Showing marginal product of capital is independent of the scale of production

Question

The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.

I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.

It's either I'm not partial differentiating correctly or the whole theory is wrong.

I don't see anything wrong with what I've done but why are they not the same?

Answer 1

There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).

To compute the MPK, we must differentiate the production function with respect to the current level of capital: $\partial F/\partial K$.

But in your final line, you are not differentiating with respect to the current level of capital (which is $\tilde{K}=\lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/\lambda$ of the current amount of capital.

If we compute the derivative with respect to $\tilde{K}\equiv \lambda K$ instead of just $K$ then everything works as it should:

$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$

This does not depend on $\lambda$ so MPK is indeed independent of the scale of the economy. QED