How do you convert or move from a linear cost function to a quadratic cost function?

Question

I am reading a book on electricity cost modelling. I understand equation 2.7 below, which indicates that the total cost for an ith plant is a function of fixed cost(FC), fuel cost(FL), plant efficiency (af) and quantity of electricity produced (Q).

Equation 2.9 is a two-step piecewise cost function which describes the existence of two possible ranges of operation, and that producing above a threshold implies that there is an increase in the variable costs. I understand this too.

Equation 2.10 provides a more general functional form that represents a continuous and smooth version of a multiple-step piecewise linear cost function through a quadratic function. However, I do not understand how this was derived. The only difference between equation 2.7 and 2.10 seems to be the squaring of Q in equation 2.10.

My questions are:

1. Why was Q squared?
2. How was this done? What is the broader concept/principle through which this was done?
3. In what situations can this be applied as I build new models in the future?

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Answer 1

Your eq (2.10) is not more general than (2.9), but corresponds to an alternative specification. A more general version would be: $$ C_i(Q_i)=FC_i+a_{1,i}Q_i+a_{2,i}Q^2_i. $$ This specification allows marginal cost to be constant as in (2.9) if $a_{2,i}=0$ or nonconstant as in (2.10) for $a_{1,i}=0,a_{2,i} \neq 0.$ It is more general because it is compatible with an increasing and decreasing marginal cost over some range of the production level. In contrast, (2.9) implies that if the cost function is increasing in $Q_i$, then is also has to be convex: $C''>0.$