# Optimization for minimum cost, with the total cost function $C=TQ^{1/a} + F$

I have the function $C=TQ^{1/a} + F$. Where C is total cost, Q is output, a is a positive parametric constant, F is fixed cost, and T measures the technology available to the firm (Parameter). We also know that $T > 0$.

A hint is to use calculus of optimization, and to check our second-order condition.

I got the answer of it being convex with the function of $T(1/a)(1/a-1)Q^{1/a-2}$

However I can't figure out the rest:

Assuming $a<0.5$, and by using optimization what is the value of q that minimizes the average cost?

Hint: the answer for $q$ will be in the parameters of the model, be sure to check the second order condition.

• What is $q$? Is it the same as $Q$? It would be useful to have the problem in standard form where it becomes clear what are the constraints in your optimization. Nov 17, 2014 at 11:29
• Sorry, q is the same as Q. The only constraints i'm given are that T>0, and a<0.5 Nov 17, 2014 at 18:54

a) Look up what it means to be concave/convex. Sketch a graph of C. Based on the graph, does it look like C is concave/convex? How might you prove it?

b) What is the definition of "average cost"? Now this is a standard calc 1 exercise: Find the global minimum of the average cost.

• – Neal
Nov 21, 2014 at 4:57

$$AC=\frac{C}{Q}=TQ^{1/a-1}+FQ^{-1}$$

$$AC'=T(1/a-1)Q^{1/a-2}-FQ^{-2}=0 \Rightarrow Q=\left(\frac{F}{T(1/a-1)}\right)^a$$

$$AC''=T(1/a-1)(1/a-2)Q^{1/a-3}+2FQ^{-3}>0,$$

because $a<0.5$.