Understanding the law of supply In high school, I've never really questioned the law of supply, taking it as a given. 

As I begin an introductory economics course in university, "relearning" the fundamentals of Economics, I realise I'm coming up against some difficulties trying to genuinely understand a fundamental concept: the law of supply. I have searched for similar questions here that concern this but none address the specific nature of my question.

I was initially confused at the direct link between price and quantity supplied, given by my textbook, which states that at higher prices, producers will find it more profitable to produce more and will thus do so. At this point in the text, concepts of MC, MR, cost curves etc are a few chapters away, and I am somewhat clueless about them at the moment, but do feel free to use that in your explanation if needed - I know that ultimately, marginal cost has something to do with this but I'm not sure what (I've forgotten most of my high school stuff).

I guess a key concern is this: if more is produced, then total cost of producing also increases right and it may not always be more profitable to have more quantity supplied, unlike what my textbook says. So does the supply curve already account for this? I.e. at every point, the revenue earned will definitely be more than the cost? How can we just assume this?

The interesting thing is, I'm actually able to sort of understand it by reading values off the supply curve by looking at the x-axis (quantity) first (which is a valid approach) then y-axis (price), which is that if producers are to supply more, they will need greater revenue to cover greater costs of production to allow sustained profits (preferably increased profits, maybe same amount - but less important at this point).

In fact, this "inverted" law of supply is what I'd prefer. When I then try to "re-invert" this to try and understand the actual law of supply whereby price change is the cause and quantity supplied is the effect, I'm confused once more. Was hoping somebody could explain how exactly this works, preferably in relation to the explanation for the "inverted" law of supply. 

Was wondering also if there's a "mathematical proof" for this - that'd be great, though the intuition behind it should preferably still be explained in plain simple words.
As you noted correctly, it has something to do with the costs.  An important point here is a cost of producing an additional unit, and not average cost.


Let me give the following example.  Suppose that you are an owner of a bakery, and you have a single employee who works $8$ hours and produces $8$ breads a day ($1$ bread an hour). Assume you have to pay your employee $\$1$ an hour for the first $8$ hours and $\$1.50$ (overtime) for each additional hour (after $8$ hours), and you sell each bread at $\$1.20$.  

One day you think to yourself: why not to produce and sell $9$ breads (you come to this thought by the same reasoning as in your question)?  Then, you realize that in order to produce the $9$th bread, you need to keep your employee working for an additional hour and have to pay overtime, namely $\$1.5$ for the $9$th hour.  However, you will still sell the $9$th bread for $\$1.20$, which means you will lose $\$0.30$ from producing and selling the $9$th bread.

Suppose that the price of bread has increased to $\$1.60$.  Now, you surely want to keep your employee for the $9$th hour (or even more), since for each additional bread produced (after $8$ breads), you will make $\$0.10$ profit per bread.  So, you can see that an increase in price induced you to supply more.