I tried to do following question but I am not sure about my solution. Please tell me your opinions
Since gross profit is strictly concave, I can say that $R’’(y)-C’’(y) <0$
Now I maximize the net profit and
FOC is $R’(y)-C’(y) -t=0$
SOC is hold because of strict concavity in y.
That is $MR(y)=MC(y)+t$ from FOC.
Now show that the implicit relationship btw y and t can be solved for the explicit choice function
FOC with respect to y combine with above function
Derivative wrt t
$$R’’(y) (dy^*/dt)-C’’(y) (dy^*/dt)-1=0$$
which means that the output will decrease as the tax goes up.
Think again gross profit
I am not sure about my solution after this point especially.
Suppose the firm is price taker then $R(y)=py$
So $$G\pi =py^*(t)-C(y^*(t))$$
That is, since the output will decrease as tax rises, revenue $R(y) $ will decrease as well. Accordingly gross profit will decrease.
I am not sure about the last part especially. Thank you.