Question

Please answer the Problems on the second page (#1-4).

Tax rates and concavity In this activity, we will use some simple ideas about taxes and tax rates to investigate some basic properties about functions. Economists use the term marginal taz nate to refer to the additional tax paid on the next (taxable) dollar earned. For example, suppose that under some tax plan the tax on $34,567 is $6897.76 while the tax on $34,568 is $6898.04. Then, for a taxpayer earning $34,567, the tax paid on the next dollar is $0.28, so the marginal tax rate is $0.28/$1 = 28%. (Note that this is not directly related to $6897.76/$34,567 = 19.95%, the percentage of taxable income that goes to tax.) Let t(z) denote the amount of tax that is paid on $z of taxable income. 1. Economists refer to a tax as progressive if the marginal tax rate increases (as taxable income increases). If t(z) is a progressive tax, what will the graph of t(z) look like? Sketch t(z) for a progressive tax. 2. At a few points on your graph, sketch the tangent line. What is the relationship between the graph and its tangent lines? 3. Explain how the marginal tax rate is related to t'(z). (Hint: Take Az = $1 in the definition of t'(z).) 4. If t has a second derivative, what property will t" have?

Answer 1

1. If $\small t(x)$ denotes the amount of tax that is paid on
то
of taxable income, then
2) = I - (2
denotes the income left after tax.

2. As u(x) denotes the income left after tax, depending on whether the marginal tax rate, t(x) increases/decreases as taxable income increases, then the rate of change of u(x) will decrease/increase, i.e the graph of u(x) will be concave down/up, that is, u(x) will be below/above it's tangent line. This can be shown mathematically as
$\small u'(x)=1-t'(x),u''(x)=-t''(x)$
so, if the second derivative of t(x) is positive, then the second derivative of u(x) is negative and vice versa. For example, if $\small t(x)=0.6x^2$ (progressive tax system, second derivative of t(x) is positive)

where red denotes t(x) and blue denotes u(x).. notice how the slope of u(x) keeps getting lesser and lesser as x increases.

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3. If a function is concave down, the the slope of the function will keep on decreasing, and ultimately try to point vertically downwards.
Now, if the function is given to be positive and unbounded, then it cannot cut the x-axis at any point. However, if the slope of the function keeps decreasing then the slope is bound to point towards the x-axis at some point and eventually cut it. The only way to not violate the positivity constraint would be to straighten the slope up horizontally, which would mean the slope would increase, which violates the concavity. Thus, a function cannot positive, unbounded and concave downwards.

4. If the marginal tax rate $\small t'(x)$ is constant over some interval, then it means that at any income x inside that interval, the increase in tax is constant, hence, the graph will simply follow the slope, i.e. the graph will be a straight line. For example, if
$\small t'(x)=0.28$ then one possibility of t(x) is $\small t(x)=0.28x$ and it's graph looks like

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