Incidence is all about who suffers from a tax. From the standpoint of politicians, lawyers, or accountants, this is a simple question. Who does the law say has to pay the tax? This is called statutory incidence. However, the issue is much more complex than those knuckleheads realize. There is what is known as tax shifting which is when the statutory burden of a tax is pushed onto someone else. These people who get hit with the true burden of the tax suffer its economic incidence. So those employer mandates for health insurance Clinton talked so much about may have had a statutory incidence on firms but an economic incidence on workers.

A unit tax is a tax of a set amount on a unit of output. An example would be a tax of 5 cents on a can of beer. The tax would be the same on a can of Pabst's Blue Ribbon (dirt cheap with emphasis on the dirt) as it would be on a can of expensive Beck's. An ad valorem tax is like a sales tax. If it was 10%, then if Pabst's sold for 10 cents before the tax, then with tax it would sell for 11 cents with the tax (assuming inelastic demand). If Beck's sold for $1.50 per can, with tax it would sell for $1.65. A unit tax on suppliers raises the supply curve by the amount of the tax. An ad valorem tax on suppliers rotates the supply curve upward from the point where it hits the horizontal axis. For graphing simplicity, I will generally work with unit taxes since it's easier to draw a parallel shift. (The Pabst verse Beck's example is a little misleading because they are supposedly different goods, although both taste terrible to me.)

As you may recall from previous studies in economics, elasticity of supply and demand figure in here somehow. As a rule of thumb, the more inelastic is supply or demand, the more of the burden of the tax will be borne by the inelastic sector.

Note however, that our analysis here assumes "partial equilibrium." That is, we are thinking about only one market, but this can be very misleading. It could well be that introducing a tax on beer will cause me to give up drinking beer and switch to drinking orange juice. Here I avoid paying any tax, yet since I clearly bear some incidence of the tax, that is I'm worse off, because I prefer to start my day with Pabst rather than orange juice.

Supply and Demand and Taxes

I swear to God that the effect of a tax on suppliers was the same as a tax on demanders. Yet some people don't want to believe. Let's see if I can make some converts.

What is a demand curve? A relation between the price and the quantity demanded, usually downward sloping in price-quantity space. What is a supply curve? A relation between price and quantity supplied, usually upward sloping in price-quantity space.

Let P_D and P_S be functions of quantity and be, respectively demand and supply curves. That is, P_D is the price to demanders of buying a unit of the good, and P_S is the net price suppliers receive from selling a unit of the good.

In equilibrium with no taxes, P_D(Q₀) = P_S(Q₀) for some Q₀.

100 - 0.5*Q₀ = 10 + 0.4*Q₀ => Q₀ = 100, P_D(Q₀) = P_S(Q₀) = $50

Let's put a unit tax on suppliers of $9. This increases the price they would demand to provide each quantity supplied.

In equilibrium with this unit tax on suppliers,

Before the tax, 100 units were sold at a price (to buyers and sellers of $50. After the tax, only 90 units are sold at a market price of $55 dollars (paid by buyers and received by sellers but then the sellers had to pay the $9 tax per unit sold). The after tax payment received by the sellers was only $46.

The total dead weight loss (DWL) triangle was 1/2*t*dQ = 1/2*$9*10 = $45 (the units of dead weight loss are dollars because the tax is in units of dollars per unit and the units on dQ are units so their product comes out dollars). The DWL coming out of consumer surplus is 1/2*dP_D*dQ = 1/2*$5*10 = $25. The DWL coming out of producer surplus is 1/2*dP_S,Net*dQ = 1/2*$4*10 = $20. The tax revenue is t*Q₁ = $9*90 = $810. The tax revenue effectively extracted from demanders is dP_D*Q₁ = $5*90 = $450. The tax revenue effectively extracted from suppliers is dP_S,Net*Q₁ = $4*90 = $360. The total burden on consumers from this tax (DWL and tax revenue) is $25 + $450 = $475. The total burden on suppliers from this tax (DWL and tax revenue) is $20 + $360 = $380. Clearly due to the DWL, the total burden on society from this tax exceeds the tax revenue. Another way to think about this is that in this case, the ratio of DWL to taxes raised was $45/$810 = 5.6%, so for every dollar of revenue raise here, welfare falls by a bit over 5 cents. I hope these tax revenues are spent very wisely.

What about the case of a $9 unit tax put on demanders? After paying the tax per unit, this will effectively decrease the amount the demander will choose to buy at any given price set by the supplier, as can be seen by rewriting the demand curve as Q_D on the left hand side as a function of P_D and t.

In equilibrium with this unit tax on demanders, P_D(Q₂) = P_S(Q₂) for some Q₂. (Note that this is just the same case as before except that instead of the tax appearing positively on the right hand side, it is now negative and on the left. Thus the two cases are perfectly equivalent.)

100 - 0.5*Q₂ - t = 10 + 0.4*Q₂ => Q₂ = (90 - t)/0.9 = (90 - 9)/0.9 = 90.

P_D,Net(Q₂) = 100 - 0.5*Q₂ - t = 100 - 0.5*90 - t = 100 - 0.5*90 - $9 = $46. (Price to Demanders after paying the unit tax)

P_S(Q₂) = 10 + 0.4*Q₂ = $46 (Price to Suppliers)

Before the tax, 100 units were sold at a price (to buyers and sellers of $50. After the demanders paid the tax, only 90 units are sold at a market price of $46 dollars (paid by buyers and received by sellers). The total dead weight loss (DWL) triangle was 1/2*t*dQ = 1/2*$9*10 = $45. The DWL coming out of consumer surplus is 1/2*dP_D,Gross*dQ = 1/2*$5*10 = $25. The DWL coming out of producer surplus is 1/2*dP_S*dQ = 1/2*$4*10 = $20. The tax revenue is t*Q₂ = $9*90 = $810. The tax revenue effectively extracted from demanders is dP_D,Gross*Q₂ = $5*90 = $450. The tax revenue effectively extracted from suppliers is dP_S*Q₂ = $4*90 = $360. The total burden on consumers from this tax (DWL and tax revenue) is $25 + $450 = $475. The total burden on suppliers from this tax (DWL and tax revenue) is $20 + $360 = $380. Clearly due to the DWL, the total burden on society from this tax exceeds the tax revenue. Also clearly, the cases where the unit tax is up on demanders and when it is put on suppliers are the same.

So, in the real world, does it matter where the tax is put? It may, if the collection and transactions costs differ by type of tax. It may be much more difficult to collect from consumer than firms.

[Note: The following brief discussion of ad valorem taxes is just included for completeness. I’m not going to ask any questions about it. You can skip over it to get to elasticity issues.]

What about an ad valorem tax on suppliers? Let the ad valorem tax rate be, oh, about 19.56522%.

P_D = P_D(Q) = 100 - 0.5*Q (Demand Curve)

P_S = P_S(Q) = (10 + 0.4*Q)*(1 + τ) (Supply Curve with ad valorem tax)

P_S = P_S(Q) = (10 + 0.4*Q)*(1.1956522) (Supply Curve with ad valorem tax)

In equilibrium with this ad valorem tax on suppliers, P_D(Q₃) = P_S,Gross(Q₃) for some Q₃.

100 - 0.5*Q₃ = (10 + 0.4*Q₃)*(1 + τ)

=> Q₃ = (100 - 10*(1+τ))/(0.5+0.4*(1+τ)) = 90 (Yech, is it any wonder I prefer working with unit taxes?)

P_D(Q₃) = 100 - 0.5*Q₃ = 100 - 0.5*90 = $55 (Price to Demanders)

P_S,Gross(Q₃) = (10 + 0.4*Q₃)*(1 + τ) = (10 + 0.4*90)*(1.1956522) = $55

(Gross Price to Suppliers)

P_S,Net(Q₃) = 10 + 0.4*Q₃ = P_S,Gross/(1 + τ) = $55/1.1956522 = $46

(Net Price to Suppliers)

The DWL, tax revenue, and everything else works out just the same as before. Remarkable.

An ad valorem tax on demanders? Let the ad valorem tax rate again be tau = 19.56522%.

P_D = P_D(Q) = (100 - 0.5*Q)/(1 + tau) (Demand Curve with ad valorem tax)

P_D = P_D(Q) = (100 - 0.5*Q)/(1.1956522) (Demand Curve with ad valorem tax)

P_S = P_S(Q) = (10 + 0.4*Q) (Supply Curve)

In equilibrium with this ad valorem tax on demanders, P_D(Q₄) = P_S,Gross(Q₄) for some Q₄.

(100 - 0.5*Q₄)/(1 + tau) = (10 + 0.4*Q₄)

=> Q₄ = (100 - 10*(1+tau))/(0.5+0.4*(1+tau)) = 90

This step is just the same as when the tax was on the suppliers. So again the problems are perfectly equivalent and so I can stop.

Elasticity, Tax Burdens, and DWL

Imagine that demand for some product is perfectly inelastic.

In equilibrium, with no taxes, P_D(Q₅) = P_S(Q₅) for some Q₅. However, we know that Q_S is 100 so P_S must be 50.

Now let's put a $9 tax on demanders. Q_D = 100, P_S = 50, and P_D,Net = $50, P_D,Gross = $50 + t = $50 + $9 = $59. There's no DWL since there's no change in quantity exchanged. Tax revenue is t*Q₅ = $9*100 = $900, the burden of which is entirely born by the demander.

What if the $9 unit tax had been on suppliers?

In equilibrium, with no taxes, P_D(Q₆) = P_S(Q₆) for some Q₆. However, we know that Q₆ is 100 so P_S,Gross must be $50+$9 = $59, as must, by extension, Q_D.

There's no DWL since there's no change in quantity exchanged. Tax revenue is t*Q₆ = $9*100 = $900, the burden of which is entirely born by the demander.

What if the supply of the product was perfectly inelastic?

Everything turns out the same as above with the inelastic demand. What if a $9 unit tax is put on? Everything is the same except that the total burden is on the supplier.

What about a perfectly elastic demand curve with a $9 tax or supply or demand?

What about a perfectly elastic supply curve?