Assume that a profit maximizing monopolist faces an inverse demand function give

Question

Assume that a profit maximizing monopolist faces an inverse demand function given by p(.) where p'(y) < 0, and a total cost function c(.) given by c(y), where c'(y) > 0.

Prove that the imposition of a lump sum tax T > 0 does not affect the profit maximizing price and output of the monopolist. Mathematically.

I cannot figure this one out or how to even set it up...please help me out! Thanks!

Explanation / Answer

Assume that a profit maximizing monopolist faces an inverse demand function given by p(.) where p'(y) < 0, and a total cost function c(.) given by c(y), where c'(y) > 0. Prove that the imposition of a lump sum tax T > 0 does not affect the profit maximizing price and output of the monopolist. Mathematically. So, your profit function that you maximize over y is: Profit=p(.)-c(.) And then to max profit you differentiate with respect to y: d(profit)/dy=p'(y) -c'(y) This is the First Order Condition that you solve to find the output that maximizes profit. Now, consider the new profit function with the lump sum tax: Profit=p(.)-c(.)-T But when you find the first derivative (FOC) of the new profit function, you still get the same result as above: d(profit)/dy=p'(y) -c'(y)+0 This is due to the fact that the first derivative of a constant is zero. Graphically, if you look at the profit function, the tax (T) is just shifting the graph up and down. Although this changes the level of maximum profit (the new profit will be less), the maximum of the profit function (highest point on the vertical axis) does not change along the quantity (horizontal) axis.

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