billigen
to approve
To officially approve, endorse, or agree to something. Often used in formal or official contexts.
Der Stadtrat hat den Bau des neuen Kindergartens gebilligt.
The city council approved the construction of the new kindergarten.

to approve / condone
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to approve
To officially approve, endorse, or agree to something. Often used in formal or official contexts.
Der Stadtrat hat den Bau des neuen Kindergartens gebilligt.
The city council approved the construction of the new kindergarten.

The verb 'billigen' (to approve / condone) is related to the adjective 'billig' (cheap). 'Billig' originally meant 'fair' or 'appropriate'. From this older meaning, 'billigen' developed in the sense of 'to recognize something as fair' or 'to approve'.
Die Aktionäre billigten den Vorschlag des Vorstands.
The shareholders approved the board's proposal.
In the meaning of 'genehmigen' (to approve), 'billigen' belongs to formal official or business language. You often find it in minutes, contracts, or official announcements.
Das Parlament muss das neue Gesetz noch billigen.
Parliament still has to approve the new law.
Formal plans or decisions are typically approved. Common objects are: 'einen Antrag billigen' (to approve an application/motion), 'ein Budget billigen' (to approve a budget), 'einen Plan billigen' (to approve a plan), or 'ein Vorgehen billigen' (to approve a course of action).
Die Geschäftsführung hat die Investition gebilligt.
Management approved the investment.
In this meaning, 'billigen' (to condone) has a negative connotation and is often combined with the adverb 'stillschweigend' (tacitly). It describes not actively preventing or criticizing problematic behavior and thereby tolerating it.
Er hat die unfairen Kommentare seines Kollegen stillschweigend gebilligt.
He tacitly condoned his colleague's unfair comments.
to allow
'erlauben' and 'billigen' are close in meaning because both can express that someone agrees with something. 'erlauben' mainly means that someone gives permission for another person to do something. 'billigen' sounds more formal and more often means that you approve of or accept a plan, a decision, or a behavior. It can also mean that you tacitly accept something questionable. Learners may confuse the words because both show approval or agreement. In everyday language, 'erlauben' is much more natural when it is about being allowed to do something; 'billigen' fits better with official decisions or evaluations.

Nach dem Gespräch billigt Hannes den Vorschlag seines Teams.
After the discussion, Hannes approves his team's proposal.
Für die Prüfung erlaubt der Lehrer nur einen blauen Stift.
For the exam, the teacher only allows a blue pen.