local authority vs agency

local authority and agency both are nouns.

Word	Noun	Adjective	Verb	Adverb
local authority	Yes	No	No	No
agency	Yes	No	No	No

As nouns, agency is a hypernym of local authority; that is, agency is a word with a broader meaning than local authority:

Other hypernyms of local authority include authority, bureau, federal agency, government agency, office.

local authority (noun)	agency (noun)
an administrative unit of local government	to have the ability to produce a particular effect or achieve an end
	an administrative unit of government
	a business or organization that provides a particular service, especially the mediation of transactions between two parties
	the state of serving as an official and authorized delegate or agent
	the state of being in action or exerting power