# Could Depreciation of Capital $=\delta(K(t))^{\theta}$ with $\theta<1$ Be a Realistic Assumption?

In the context of a long-term growth model with $K$ representing aggregate (man-made) capital, what theoretical or empirical reasons, if any, might be given for assuming that depreciation of capital $D(t)$ could be given by:

$$D(t) = \delta(K(t))^{\theta}$$

where $\delta$ and $\theta$ are fixed parameters both $> 0$ and (this is the key point) $\theta<1$?

[The question is prompted by this paper by Buchholz, Dasgupta & Mitra which considers the feasibility of constant consumption to infinity in a growth model containing a single produced good which can be either consumed or used as capital and a single nonrenewable resource. It implies (p 553) that constant consumption is impossible when $\theta=1$, reducing the above formula to the familiar $D(t)=\delta K(t)$, but is in some circumstances possible when $\theta<1$. Hence it is of interest to consider whether $\theta<1$ could ever be a realistic assumption.]

• Basically it means that large amounts of capital depreciate at a slower rate than smaller amounts of capital, right? Interesting assumption. I am not sure it is true but you could probably argue for it using economies of scope. – Giskard Oct 16 '16 at 18:09

For zero investment, $\delta = 0.05$ and $\theta =1$ and $\theta = 0.9$ we get
So it appears to be a quantitative/empirical matter. With $\theta <1$ capital depreciates slower, but the qualitative behavior is the same, and for smaller values of $\theta$ it can even mimic the Straight Line method, which, as said, is realistic for some forms of capital. So yes, we can have $\theta<1$, it does not appear to contradict any fundamental assumption.
• Taking your answer together with denesp's comment above, it seems that $\theta<1$ can be plausible in a time path of depreciation of a given capital vintage of capital, but (given fixed $\delta$) is dubious in comparing the depreciation paths of vintages of different sizes, or in comparing depreciation on aggregate capital of different sizes at different dates within a growth path.. – Adam Bailey Oct 19 '16 at 9:32