# How to derive cubic cost function from a problem of constrained optimization?

The cubic total cost function usually take the form

$TC(q)=a+bq+cq^{2}+dq^{3} \qquad a,b,d>0, c<0$ and $c^{2}<4bd$

I know that from a constraint maximization problem

$min\quad wL+vK$

subject to

$q_{0}=f(k,l)$

i can express it with the lagrangean function

$\mathcal{L}=wl+vk+\lambda(q_{0}-f(k,l))$

with some algebra for the case of a Cobb-Douglas production function $q_{0}=k^{\alpha}l^{\beta}$ i can reach

$TC=q^{\frac{1}{\alpha+\beta}}w^{\frac{\beta}{\alpha+\beta}}v^{\frac{\alpha}{\alpha+\beta}}\left(\frac{\alpha+\beta}{\alpha^{\frac{\alpha}{\alpha+\beta}}\beta^{\frac{\beta}{\alpha+\beta}}}\right)$

I could give values to $\alpha$ and $\beta$ to obtain a cubic function, but not the one I described in the preamble. It could also fix some other factor such as the size of the factory, but would not work.

Any idea?

• You write $TC(q)$ usually takes this very specific form. Can you please post a reference/source? I have never seen these parameter restrictions before. – Giskard Jun 6 '15 at 23:37
• The texts do not usually specify this. But, in practice, these restrictions $a,b,c>0, \quad c<0$ and $c^{2}<3bd$ make sense to not have solutions in the field of complex numbers. a text that uses this (page 2): ijecm.co.uk/wp-content/uploads/2014/02/2214.pdf – Héctor Garrido Jun 6 '15 at 23:48

A note on the parameter constraints: $$TC(q)=a+bq+cq^{2}+dq^{3} = a + q(b+cq+dq^{2})$$ It appears that what we want is that for the second degree polynomial to not take negative values. This would require for it to not have real and positive roots. Since $c<0$, then if there are real roots one of them will be positive. This would imply that for an interval of positive values of $q$ the 2nd degree polynomial will take on negative values, and marginal cost will turn negative. So we need the discriminant to be negative, and so $c^2 - 4bd < 0 \implies c^2 < 4bd$. I don't see whey they write "$3$" instead of "$4$".

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As for obtaining such a cost function rigorously from a production function:
The reference is Silberberg. E (1990), "The Structure of Economics" (2nd ed), ch. 9.

A) When the production function is homogeneous of degree $r$, then the cost function has the form

$$C(q,\mathbb w) = q^{1/r} \cdot h(\mathbb w)$$

where $h(\mathbb w)$ is a function of prices (of input factors), and it is homogeneous of degree one (or "linearly homogeneous").

B) When the production function is homothetic, which can be represented as a monotonic function of a homogeneous of degree one function, then the cost function remains multiplicatively separable in output and prices for some function $J(q)$:

$$C(q,\mathbb w) = J(q)\cdot h(\mathbb w)$$

But again, since homotheticity is a monotonic transformation of homogeneity of degree one, $J(q)$ should not be expected to take any polynomial form like the one you seek.

So none of the usual functional specifications for production functions will give a cost function like the one you want to arrive at. And indeed in papers with such cost functions I have never seen a derivation of the underlying production function.

If any positive result comes up, I will return.

Given a Cobb-Douglas production function your cost function will be cubic exactly if $\alpha + \beta = \frac{1}{3}$, and in this case you will have $a = b = c = 0$. There is no going around this. If you want to have the kind of cost function you describe you will have to slightly alter your Cobb-Douglas function, e.g. $$f(k,l) = (k-1)^{\alpha} \cdot (l-2)^{\frac{1}{3} - \alpha}.$$