Step 1 :Let $x$ be the number of hats produced each week. The cost of producing one hat is $2.03. However, there are also overhead costs of $4000 per week. Therefore, the total cost of producing x hats is $2.03x + $4000.
Step 2 :The average cost of producing one hat is then this total cost divided by the number of hats, x. This gives us the function $c(x) = (2.03x + 4000) / x$.
Step 3 :Simplify the function to get the final answer: $c(x) = 2.03 + \frac{4000}{x}$
Step 4 :\(\boxed{c(x) = 2.03 + \frac{4000}{x}}\)