# Deriving long-run cost functions from production function

Suppose that I have a production function $$(aK + bL)^3$$ in a perfect competition where a and b are constants. I am confused on how to obtain the long-run cost function from this production function without using Lagrange. I have obviously tried $$MRTS=w/r$$, where w is wage and r is rent for capital, to express L in terms of K or vice versa. This does not yield to anything as the variables K and L just get cancelled in the MRTS. I am utterly lost. How do I go about this?

• Can you show what you have tried so far? – Brennan Mar 3 '20 at 23:33
• Well, the usual way to proceed is to express L in term of K and plug that into the production function. The way to do this is by definition that MRTS = w/r. However, MRTS in this case is (3b(aK + bL)^2)/(3a(aK + bL)^2). So (aK + bL)^2 would just cancel and I cannot find a definition to utilize. – Magic Mar 4 '20 at 4:50

Look again at the production function. Perhaps draw out isoquant curves (for the same output, how could you vary $$K$$ and $$L$$). How are the isoquants for this production function different from those of standard Cobb-Douglas production function?