An electronics manufacturer has two brands that they wish to release into the market, the Xtreme Sound Radio and the You-Play CD System. Suppose that the quantities of each brand produced by the manufacture are labelled x and y respectively, and the market prices,
labelled ~ and ~,arefound to be given by ~(x)=250-x
~(y)=150-y
and the total cost to manufacture the two products IS given by
C(x,y)= 10(x+y)+200.
(a) Write down the total profit equation for the manufacturer in terms
ofx andy.
(b) Find any stationary points (x , y values) for the profit equation you obtained in part (a).
(c) Show that there is a single profit maximizing point, and calculate the total manufacturing costs, market prices, and profit at that point.
Suppose now that a regulator has decided to restrict the production of the two electronics systems by the manufacturer, such that now the total quantity of sound systems produced by the manufacturer must not be greater than 80.
Assuming that the manufacturer produces exactly 80 sounds systems in total,
(d) Write down the Lagrangian needed to solve this new optimization
problem.
(e) Find the single stationary point for the new problem by solving the first-order conditions for the Lagrangian you obtained in part (d).
(f) Show that the point you obtained in part (e) is a profit maximizing point.
(g) How would your answer to part (f) change if the total cost to manufacture the two products was instead given by
C(x,y)= 10(x+y)+250? Briefly explain your answer.