to buy / bribe
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to buy
To obtain something, often something immaterial such as freedom, success, or peace and quiet, through great effort, sacrifice, or a high price.
Den Frieden mussten sie sich mit schmerzhaften Kompromissen erkaufen.
They had to buy peace with painful compromises.
to bribe
To get someone to act in one's own interest by giving them money or other advantages; to bribe.
Er versuchte, das Schweigen des Zeugen mit einer hohen Geldsumme zu erkaufen.
He tried to buy the witness's silence with a large sum of money.
'Erkaufen' (to buy / bribe) is an inseparable verb. The prefix 'er-' emphasizes that something is successfully obtained through the act of buying, though often at a high price that is not only financial.
Er hat sich seinen Erfolg teuer erkauft.
He bought his success at a high price.
You can distinguish the two meanings by the object. If you 'erkauft' (buy/bribe) a person, you bribe them. If you 'sich etwas erkauft' (buy something for oneself), you obtain an abstract good through sacrifice.
Der Politiker versuchte, die Journalisten zu erkaufen.
The politician tried to bribe the journalists.
In this meaning, the verb is almost always used reflexively: 'sich (dative) etwas (accusative) erkaufen' (to buy something for oneself). The dative pronoun shows for whom the benefit is obtained.
Ich musste mir die Ruhe mit einem Umzug aufs Land erkaufen.
I had to buy myself peace and quiet by moving to the countryside.
The 'price' or sacrifice one pays for what is achieved is often expressed with the preposition 'mit' (with) and the dative.
Das Unternehmen erkaufte sich den Marktvorteil mit einem hohen Risiko.
The company bought itself the market advantage with a high risk.
to bribe / impress
Bestechen directly refers to trying to persuade someone to act dishonestly or unlawfully by offering them an advantage. Erkaufen, by contrast, presents the desired help or approval as something obtained by paying a high price.
Der Unternehmer erkaufte sich mit teuren Geschenken die Hilfe des Beamten.
The businessman bought the official's help with expensive gifts.
Der Unternehmer bestach den Beamten mit teuren Geschenken.
The businessman bribed the official with expensive gifts.
