Does a per-unit tax shift the demand curve too when dealing with an inelastic supply curve?

I’m studying Tax Incidence from Stiglitz’s ‘Economics of the Public Sector’, and it says that the tax incidence doesn’t change whether I use a shift in the demand curve or one in the supply curve. However, they also talk about producers bearing the entirety of the tax when the supply is perfectly inelastic. My question is - in the top panel, can’t I shift the demand curve downwards and have the price reduce? Why does my book say that the price remains unchanged, and therefore, the producer bears the entire burden of the tax?

• You can also refer to this answer for clarity: qr.ae/pG0F9y
– Amit
Nov 2, 2023 at 9:10

In the diagrams, the price being measured on the vertical axis is the consumer price. The consumer price differs from the producer price because their is a tax.

So, the short answer is that if you instead measured the producer price on the vertical axis, then you would move the demand curve down by the amount of the tax as you suggest. The new equilibrium producer price would go down by the amount of the tax. The new equilibrium consumer price would not change because the consumer price is equal to the producer price plus the tax.

For a longer more mathematical answer, see below.

Let $$D(p)$$ denote the demand. In the first diagram, the supply function is given by $$S(p)=Q_0$$ for all $$p$$.

Before the tax, the equilibrium (consumer and producer) price is defined by the equation

$$D(p_0)=S(p_0) \iff D(p_0)=Q_0$$

When there is a specific tax, the equilibrium consumer price $$p_1^d$$ and producer price $$p_1^s$$ are related by the equation $$p_1^d=p_1^s+t.$$

The equilibrium prices are defined by the equation

$$D(p_1^d)=S(p_1^s)\tag{E}$$

In terms of the consumer price this can be written as:

$$D(p_1^d)=S(p_1^d-t).$$

So, if the vertical axis is measuring the consumer price, then the demand curve does not move because $$t$$ is not in the argument of the function $$D$$. Although $$t$$ does enter into the argument of $$S$$, the supply curve does not move either because supply is inelastic. Since none of the curves move, we have $$p_1^s=p_0$$ and so

$$p_1^d=p_1^s-t=p_0-t.$$

On the other hand, if we write $$(E)$$ in terms of the producer price we have:

$$D(p_1^s+t)=S(p_1^s)$$

So, if the vertical axis were instead measuring the producer price, then the demand curve would move because $$t$$ is in the argument of the function $$D$$. The supply curve does not move (regardless of whether it is inelastic or not) as $$t$$ does not enter into its argument. The demand curve would shift down by the amount of the tax so that $$p_1^d=p_0-t$$ and so $$p_1^s=p_1^d+t=p_0$$.

You may find this more general illustration of these two different ways of representing the effects of a specific tax useful: tax imposition on supply and demand curve

• Thank you! Extremely insightful
– Anu
Dec 8, 2023 at 14:28