# Constant Returns in a Production Function $\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$ ($R$ = Resource)

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 $\alpha+\beta<1$.

• If you are dividing through by $L$ everywhere then you do not need $\alpha+\beta=1$ or $\alpha+\beta+\gamma=1$ to get constant returns to scale. This summing to $1$ to get constant returns to scale only matters when your expression is something like $Y = K^{\alpha}R^{\beta}L^{\gamma}$ Dec 6 '16 at 22:41
• @Henry Is your point simply that if $K$, $R$ and $L$ all increase by the same proportion, then $K/L$ and $R/L$ are unchanged, so $Y/L$ will also be unchanged? Or is there more to it? Dec 9 '16 at 9:58
• Adam: that is precisely my point, as $Y$ increasing by the same proportion as $K$, $R$ and $L$ is the definition of constant returns to scale Dec 9 '16 at 11:40

The purpose of the paper under consideration is to examine/show the "investment rule" that leads to "intergenerational equity", which with constant population translates into constant consumption.

The investment rule under examination is (last line of p. 973) "invest all net returns from exhaustible resources in reproducible capital" (and consume the rest).

Earlier, eq. $(1)$ of the paper (the law of capital accumulation) tells us that gross returns to exhaustible resources equal $f_yy$: this implies that we assume that the return per unit of exhaustible resource equals its marginal product.

But this in turn implies pricing and the existence of markets. So there must be a market for capital also. If the capital market is also characterized by marginal pricing, then , if we assume that the production function has decreasing returns to scale in capital and exhaustible resource ($\alpha + \beta <1$), then

$$f_kk + f_yy < x$$

and some part of output would have been left unaccounted for.

So the author assumes constant returns to scale in these two so that he can also assume competitive markets and marginal pricing, and per capita output exhausted in the rewards to these inputs.

This of course begs the question: what happens to the labor market? Well, we can get away with murder making the following assumption: There is no leisure-labor choice, labor is offered inelastically, and moreover, there is no market for labor, it is subsumed to the other factors of production, i.e. it is offered together with them and it is not paid separately: think capital owners that also work in their business without paying themselves a wage.

This of course means that the formulation with unitary labor and an irrelevant exponent $\gamma$ is sloppy and problematic, it should be absent (it would not affect the paper), and it rightfully led to the OP's question.