Tax Incidence II - Ec10 model

The previous post on tax incidence made the usual assumption of a competitive market. In some sense this is an extreme assumption, since markets are rarely completely competitive. What about if we make an assumption at the opposite extreme? Suppose that we have a monopolistic supplier of oil. Suppose also that the cost of producing oil is relatively low, so we can approximately describe the supplier's behavior as trying to maximize his revenue. (I've heard it said that it costs Saudi Arabia $2/barrel to pump oil.) Let D be the demand as a function of price p. Since our idealized supplier can determine the quantity supplied and can set the price, he seeks to maximize his revenue R as a function of p and the tax rate t:

R=(p-t) D

Taking the derivative with respect to p and setting the result to zero gives the equations

D+(p-t)D_p=0

t=p+D/D_p

which the supplier can use to determine p given t. Here the subscripts denote differentiation. How does p change with a change in t? Treating t as a function of p in the last equation, and taking another derivative with respect to p gives

t_p=1+(D/D_p) _p

p_t =1/(1+(D/D_p) _p)=1/(2-DD_pp/(D_p )²)

where p_t is the rate of change of p with respect to t. If we assume that the slope of D is increasing (D is convex down), then D_pp>0 so the last equation tells us that p_t >1/2. We conclude that at least half of an increase in tax will show up as an increase in price. How big can p_t be? There is no limit. It is possible for p_t to be greater than one, so for example an increase in the tax by $0.10 may result in an increase of price by $0.20. Here is an example of such a situation. The demand curve here:

looks pretty ordinary, and indeed has pretty tame elasticity:

The price increases with the tax thusly:

With p_t everywhere greater than one:

If the current price is $4.40, we can expect an increase in price of about $0.17 if the tax is increased by $0.10. So what's the overall conclusion? Beats me. I have no idea which model is appropriate. Does anyone? Can it be tested? See the reference to Jim Manzi's recent article below.