Why does quantity supplied increase with price in economics?

Question

The Law of Supply is my worst enemy in economics because I could never truly understand it, and as a result, the stuff I learned after that was built on a weak foundation. The Law of Demand is totally different though, it makes perfect sense to me. I have spent hours thinking about this and I've figured out exactly what I don't understand.

The Law of Supply would make perfect sense to me if price was substituted with revenue. With each good supplied, revenue would increase in a linear manner. However, as I realized, it is not revenue we are dealing with. The way the Law of Supply works, revenue would increase in an exponential manner if we took a supply schedule and multiplied price and quantity for each price level.

I don't understand why price has to increase if quantity increases. Shouldn’t the increase in quantity supplied already generate more revenue to cover the extra costs of production? Why should the price be spiked up to further increase revenue? Why wasn’t the price that high in the first place if the good could be sold like that?

The Law of Supply seems counterintuitive to me in some ways. If you have only a small amount of something to supply, shouldn't you make the price HIGH so that those rare fools willing to pay for overpriced things (the ones at the top of the demand curve) would clear your stock and maximize your revenue?

One explanation that almost made sense is that the more you tried to produce, the higher the costs of production would get. A producer would have no choice but to raise prices if costs of production were that high. But this only makes sense if costs of production increase EXPONENTIALLY, which I don't understand why would happen with EVERY good! Why is it always assumed that marginal cost is increasing and not remaining constant?

On a side-note: how would the Law of Supply work in the digital realm, where stock is basically infinite? What would a supply and demand graph look like there?

On a side-side-note: why are the supply and demand graphs really considered CURVES when they are almost always represented by lines?

Maybe there's something fundamentally wrong about my understanding of this. It's supposed to be easy to understand, but for me it's not. I would be so grateful if someone could clear this up for me!

Answer 1

Not all (current and potential) production has the same costs.

Some production has very low additional cost: maybe all the factories and workforce are already in place, they're close to where the product is sold, and it's very little effort to start production and get new product to market. Other production has higher costs.

When the price is very low, then in general only the lowest-cost production will happen, as any other production would generate a loss, not a profit.

As the price rises, then additional forms of production become profitable. It becomes worthwhile for new investors to move into the sector, and for workers to re-train into that industry, for new factories to get built even on more expensive land, and so on and on.

So, when the price is high, all the lowest-cost production happens, as before. AND lots of the higher-cost production happens, too. So the quantity supplied, increases.

In a well-functioning market, no one is a price-setter - no supplier, no demander; the price arises automatically from the collective responses of all of the participants. So if a producer has only a small quantity to sell, they can't just set a high price, and reap excess profits. If they tried to do that, then someone else would see the excess profits on offer, and go in and undercut the incumbent supplier, driving them out of business. Sometimes, we do see cases where a supplier can set an excessively high price. Then, either new investors do indeed come in, maybe after a year or two; or the industry gets investigated for anti-competitive practices, and measures are taken to restore the market's competitiveness.

It's important to remember that this happens in theory and in practice. It's been observed countless times over many centuries, for just about every product and service that has a functioning market.

Answer 2

I believe that I have found the problem in this line:

I don't understand why price has to increase if quantity increases.

The concept is actually the other way around.

Think of it in the same way as the Law of Demand.
If you go to the grocery store and you see a food that you like selling for \$0.25/lb, you would buy a whole lot of it before the price rises.
Conversely, if you go to the grocery store and you see a food that you like selling for \$100/lb, you would probably wait to buy this item until it is cheaper or at least buy a small amount of it.
In economics, the price drives the quantity demanded by the consumer.

Now let us look at the Law of Supply.
Imagine that you are the owner of a company. You go to the store, and you see that the item you are producing and the similar items produced by your competitors is selling for \$0.25. You would not necessarily want to produce a lot of the product because the margin (profit) between the selling price and the production costs is (presumably) small.
Conversely, imagine going to the store and seeing that the item you are producing and the similar items produced by your competitors is selling for \$100. You would want to produce a lot of the product because the margin between the selling price and the production costs is (presumably) large.
In this case, as in the other case, the price drives the quantity produced by the supplier.

Answer 3

In fact, the law is quite easy to prove (and holds under very general assumptions). Consider a firm that chooses which quantity $q \geq 0$ to supply taking the price $p > 0$ as given. Let $C(q)$ denote the firm's total cost from supplying $q$ units so that the firm's total profit can be written $pq - C(q)$. Assume that the firm chooses $q$ to maximise its profits; and let $q^*(p)$ denote the firm's optimal supply when the price is $p$. We then have the following:

Proposition [Law of Supply]. If $p > p'$, then $q^*(p) \geq q^*(p')$. That is, the firm's supply of the good is weakly increasing in its price.

Proof: Since the firm maximises profits, supplying $q^*(p)$ must be at least as profitable as supplying $q^*(p')$ when the price is $p$. That is,

$$ pq^*(p) - C(q^*(p)) \geq pq^*(p') - C(q^*(p')).$$

Similarly, profit maximisation implies that supplying $q^*(p')$ is at least as profitable as supplying $q^*(p)$ when the price is $p'$. That is to say,

$$ p'q^*(p') - C(q^*(p')) \geq p'q^*(p) - C(q^*(p)).$$

From these two inequalities, it is easily inferred that $p[q^*(p) - q^*(p')] \geq p'[q^*(p) - q^*(p')]$. So if $p > p'$, it must be that $q^*(p) \geq q^*(p')$. QED.

A few remarks on the above:

• The derivation just given concerns a single firm. However, if every firm's supply is weakly increasing in price, then total supply must be weakly increasing in price.
• As the derivation makes clear, the law of supply does not rely on the assumption that $C''(q)>0$. However, if you want to ensure that supply is strictly increasing in the price, you will want to assume strictly increasing marginal cost.
• Unlike the law of demand, the law of supply is very general. In contrast, it is easy to construct cases in which the solution to utility maximisation problems violates the 'law' of demand.
• Finally, we should remember that the concept of supply is only well defined under the assumption of price taking (i.e. firms choosing $q$ taking $p$ as given). So while the law of supply holds under very general conditions, the conditions in which it is meaningful to even speak of supply are far more limited.

Edit: It may also be helpful to provide a proof of a stronger law of supply. Unlike the previous proof, this does rely on increasing marginal cost:

Proposition [Strong Law of Supply]. Assume that $q^*(p) > 0$ and $C''(q) > 0$ for all $p > 0$ and $q > 0$. Then if $p > p'$, then $q^*(p) > q^*(p')$. That is, the firm's supply of the good is strictly increasing in its price.

Proof: Since optimal quantities are strictly positive,

\begin{equation*} \frac{d\pi}{dq}\biggr\rvert_{q^*} = p - C'(q^*) = 0 \implies p = C'(q^*) \end{equation*}

Hence, if $p$ increases, $C'(q^*)$ increases. Furthermore, we have assumed that $C''(q) > 0$ (i.e. $C'(q)$ is strictly increasing in $q$). Thus, if $p$ and therefore $C'(q^*)$ increases, it must also be that $q^*$ increases. QED.

Answer 4

I was just trying to understand this myself, and I think I get it now. Let's think of it as two subquestions: 1) why should any firm ever make intermediate amounts of any good (as opposed to none, or as much of it as possible)? 2) why does the intermediate amount they should make, increase with its price?

So at first I was also confused. Suppose you're selling bicycles. You want to maximize your profit. At any price, the more bicycles you sell, the more revenue you get from selling bicycles. So shouldn't you always want to make and sell as many bicycles as you possibly can, no matter what the price?

For any one particular good (like bicycles): no, you shouldn't.

The first insight is that making bicycles costs money, which you could have used to make and sell something else instead (or which you could have invested).

That's not enough on its own, though - maybe you should just either make as many bicycles as possible, or not make any bicycles and instead make as many as possible of whichever other thing is more profitable to make and sell, i.e. has a higher price-to-cost-of-production difference.

The second insight is that the marginal cost of production for a good - how much it costs to make "one more" of it - can change with how many of the good you've produced. The 10th bicycle you make, might cost more or it might cost less to make, than the first did.

Economists either assume, or argue, that for many goods the marginal cost of production increases with quantity: it costs more to make the 1000th bicycle, than to make the 100th. It's not all that clear to me why, but let's take it as a given.

To visualize why marginal cost of production might be important, first forget price and revenue and profit, and let's just say that for some reason you want to make as many "units" of stuff as possible. Assume for simplicity that you only know how to make bicycles and tricycles.

It could be that making your first bike is cheaper than making your first trike. But assume, as economists do, that the marginal cost of production increases, for both goods. As you keep making bikes, eventually there comes a time when instead of making "one more bike", you find it more economical to switch and start making trikes instead. So if you have a fixed budget, you won't necessarily want to make no bikes at all, or make as many as possible. You'll make an intermediate number of bikes (as well as an intermediate number of trikes, such that their marginal costs of production are the same).

You don't actually want to maximize the number of goods, you want to maximize profit. Define marginal profit to be the difference between price, and marginal cost of production. If the price of bikes is fixed, and the marginal cost of production increases with quantity produced, then of course the marginal profit decreases with quantity produced. You want to keep making bikes, until the marginal profit of making "one more bike" is lower than the marginal profit of making a trike instead.

The higher the price of bicycles, the higher the marginal profit for bicycles, at all quantities. Therefore the higher the price of bicycles, the more you can make before the marginal profit decreases to the point where you should stop and make a tricycle instead. Therefore the higher the price of bicycles, the more bicycles you will find it profitable to make.

Answer 5

Lots of people have good, comprehensive answers, but here’s a very short one:

First, you’re assuming the firm is a price setter. In competitive markets, both producers and firms are price takers. When we draw the demand curve, we assume a consumer faces a given price; similarly, when we draw a supply curve, we assume the firm faces a given price.

Second, the law of increasing marginal cost suggests that the cost function is increasing exponentially—that is, the second derivative of the cost function is greater than zero. This isn’t always true, but it generally is—which is why the Law of Supply isn’t true as often as the Law of Demand; in fact, Perloff’s textbook goes as far as to say: “Although the Law of Demand states that the demand curve slope downward, we have no ‘Law of Supply’ that requires the market supply curve to have a particular slope.” But generally, marginal costs are increasing and $C_{qq} > 0$. The intuition behind this is diminishing marginal product. If, for example, a firm operates with a given input $x_i$, with a production function $f$ such that

$$\frac{\partial^{2} f}{\partial x_i^2} < 0$$

then, to produce the same units of output with additional inputs, the firm pays more for the same output, so the firm’s marginal cost is increasing—i.e., the total cost curve is increasing exponentially. In fact, the supply curve can be thought of as the marginal cost curve of many firms combined, and so has a positive slope.

Answer 6

You are a supplier of magic crystals. You dig them out of a plot of ground you own, and, aside from your effort and the fuel you use for your digging machine, they are "free".

Some are not buried too deeply and take only 5 minutes and 10 cents worth of fuel to dig out, but you've already dug those up, so you have to dig deeper -- they take twice as long and need twice as much fuel. And after you've dug those then you will have to go deeper still.

Answer 7

look from a businessman's point of view.Higher prices mean higher profits(assuming same costs).Hence people from other industries will want to enter into the industry where prices of the product has risen.Hence the no. of suppliers will increase and so will quantity supplied.This reasoning isn't exhaustive as arguments such as the incumbent firms in that industry increasing their production is also legit. hope that helps!

Answer 8

I have been pondering this while enjoying a bowl of cereal as you will soon discover, and I think it is important to mention that this question is probably applied to a 'Short Run'..... yes prices & quantity supplied increase for now..., this is made easier to understand with this example,

Immediate- right now, how many bowls of cereal can you make.... and what would it cost you?(revenue - House, Bowls, fridge running cost, etc..)

Short Run- if I gave you an hour, How many bowls of cereal can you make, giving you time to go to the shop find money to buy ingredients, how much can you fit in the car etc... How much would it cost on top of original costs (& could i start taking a bit more profit at this point??)

'Long Run' If I gave you a year and you bought a van, big fridges, many many bowls etc.. eventually the large investments in reaching market capacity/equilibrium will have levelled out and economies of scale will begin effect creating a gradual downward sloping supply curve...

So this should answer the question simply with, price increases with quantity sold in order to facilitate manufacture and costs in the short run (and of course some taking advantage in profit maximisation), but this is not true for the immediate or long run scenarios..