# Constant returns to scale and diminishing marginal returns in the Solow model

My introduction to economics course had a section on the Solow model which I'm revising for next year. I'm a little confused by two of the assumptions which our lectuer taught us are made by the Solow model: a) that it exhibits constant returns to scale, and b) that increases in labour and capital are subject to diminishing marginal product.

$$Y=A\times f(K,L)$$

Initially I thought this was contradictory, but from what I've been able to pick up, an equal increase in both factors leads to constant returns to scale, but an increase in just one of the factors leads to diminishing marginal product.

Is this correct, or have I misunderstood something?

Thanks!

These two assumptions are not necessarily contradictory. Just check whether the assumptions are satisfied by any candidate function. For example, take $$F(K,N) = K^{\alpha}N^{1-\alpha}$$, with $$\alpha \in (0,1)$$.

Constant returns to scale:

For any scaling factor $$c \in (0, \infty)$$:

$$F(cK,cN) = (cK)^{\alpha}(cN)^{1-\alpha} = c^{\alpha}c^{1-\alpha} K^{\alpha}N^{1-\alpha}=cK^{\alpha}N^{1-\alpha}=cF(K,N)\quad \checkmark$$

Diminishing marginal return (product):

This means increasing returns, but at an ever slower rate. So the first derivative of needs to be positive, and the second one negative. $$\frac{\delta}{\delta K}F(K,N) = \alpha K^{\alpha-1}N^{1-\alpha} > 0 \quad \checkmark$$ $$\biggl(\frac{\delta}{\delta K}\biggr)^{2}F(K,N) = (\alpha-1)\alpha K^{\alpha-2}N^{1-\alpha} < 0, \quad \textrm{because} \quad 0< \alpha < 1 \quad \checkmark$$

You can verify in the same way that $$F$$ satisfies diminishing returns for the labor input $$N$$.

So our candidate function $$F$$ satisfies both assumptions, and there is no contradiction.

• So I can follow the maths and that all makes sense - thanks! How would I think about this conceptually though? I can see that if I double both inputs (capital and labour), I will end up with double the national output, and capital per worker will stay the same. Is it that the concept of diminishing marginal returns only apply to changes in capital per worker?
– Evan
Jul 28 at 16:04
• The concept tries to model the fact that if you increase only one single input , output will increase less than proportionately and that increase will be less and less the more you increase that input. For example, a worker may be able to handle one machine well, but if you give her two, output may still increase, because she can watch that additional machine, but sometimes it will run idle, so output is less. Conversely, if you employ a second worker at one single machine, more than required, they may still be able to increase output, but only a little, and even less so with three workers.
– BrsG
Jul 28 at 16:51
• In short, it applies to each input, individually. I have added a corresponding line in my answer. In the example I have used, you cannot take the derivative with respect to capital per worker, unless $\alpha=0.5$. Only in that case it would also apply to that ratio, but not generally. There might be functional forms where it does, but that's not the main conceptual feature.
– BrsG
Jul 28 at 17:03