In his 1977 article (from which has developed a considerable literature on the Hartwick Rule for maintaining long-term constant consumption given depletion of non-renewable natural resources), Hartwick uses (on p 973) this aggregate production function:
$$x = k^{\alpha}y^{\beta}1^{\gamma}$$
Here (see p 972) $x$ is per capita output, $k$ is per capita reproducible capital, and $y$ is per capita use of an exhaustible resource. $1$ is just the number one, labour being assumed constant (so the term $1^{\gamma}$ seems redudant). So in more familiar notation (output $Y$, capital $K$, use of exhaustible resource $R$, labour $L$), and glossing over the difference between population and labour, this is:
$$\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$$
Hartwick then assumes (p 972) constant returns to scale in the form (explicitly on p 973) $\alpha+\beta=1$.
Question: What reasons might justify the assumption of constant returns in the above form? Isn't it more plausible to assume constant returns when all factors are increased, that is to assume $\alpha+\beta+\gamma=1$ in a production function of the form:
$$Y=K^{\alpha}R^{\beta}L^{\gamma}$$ This does imply the above function since:
$$\frac{Y}{L} = \frac{K^{\alpha}R^{\beta}L^{\gamma}}{L^{\alpha}L^{\beta}L^{\gamma}}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$$
However, given $\gamma>0$, it is inconsistent with $\alpha+\beta=1$: instead $\alpha+\beta<1$.