Axel Leijonhufvud at an Institute for New Economic Thinking Conference

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 of  everyone’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=1/2. 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 = 1/2.    (GDP Per Person Under a Flat Tax)

… 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.