A firm's managers aim to maximize revenue rather than profits. At the same time, shareholders stipulate that an acceptable profit of m should be achieved. (Suppose the managers set to achieve this exactly.) The firms revenue R is a function of its output q and advertising expenditure a such that: R(q , a) = q2 + 2a
Other than advertising costs, the firm incurs production costs. The unit variable production cost is given by v, and the fixed cost is f.
i) Use a Lagrangian function to find the manager's optimal output and optimal advertising expenditure. [Hint: You should be able to figure out your endogenous variables from the sentence I just gave you for this part.]
ii) Check to see if your second-order condition is satisfied.
iii) Without performing another optimization problem, find the impact on the firm's revenue if the constraint is relaxed by one unit. (Be sure to clearly explain what you understand by "relaxing the constraint by one unit.") Provide an explanation of your answer in plain English.
iv) What is the impact of a rise in shareholders' acceptable profit on optimal advertising expenditure in this firm? (I need an answer derived mathematically and explained in words - using plain English.) [Hint: Here I am asking you to perform comparative statics; use your reduced form solution for the variable 'a' from your answer to part i).]