# Marginal Product of Capital in the Solow Model

In the classic form of the Solow Model:

$$Y=K^\alpha (AL)^{1-\alpha }$$

Describe circumstances in which the marginal product of capital could rise over time, at least for a temporary period.

I've calculated:

$$MPK = \frac{dY}{dK} = \alpha K^{\alpha -1}(AL)^{1-\alpha }$$

The thing is, I thought one basic assumptions of the Solow Model was diminishing marginal returns, so how could there ever be a case when MPK is increasing?

## 2 Answers

According to your calculations MPK is not increasing in $K$. The Solow model assumes $0< \alpha < 1$, thus $\alpha - 1 < 0$ and $K^{\alpha - 1}$ is decreasing in $K$.

• I agree with that, but the question makes it sound like there are certain cases where MPK could rise? – James Baker Jan 13 '16 at 14:19
• @JamesBaker I take your word for that but since I only see your question not 'the question' I cannot comment further. – Giskard Jan 13 '16 at 15:33

Normally, in the Solow model, there are not increasing marginal returns because by construction, Solow model assumes that marginal return of capital is decreasing. In fact this feature ensures the catch up between countries? which is the main insight of this model.

There is another branch of literature on endogeneous growth models which has taken into account the constant and increasing marginal returns of capital due to different externalities coming from education, infrastructure, accumulation of knowledge etc.

To reply to your question, technically the marginal return of capital is increasing if you assume $\alpha-1>0$. Because in this case, for each additional unit of capital that you have, the marginal productivity will increase.

In the case with decrasing MPK (case with $\alpha-1<0$ ), you will see that when $lim_{k\rightarrow\infty}MPK=0$

• So within the Solow model, there should be no case where the MPK rises, but in reality because of exogenous variables (education, skill, infrastructure etc.) there are some cases in which it would rise. – James Baker Jan 13 '16 at 14:21
• Yes, in the basic Solow model, there is the assumption of decrasing MPK. Constant or rising MPK cases are treated in the endogeneous growth models where people thought that there could be something in economy like education, skills and infrastructure could prevent the economy from decrasing MPK. You are right, the endogeneous models fits more reality. – optimal control Jan 13 '16 at 15:01