"The power of contempt is the means by which the court enforces its fundamental authority.
. . .
[T]he failure to pursue blatant perjury and clear disobedience of court orders rapidly leads to a wholesale disrespect for the law."

The power of contempt is the means by which the court enforces its fundamental authority. Courts have no armies to command and no taxes to raise and spend. In a very fundamental sense, the judicial power is the contempt power; and the failure to exercise it becomes a failure of the judiciary.

C. Contempt of Court is Critical to Maintaining our
Republican Form of Self-Government

Another essential, but often overlooked, vital characteristic of the contempt power is the maintaining the republican form of self-government. In America, the people are sovereign. The people have delegated their authority to the three branches of government.
. . .
Hence, the failure to obey the duly executed orders and judgments of the courts, or acts or omissions that impair the orderly administration of justice in those courts, is a direct affront to the republican government.
. . .
[quoting a Colorado case] "It was said in argument by counsel for respondents 'that by the common law every judge was regarded as the direct representative of the sovereign, and upon this fiction the power to punish for contempt was based.' With us the people have been substituted for the crown. The courts are created by the people, and are dependent upon the popular will for a continuation of the powers granted. They are the people’s courts, and contemptuous conduct toward the judges in the discharge of their official duties tending to defeat the administration of justice, is more than an offense against the person of the judge; it is an offense against the people’s court, the dignity of which the judge should protect, however willing he may be to forego the private injury."

Put another way, the failure to invoke the power of contempt when appropriate not only undermines the administration of justice and the rule of law, it strikes at the heart of our republican form of government.