Tax Revenue and Deadweight Loss

So far in our videos, we've looked at the effect of taxes on market prices, but we haven't said much about why government levies taxes in the first place, namely to get revenues. So let's look at that and also at the cost of raising revenues, which is deadweight loss.

We can show pretty much everything we need to show with a single diagram. So here is our initial equilibrium. The price with no tax is $2 and the quantity exchanged with no tax is 700 units. Now, let's recall that consumer surplus is the consumer's gain from exchange, and it's this green area here, the area underneath the demand curve and above the price, up to the quantity exchanged. So it's the area above the price of $2 and up to the quantity exchanged of 700 below the demand curve -- this area right here. Producer surplus is the producer's gain from exchange, and it’s the area above the supply curve, up to the quantity exchanged and below the price, below the producer's price. Now, you may also recall that a free market maximizes consumer plus producer surplus. What we're going to show is that when we have a tax, this is no longer true. The intervention into the free market means that consumer and producer surplus are not maximized.

Let's take a look. So suppose we have tax of $1, and using our wedge method, we can find what the new price is going to be for the buyers. It's going to be here. So the new price for the buyers is say, $2.50. Notice now, the consumer surplus is not this large green area since the price is now higher and the quantity exchanged has fallen. The quantity exchanged falls from 700 units to 500 units. So, the consumer surplus with the tax is this smaller green area here. Again, it's the area above the buyer's price, up to the quantity exchanged, and below the demand. So exactly the definition hasn't changed, but because of the tax the price to the buyer changes, and the quantity demanded exchanges, so the consumer surplus changes as well. In this case, it gets a lot smaller.

What about producer surplus? Well, again, the price which the sellers receive falls. So producer surplus is no longer this large blue area, but is now just this much smaller blue area. So the tax reduces consumer surplus and it reduces producer surplus.

Now, what about this area in the middle? Well, fortunately, that's not wasted. That, in fact, is tax revenues. So notice that the tax -- the height of the tax here -- is $1, and there are 500 units exchanged, so the government gets $1 for each of those 500 units. So this revenue, tax revenue, is the area. It's the height of this box times the width, and the height is the tax, the width is the quantity exchanged. So this is tax revenue.

Now, what about this final bit over here? That used to be consumer and producer surplus, but now it's deadweight loss. Nobody gets that. That is lost gains from trade. So remember, people used to trade 700 units. Now they're only trading 500 units. Those units were benefitting people, but they're not anymore because these trades are not occurring. I'm going to explain that in a little bit more detail in the next slide. For now, just be sure that you understand how to label these areas. So this is the new consumer surplus, tax revenues, the new producer surplus, and this area is deadweight loss. Okay, let's explain deadweight loss in a little bit more detail.

Here's the way to think about deadweight loss. Suppose that you're planning a trip to New York and you're going to take the bus. The benefit of the trip to you, the value of seeing the sights in New York is $50. The cost of the bus ticket is $40. So do you take the trip? Is it a value? Yes, you take the trip. The total value of the trip is $10, it's a positive, so you decide to take the trip. Trips is equal to one. You make the trip. Okay, no problem. Now, suppose there's a tax of $20 on bus fares and let's suppose that raises the cost of the trip from $40 to $60. It doesn't have to raise it by exactly that amount, by exactly the $20, but let's suppose it does. Okay, so the cost of the trip is now $60. The benefit is still $50. So do you take the trip? No. The benefit is less than the cost. So now, no trip. Trips are equal to zero. Does the government raise any revenue from you? No. Since you don't take the trip, the government makes no revenue. Is there a deadweight loss? Yes. You have lost the value of the trip. You used to, when there was no tax, you took the trip, it was worth $10, so the world was better off by that $10 of value. Now with the tax, you don't take the trip, so that $10 is a deadweight loss. It's gone. And notice that it's not made up for by revenue. There's no revenue. So deadweight loss is the value of the trips not made because of the tax, and there's no revenue on trips which aren't made. Government only makes revenue on the trips which continue to occur. So deadweight loss is the value of the trips not made because of the tax.

Now, to return this to a more general case, instead of trips, let's just replace that with trades. Deadweight loss is the value of the trades not made because of the tax. Very quickly, here's our diagram again. Before the tax, there were 700 trades. After the tax, there were 500 trades. So these are the 200 trades which are not made because of the tax. And the value of those 200 trades occurs because for these trades, the demanders value them more than it costs the suppliers to provide those trades. So the demanders value the trades as given by the demand curve, the height of the demand curve. The suppliers are willing to supply those trades -- the cost to them is given by the height of the supply curve. So the value, the value minus the costs, if you like, is given by this triangle. Because those trades no longer occur, that value is no longer produced -- that's deadweight loss, the value of the trades which don't occur because of the tax.

Here's one more important point about deadweight loss. Deadweight losses are larger the more elastic the demand curve holding revenues constant. So for example, which of these goods would we more like to tax -- the one on the left where the demand curve is elastic or the one on the right where the demand curve is more inelastic? Notice that tax revenues are the same. So if we have a choice, which good do we want to tax? Well, pretty clearly, we want to tax the good with the inelastic demand because the deadweight losses, the lost gains from trade, are much smaller over here than they are over here. So the tax on the good with the elastic demand -- it's creating a lot of waste in order to get this revenue. Over here, the tax on the good with the inelastic demand -- there's only a little bit of waste for the same amount of revenue.

The intuition here is pretty simple. If the demand curve is inelastic, then a tax won't deter many trades. And that's what we don't want. We don't want to deter a lot of trades, because it's the lost gains from trade which create the problem. We don't get any tax revenue when we deter a trade. There's no tax revenue when you deter an exchange. So we want to make sure that we deter as few exchanges as possible and that will maximize our revenue compared to our loss. Now, sometimes economists are laughed at or derided because this implies, for example, that you ought to tax insulin, a good with a very inelastic demand. Now, there are many reasons for taxing some goods and not other goods, depending upon who uses the insulin, whether it's poor people or rich people or how important health is and so forth. Nevertheless, as a general rule, it is better to tax goods with an inelastic demand than goods with an elastic demand. That's important, and let me give you an illustration of that.

Here's something which you might think would be a good idea to tax --- luxury yachts. They're only bought by the rich, so you're not really harming people very much, right? Well, maybe so. However, in 1990, the federal government actually applied a 10% luxury tax to many luxury goods, including pleasure boats or yachts with a sales price above $100,000. They expected tax revenue of $31 million. The reality, however, was quite different. The tax revenues were only $16.6 million. That was because sales of yachts fell tremendously. Perhaps the yacht buyers decided, well, they could wait a year or two before buying their yacht -- see what happens. Or maybe they decided they could buy their yachts in other countries. Yachts are pretty easy to move around the world. After all, that's what they're for. The net result, in fact, was a loss of 7,000 jobs in the yacht industry. Indeed, the federal government ended up paying out more in unemployment benefits to unemployed yacht workers than it collected in tax revenues from yachts. Because of this, the federal tax was repealed in 1993. The lesson here -- don't tax goods which have really elastic demands. You're not going to get a lot of revenue, you're going to deter a lot of trades, and that will create a lot of deadweight loss, and perhaps, secondary losses for other people, such as the workers.

That's it actually for taxes. The only thing we have left to do is subsidies. We can actually do that in the next lecture pretty quickly because subsidies are just negative taxes. So everything we've said about taxes, with just a few changes to our language, we'll go through with subsidies as well. Thanks.