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From a US income tax perspective it would be ideal if the PayPal account was in the name of the business, not in your personal name. It really doesn't matter that the account used to be a personal account. I don't see any issues whatsoever in upgrading a personal account to a business account so long as you don't mix personal transactions once business has started.
The existing $1,000 would be considered "additional paid in capital" you would debit cash and credit shareholders equity to record the contribution the shareholder to the company.
Let me know if something is not clear and I will get back to you as soon as I can.
How can we show the paid up capital in accounting? - You would make journal entry on the date business started to record the $1,000 shareholder contribution. You would Debit the cash account $1,000 and Credit \the authorized share capital account by the same $1,000
should this also be added in company papers and modify authorized share capital? - Basically you would just make this accounting journal entry in your accounting software or records.
Correct, you would still record this the journal even though cash does not get transferred to your bank account. Basically the shareholder contributed the cash and the company is using that cash to pay expenses. The company is paying for the expenses out of the Paypal account instead of the it's cash account.