Tax Incidence I - Ec10 model

Let's do some tax-incidence analysis a la EC10. I'm rusty, maybe this will do me some good. Let p^d be the per unit price consumers pay for oil, let D(p^d) be the quantity of oil demanded as a function of price, let p^s be the per unit price suppliers get for their quantity supplied S(p^s), and let t be the tax per unit that the government collects. It doesn't matter if it collects it from suppliers or consumers. In order to reach equilibrium in a competitive market, we need demand to equal supply:

D(p^d)=S(p^s)

and payments by consumers to equal receipts by the suppliers plus receipts by the government

p^d=p^s+t

What is the effect on p^d of a change in the tax? Substituting p^d-t for p^s in the first equation, taking the derivative with respect to t, and solving for dp^d/dt, we get:

dp^d/dt=1/(1-D'/S').

Here D'=dD/dp^d (the slope of the demand curve, as seen from the p axis) and S'=dS/dp^s (the slope of the supply curve as seen from the p axis). So when D'=0, meaning D doesn't change much when p^d changes, dp^d/dt =1, so the increase in tax just increases the price to consumers by that same amount. Consumers foot the bill. The situation is similar to this:

Ignore that stuff about shifting S. Not sure what that's supposed to mean. On the other hand, as S' goes to zero, dp^d/dt heads to zero. In this case, a tax increase will be paid by the supplier. The situation is similar to this:

So which graph represents our oil market? Probably neither. But politicians who want to raise taxes on oil companies or at the pump better hope it’s not the first. I got these images from here.