As Axel Leijonhufvud wonderfully spoofed in “Life Among the Econ,” one of the bread-and-butter tasks of a working economist is to build and study economic models. Some of these models are meant to be a reasonable representation of some aspect of the economy, while others are meant to be only what one of my favorite philosophers, Daniel Dennett would call “intuition pumps”—parables that give us the pregnant analogies and intellectual workouts needed to raise our economic IQ for thinking about the real world. In this post, I’ll tell you a tax parable. It is not meant to have any immediate moral for the real world, but only to provide food for thought. 

Economic models are inhabited by creatures called “agents.” Agents are stand-ins for people. The key ingredients in economic models are

1. what the agents want,
2. what the agents know, and
3. what is possible for the agents to do.

In this model, the agents are all identical (more identical than real-life identical twins), and want three things: consumption C, leisure time L and a public good (think of city parks or Mars landers) labeled G to stand for government purchases. They know everything going on in the model, and the key thing they can do is divide up their time between making consumption goods C, enjoying leisure L, and making the public good G. Let’s measure the amount of each good by how much time it takes to make it or enjoy it. All the different uses of time have to add up to all the time the agents have. Since all the time they have is 1 day per day, measuring everything in time equivalents enables us to say that   

          C+L+G=1.                  (Only So Much Time in a Day Equation)

You might want to think of this as waking time being divided up, with sleep time off limits to the model. You also might find it helpful to imagine that a stripped-down central bank has no other job than to keep the wage at $1 per (waking) day.

The details of how much the agents want each of C, L and G are governed by the preferences and utility function most beloved of economists: Cobb-Douglas preferences, represented by a Cobb-Douglas utility function. Because I, like many economists, love logarithms (in particular the natural kind), I will use logarithms to write down the utility function.  But since many readers will not have the same love for logarithms that I have, I am giving you fair warning to avert your eyes from the following utility function:

          (1/3) log( C ) + (1/3) log(L) + (1/3) log(G)             (Utility Function)

You can escape having to think any more about logarithms, if you know just one thing: Cobb-Douglas preferences make agents want to devote a fixed fraction of their spending to each individual good. (That is the way to maximize utility.) In this case, if C, L and G were ordinary goods, the agents would want to devote 1/3 of their spending to each. Update: If you want to learn more about logarithms and why this utility function tends to lead to equal amounts spent on each good, see my posts “The Logarithmic Harmony of Percent Changes and Growth Rates” and “The Shape of Production: Charles Cobb’s and Paul Douglas’s Boon to Economics.”

The closest we can come to treating consumption, leisure and the public good in this model as ordinary goods is if we imagine a social planner. In the real world, free-market economists are not at all fond of social planners, seeing them as chauffeurs on Friedrich Hayek’s Road to Serfdom. But as long as they are confined inside of economic models, free-market economists love social planners best of all. The reason is that–as long as there are no distortionary taxes or other more complex distortions–the free market delivers the same outcome as a wise and benevolent social planner. In other words, the social planner I am talking about is not a fallible human, but the Invisible Hand. The one limitation to the benevolence of the Invisible Hand is that the Invisible Hand sometimes favors some individuals over others, making some rich and some poor. But in this model, all are alike to the Invisible Hand, and there is equality.  But there is another complication for the Invisible Hand. The Invisible Hand knows just how hard the agents should work once the amount of the public good G has been decided, but needs a government to decide how much of the public good G to make. Fortunately, since everyone is identical in this model, as long as the principle of equality is maintained, there are no political disagreements in this model, and the Invisible Hand plus a democratic government would yield the same result as an all-wise, benevolent social planner committed to equality: each agent would spend a 1/3 of her time making consumption goods, 1/3 of her time enjoying leisure, and 1/3 of her time making public goods. (“Agents” are traditionally female ever since the dawn of political correctness, and I will hew to that tradition here.)

Recall now that to bring forth the Invisible Hand in all its power from Aladdin’s lamp, the taxes must be non-distortionary taxes.  Non-distortionary taxes are taxes that do not create perverse incentives, which means they need to be taxes where the amount does not depend on what the agents do.  Since all the agents are being treated equally, that means a head tax: each agent pays exactly the same amount, regardless of how much she earns.  To show that with a head tax the Invisible Hand does the same thing as a benevolent, all-knowing social planner, think of things this way. Once the political decision is made to have the amount of the public good that 1/3 ofeveryone’s time can produce (financed by the head tax), each individual then wants to divide up the remaining time equally between the two private goods: consumption and leisure.  Since 2/3 of the time remains after the public good G is produced, dividing it equally between consumption and leisure leads to 1/3 of everyone’s time being devoted to consumption and 1/3 to leisure, exactly as the social planner would have done.   

Whatever time the agents are not spending at leisure, they spend working. So spending 1/3 of their time at leisure means they spend 2/3 of their time working. With the amount work produces worth $1 per waking day of work, that means that output (GDP) is 2/3 of a dollar per person per day, or 66 and 2/3 cents per day:  

          Output = C+G = 1-L = 2/3.    (GDP Per Person Under a Head Tax)

So far so good. Now, suppose that in order to have a tax system more like other worlds, this economy switches over to financing the public good by a flat tax on labor income. (The government is only going to raise enough taxes to pay for the public good G.) Think of the agent’s decision of how much time to spend working under the flat tax. She is working when she isn’t at leisure, so she is working 1-L per day.  If the tax is at rate t, then she takes home (1-t)(1-L) dollars each day after taxes if the wage is $1 per day. Firms have to pay the whole before-tax wage of $1 per day, so with competition, prices end up at $1 for a day’s worth of consumption goods and $1 for a day’s worth of the public good.

We haven’t yet figured out the best size of government (amount of the public good G) is when there is a flat tax. But whatever G is, if the agent sees G as fixed, to maximize the rest of her utility she will want to “spend” an equal amount on the other two goods: leisure and consumption. But what is the “price” of leisure she sees that we should use in figuring out how she thinks of herself as spending on leisure? If she works a bit less, she only has to sacrifice the after-tax wage $(1-t) per day’s worth of extra leisure. So let’s think of her total spending on leisure as (1-t)L dollars per day. Spending the same amount on consumption as leisure (as the Cobb-Douglas preferences lead her to do) then means that

          C = (1-t)L.        (Equal Shares Equation: Flat Tax)

But she also has to be able to pay for the consumption. Since she takes home (1-t)(1-L) dollars per day, the amount she can afford is

           C=(1-t)(1-L).     (Budget Constraint: Flat Tax)

There has been a debate online about how much algebra students should be taught in high school. (One of the best pieces in that debate is this one.) One of the arguments in favor of learning algebra is that one never knows when the urge to analyze an economic model might strike. Combining the Equal Shares Equation and the Budget Constraint, 

          (1-t)L = C = (1-t) (1-L)       (Combination Equation: Flat Tax)

The tax rate is less than 1, that is, less than 100%, so (1-t)>0 and we can divide by it to find that 

          L = 1-L.          (Equation to Solve for Leisure L: Flat Tax)

This one is easy to solve: L=½. That is, under the flat tax, an agent always spends half of her waking time at leisure and the other half of her time working.  

Being at leisure half the time means the agents work the other half of the time. So

          Output = C+G = 1-L = ½.    (GDP Per Person Under a Flat Tax)

That is, GDP under the flat tax is 50 cents per person per day under the flat tax instead of the 66 and 2/3 cents per person per day it was under the head tax. Since ½ is equal to ¾ of 2/3, imposing a flat tax instead of a head tax has caused GDP to fall by ¼ or 25%.   

It is time to think about the size of the government–in this case, the amount of the public good G.  Let’s imagine that the government in this world experiments with various levels of G for a little while before they settle on the best amount.  The agents in this world will notice something interesting: the size of the government has no effect on GDP! As long as the budget is balanced, with the flat tax just paying for G, time worked, and therefore output, will stay constant at ½.  What is going on? As I wrote in two passages of my post

“Can Taxes Raise GDP”: 

… what about a consumption tax that is a certain percentage of everyone’s consumption?  On the one hand this makes people feel poorer so they want to work more, but on the other hand, what someone can buy with an extra hour of work is less, so they want to work less.  The standard view is that these two effects will roughly cancel each other out.  So the amount people want to work—and thus GDP in this simple model—will stay about the same. …The basic argument for the standard view is that to households, a consumption tax looks a lot like a wage cut.  And we have a lot of information about what higher or lower wages do to desired work hours.  Among people who have to live on their own wages, there is surprisingly little difference in how many hours people want to work based on whether they have high wages overall or low wages overall over the course of their lives.  

In the model at hand, it is easy to see how a tax increase looks like a wage cut: the after-tax wage is 1-t.

With G having no effect on the fraction of time worked and therefore no effect on GDP, the  amounts of consumption and the public good have to add up to the constant GDP of ½:

          C+G = ½.

Equivalently,

         C = ½ - G.

Thus, more of the public good G comes at the expense of consumption, one for one. So even though a larger government does not reduce GDP, it does cost something.  It has an

opportunity cost

in reduced consumption.

What is the right size of government if it is stuck with financing G with a flat tax? Again, the Cobb-Douglas preferences help. Holding leisure fixed at ½, Utility is maximized by equalizing the amount spent on consumption and the amount spent on the public good. Since the two amounts add up to ½, to equalize them, the quantity of consumption and the quantity of the public good must both be ¼. Here again, as under the head tax, this happens through the political process, but with everyone identical and treated equally, there is no reason for any disagreement. Since half of output was to be used for the public good, the flat tax rate had to be 50%.   

As in all standard economic models, all of the agents in this model understand the economics of the model perfectly. But let’s pretend that some of the children, too young to be agents making economic decisions in their own right–while eating their portions of their mothers’ consumption–ask questions about taxes: “Mom, in the olden days when our people experimented with different sizes of government, it seemed that the size of government and the level of taxes had no effect on output. Does that mean that taxes don’t matter?” And the mothers all carefully explain to their children that in the truly ancient days (before they had begun to follow the customs of other worlds), when there was a head tax instead of a flat tax, output had been higher: 66 and 2/3 cents per person per day, instead of 50 cents per person per day. And in those truly ancient days, agents had had both more consumption and more of the public good (with the same 50-50 shares of output in C and G as now). So, they told their children “Taxes do matter for output.”

But one bright and brave girl said “Mom, in those truly ancient days, people may have had more consumption and more of the public good, but they had less leisure. Are you sure they were better off?” That led to an algebra lesson. Her younger sister averted her eyes in fear of the equations. But this bright and brave girl learned (with her mother writing with big spaces in between equations to make things less fearsome) about utility functions. She learned that utility functions only had more and less but otherwise had no meaning, so that any increasing function of a utility was as good as the original utility function. And she learned that 

exp(3*[ (1/3) log( C ) + (1/3) log(L) + (1/3) log(G)]) = C L G,

so that one could compare how well-off people were in the truly ancient days to how well-off they were now by looking at C L G just as well as by looking at

(1/3)log( C )+(1/3)log(L)+(1/3)log(G).

In the truly ancient days, under the head tax, C L G was

(1/3)(1/3)(1/3) = 1/27.

Now, under the flat tax, C L G was 

(¼)(½)(¼) = 1/32.

In order to compensate for the effect of the flat tax, current consumption C would have to be increased by the factor 32/27, holding L and G constant, in order to make agents as well-off as in the truly ancient days under the head tax. Since 32/27 is about 1.185, it would take an 18.5% improvement in the technology for producing consumption in order to have the same effect on agents’ welfare as it would to switch back to the head tax, her mother told her.    

But the bright and brave girl was not finished with her questions. “Mom, what if when I grow up I am not as good at making things as the other agents and so have a lower income? Would I still be better off under the head tax than under the flat tax?” Her mother first reassured her “In all of our world, no one has ever been any different from anyone else in productivity once they were grown. But let’s look at your question as a math problem.” After setting up the problem as a cubic equation, the mother left her daughter to do the problem as math practice. Assuming she was less productive than others (A<1, where A was her relative productivity) the bright and brave girl could see that, under the flat tax she would pay only A/4 in taxes (and others would pay ¼), while under the head tax she would pay 1/3.  But under the flat tax, G would be only ¼, while it was 1/3  under the head tax. Also, under the flat tax, the annoying incentive to work less and the smaller income effect from taxes would mean she would have leisure of ½ and consumption of A/4, while under the head tax, both her consumption and leisure would be [A-(1/3)]/[1+A]. Though the bright and brave girl was not completely certain her answer was correct, the number she came up with was that as long as she was at least 86.4% as good at making things as all the other agents when she grew up, she would still be better off under the head tax.  

But even then, once she had solved the cubic equation, the bright and brave girl had one more question: “Mom, why do we have to follow the customs of other worlds and have a flat tax instead of a head tax?”