What kind of production function would give a cubic-shape cost function?

I would like a production function that gives a cost function with the following shape:

The figure was taken from “Microeconomic Theory: Basic Principles and Extensions, 12th edition”, on Chapter 10, section 10.4.3

The cost function $ c $ should satisfy the following requirements:

1. $ c(q) $ is defined and continuous for all $ q \geq 0 $
2. $ c(q) $ is twice continuously differentiable for all $ q > 0 $
3. $ c(q) \to \infty $ as $ q \to \infty $
4. For all $ q > 0 $ , we have $ c’(q) > 0 $
5. There exists $ q_1 > 0 $ such that $ c’’(q) < 0 $ for all $ 0 < q < q_1 $ and $ c’’(q) > 0 $ for all $ q > q_1 $

I don’t need [Math Processing Error] $ c $ to be literally a cubic polynomial (i.e. $ c(q) = a_0 + a_1 q + a_2 q^2 + a_3 q^3 $ ). I just need it to look like the shape in the figure and satisfy the above requirements.

As your cost function exhibits non constant first and second order derivative, a third degree polynomial is a good starting point.

There are infinitely many production functions behind a cubic cost function. Below there is just one example, using a trick from Gorman, which consists to interpret a cost function whose expression is the sum of three (or four) parts, as if it was obtained from a single firm using three machines (or three plants).

Machine (or plant) number $ j=1,…,3 $ , produces a constant share [Math Processing Error] $ s_j $ of total output [Math Processing Error] $ y $ according to its own production function [Math Processing Error] $ h_j $ such that

$$ s_j y = h_j(x_j-k_j), $$ where each $ h_j $ is homogeneous of degree $ 1/\alpha_j $ in $ (x_j-k_j) $ , and $ k_j \geq 0 $ denotes the minimum input requirement for production: $ h_j(x_j-k_j)=0 $ for $ x_j \leq k_j $ . Output is denoted by $ y $ and the input price vector by $ w $ (it can be considered as constant here). Then the optimal input demand vector takes the form (left as exercise): $$ x_j^(w,y)=k_j + b_j(w)y^{\alpha_j} $$ and the corresponding cost function is [Math Processing Error]$$ c(w,y) = \sum_{j=1}^3 w^T x_j^(w,y) = a_0(w) + a_1(w)y^{\alpha_1} + a_2(w)y^{\alpha_2} + a_3(w)y^{\alpha_3} $$ which is the expression of the cubic cost function for $ \alpha_1=1,\alpha_2=2, $ and $ \alpha_3=3 $ .

Further restrictions on the [Math Processing Error] $ a_j $ function yield the above figures.

引用自：https://economics.stackexchange.com/questions/48185