The income and substitution effects of a tax cut

Benjamin Franklin once famously wrote that "in this world nothing can be said to be certain, except death and taxes". Taxes may be certain, but the tax rate is not. Tax rates vary widely across the world, and have varied widely over time for each country. Economists are concerned about tax rates because, among other things, they affect the incentives to work.

However, it isn't just any tax rate that affects work incentives. We need to take into account all of the changes that affect a worker's income, as a result of working more. Working more results in more income tax payable, but might also reduce a worker's entitlement to government transfers (social security benefits, student allowances), or reduce their entitlement to subsidies (for example, subsidised housing or healthcare) or rebates (for example childcare rebates). All of those changes need to be taken into account.

The effective marginal tax rate (EMTR) is the amount of the next dollar of earnings that a person loses to taxes, to decreases in entitlements to government transfers, and to decreases in subsidies and rebates. It is the EMTR that best captures the incentive effects of the tax and transfer system. When a worker faces a high EMTR, there is a disincentive to work more. When the EMTR is lower, there is more incentive to work.

To see why, consider what happens if the EMTR decreases. For example, if the government offers an income tax cut, this will decrease the EMTR. The after-tax-and-transfers reward for working increases, making work more attractive. The opportunity cost of leisure time (measured as the after-tax-and-transfers wage) increases, making leisure time less attractive. The worker decides to work more. This is an example of the substitution effect. The relative price of working compared with leisure has increased, encouraging a shift to more work and less leisure.

However, there is also an income effect. The higher wage increases the worker's income, and they use that income to consume more normal goods. Leisure time is a normal good, so the worker wants to consume more of it (and work less). Notice that the income effect works in the opposite direction to the substitution effect here. They offset each other.

This leads to an interesting implication of a tax cut. For some workers, especially those on low wages, the substitution effect (work more) is larger than the income effect (work less). In general, a tax cut encourages those on low wages to work more. However, for other workers, especially those on high wages, the substitution effect (work more) is smaller than the income effect (work less). In general, a tax cut encourages those on high wages to work less. Why might the high wage workers work less? Those workers may realise that they can continue to earn the same amount as before, while working fewer hours. This allows them to continue to spend the same amount as before, and have more leisure at the same time. In fact, some workers may be able to both earn and spend more, and have more leisure time.

The income and substitution effects of a tax cut do not necessarily lead all workers to work more. Some workers will respond by working less. In fact, if we look at work and leisure time over the long run (as in this post), we see workers both earning more income, and spending more time on leisure (and working fewer hours). Changes in tax rates change the incentive to work, but they don't necessarily affect all workers in the same way.

Read more:

• How increased wages over the last century reduced the incentive to work

Carbon taxes are not a costless way to boost the economy

Economists recognise that any choice we make comes with an opportunity cost - for every thing that we choose to do, we are giving up the opportunity of doing something else. This is memorably captured in the acronym TANSTAAFL - There Ain't No Such Thing As A Free Lunch. So, I was interested to read  this article in The Conversation last week, by Mona Mashhadi Rajabi (University of Technology Sydney), about carbon taxes:

A new study has found that a carbon tax, accompanied by “revenue recycling”, can produce both environmental and economic benefits for Australia.

Revenue recycling means money reaped from a carbon tax would be redirected back into the economy. This means the money accumulated from the tax would be redistributed between different stakeholders in the economy without increasing the government’s revenue.

To be more specific, the tax income should be used to support consumption, invest in new research and development projects, and subsidise energy saving and pollution reducing programs. Using this approach makes the tax more politically appealing to companies and other opponents.

So far, so good. But this bit is more than a little surprising:

My study recommends that in the first year, all the tax revenue would be used to support consumption. However, the amount of money allotted to investment would rise from the second year as the carbon tax rate increases. It is estimated that about $57 billion would be available for new technologies over 13 years under this carbon tax.

Imposing a carbon tax would also provide a financial incentive for industry to reduce its fossil fuel use. It would motivate the sector to shift to low-carbon technologies as they would bear a smaller tax bill and reap larger profits.

My study concludes Australia could reduce carbon emissions by 35% while GDP would increase by 0.286% by 2035 and new jobs would be created in research and development. Following the recommended carbon tax design, Australia’s transition to a low-carbon economy would be accelerated, which would benefit both the economy and the environment.

This appears to describe a policy choice with no trade-off. That is, there is no opportunity cost. If the government can tax carbon, pump that money back into the economy to support consumption, and increase consumption and output (GDP), then someone ought to be shouting this from the rooftops. Rajabi is describing a route to infinite consumption. Because, having increased consumption and GDP, the government can then raise taxes again, increasing consumption and GDP by more, and then raise taxes again, increasing consumption and GDP by even more. Repeat this process over and over again, all the way to infinite GDP. Of course, that makes no sense.

To see why, let's consider what happens when you tax a product, and give all of the tax revenue back to consumers as additional income. We'll start with the basic consumer choice model (or the constrained optimisation model of the consumer), as shown in the diagram below. We are using some made up numbers, but the example works the same with other combinations of numbers. The diagram shows the budget constraint for a consumer with income of $1000, choosing between buying carbon goods (with a price of $1), or all other goods (AOG, also with a price of $1). The consumer can choose to consume anywhere on their budget constraint or below it (this is the feasible set). The budget constraint runs from a bundle of goods with no carbon goods at all, and the consumer spending all of their income to buy 1000 AOG, to a bundle of goods with no AOG at all, and the consumer spending all of their income to buy 1000 carbon goods. The slope of the budget constraint is equal to the relative price of the goods (the price of carbon goods, divided by the price of AOG), which is equal to 1 ($1/$1). The consumer is trying to get to the highest possible indifference curve, which is the indifference curve I0 (for more on indifference curves, read this post). They buy the bundle of goods E0, which contains 500 of AOG (costing $500) and 500 carbon goods (costing $500).

Now consider what happens when carbon goods are taxed. Let's assume that the tax is so high that it doubles the price of carbon goods [*]. The effect of this change is shown in the diagram below. The consumer's budget constraint pivots inwards to the red line, and becomes steeper. The slope of the budget constraint is now equal to 2, which is the new relative price of the goods (the price of carbon goods, divided by the price of AOG, which is now $2/$1). If the consumer were only buying AOG (and no carbon goods), they could still buy 1000 AOG. However, if they were only buying carbon goods (and no AOG), they could now only buy 500 AOG. The consumer can no longer afford the bundle E0 - it is outside the feasible set (outside the budget constraint). Instead, the consumer will consume the bundle of goods on the highest indifference curve that they can reach on the new budget constraint. That is the bundle E2, which is on the indifference curve I2, and contains 600 of AOG (costing $600), and 200 carbon goods (costing $400). Notice that the consumer is still spending all of their income, but they have been made much worse off (they are now on a lower indifference curve, meaning that they get less utility from their consumption).

Now let's see what happens if we recycle the tax revenue back to the consumer. The budget constraint remains steep (because the relative price of carbon goods and AOG is still equal to 2), but moves outwards parallel to the previous budget constraint. It doesn't move out all the way to E0 though. At E2, the consumer was paying $200 in carbon tax, so their income goes up by $200, to $1200. The new budget constraint, shown in blue, extends from the point where they spend all $1200 on AOG, buying 1200 AOG, to the point where they spend all $1200 on carbon goods, buying 600 carbon goods. The consumer can now do better than the bundle of goods E2, and reach a higher indifference curve I1, by buying the bundle of goods E1. That bundle contains 700 AOG (costing $700), and 250 carbon goods (costing $500). However, notice that even after the tax revenue has been recycled to them, the consumer is not as well off as they were before the tax was introduced. They may be spending more ($1200 instead of $1000), but their utility is lower.

This demonstrates that there is a cost to this policy. The consumer pays it in the form of lower utility. To be fair, in a sense Rajabi is correct. The consumer is now spending more than before ($1200 instead of $1000), so consumption has increased. But that is an increase in nominal terms. The price of carbon goods has gone up. If we think about the value of the goods that the consumer is buying, at the original (non-taxed) prices, they are only buying $950 worth of goods now (700 AOG worth $700 and 250 carbon goods worth $250), compared with $1000 before the tax was introduced. Or if we think about the value of the goods that the consumer was buying before the tax was introduced, but valued at the new prices, they were previously buying $1500 worth of goods (500 AOG worth $500 and 500 carbon goods worth $1000), but now they are buying $1200 worth. Either way, they are now buying a total bundle of goods that is worth less - the consumer's real income has decreased.

Of course, most people who advocate for 'revenue recycling' are not arguing that every taxpayer would receive back exactly what they paid in the tax. Many prefer that the 'carbon dividend' would be paid only to low income households. That would change the analysis somewhat. First, there would be high-income households for which the analysis looks much like the second diagram - a carbon tax with no offsetting dividend. These households would be unambiguously worse off with the tax. Second, there would be low-income households for which the analysis looks like the third diagram, but with the increase in income from the revenue recycling being much larger. In theory, these low-income households would be better off as a result of the carbon dividend. In other words, the tax would redistribute income from high-income taxpayers to low-income taxpayers. However, it still wouldn't increase the real value of consumption or GDP overall. It would simply redistribute who is doing the consumption spending.

Overall, we can conclude that it isn't possible to simply pump up the economy endlessly by taxing carbon goods and recycling the tax revenue to consumers as additional income. Any analysis that shows this is possible is clearly missing something important. There ain't no such thing as a free lunch.

*****

[*] In reality, this would require a tax of more than 100 percent, because a tax raises the price that consumers pay, but producers share the burden of the tax, so the whole tax amount is not passed onto consumers as a higher price. For more on this, see this post. However, for simplicity we will assume that the entire tax is passed onto the consumer in the form of a higher price. This assumption would affect the exact numbers from the example, but not the overall conclusion.

Book review: From Here to Economy

I enjoyed Todd Buchholtz's book New Ideas from Dead Economists (which I reviewed here a couple of years ago), so I was looking forward to reading his 1995 book From Here to Economy. The subtitle promises that it is "a shortcut to economic literacy", and it mostly delivers. However, the problem with books based on a collection of contemporary economic and policy issues is that they don't necessarily age well. And sadly, that is the case for this book. The underlying economics is fine, but a reader in the 2020s is not necessarily going to get as much out of the examples as a reader in the 1990s would have.

That said, Buchholtz is clearly a fan of small government, and the book presents that as a position that most economists agree with (which it isn't, and wasn't even in the 1990s). This shines through in his selections at the end of the book of the five greatest economists of all time. Buchholtz nominates Adam Smith, David Ricardo, Alfred Marshall, John Maynard Keynes, and Milton Friedman. I have no argument with the first four, but Friedman would be way down my list, behind at least Paul Samuelson, Arthur Pigou, Gary Becker, Ronald Coase, and Kenneth Arrow (not necessarily in that order).

The other thing that marks an older book is the predictions it makes. Buchholtz does fairly well on this score. He is skeptical that there would be agreement on adopting the Euro until the 21st Century (the Euro was adopted in 1999), and he predicts Nobel Prizes for Amartya Sen (who won in 1998) and (who won in 1995, as the book was being finalised). His third predicted Nobel winner, Martin Feldstein, never won but probably deserved to.

Putting aside its datedness, the book is a lot of fun. Buchholtz can be hilarious, as these excerpts show:

If virgin wool sweaters go out of style because they itch too much, their prices would fall, so that both manufacturers and sheep get the message: stop making so much.

Perhaps sheep are utility maximisers? And on the Efficient Markets Hypothesis:

You would probably do just as well choosing a stock by throwing a stockbroker at a dartboard as listening to his advice - and you would save money.

And on the safety of mutual funds:

The word mutual certainly does not by itself mean that the fund is especially safe. Think of the Titanic as a mutual vacation.

Having started by explaining some basic macroeconomics and microeconomics, the book then moves to personal finance, which is a different approach to that taken in most pop economics books. And the end of the book is devoted to economic history, which recalls Buchholtz's earlier book. The personal finance section remains useful, although the examples are obviously dated. The economic history is, as I noted in my review of New Ideas from Dead Economists, very welcome.

Overall, I enjoyed the book, but for current readers, there are others that I would recommend before it. This is not because of the quality of the writing and explanations, but only that the datedness of the examples limits their usefulness.

The deadweight loss of free beer

My excellent (and sports-mad) colleague Shaen Corbet shared with me a story about the Nebraska-Northwestern college football game played in Ireland last year. As reported in the Irish Mirror:

But what couldn't be predicted was the events in the Aviva Stadium on Saturday night as technical glitches saw thousands upon thousands take advantage of free food, drink and alcohol.

There was always going to be a party atmosphere for the first Aer Lingus College Football Classic since 2019, a momentous occasion to remind us just how lucky we are to have these events back.

But the Aviva Stadium was rocking like a Harry Styles gig from just a few weeks prior as queues went a dozen deep as match attendees fleeced the concession stand and bar in a one time only offer of everything being free.

A Twitter (ok, X, but it was Twitter then) user posted this video of the queues for beer, where you can see that the entire foyer area in front of the bar is jam-packed with spectators looking for free beer. Usually, if the price is reduced to zero, we would expect to see a shortage. That's because the sellers would want to sell less (because it is less profitable) at the same time that the buyers are wanting to buy more.

However, in this case, the government chose to subsidise the beer (Shaen tells me it was to reduce the chance of unruly fans getting out of control). We can see the effect of this subsidy, reducing the price to zero, using a supply and demand model as shown below. If the beer market was operating in equilibrium, the price would have been P0, and the quantity of beer traded Q0. Instead, the government paid a subsidy to the beer sellers. We demonstrate this on the diagram with a new curve, S-subsidy, which is below the supply curve S by the amount of the subsidy (which was exactly enough to lower the price from P0 to zero). The effective price for the beer sellers increases to PP, which is the zero price they receive from the spectators, plus the per-unit amount of the subsidy. The quantity of beer demanded increases to Q1, and so does the quantity of beer supplied. There is no shortage of beer.

It is worth considering the impacts on economic welfare of this subsidy though. Consumer surplus is the difference between the amount that consumers are willing to pay (shown by the demand curve), and the amount they actually pay (the price). In the diagram, at the equilibrium price and quantity, consumer surplus is the triangle AEP0. Producer surplus is the difference between the amount the sellers receive (the price), and their costs (shown by the supply curve). In the diagram, at the equilibrium price and quantity, producer surplus is the triangle P0EB. Total welfare is the sum of the two areas (consumer surplus and producer surplus), and is equal to the triangle AEB.

Once the subsidy is introduced, the consumer surplus increases to AQ1O, while the producer surplus increases to the area BCQ1O. The government loses the area of subsidy, which is the rectangle PPCQ1O (this rectangle is the per-unit amount of the subsidy, multiplied by the quantity of subsidised beer). Total welfare is the sum of consumer surplus and producer surplus, minus the subsidy (the subsidy is subtracted because it has an opportunity cost of lower government spending in other areas), and is equal to the area AEB-ECQ1 [*]. In other words, total welfare is lower by ECQ1 as a result of the subsidy. This is the deadweight loss of the subsidy.

To add insult to injury, even though the price of beer may have been zero, the cost of beer was not free. That's because you have to factor in the cost of the time spent waiting to be served (which will be much higher when the queues are longer), as well as the loss of enjoyment of missing part of the game while waiting for beer. Plus, there are external costs of over-drinking imposed on other fans. Shaen, who was at the game, tells me that there were spectators who vomited all over other spectators after over-indulging in 'free' beer. So, free beer wasn't necessarily a good deal for everyone, least of all for Irish taxpayers and for those who needed a dry-cleaner (and possibly a counselling session) after the game.

[HT: Shaen Corbet]

*****

[*] The overlapping areas of consumer surplus, producer surplus, and subsidy make this tricky to see. However, there is a shortcut. The area of total welfare is the area that is in-between marginal social benefit (MSB) and marginal social cost (MSC) out to the quantity that is traded (in this case, Q1). When MSB is greater than MSC, this represents positive welfare (the area AEB). But when MSB is less than MSC, this represents negative welfare (the area BCQ1).