Question

You might think that when a production function has a diminishing marginal rate of technical substitution of labor for capital, it cannot have increasing marginal products of capital and labor. Show that this is not true, using the production function Q = L2K2.

Answer 1

Production function is Q = L^2 K^2

|MRTS| = MPL/MPK

= 2LK^2/2KL^2

= K/L

dMRTS/dL = -K/L^2 < 0

As L rises MRTS falls so MRTS is a decling function.

Now MPL = 2LK^2 and dMPL/dL = 2K^2 > 0 which implies MPL is an increasing function. As L rises, MPL is also rising. Similarly, MPK = 2KL^2. d2KL^2/K = 2L^2 > 0 which implies MPK is an incraesing function of K. As K rises MPK also rises.

When a production function has a diminishing marginal rate of technical substitution of labor for capital, it can also have increasing marginal products of capital and labor.