After reading some comments, I thought it would be helpful to go back to the basics of tax incidence analysis. Once again, it’s important to keep in mind the difference between tax incidence and tax collection. If we mix up those two, we won’t get anywhere.

Look at the economy right before and right after the IRS cashes the tax check. Whose pools of money have shrunk? That’s tax collection. A corporation’s income properly belongs to its shareholders and so when the corporate income gets taxed, the collection is against the shareholders.

Now, compare two economies. One in which the tax exists and the other where the tax does not exist. The difference between the two is the tax incidence. If I choose to work less because the higher tax makes it less worthwhile for me, that’s tax incidence. This is what we are looking at here. And ultimately, when making tax policy, this is what we are interested in.

So let’s look at a very simple market for “widgets”. The supply for widgets is S = 0.5P where S is how many widgets firms are ready to produce at a price P. The demand for widgets is D = 100 – 0.5P where D is the number of widgets people are ready to buy at a price P.

If the quantity of widgets people are willing to buy is greater than the quantity of widgets people are willing to sell, the buyers will bid up the price until the quantities are equal. If the quantity of widgets people are willing to sell is greater than the quantity of widgets people are willing to buy, the sellers will bid down the price until the quantities are equal. That’s the equilibrium where S=D.

So we can solve for this easily:

0.5P = 100 – 0.5P

P = 100 and S = D = 50.

50 widgets are produced and sold at $100 each. So now let’s calculate the surplus. The consumer surplus is how much the item was valued at minus how much it was purchased for. We get that from the demand curve. Consumers were willing to pay $198 for the first widget, but they paid $100 for each widget, so the surplus from the first widget is $98 and so on and so forth. (Actually, we assume continuous quantities, so we integrate the area under the demand curve and subtract the price paid times the quantity. Look at the diagrams for the consumer surplus and producer surplus triangles if the math-speak is confusing you) The total consumer surplus is $2,500. You can do the same for the producer surplus. Producers were willing to be paid $2 for the first widget, but they were paid $100 for every widget so the producer surplus on the first widget was $98. As it turns out, the total producer surplus is also of $2,500. So we have $5,000 of economic value that has been produced through these trades.

Now let’s introduce a $20 tax per widget. We will say that the tax is collected on the buyers of widgets. In other words, the quoted price is a before-tax price. So now we have to adjust the demand function. After all, buyers don’t care about the before-tax price. They care about the money that enters and leaves their wallet whether it’s because of taxes or any other reason. Consumers are now ready to pay $20 less per widget to the producer since they then pay an extra $20 to the government. So we have to adjust the demand curve.

D = 100 – 0.5(P + 20) we can simplify this D = 90 – 0.5P

[Updated Thanks Michael. The math is correct, I simply made a typo in the above-equation]

Solving for the new equilibrium

0.5P = 90 – 0.5P

P = 90 S = D = 45. In other words, now, only 45 widgets are made and the producers are paid $90 for each widget only. Of course, the consumer has to pay the $20 on top which means the consumer is now paying $110 per widget. The total tax collected is $900.

So now once again let’s calculate the surplus for consumers and producers. Well, for the first widget, the consumer was willing to pay $198, but they paid $110 including the tax, so their surplus for that first widget is of $88. The total consumer surplus is $2,025. For the first widget, the producers were willing to accept $2 but they got $90 for every widget so they got a surplus of $88 on the first widget. The total producer surplus is $2,025 for a total economic value created of $4,050. That’s $950 less than the $5,000 created without the tax.

Note that in this simple example, the seller and buyer share equally in the loss from the tax despite the consumer writing the check to the IRS. You could re-do the math having the producer send the check. The result is the same. I will follow up with another post detailing how in different markets, the burden might be shared differently between producers and consumers.

Note that this works the same way whether we are selling widgets to consumers or selling work to our employers. If you tax something, it will cost more to those who buy it, bring in less to those who produce it and less of it will be bought, sold and produced.