local authority; it will shape how the organisation is led and managed for the term of the appointment. The Local Government Act requires a council’s governing body to make this decision in recognition of its significance. It must not delegate the decision to a smaller group.
The appointment process should be reasonably straightforward if it is managed carefully and systematically and supported with appropriate external advice. The Office of the Auditor General publication Guide for Local Authorities on Hiring and Managing Relationships with the Chief Executive provides useful advice.
It is important for elected members to remember they have legal obligations to be good employers and provide a good and safe working environment. It is unlikely to be appropriate to raise concerns about a chief executive's performance or the terms of his/her employment in public or to make it a matter of political debate. The LGNZ guide comments that:
Confidentiality is paramount. It is important to remember in both recruitment and performance management processes that the local authority is dealing with the personal and professional lives of individuals. At times there is a tension between the good employer and confidentiality requirements and the political roles of councillors. (Clause 6.25)
Relative responsibilities of chief executives and governing body members are covered in Section 3: Elected members.