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?
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$\begingroup$ Can you show what you have tried so far? $\endgroup$– BrennanMar 3 '20 at 23:33
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$\begingroup$ 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. $\endgroup$– MagicMar 4 '20 at 4:50
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Don't get stuck with math.
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?
Hint: perfect substitute.
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1$\begingroup$ Thank you for this! Now I feel stupid. It was such a simple solution. Thank you so much! $\endgroup$– MagicMar 4 '20 at 5:19