# Profit tax analysis

I don't really understand this. Why didn't the author multiply out 1-c then find the first order condition? And why does the 1-c term just disappear?

1. The pre-tax profit is given by $$TR-TC$$. Since the firm has to pay profit times $$c$$ in taxes, the net profit equation is given by $$(TR-TC)(1-c)$$. Since this is what the firm actually keeps, this is what the firm maximizes.
2. The choice variable for the firm is $$q$$. So, the firm wants to choose $$q$$ that maximizes $$\pi$$. This is done by taking the derivative of $$\pi$$ with respect to $$q$$ and setting that equal to zero. If this step is confusing, do an internet search for something like "calculus maximization".
3. The author says that $$(1-c) \ne 0$$. This is important so that we can solve for $$q$$. Divide both sides of $$(360-5q)(1-c)=0$$ by $$(1-c)$$ to get $$360-5q=0 \implies q=72$$. You may want to review the mechanics of solving algebraic equations so that you can focus on the concepts in your class.
• Why would the author want to complicate the algebra unnecessarily? This shows that the optimal $q$ is independent of $c$ for $c \in [0,1)$. Commented Apr 14, 2023 at 18:47
• @sffffii You would presumable accept that $\frac{d}{dx}k\,f(x)= k f'(x)$. Here $k=1-c$, a constant that does not depend on $x$ Commented Apr 14, 2023 at 22:21
Profit is made up of two functions, total revenue $$(TR = f(q) = 360 - 2.1q^2)$$ and total cost $$(TC = 50 + 0.4q^2)$$. In this case, $$1-c$$ is the proportion of profit allocated to taxation. In the first equation, the profit equation was simply rearranged to facilitate algebraic manipulations. The following two equations are the first and second order conditions, respectively. Next, the term $$(1-c)$$ disappears because the example assumes a $$100\%$$ tax on profit, that is, $$c = 1$$, which makes $$1-c = 0$$, resulting in an optimal production of 72 units. In short, since $$0\leq c \leq 1$$, then $$1-c$$ only disappears when $$c=1$$.
• "$1-c$ only disappears when $c=1$." this is false, it disappears due to algebraic manipulation, so (-1). Commented Jan 21 at 8:49