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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.

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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.

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Assume a profit maxmizing but price-taking firm. It solves

$$max_q \{pq - c(q)\}$$

and the first order condition is

$$q^*: p=c'(q^*)$$ and this will be a global maximum if $c''>0$ (convex, i.e. accelerating costs)

Inverting

$$q^* = (c')^{-1}(p)$$

The last equation is the supply curve of the individual firm, i.e. the profit maximizing quantity at each price level (we can postulate the existence of a price so low that maximum profits are negative and so production will be zero in the long run. But we are mostly concerned with what happens as price rises).

A first question is, since we already have assumed accelerating costs as quantity rises, what guarantee do we have that profits will stay positive and rising at high levels of prices?

The profit function is

$$\pi = pq^* - c(q^*) = p\cdot [(c')^{-1}(p)] - c[(c')^{-1}(p)]$$

and we want positive and rising profits, namely

$$\pi > 0 ,\;\;\; \frac{\partial\pi}{\partial p} > 0$$

But if we have $\frac{\partial\pi}{\partial p} > 0$ it suffices to obtain a single price level where maximum profits are positive, and then we know they will be positive for any higher price level. But if there is no price level with positive maximum profits, the market won't exist. So let's concentrate on the derivative calculation.

It seems complicated but it is not so much. And this is because

$$\frac{\partial\pi}{\partial p} = \frac {\partial}{\partial p}\big(pq^* - c(q^*)\big)$$

$$ = q^* + p\frac{\partial q^*}{\partial p} - c'(q^*)\cdot \frac{\partial q^*}{\partial p}$$

$$=q^* +[p-c'(q^*)]\cdot \frac{\partial q^*}{\partial p}$$

But the profit function is characterized throughout by the profit maximizing relation $p=c'(q^*)$,so we conclude that

$$\frac{\partial\pi}{\partial p} = q^*>0$$

So we know that as price increases the price-taking supplier can increase her profits by following the rule "quantity such that marginal cost equals price".

But does this imply that the maximum profits will increase by increasing quantity, which is what we want to prove?

We realize that for this we need to show also that $\frac{\partial q^*}{\partial p}>0$

Here we need to apply the inverse function theorem. Some math properties of the cost function will ensure that $(c')^{-1}(p) = (c^{-1})'(p)$ and then

$$\frac{\partial q^*}{\partial p} = \frac{\partial (c')^{-1}(p)}{\partial p)} = \frac{\partial (c^{-1})'(p)}{\partial p} = \frac {1}{c''(p)} >0 $$

and we just obtained the "Law of Supply" (upward sloping supply curve).

We see that accelerating costs not only do they not create troubles to the Law of Supply, but on the contrary, guarantee that it holds. Intuitively, since the producer must set quantity so as for marginal cost to equal price, if price increases, it follows that production will be at a level where marginal cost is higher than before. So if costs are accelerating in quantity (positive 2nd derivative) this will be achieved at higher output level.

Note: beware that the title "Law of Supply" is also used for Say's Law.

ADDENDUM
With constant returns to scale in the production function and price-taking behavior, we know that costs are linear, i.e. marginal cost is constant and $c''=0$. In such a case, the "Law of Supply" is defended in the following way: since the profit maximizing problem is now not well-defined, the size of each firm is indeterminate. Then we invoke, although not explicitly model, size constraints (which is reasonable). Then a higher price is an indication that quantity demanded exceeds quantity supplied, and due to size constraints the current firms won't be able to fulfill the excess demand. This will induce new firms to enter the market, and so overall quantity supplied will increase.

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