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Let’s say a producer supplies 100 units of some good but consumers will only demand 100 units if the price is zero. For every dollar the producer charges per unit, 1 consumer is lost. So if he priced his units at 1 dollar per unit, he would only be able to sell 99 units. In this case, should the producer just give away all the units for free since the quantity supplied and quantity demanded meet at 0 dollars? Or should he just price it at the highest possible price and accept he won’t be able to sell every unit?

The reason I’m asking this question is because I read that if there were no scarcity in the world, then everything would be free, but then I also read that something is deemed scarce when its demand exceeds its supply at a price of zero. But I can’t help but feel that even if the quantity demanded for something equaled its quantity supplied at the price of zero, the supplier would still probably charge a price for it.

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Scarcity has nothing to do with what you ask.

To make this interesting, suppose the supplier did not know before hand of the demand curve they would face (because if they did, they would produce less).

The demand function is $$Q^d = 100- P.$$ The profit function of the supplier is $$\pi = P\cdot Q^d - C(100) = 100P - P^2 - C(100).$$

Costs do not matter, because they have already been incurred in their totality. So essentially the supplier seeks to maximize sales revenue, not profits.

The f.o.c and optimal price is $$ 100 - 2P = 0 \implies P^* = 50$$.

The supplier will receive 2500 Revenue, against their production costs, which is better than nothing.

What the producer will do with the other 50, unsold units is a matter of marketing strategy, whether the good is perishable or not, whether this is a one-off situation or has a dynamic (repetitive) aspect, etc.

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