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

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?

• The trick is to show the aggregate production function is homogenous of degree 1 – Pedro Cavalcante Oct 21 '18 at 20:01

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