der Erwerb
acquisition
The process of buying or taking possession of something.
Der Erwerb von Wohneigentum ist eine große finanzielle Entscheidung.
The acquisition of residential property is a major financial decision.

acquisition / livelihood
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acquisition
The process of buying or taking possession of something.
Der Erwerb von Wohneigentum ist eine große finanzielle Entscheidung.
The acquisition of residential property is a major financial decision.

'Der Erwerb' (acquisition) is the noun corresponding to the verb 'erwerben' (to acquire). The prefix 'er-' often expresses that a result is achieved through an action or effort. 'Erwerben' therefore means to get something through purchase, work, or learning.
Sie hat sich im Ausland wertvolle Sprachkenntnisse erworben.
She acquired valuable language skills abroad.
The word 'Erwerb' (acquisition) is used mainly in written language and in formal or official contexts. It sounds more elevated and more formal than, for example, 'der Kauf' (the purchase).
Der Erwerb der deutschen Staatsbürgerschaft ist an Bedingungen geknüpft.
The acquisition of German citizenship is subject to conditions.
In the meaning of 'Kauf' (purchase), 'Erwerb' (acquisition) is often combined with the preposition 'von' and a noun in the dative to say what is being acquired. Typical examples are 'der Erwerb von Immobilien' (the acquisition of real estate), 'der Erwerb von Aktien' (the acquisition of shares), or 'der Erwerb von Kenntnissen' (the acquisition of knowledge/skills).
Wir planen den Erwerb von neuen Maschinen für die Produktion.
We are planning the acquisition of new machines for production.
The meaning 'Erwerbstätigkeit' (gainful employment) is rarer today and is found mainly in legal or sociological texts. A typical expression is 'seinem Erwerb nachgehen', which means 'to work in order to earn money'.
Die Statistik erfasst alle Personen, die einem regelmäßigen Erwerb nachgehen.
The statistics record all people who pursue regular gainful employment.
When 'Erwerb' (acquisition) refers to something purchased or obtained, the word is countable and can be used in the plural. This usage is typical for museums, libraries, or collections that talk about their 'neuen Erwerbe' (new acquisitions).
Die Bibliothek präsentiert ihre wertvollsten Erwerbe des letzten Jahres.
The library is presenting its most valuable acquisitions from last year.
purchase
'der Kauf' and 'der Erwerb' can both mean that you get something and then own it. 'der Kauf' is narrower: it is usually clearly about paying. You use it in everyday language for things you buy. 'der Erwerb' sounds somewhat more formal and broader. It can also generally mean taking possession of something, not just normal shopping in a store. When referring to the thing itself, 'der Kauf' is more colloquial, while 'der Erwerb' sounds more elevated or written.

Für den Erwerb des Grundstücks musste die Firma viele Unterlagen prüfen.
For the acquisition of the property, the company had to review many documents.
Nach dem Kauf des Fahrrads bekam Lisa eine Rechnung.
After buying the bicycle, Lisa received a receipt.