甲斐（かい）, a noun meaning ‘worth’ or ‘effect’, will often be used following verbs or nouns, in order to show that (A) was or will be worth doing. In many cases, this grammar point will be seen in hiragana, and will appear as either かい or がい, depending on how it is conjugating with the previous word. Possible translations include things like ‘it is worth doing (A)’, or ‘(A) will pay off’.
が and the う-Verb ある ‘to be’ will follow かい. This states that the ‘worth of (A) exists’. Occasionally, the が will be omitted. When following verbs, がい will be used after the ます-stem, or かい may be used after any standard form.

• この食（しょく）パンを買（か）えて、朝（あさ）の4時（じ）から並（なら）んだ甲斐（かい）があった。
  It was worth standing in line since 4 a.m. as I was able to buy this bread.
• こういう景色（けしき）を見（み）ると生（い）きている甲斐（かい）があると思（おも）う。
  This kind of scenery makes me feel that life is worth living.
• 彼女（かのじょ）は何回（なんかい）注意（ちゅうい）しても上達（じょうたつ）しないから教（おし）えがいがない。
  She is not worth teaching because she does not improve no matter how many times I give her advice.

The primary difference between these conjugation patterns is that there is more focus on the tense of the verb with かい, whereas がい will frequently have the nuance of it ‘always being worth (A)’.
When pairing with nouns, の will be required before かいがある.

• みんなの努力（どりょく）の甲斐（かい）があり、この大会（たいかい）で優勝（ゆうしょう）する事（こと）ができました。
  Everyone’s hard work paid off, and we were able to win the tournament.
• 看病（かんびょう）の甲斐（かい）があり、父（ちち）がどんどんと元気（げんき）になってきた。
  Nursing my father paid off and he is rapidly getting better and better.

Alternatively, かいがない may be used in order to indicate that it is ‘not’ worth doing something.

• しょっぼ、なにこれ。こんなもんのために東京（とうきょう）からきた甲斐（かい）がない。
  What the heck is this, this is it? It was not worth coming all the way from Tokyo for something like this.

かいがある is primarily used when the speaker wants to convey that a positive result will come from something that is either difficult or unpleasant.