There's no formula to figure out a value of such a thing. However, you can estimate it based on the revenue generated through or by the Instagram account and other relevant metrics. Consult with a business attorney who may be able to act as a sort of mediator here.
There is most definitely a value and you can seek out a business valuator even for this type of business. Your partnership-operating agreement should adsress this, but if it doesn't or you didn't make one, then then you might want a "break up agreement at this point to put all l the agreements and obligtions in writing now. For ex, you would agree to split the costs of the valuation, and agree to the amount or if you don't agree there is some process for further valuations and costs bearing etc...........Also, you don't have to dissolve the LLC either, the party keeping the business and "buy" out the other partner's membership units; it is similar to a stock purchase agreement, then the LLC status would remain the same just there would be one less owner, and the CPA would handle the K-1s accordingly. Then all the accounts in the LLC name do not need to be changes, just the contact person and password.
The foregoing is for informational purposes only and may not be relied on as attorney-client advice.
Get a business valuator to value the account and have partner who wants it to pay the partnership for it, then dissolution can go ahead normally
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