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YES, "people paying 3 means the item is actually objectively worth 3", because labor is not the value of the cake minus other costs.

You can spend exactly the same labor building the cake or smashing trash, but the smashed trash is not worth the same as the cake despite spending the same labor, so the labor value doesn't explain the value of the product.

You pay 3 for a. Ake only if you value what you buy more than what you pay, so you win by buying. Otherwise you would not buy. The seller had no power on how much you value it, so you decide how much is the cake valued when you decide to pay or not pay for it. That's independent of the cost of manufacturing. You don't care how much it had cost something you buy in the supermarket. You don't even know. You decide by judging if you are better buying it or something else you could get with that money.

Prices are created by consumers and represent priorities, not absolute value.

Because the value of things is decided by consumers, the value is independent of the cost of production, and for that reason is a lie that the difference between value and cost is the labor.

If labor were worth the difference between product value and cost, it would not be justified to produce anything, because at the end of day, nothing would be won by working. The final value would be the same than at the start of the day. It only makes sense to work if the final product is more valuable than what was spent, including labor.

Labor has limited value. You decide if you spend your labor cleaning your room, or if you are better doing nothing, or playing a videogame, because your labor has a limited value. You would not do any work if the benefit is smaller that the value of your work.

If labor were worth the final value . Minus other cost, you could not decide were to work. You would spend the same labor building a pile of trash or something more productive. A car would be worth the same as a pile of trash, despite requiring the same labor.