# The economic staff of the U.S. department of the treasury has been asked to recommend a new tax...

## Question:

The economic staff of the U.S. department of the treasury has been asked to recommend a new tax policy concerning the treatment of the foreign earnings of U.S. firms. Currently the foreigh earnings of U.S. multinational companies are taxed only when the income is returned to the United States. Taxes are deferred if the income is reinvested abroad. The department seeks a tax rate that will maximize total tax revenue from earnings. Find the optimal tax rate if:

A) {eq}B(t) = 80 - 100t {/eq}

B) {eq}B(t) = 80 - 240t^2 {/eq}

C) {eq}B(t) = 80 - \sqrt{t} {/eq}

where {eq}B(t) {/eq} is the foreign earnings of U.S. multinational companies returned to the United States and {eq}t {/eq} is the tax rate.

## Finding Minima & Maxima:

The maximum value of the function that is decreasing is generally at the highest point of the curve and if the function is increasing, then the minimum value is at the lowest point of the curve. We use differentiation to find increasing or decreasing function.

We have to find the optimal tax rate (t) when the revenue is maximum. So the revenue function is given as:

A) {eq}B(t) = 80 - 100t {/eq}.

So here we will differentiate the function wrt t, as follows:

{eq}\Rightarrow B'(x)\\ \Rightarrow \frac{d}{dt}\left(80-100t\right)\\ \Rightarrow -100 {/eq}

So now we see that here the derivative is always negative and hence this is the decreasing function. So the optimal value of B(t)is when it is zero.

{eq}\Rightarrow B(t) = 0\\ \Rightarrow 80 - 100t=0\\ \Rightarrow t=0.8~ {/eq}%

is the optimal tax rate.

B) {eq}B(t) = 80 - 240t^2 {/eq}

On differentiating, we have:

{eq}\Rightarrow B'(t) \\ \Rightarrow \frac{d}{dt}\left(80-240t^2\right)\\ \Rightarrow -480t\\ {/eq}

So this is again the decreasing function. So the optimal value of B(t)is when it is zero.

{eq}\Rightarrow 80 - 240t^2 =0\\ \Rightarrow t=0.57~ {/eq} %

is the optimal tax rate.

C) {eq}B(t) = 80 - \sqrt{t} {/eq}

On differentiating, we have;

{eq}\Rightarrow B'(t)\\ \Rightarrow \frac{d}{dt}\left(80-\sqrt{t}\right)\\ \Rightarrow -\frac{1}{2\sqrt{t}}\\ {/eq}

So this is again the decreasing function. So the optimal value of B(t)is when it is zero.

{eq}\Rightarrow 80 - \sqrt{t}=0\\ \Rightarrow t=6400~~ ~ {/eq} %

is the optimal tax rate.