# 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

## 1 Answer

For zero investment, $\delta = 0.05$ and $\theta =1$ and $\theta = 0.9$ we get

We see that it is still a slowing depreciation, which reflects the assumption that capital is more productive in its first years of usage (since depreciation is the mirror image of productive contribution), an assumption good enough for capital equipment with moving parts (and for human capital), and perhaps also for roads, but not necessarily for buildings and things like furniture (here perhaps the Straight-Line method, widely used in Accounting, would be more appropriate -but it creates the need to keep track of capital vintages, since depreciation in that case is a fixed percentage of initial acquisition cost).

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