What would a stock be worth if dividends did not exist?

Unrealistic assumption, but I'll play along.

  1. Ultimately, dividends would exist because some innovative shareholder of some company, at some time, would desire income from their investment and could propose the idea of sharing the profit. Like-minded investors also desiring income could vote for dividends to come into existence — or, rather, vote for a board of directors that supports enactment of the idea. (In your fictitious world, shareholders do still control the corporation, right?)

    In this world, though, dividends wouldn't be called "dividends", a terrible name that's too "mathy" for the inhabitants of that world. Rather, they would institute a quarterly or annual shareholder profit share. Governments would enact legislation to approve of—nay, encourage such an innovation because it becomes a new source of recurring income they can tax.

  2. Alternatively, even if the idea of a cash dividend didn't occur to anybody in that world, investors would realize the stock price is depressed and could propose and vote for the board to institute share buybacks.

    The company repurchasing some portion of shares periodically would provide income to shareholders participating in the buyback. If the buyback were oversubscribed, they could structure it fairly (pro-rata participation, etc.)

  3. Alternatively, shareholders would pressure the board (or fire them and vote in a new board) to put the company up for sale and find a larger buyer, who would purchase the shares for cash. This can't scale forever, though, so the pressure will increase for solutions like #1 and #2.

The company can be dissolved

In the unlikely case that noone finds a way to extract resources from the company and distribute them to shareholders periodically in a way that's de facto equivalent to dividends, any company can be dissolved.

The assets of the company would be sold for their market value, the liabilities would have to be settled, and the net result of all this (company cash + sale results - liabilities) would be distributed to shareholders proportionally to their shares.

The 'liquidation value' is generally lower than the market value of a company as an ongoing concern that's making business and earning profit, but it does put a floor on it's value - if the stock price is too low, someone can buy enough stock to get control of the company, vote to dissolve it, and make a profit that way; and the mere fact that this can happen props up the stock price.

Companies could even be created for a limited time period in the first hand (which has some historical precedent with shareholders of 'trading companies' with lifetime of a single trade voyage).

Imagine that there is some company Megacorp2015 where shareholders want to receive $1M of its cash as "dividends". They can make appropriate contracts that will form a new company called Megacorp2016 that will take over all the ongoing business and assets except $1M in cash, and then liquidate Megacorp2015 and distribute it's assets (shares of Megacorp2016 and the "dividend") among themselves. The main difference from normal dividends is that in this process, you need cooperation from any lenders involved, so if the company has some long-term debts then they would need agreement from those banks in order to pay out "dividends". Oh, and everyone would have to pay a bunch more to lawyers simply to do "dividends" in this or some other convoluted way.

As a thought experiment I suppose we can ask where dividends came from and what would be different if they never existed. The VOC or Dutch East India Companywas the first to IPO, sell shares and also have a dividend. There had been trade entrepot before the VOC, the bulk cog (type of sea-going ship) trade in the Hanseatic League, but the VOC innovation was to pool capital to build giant spice freighters - more expensive than a merchant partnership could likely finance (and stand to lose at sea) on their own but more efficient than the cogs and focused on a trade good with more value. The Dutch Republic became rich by this capital formed to pursue high value trade. Without dividends this wouldn't have been an innovation in seventeenth century Europe and enterprises would be only as large as say the contemporary merchant family networks of Venice could finance. So there could be large partnerships, family businesses and debt financed ventures but no corporations as such.