Question

Use the total differential to find the MRTS for the production function

y = [0.3x1^(-3) + 0.7x2^(-3)]^(-1/3)

Show that its isoquants are strictly convex to the origin.

Answer 1

In Marginal Rate of Technical Substitution (MRTS),  

- dK/dL (at q=q0) = MPl/MPk

This can be easily found out by considering the total derivative of the production function:

dq = (∂f/∂K)dK + (∂f/∂L)dL

Along an isoquant, dq= 0, so

dq = 0 = (∂f/∂K)dK + (∂f/∂L)dL

=> - dK/dL = MPl/MPk

Therefore, in given question,

MRTS= 3(x2)^4 / 7(x1)^4 . This gets smaller as one moves along an isoquant from left to right - as x1 rises and x2 falls. Thus the isoquants are strictly convex to the origin. To see this more formally, we can write the equation for an isoquant in the form x2 = g(x1) and then finding out dx2/dx1 and then d²x2/d(x1)².