Economic problem from my textbook (here is my translation from Russian):
There is a firm that is both monopoly and monopsony. It's monopoly on market of its product and it's monopsony on labor market (i.e. one and only employer). Demand function for its product is $ P=160-Q $ , its production function is $ Q=2L $ . Labor supply is $ L=W - 140 $ , where $ L $ is number of person-hours and $ W $ is price of one person-hour. You need to find such $ L $, that it would maximaze profit of the firm.
I tried to solve the problem, but it seems like something is wrong with my solution. I tried to use $ MRP_L=MC_L $ and solution in the book uses the same formula. $ MC_L $ is the first derivative of $ TC_L $ , and $ TC_L $ is $ L * W $. Thus $ TC_L=L * W=L *(L+140)=L^2+140L $ (I used the labor supply formula to express W). Therefore $ MC_L=(TC_L)'=(L^2+140L)'=2L+140 $. The solution in my texbook agrees that $ MC_L=2L+140 $
Things started to be messy when I tried to calculate $ MRP_L $. My textbook says that $ MRP_L=MR * MP_L $
I found $ MR $ as the first derivative of $ TR$, $TR=P*Q=(160-Q)*Q=160Q-Q^2$ therefore $ MR=(TR)'=(160Q-Q^2)'=160-2Q $
As for $ MP_L $, I found it as the first derivative of the production function, $(2L)'=2$. Thus $ MP_L=2 $
From this I can conclude that $ MRP_L=MR*MP_L=(160-2Q)*2=320-4Q=320-8L $
Yet in the solution it's said that $ MRP_L=160-8L $
I have no idea how they got this.