Pls take a look at my solution, as I don't have the model answer.
Company headquarters (HQ) holds 100 units of a raw material; it proposes to sell this material at a price per unit to its three area factories in the North, the Midlands and the South of the country. Each factory can turn the amount of the raw material into an output . Local variations in prices are such that the selling prices are:
Compute in terms of how much raw material each factory should buy from company HQ so as to maximise the factory's profit. How should HQ select the price to maximise the company's total divisional revenue:
where are the optimal amounts which the factories buy from HQ. [Hint: Use the Lagrange multiplier method].
I see the first part of the problem as a maximisation of a function of a single variable. Eg for North,
the profit function is , so setting its first derivative to zero we get
The second derivative is negative, therefore, this is indeed the maximum of the profit function.
Similarly, for Midlands
For the second part, I will use the Lagrange multiplier method to find common tangent points of the profit function f(x,y,z) and the 'budget' function g(x,y,z)=x+y+z=100.
which I solve for x, y and z using the fact that
Interpretation: these quantities will maximise the total revenue of the division after intercompany sales of the 100 units of the given material. Question: common sense would suggest that all this quantity should be sold to South, which enjoys the largest profit on sale of the end product. Where did I go wrong?
Then if I substitute any of these into a solution for x, y or z in part one, I can find , which is the same if I use x, y or z:
Your comments are welcome.