# Question about the Law of Supply

I have a contention with the Law of Supply's dictum "if the price of a good falls, then the quantity supplied of that good falls, ceteris paribus."

Imagine a market for widgets produced by multiple factories with colossal machinery; the average cost differs among the factories.

If the price of the good decreases, then, according to the Law of Supply, quantity supplied decreases. The explanation I'm supposed to give in examinations goes along the lines of "the profit margin decreases, hence producers are incentivized to produce less".
Yet I would argue that, provided the drop in price is not drastic, every factory should produce the same amount as before, even those for which the price has dropped below the average cost, as this scenario would leave their owners better off compared to paying the fixed costs without revenue. Thus, the same quantity of widgets would be produced after the price drop.

The holes in my contention that are obvious to me are:

• If the decrease in price was drastic, then some owners would be obliged to shut down their factories.
• If the same machinery could be used to produce bananas instead of widgets, then the factories would switch to banana production if it yields more profit, thereby reducing the quantity of widgets supplied.

I also think that my argument is only valid in the short run: then, the marginal costs of individual firms would be more or less constant, and negligible changes in it would probably be brought about by small-scale laying off of employees. In the long run, owners could sell machinery and buildings and the marginal cost would be more variable.

So, taking account of all assumptions so far, i.e. the decrease in price is not drastic, the machinery is not good for anything other than producing widgets, and we are only considering the short run, then decreases in price should effect only slight changes in the quantity produced: this would be represented by a very inelastic supply curve. But, if interpreted as a marginal cost curve, this curve also says that for tiny nudges in the quantity produced, the marginal cost varies significantly, which is contrary to my earlier understanding that the marginal cost is approximately constant because factories cannot dump machinery easily in the short run.

What am I missing?

The law of supply holds under fairly general conditions. Essentially, it states that "quantities respond in the same direction as price changes." Formally, $$(p-p')(q-q')\geq 0,$$ where $$p$$ is price, $$q$$ is quantity and primed and unprimed reflect two different states. You see that the inequality is weak such that a constant range is not excluded. However, a decreasing supply curve would violate the law of supply.

Moreover, your argument appears to be a bit fuzzy. Sure enough, it implies that for a given price-quantity pair $$(p,q)$$ that $$S(p) = S(p-\epsilon)=q$$, when $$\epsilon$$ price changes are not drastic. However, then consider price-quantity pair $$(p-\epsilon,q)$$ and another non-drastic price drop. Your argument appears to imply $$S(p-\epsilon)=S(p-2\epsilon)=q$$ and so on culminating in a constant demand function, which --it seems-- you do not want to apply. It seems more reasonable to suggest that a marginal price drop also has a marginal effect on supply such that $$S(p)=q$$ and $$S(p-\epsilon) = q - \varepsilon$$, where both $$\epsilon$$ and $$\varepsilon$$ are tiny such that a "non-drastic price drop" implies a "non-drastic quantity reduction."

• I've realized that the extrapolation of "non-drastic price causes negligible quantity change" to an entire supply curve was incorrect, so the supply curve would not be "very inelastic". But I fail to understand how, in my example, significant changes in quantity supplied could be caused by anything but shutdowns of factories due to accumulated non-drastic price drops (I do not get why there should be "non-drastic quantity reductions"); represented by a step-like supply curve. Sorry for revising my original argument. Nov 12 '20 at 18:32
• Perhaps it is incorrect to consider the act of "shutting down" as taking place in the short run? Nov 12 '20 at 18:58
• Of course you don't have to shut down an entire factory, but you can send home one temporary worker for a day or two. It is perfectly fine if the effect of a tiny price drop on quantity is infinitesimally small but non-positve. Nov 12 '20 at 19:33

The law of supply is only valid for continuous and strictly monotonically increasing supply function. It does not apply to any arbitrary supply function. For example, consider trivial example of perfectly inelastic supply.

As you can clearly see with this perfectly inelastic supply no matter what the price is the supply will be some fixed quantity $$\bar{Q}$$. You can also derive other examples of supply functions where law of supply would be violated (e.g. backward bending supply).

There also exist 'step-like' supply curves such as:

where supply changes only once price raises above some threshold. For example, consider situation where every person has 1 house. At price $$p=100$$ only person A is willing to supply house to the market, at price $$p=200$$ person B is also willing to supply their house to the market so supply jumps, but in between it remains constant.

However, the reasoning you provide is not generally valid.

1. Even in presence of fixed cost or constant marginal costs supply is not necessary inelastic. As a matter of fact it usually won't be save for some special cases.

2. There is no a priori reason to think that marginal costs should be constant even in short run. Sure, in some special cases this might be empirically true but it is not generally valid. It is not true that ability of firm to get rid of machinery in short run would determine whether marginal costs are constant or not. Firm's cost function does not depend just on capital but also on material costs, labor costs and other factors. Increasing production will also eventually put upward pressure on prices in factor markets at some point.