Home > A Theory of the State > A Model of the State (What Would a Dictator Do?)

A Model of the State (What Would a Dictator Do?)

Let us begin with a model of dictatorship.  This follows from the basic story layed out in The Role of the State.  We begin with a dictator who has established control over some group of subjects.  The dictator incurs some cost in order to protect the property of his subjects.  This is assumed to be an increasing function of the amount of wealth created and will be denoted G(Y) where Y is total output.  So if the population is fixed and equal to N, then each individual (they are assumed to be identical) produces y=Y/N.  Also, the dictator decides what percentage of output to seize (tax) from his subjects.  Let this percentage be denoted by t.  This means he will receive income equal to tY.  The problem faced by the dictator will be to choose t in order to maximize his profit given by tY-G(Y).

In order to do this, the dictator will have to consider the maximization problem faced by his subjects.  The issue, in a nut shell, is that the higher the tax rate t, the less incentive there is for people to produce because they get to keep a smaller percentage of their output.  This can be seen easily by imagining that each member of the economy produces output according to the production function y=l where l is the quantity of labor an individual devotes to production.  Also let us assume that each member faces a cost of labor c(l) which is increasing and convex (increasing at an increasing rate).  The amount of output that a worker gets to keep will be (1-t)l-c(l). (For simplicity I am measuring the cost as the value of foregone leisure in terms of output, so in other words I am counting that as lost goods which means the proper way to interpret this is as the amount by which the workers wealth increases from their initial state where they have all leisure and no production goods.  This is somewhat simpler, and in my opinion no less accurate, than dealing with a worker with a utility function over consumption and leisure)  So the worker’s first order condition for the maximization of this expression will be

c'(l)=1-t                           

From this we can easily see that when t increases the marginal cost of labor will have to decrease which means he will produce less (since c”(l)>0).  Going forward let us assume that c(l)=l^2.  This will make the above equation

2l=1-t

which means l and y will be given by

y=l=(1/2)(1-t).

Plugging this into the dictator’s profit gives us

t(1/2)(1-t)N-G[(1/2)(1-t)N]

Now, to make is simpler, let’s let G(Y)=Y/5.  This means that the first order condition for the dictators maximization problem will be (notice that the Ns cancel out)

1/2-2t*+1/10=0

This will give us

t*=.3     y=.35   

The profit to workers will be .35(1-t)-.35^2=.1225.

The profit to the dictator will be .3(.35)N-.2(.35)N=.035N.

Now let’s imagine a government of some form which makes decisions in order to maximize the prosperity of its citizens rather than of the dictator.  In this case, the government will need to raise just enough money to pay the expense of protecting the economy’s output.  Mathematically, we can impose the constraint

tY=(1/5)Y              which implies t=1/5

In other words, we can just set the tax rate equal to the marginal cost of protection (if G( ) were not linear this would work out a little different).  In this case the problem for each individual worker will be to maximize

.8l-l^2

which gives the first order condition

.8-2l=0

so l* (and y*) will be .4 and the profit to workers will be .8(.4)-.4^2=.16.

There are two important things to notice here.  First is that this benevolent government makes citizens richer by transferring the profit formerly accrued by the dictator to them.  Second, this is more efficient.  This can be seen simply by noting that output increases when the dictator is removed (in this case from .3 to .4).  This also makes the citizens better off.  This happens because the dictator, in his attempt to capture as much wealth from the society as possible, damages the incentive to produce.  He will not destroy it entirely because this would also destroy his source of wealth, but he will do so to an extent which is inefficient.  It is in an attempt to acquire both of these benefits for the people, that men endeavor to establish free rule of law societies.  The rule of law allows men to get the benefits of secure property rights without surrendering to a dictator the ability to loot their property to whatever extent he desires.

Now with this in mind consider the Laffer curve.    This is a theoretical curve showing the total amount of tax collected as a function of the tax rate.  For low levels of the tax rate it is increasing and for high levels it is decreasing.  This was the argument used in the Reagan administration to justify lowering the tax rate.  The position of that paragon of small government conservatism was that the tax rate was so high that it was on the downward sloping section of the Laffer curve so we could actually get more revenue by lowering the tax rate.  And this is the same argument going on now.  In a brilliant sleight of hand, Democrats are now claiming that we need to raise taxes in order to reduce government deficits and Republicans are saying that we need to lower taxes to “stimulate the economy” in order to lower deficits.  The entire debate amounts to an argument of where the maximum point of the Laffer curve is.  In other words, what tax rate maximizes government revenue?  Or in still other words, what would be the appropriate tax rate for an absolute dictator to impose on the economy?

You see the argument made by the Reagan administration was not that the government had no right to confiscate your property to whatever extent it desires.  The argument was that the government could actually confiscate more of your property if it lowered the rate of confiscation.  This argument would have been no less compelling had it been made to Castro.  And this is the closest thing we’ve had to a small government republican administration in… I don’t know let’s say fifty years (Ike wasn’t that bad I guess…). 

So how did we get here?  We kept convincing ourselves that we could get more stuff from the government without paying for it.  Now we have such a massive government (and government debt) that the maximum amount they can possibly confiscate is barely enough to pay the expenses of that government (and actually it’s probably far short of that amount).  This lack of foresight on our part has not only allowed the argument to become “what would a dictator do?” but it has allowed them to have the argument in the name of “fiscal responsibility.”

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  1. Monkeyfodder
    August 1, 2010 at 2:08 am

    The dictator of Singapore has a forced savings requirement. What would this do to the dictators maximization problem if included in your model?

  2. Free Radical
    August 1, 2010 at 10:25 pm

    What do you mean a forced savings requirement?

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