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

Question

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!

Answer 1

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.