# Derive a cost production function give prod f, only K

In this question, only K is included and L is excluded how would I go about deriving it?

Total cost= Fixed costs + Average costs.

Since the variable input costs r per unit, the variable costs is r times the number of units rQ, hence $VC= rK^\alpha$.

Thus $C(Q)= c_0 + rK^\alpha$

Is the above solution correct? IS there a more mathematical way of doing it, one that involves grpahs etc.

• Please show your work thus far or this is likely to be closed. – BKay Apr 4 '15 at 17:28
• Last time I checked, univariate problems were easier to solve than multivariate... – Alecos Papadopoulos Apr 4 '15 at 19:31
• If both K and L were present, it would have been much easier to derive it, however with L not being part of the production function, it gets a bit tricky! – blzox Apr 6 '15 at 10:02

Here you have to express $K$ in terms of $Q$, since the cost depends on the number of units of capital employed and not on the number of products produced. You have $$Q = K^a \Leftrightarrow K = Q^{\frac{1}{a}}$$ so your cost function will be $$C(Q) = c_0 + rK(Q) = c_0 + rQ^{\frac{1}{a}}$$

• Great thanks, sometime I complicate things for myself! – blzox Apr 6 '15 at 11:09
• What would the 1st derivative of that be? (i.e MC) – blzox Apr 6 '15 at 11:10
• Easy, it would just be MC = (r/a)*Q^((1-a)/a) – tadejsv Apr 6 '15 at 11:11
• perfect thanks, now how would you interpret that in an economic context? Thanks a mill for your help! – blzox Apr 6 '15 at 11:17
• @blzox it seems like you're expecting him to solve the whole problem set? Be cautious with the number and type of followup-questions you pose in comments to someone who already answered your initial question. – FooBar Apr 6 '15 at 11:28