14.7 Optimization in R

The method of maximum likelihood requires an optimization routine.
For a few very simple probability models, a closed-form solution exists and the MLE can be derived by hand. (This is also potentially the case for OLS regression.)
But, instead lets use some machine learning to find the estimates that maximize the likelihood function.
There are many optimizers (e.g. optimize, and optim). optimize is the simplest to use, but only works in one dimension.

14.7.1 Optimization Example: Optimum Price

Suppose that a firm’s profit from selling a product is related to price, \(p\), and cost, \(c\), as follows:

\[ \text{profit} = (p - p^2) - c + 100 \]

Explain how you would use calculus to find the maximizing price. Assume that cost is fixed.
What is the firms revenue as p=0, cost = 2? What is it at p=10, cost = 2?
Create a plot with the following characteristics:
- On the x-axis is a sequence (seq()) of prices from [0, 10].
- On the y-axis is the revenue as a function of those prices. Hold cost constant at c=2.
- What does the best price seem to be?
Solve this numerically in R, using the optimize() function.
- Take note: using the default arguments, will optimize try to find a maximum or a minimum?
- Check into the help documentation.

Code

profit <- function(p, c) { 
  r = (p - p^2) - c + 100
  return(r) 
  }

Code

profit(p=2, c=2)

## [1] 96

Code

best_price <- optimize(
  profit,                    # profit is the function
  lower = 0, upper = 1000,   # this is the low and high we consider
  c = 2,                     # here, we're passing cost into profit
  maximum = TRUE)            # we'd like to maximize, not minimize
best_price

## $maximum
## [1] 0.5
## 
## $objective
## [1] 98.25