Optimal Price
The cost of publishing a book is a fixed $1200 (initial cost) plus $9 per copy. At $15 selling price, it is expected the number of copies sold will be 300, and that each increment of the price by $1 will reduce the number of copies sold by 10. What is the optimal selling price of the book (that will maximize the profit of selling the book) ?
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2 solutions
Let
- p be the selling price
- C be the cost
- x be the number of copies
- R be the revenue
C = 1 2 0 0 + 9 x x = 3 0 0 − ( p − 1 5 ) 1 0 = 4 5 0 − 1 0 p R = p x − C = p x − 1 2 0 0 − 9 x = ( p − 9 ) x − 1 2 0 0 R = ( p − 9 ) ( 4 5 0 − 1 0 p ) − 1 2 0 0 = 1 0 ( ( p − 9 ) ( 4 5 − p ) − 1 2 0 ) R = 1 0 ( − p 2 + 5 4 p − 5 2 5 )
max R ⟹ max ( − p 2 + 5 4 p − 5 2 5 )
− p 2 + 5 4 p − 5 2 5 is concave downward.
Its maximum when p = 2 α + β = 2 5 4 = 2 7
Taking the revenue as Sales - Costs, we can write the following function:
R ( x ) = ( 3 0 0 − 1 0 x ) ( 1 5 + x ) − [ 9 ( 3 0 0 − 1 0 x ) + 1 2 0 0 ] = − 1 0 x 2 + 2 4 0 x + 6 0 0 = − 1 0 ( x 2 − 2 4 x ) + 6 0 0 = − 1 0 ( x − 1 2 ) 2 + 2 0 4 0
which attains its maximum value of $ 2 , 0 4 0 at x = 1 2 , or 3 0 0 − 1 0 ( 1 2 ) = 1 8 0 copies. This yields at optimized selling price of 1 5 + 1 2 = 2 7 dollars.