22 Jan 2019

## Why a Revenue-Neutral Tax Dividend Check Doesn’t Negate the Deadweight Loss of a Tax

I can see in the comments here, regarding my article about the economists writing a pro-carbon tax letter to the WSJ, that several of you aren’t seeing the big picture on the dividend stuff.

So in this post, let me give you a quick numerical example to show the standard way economists think about taxes, and how they distort behavior to cause “deadweight loss.”

Suppose there are two goods, X and Y. The consumer has a utility function:

u = ln(X) + ln(Y),

where ln(*) is the natural logarithm.

Further suppose that the price of X in the market is 1, while the price of Y is 2. Finally, suppose the consumer has a budget of 3000.

Without any taxes, the consumer optimizes by maximizing utility subject to the budget constraint. Typing this through WordPress I’m going to have trouble, but hopefully you can follow the argument, and I’m going to switch to lowercase variables now:

From the budget constraint we have

x + 2y = 3000

which means

y = (3000 – x) / 2.

Substitute that into the utility function and you have

u = ln(x) + ln[(3000-x)/2].

Take the derivative of u with respect to x and set to 0, in order to find the local maximum:

du/dx = 1/x + [2/(3000-x)]*(-1/2) = 0

Then use algebra to find

x* = 1500.

You can use a similar procedure to find y* = 750.

Notice that the consumer ends up spending the same portion of income (i.e. half) on either good. That’s not a coincidence; that’s what happens when you choose ln(*) for your utility function with the same coefficient on each term like this. (Someone who knows what he or she is talking about: Please confirm this for me. I want to make sure I’m not generalizing too much.)

OK, now suppose the government imposes a \$1 tax on good X, so that the after-tax unit price on X is now 2, and the price on Y (from the market) is still 2.

Further, the government is going to make the tax revenue neutral. Specifically, the government will take the incoming tax receipts and give them back, lump sum, to the household.

THIS IS IMPORTANT: The household doesn’t optimize in the new situation thinking that its own tax payments are coming back. Rather, the household takes the dividend check from the government as a fixed dollar amount, which is the same regardless of the household’s behavior. However, in equilibrium it has to be true that what the house pays in taxes on good X, it gets back “dollar for dollar” in the dividend check from the government.

(If you want to make it more realistic, assume there are millions of identical households paying into one giant pot, and then the government sends the same lump-sum refund to every household. That way no individual household’s spending on X can affect its dividend check very much.)

OK so from the household’s perspective, its budget is now 3000 + R, where R is the total receipts from the new tax. (But remember, when doing the calculus, the household treats R as a constant, not as a variable.)

So the household’s new budget constraint is

2x + 2y = 3000 + R.

To optimize, the household allocates half of its total budget to good x, and the other half to good y, just as before.

So we can write that in the new equilibrium, after this tax has been placed on good X, that the optimal x* is

x* = (1500 + R/2) / 2,

i.e. half of the total budget divided by the new (after-tax) price of X.

We also know, however, that the total Receipts R will be equal to \$1 times the number of units of X purchased. I.e. we know

R = x

Substituting and solving, we find

x* = 1000.

Knowing that total tax receipts are \$1000, we now know that the total budget is 3000 + 1000 = 4000, which means the household allocates 2000 to good Y, meaning

y* = 1000.

So, in the new equilibrium, the household gets a \$1000 dividend check from the government that bumps its total budget up to \$4000, and then–facing equal (after-tax) prices on goods X and Y–the household spends \$2000 on each good. Since each good (after tax) has a price of \$2, that means buying 1000 units of each good. And yep, it checks out, that buying 1000 units of X causes the household to hand over \$1,000 in tax receipts to the government.

So notice that the household hasn’t been “hurt financially” by the tax. The tax by construction was “revenue neutral.” The household got back in a dividend check “exactly what it paid in higher prices on the taxed good.” So according to the WSJ letter signers, you would think the household should be indifferent to the tax.

But nope, that’s wrong. If you substitute the two different outcomes into the utility function, you see that the household gets more utility from the first scenario.

Specifically,

ln(1500) + ln(750) > ln(1000) + ln(1000).

This is what economists have in mind when they talk about taxes having a distortionary effect. Economists don’t really “follow the dollars around”; the reason taxes (in general) are “bad for the economy” is that they alter behavior. In this example, the household is led to buy fewer units of X and more units of Y, in a way that makes the household worse off.

This is standard stuff. I wouldn’t get into this type of numerical example in a Principles class, but I would definitely do it in an Intermediate Micro class for undergrads.

Of course, the economists who signed the WSJ letter know all of the above. It’s also true that they think the negative externality of climate change is something to counterbalance against resource allocation in the pure market place. But my simple point is that they are being incredibly misleading by saying most households will “financially benefit” from a carbon-tax-and-dividend scheme.

To repeat: In the example above, the household was held “harmless financially” if that’s the way you want to think about it. But that hardly means the tax had no impact. Now if I told you that good X was actually “electricity from a coal-fired power plant,” while good Y was “transportation in a solar-powered train,” then you could say that the tax was a good move ALL THINGS CONSIDERED. But it would be wrong to claim that the environmental benefits were good and that the dividend check compensated the household for the higher price on coal-provided electricity. No, the higher price on good X leads to a definite loss of utility in terms of the direct services provided by X and Y, which maybe are justified by the gain in environmental quality (not modeled here). But the dividend check doesn’t render the tax costless.

This is the stuff Tyler Cowen surely had in mind when he wrote (italics in original):

[The lump-sum dividend rebate] tries to make a carbon tax a free lunch, which it is not, no matter how great the longer-term gains.  I don’t believe in economists tricking people, even though I will admit tricking people can be useful.  The tricking is somebody else’s job!

#### 37 Responses to “Why a Revenue-Neutral Tax Dividend Check Doesn’t Negate the Deadweight Loss of a Tax”

1. David R. Henderson says:

Good job, Bob.
You might want to do a separate blog post on this also: https://worthwhile.typepad.com/worthwhile_canadian_initi/2019/01/marginalists.html

• Bob Murphy says:

Thanks David, I can’t explicitly deal with it anytime soon, but I’ll try to remember to link when I write for IER on this again.

2. Transformer says:

Your post is an excellent description of the dead-weight loss from a tax. However I am going out on a limb here and claiming that taxes on negative externalities do not generate dead weight loses and therefore your post in not relevant to the matter at hand. Taxes on negative externalities cause prices to more accurately reflect costs and increase rather than diminish consumer surplus.

If one assumes that the proposed carbon tax would effectually tax an externality then I do not see how it is the least bit misleading to claim that ‘The majority of American families, including the most vulnerable, will benefit financially by receiving more in “carbon dividends” than they pay in increased energy prices’ . They benefit from the increased utility brought to them from the carbon tax, and they get a rebate that exceeds the carbon tax they pay as well !

• Dan says:

So your argument is that after the tax the majority of families will be able to afford to buy even more stuff than they currently can buy?

• Transformer says:

– Everything Bob says above is equally true for a subsidy as for as a tax. A subsidy creates a dead-weight loss.

– An externality is functionally the same as a subsidy

– A correctly applied tax on an externality removes a dead-weight loss. If GoodX in Bob’s example is “electricity from a coal-fired power plant” then the household would get greater utility after the tax is applied and if the tax is set to the correct level.

I don’t think I’m missing anything here – but please let me know if I am.

• Transformer says:

@Dan,

They will be able to afford better stuff, not necessarily more stuff.

• Bob Murphy says:

Transformer, you’re still not getting it.

In the current example in this blog post, the household is clearly hurt, even though he is “made whole” financially by the rebate check. So the compliance cost of a tax is not simply the net reduction in dollars.

Now when it comes to policies that mitigate climate change, there are benefits and costs. When you keep saying, “So long as the carbon tax is no higher than the actual externality, then the household’s utility goes up,” all you are saying is the benefit > cost. OK fine, but still the WSJ is leading readers to believe that the cost = 0 (at least to them personally). And no, as this example shows, that’s not true.

• Transformer says:

I think it would be fairer to say not that “I am not getting it” but that I am disagreeing with you!

At the risk of falling flat on my face – here goes:

You are framing the issue in terms of the text book view in which a tax causes a deadweght loss. It seems like you are saying “on the one hand we get the deadweight loss from a taxation. but this may be outweighed by the benefits to the environment of the tax”.

I think this is the wrong way to look at it as the text book case really does not apply to taxes on externalities. The negative externalty causes a deadweight loss and the tax corrects that loss.

If the carbon tax leads to the price of carbon-related goods internalizing the externality then this alone will lead to household utility increasing with no need to look to some secondary benefits existing outside of the market’s free operation and that somehow counter the effects of the tax.

Bottom line: American Families will (in theory,
anyway) benefit from the increased utility brought to them from the carbon tax, and they get a rebate that exceeds the carbon tax they pay as well !

• Transformer says:

Or to say it slightly differently:

If there are some economic activities that for a typical American household cause more dis-utility in terms of environmental damage than they add in terms of utility from consumption then a carbon tax will directly add to the utility of that typical American household.

Any rebates that exceeds the tax they pay will be a additional boost to their utility.

• Harold says:

Transformer, I get what you are saying now, I think.

If we start from a standard supply demand curve, we have the market equilibrium point where they cross. This determines the market price and efficient level of production.

This is the point Bob is starting from. The tax creates the deadweight loss triangle on the left. Resdistributing the tax revenue cannot restore this as it is redistribution of a smaller pie. Households can still be damaged even if they pay no more tax.

Transformer says that this should not be the starting point. The negative externality shifts the price down and creates deadweight loss triangle to the right of the market equilibrium point. The tax then shifts the price back to the market equilibrium price and removes the deadweight loss.

On this viewing there is no deadweight loss from the tax so Bob’s analysis is not relevant.

Which is the more appropriate starting point depends on what claim is being made by the WSJ economists.

Are they saying “even if there were no economic cost to carbon emissions then this tax will still make most people no worse off or better off”?

If this is the case then Bob is right as the “no externality” position is the right comparison to make.

Or are they saying “Given that there is an economic cost to carbon emissions, this will make our tax system more progressive and ensure the benefits are directed to the poor, so most families need not worry”?

I am not sure if they have this covered technically, but my reading is that the impression given to the layman is closer to the former.

However, it is a difficult sell that the benefits will be in increased consumer surplus whereas the cost will be in dollars. What they are trying to convey perhaps is that there will be benefits in consumer surplus and we will make sure there is no cost in dollars to most people. In the same way the tax revenue redistribution could not restore the previous position because the pie was smaller, the correct environmental tax ensures the pie is bigger, so -redistribution can in principle make sure most people are not worse in dollar terms.

• Transformer says:

Good analysis, Harold.

You capture my point well in the first half and I suspect you are right and Bob is assuming the 45 WSJ economists mean “even if there were no economic cost to carbon emissions then this tax will still make most people no worse off or better off” – because that would be consistent with everything he has said in last 2 posts.

I think that would be an out-of-context thing for them to mean though.

They make a single statement on the matter ‘The majority of American families, including the most vulnerable, will benefit financially by receiving more in “carbon dividends” than they pay in increased energy prices’. Even Bob’s analysis agrees this is true – but he deems it ‘sleight of hand’ because he claims there will be a utility loss hidden behind that financial gain. But in the context of a discussion on a negative externality (which this clearly is) then there will in fact be a utility gain not a loss (for the reasons that you restate).

• Bob Murphy says:

Transformer, why did the economists say (paraphrasing), “Most households will receive more in dividend checks than they pay in higher prices”? Why didn’t they say something like, “Most households will receive more in dividend checks than they pay for ice cream”? After all, that would be a true statement as well.

But they didn’t say that. No, they focused the reader’s attention on the amount the household would pay in higher prices due to a carbon tax, versus the check coming from the government because of those tax payments. Clearly, the WSJ letter signers are trying to convince households in such a scenario that they will suffer no downside whatsoever from the carbon tax. And that is wrong.

In other words, Transformer, you keep saying, “#1 They benefit in utility” and “#2 They benefit from getting more in dividend than they pay in higher prices.”

But you could just as well say, “#1 They benefit in utility” and “#2 They benefit from getting more in dividend than they spend on ice cream each year.” After all, it’s better if your dividend check is higher than your ice cream expenses, rather than not. So it’s just as valid a statement as what you keep saying.

But that would be a total non sequitur if we focused on ice cream. The *reason* the “higher prices they pay because of a carbon tax” seems more relevant, is that that sounds like the cost of the carbon tax to any particular household. But, as this blog post shows, that is wrong. It is not significant to know that the dividend check is higher or lower than what the household spends on higher prices due to the carbon tax, except in the same way that getting a check higher than some dollar amount \$x is significant.

• Transformer says:

Bob,

1. Do you agree that when looking at the general case of a negative externality that could be addressed by a tax that the normal “dead weight loss from a tax” scenario that you describe in your post does not apply ?

2. Are you taking the position that the WSJ letter is claiming that even if the carbon tax would not address any externality it would still make the typical American household better off?

If the answer to both questions is ‘yes’ then we are probably having a discussion about what the WSJ45 really meant – and we just don’t know so we may never get a consensus.

Now my response:

They say: “households will receive more in dividend checks than they pay in higher prices”. Leaving aside any loss or gain in utility from the tax effects on the economy then the tax will have 2 effects on household budget
1. They will pay higher prices
2. They will get a rebate check

If the dollar value of the rebate check > (higher prices x qty bought) then the WSJ45 are right on this specific point. (IceCream sales would be totally irrelevant to the matter).

Now you say on the matter:
‘The *reason* the “higher prices they pay because of a carbon tax” seems more relevant, is that that sounds like the cost of the carbon tax to any particular household’

And I suppose it does sound a bit like that, but in the context of the whole sentence it sounds to me much more like half of the data that is needed to ascertain whether the tax itself (independent of any actual utility change caused by the tax’s more general effect) adds or subtracts from household budget constraint.

• Bob Murphy says:

Transformer wrote: “2. Are you taking the position that the WSJ letter is claiming that even if the carbon tax would not address any externality it would still make the typical American household better off?”

Great question, Transformer. I see now why you are sticking to your guns. I shouldn’t have said, “You still don’t get it,” I should’ve said, “You still keep repeating the way you are framing it, without seeming to acknowledge my equally valid framework.”

So I was tempted at first to say to you, “Yes! That’s exactly what I’m saying about the WSJ letter!” but now I see what you have in mind. If e.g. there were a tax on houses and the proceeds were refunded lump-sum, even if a typical family got more money on net than they paid in higher house prices, people might still think that their deadweight utility loss was bigger than the gain in dollars.

Am I right in thinking that’s your view?

(Just to be clear, I am still certain that the WSJ letter was misleading, Transformer. I’m just saying that I am more sympathetic to the way you are trying to analyze the issue.)

• Transformer says:

My position is more straightforward than that.

If the WSJ letter really is claiming that even if the carbon tax would not address any negative externality it would still make the typical American household better off then I would agree that everything you say in the last post and this one is correct.

However as the WSJ letter is about a tax on a negative externality it seems fair that their comments on the rebate should be taken in that context. In that case (to restate my case yet again):

– The tax will cause prices to adjust so that the typical household will now stop buying things that generate more dis-utility in the form of environment damage than the utility they provide from consumption. Leaving aside the question of how the revenue raised from the tax is spent the impact the tax has on relative prices will tend to increase typical household utility if both utility from consumption and disutilty from environment damage is factored in.

– If the revenue raised is distributed in such a way that our typical household gets more back than they pay in increased prices then this is in addition to the benefits they got directly from my first point (in your example in this post this would merely alleviate some of the dead-weight loss of the tax).

Leaving aside the probably unanswerable question of what the WSJ45 really meant I am curious if you would agree with my analysis of what the rebate would mean in the case of an optimal tax on a negative externality ?

• Transformer says:

There will be some cases (with no negative externality) where the distribution between rich and typical will be such that the rebate will more than compensate the typical household fer the dead weight loss. For example in your post if the typical household gets an additional \$100 in redistribution from the rich then their utility after the tax will be greater than before.

In suppose this would get the WSJ45 off on a technicality but was not what I was thinking off.

3. Tel says:

THIS IS IMPORTANT: The household doesn’t optimize in the new situation thinking that its own tax payments are coming back. Rather, the household takes the dividend check from the government as a fixed dollar amount, which is the same regardless of the household’s behavior.

There’s a hidden assumption that people always defect when facing a Prisoner’s Dilemma situation.

That might be quite standard … but it’s observable that sometimes people don’t behave that way, and if we believe that maintaining solidarity against the tax does improve everyone’s utility there might be good reason to expect some resistance. If most people feel they are “doing the right thing” by reducing CO2 then probably they shrug and go along with it.

(If you want to make it more realistic, assume there are millions of identical households paying into one giant pot, and then the government sends the same lump-sum refund to every household. That way no individual household’s spending on X can affect its dividend check very much.)

Not sure how realistic that is … I’d argue that the households are not identical. Some people feel strongly about the whole CAGW worry and would be spending less on X anyhow. Others feel strongly the other way. Also, budget constraints differ greatly.

• Matt M says:

“Some people feel strongly about the whole CAGW worry and would be spending less on X anyhow. Others feel strongly the other way.”

I wonder… is there any actual evidence of this being a thing?

Does expense/usage of carbon-based energy actually correlate at a statistically significant level with self-professed fear of/believe in climate change?

Anecdotally, the high profile cases (i.e. Al Gore living in a mansion, flying around in private jets, and eating meat) would seem to indicate not.

• Andrew in MD says:

I would say yes, at least to a modest level. For example, a lot of people who are very worried about carbon levels are also quite proud of themselves for driving Priuses. And I have known people who buy “carbon credits” that are supposed to help offset your carbon footprint by funding the development of new forests somewhere in the world.

There are also a few oddballs who have converted their vans to run on used cooking oil from restaurants instead of gasoline in order to reduce their reliance on “non-renewable” fuel sources. Although, I’m not sure how much, if any, benefit this creates from a carbon perspective and these people are a negligible minority in terms of environmental impact anyway.

But what I haven’t seen is someone actually inconvenience themselves by foregoing something they wanted to do. In my experience, it always amounts to “throwing money at the problem” by buying an expensive car or carbon credits and then going about their business as they would’ve otherwise. I have seen a few news stories about people who ride a bicycle or public transport in order to reduce their carbon footprint but I’ve never met one in real life. And those stories almost always mention “fringe benefits,” such as not having a car payment, getting exercise, or having more time to read, which I imagine are often a larger motivating factor than the environmental benefit.

• Matt M says:

My prediction is that if you control for wealth, believers would emit marginally less carbon, but I’d be shocked if it was more than say, 10%.

If you don’t control for wealth, I’d expect believers to emit significantly more carbon, as I think the correlation between wealth and climate change belief is quite high. In other words, if the media is right that climate skeptics are a bunch of ignorant hicks, well, ignorant hicks never get rich enough to emit a lot of carbon even if they wanted to. It’s the rich and famous who buy mansions and fly private jets.

As to your examples, I wonder how many of the common virtue signaling items really do lower carbon emissions significantly. I’ve heard that if you account for the mining of the materials for the batteries, Prius’ don’t save on emissions until you drive them for 100,000 miles. And that riding a bus with less than 10 people on it probably emits more carbon than if everyone drove. And that jogging somewhere probably emits more carbon than riding a (crowded) bus as you respirate much more.

I’m also quite surprised you know people who have purchased carbon offsets. I know quite a lot of people who are heavy environmentalists, who also have a lot of disposable income, and who also participate in lots of recreational carbon emitting activities – the exact type of person you’d expect to purchase offsets and talk about it loudly, but I’ve never once heard of anyone actually doing it. I always figured those existed just as corporate indulgences…

• Harold says:

” I think the correlation between wealth and climate change belief is quite high.”

I am not so sure.

In the USA, there is a strong correlation between belief in action on climate change and party in the USA. Republicans are much less likely to think anything should be done about it (although belief that humans are changing the climate is more even -it is just that Republicans don’t think we should take action.) I also think there is a tendency for younger people to want to do more about it. Both these factors would suggest wanting to do nothing would correlate with greater income or wealth.

” if the media is right that climate skeptics are a bunch of ignorant hicks,”
I also don’t think this is accurate. The media generally reports that political affiliation is much more important than being an ignorant hick. There is also a big difference between ignorant meaning lacking knowledge in general and being ignorant of specific things. Ignorant hick clearly means the former, whereas many media reports of “ignorant deniers” refer to ignorance about the science of climate change. For example, this quote from prof Knutti ““The kinds of statements made by Smith, the president and Pruitt are misguided. They show a woeful ignorance about science and how it works, and in particular about climate science. ”

” I wonder how many of the common virtue signaling items really do lower carbon emissions significantly.”

You are correct that many so-called carbon reduction measures are red herrings. Your jogging example appears to be wrong because respirated carbon dioxide comes from carbon taken from the air. We would need to account for the extra carbon used in growing food for the extra energy you use but I think it very unlikely this would add up to more than running a bus.

• Andrew in MD says:

I think the correlation between wealth and climate change belief is quite high. . . . It’s the rich and famous who buy mansions and fly private jets.

Those are the Boomer Climateers. They like to tell the younger generations that they “need to get their act together” while they enjoy their lavish retirement furnished by the society they helped collapse. I think the young Climateers are generally broke and true-believers of their whole leftist program because they’re desperately searching for some meaning in the destruction left behind by the materialist Boomers, who so thoroughly undermined Christianity and the American economy.

4. Tel says:

(Someone who knows what he or she is talking about: Please confirm this for me. I want to make sure I’m not generalizing too much.)

I can do the maths if that’s sufficient; the criteria for “knows what he or she is talking about” are unclear.

u = ln(X) + ln(Y)

exp(u) = X Y = u*

NOTE: utility is ordinal and has no units so, u* is as valid as u, the stationary points will be the same. Ignore u from here on in.

u* = (M_x / P_x) (M_y / P_y)

u* = (M_x M_y) / (P_x P_y)

NOTE: multiplying by constants is harmless, so move the prices across to the LHS.

u** = M_x M_y

Now prices have become irrelevant. Solve for M_x against constraint of fixed budget B.

u** = M_x (B – M_x)

u** = B M_x – M_x^2

Set the first derivative to zero to find the maximum of the upside down parabola.

B = 2 M_x

B is Budget, M_x is Money spent on X, P_x is Price of X, M_y and P_y are same for Y.

Would the same thing work in higher dimensions? u = ln(X) + ln(Y) + ln(Z)

This is left as an exercise for the reader 🙂

Specifically,

ln(1500) + ln(750) > ln(1000) + ln(1000).

That’s kind of interesting, we get 13.93 > 13.82

Although correct, it looks like there isn’t a whole lot of difference to worry about. Then again, since u has no units, it’s hard to be sure what 0.1 of a util will do to people … but hey some starving kid in Africa doesn’t even have 13.48 utils… so eat those low-carbon beans!

5. Eli says:

The easiest way to prove that the specified utility function implies the agent spends half their income on each good is to solve the general case using the same method you solved the specific case.

The general budget constraint is

P_x*X + P_y*Y = I

Solve for Y

Y = (I-P_x*X)/P_y

Substitute into utility function

ln(X) + ln((I-P_x*X)/P_y)

Take the derivative with respect to X and set equal to 0

1/X – (P_x/P_y)(P_y/(I-P_x)) = 0

Solve for X*

X* = I/(2P_x)

Substitute back into the equation for Y form the beginning (Instead of redoing whole method for Y as Bob suggested)

Y* = (I-P_x*(I/(2P_x)))/P_y = I/(2P_y)

QED

6. Toby says:

The analysis ought to include a disutility from total carbon emissions above a bliss point and a carbon emission per unit of good X and Y for completeness. Now not all costs and benefits are accounted for in the example. I am typing this on my phone so I can’t give one right now. But the conclusion depends on how carbon emissions are valued (also heterogeneity) it seems.

• Bob Murphy says:

Toby, I explicitly discussed that I wasn’t including any negative externality, and that the WSJ letter writers have that in mind. I was keeping this as simple as possible to isolate the particular flaw I noted in their discussion.

They were clearly leading readers to think, “Ah, not only will a carbon tax mitigate climate change, but if I get a bigger refund check than what I pay in higher prices, it won’t cost me anything, either! Make those rich people shoulder the cost of battling climate change.”

And so the point of this example is to show why that logic is wrong.

• Harold says:

An interesting point is that people are perhaps better at quantifying costs and benefits in dollar terms. As you correctly point out, it is quite difficult to appreciate that you will be worse off after the tax and redistribution, even if you have the same dollars in your pocket. It is quite correct to point this out and leads to greater public understanding.

On the other hand, it is also quite difficult to appreciate that you might be better off even if you have fewer dollars in your pocket. One way to avoid trying to explain this reality by derivatising utilty with respect to time is to arrange things so that people do not end up with fewer dollars in their pocket.

You can then tell them that they will not be worse off. This could be true both in dollar terms and in utility terms, although you perhaps understand that most people will only appreciate the dollar part.

I am not sure if the economists letter could be interpreted this way – do you think it possible?

7. Andrew in MD says:

I don’t get what you’re talking about, Bob. In your example, adding the tax increases the household’s budget from 3000 to 4000. That’s a huge windfall, even if they don’t get to burn so much coal. Why would anyone be against that?

. . . j/k

8. Transformer says:

I tried to reverse engineer Bob’s example to see what happens with a negative externality:

Step 1: Start from the same setup and utility function as Bob but have a subsidy rather than a tax.

Initially, price of x = \$1, price of b = \$2 and budget of 3000 gives
x* = 1500.
y* = 750.

Introduce a subsidy of \$1 on good y, paid for by a per capita tax.

Now
x + y = 3000 –T (since both x and y are now priced at \$1)

y* = (1500 – T/2)/1 (since y is priced at \$1 now)
T = y

Solving gives us:

x* = 1000
y* = 1000

So just as in Bob’s case:

ln(1500) + ln(750) > ln(1000) + ln(1000).

The subsidy causes utility loss just like a tax.

Step 2: Switch from a subsidy to a negative externality

What’s required here is to take the same budget constraint, utility function and x* and y* levels but to assume that it is a negative externality rather than a subsidy that moved the outcome from the initial condition to this one and that the optimal solution is still the one from Bob’s initial conditions.

Step 3 : Impose a tax on Y that is sufficient to take us back to the optimal starting point. This will be \$1 tax per unit. This will take us back to the initial scenario (or at least most of the way there).

ln(1500) + ln(750) is still > ln(1000) + ln(1000).

So the negative externality reduced utility and the tax on the negative externailty brought it back again. And all within Bob’s framework (sort of) and with no-one paying an extra dime in taxation!

(I probably mangled the math somewhere but hopefully not enough to change the outcome of anything important).

• Andrew in MD says:

I like what you’ve done here. I think it makes a lot of sense. The problem I have is that there isn’t a good way to price an externality the way you can with a subsidy. Like, if you have a real subsidy, you could, in theory, perfectly reverse it with the opposite tax, because you know exactly what the subsidy is. It would be a silly thing to do, because you could just repeal the subsidy instead, but still, it’s possible.

But if you replace the subsidy with this hand-wavy thing that you can’t really measure to anyone’s satisfaction, then I don’t get how you could calculate what level of tax would properly reverse it. And in the real world, in order to implement a new tax, you have to create another layer of bureaucracy that will be managed by ideologically motivated politicians rather than neutral technicians. Also, in the real world, there aren’t really any neutral technicians. Everyone has initial assumptions and confirmation bias.

• Andrew in MD says:

Another related thought: If the purpose of this revenue-neutral, carbon tax + dividend system is to reverse the undue negative effects of carbon emissions, then a universal dividend that goes evenly to every American citizen seems awfully unjust. Certainly not every American is currently “paying their fair share” for the social cost of carbon. If we’re imagining SCC as a tax that this new system is intended to reverse, then shouldn’t it be reversed in the same proportion it was payed. Do not the owner of the ski lodge, the farmers of certain crops, and the old woman who remembers fondly when her father used to drive his car out on to the lake in the dead of winter deserve a greater share of the dividend than the man who is happy that he doesn’t have to clear snow from his driveway quite so frequently and the young woman who is happy that she can swim in the ocean a half-month later than she was previously able? How can we give everyone an equal dividend and claim that we have reversed an injustice and not merely committed a new one?

• Harold says:

I think an equal rebate to everyone is very difficult to justify economically, but perhaps easy to justify politically.

• Transformer says:

Thanks for the feedback.

Yes, I agree that with a subsidy its much easier to solve the problem – just abandon the subsidy! With an externality the relevant tax cannot be scientifically calculated (even if there is really an externality may be in dispute!) and we end up with the kind of discussion about optimal levels like we have with the carbon tax.

9. Tel says:

Anyone curious to know where Australia is going… outages maked “FORCED EXTERNAL” are power rationing because it’s a hot day and they ran out of supply.

https://www.powercor.com.au/power-outages/full-outage-list/

Doing an approx scan by eye, there’s more than 10k households blacked out right now across Victoria. Probably more if I added then up carefully

10. Tel says:

http://powerhour.alexepstein.com/2019/01/24/power-hour-more-on-carbon-tax-pge-climate-change-bankruptcy-and-more/

Power Hour tackles the same issue … including a systematic evaluation of the externalities of Math education (really).

11. Don’t Be Tricked By Economists on the Carbon Tax | Raymond Castleberry Blog says:

[…] at IER, I have already directly critiqued the WSJ letter. At my personal blog, I wrote out a formal numerical example to illustrate why it was so misleading for the letter to claim that most American families will […]

12. Don’t Be Tricked By Economists on the Carbon Tax - NoPaperNews says:

[…] at IER, I have already directly critiqued the WSJ letter. At my personal blog, I wrote out a formal numerical example to illustrate why it was so misleading for the letter to claim that most American families will […]