Scaling Adjustment Costs Take a cont. diff, convex and increasing cost function $c(X)$. Say you start with a stock $K$, and want to (dis)invest $I$. Many adjustment cost functions (for example, the first example on page 2 here) then are of the type

$$ K c\left(\frac{I}{K}\right)$$

What is the purpose of scaling investment costs with $K$ here? Marginal cost of investment is

$$ c'\left(\frac{I}{K}\right)$$

It seems that this gives the property that marginal cost of investment decreases in existing capital stock. Is this observed in the real world, or does it rather simplify computations? Are there (popular) alternative formulations of adjustment costs?
The New Palgrave Dictionary of Economics Article on Adjustment Costs gives a nice overview of the use of adjustment costs in economic models. 

What is the purpose of scaling investment costs with K here?

This function form implies that adjustment costs of size X% of capital $I/k$ cost the same fraction of capital ($c(I/k)$) regardless of the amount of capital($k$) involved. It is a constant returns to scale form of adjustment cost. 

It seems that this gives the property that marginal cost of investment decreases in existing capital stock. 

I think you have the marginal cost of adjustment wrong here.

$$ \frac{\partial}{\partial K} K \cdot c\left(\frac{I}{K}\right)$$
$$ = c(\frac{I}{K})  + K \cdot c'(\frac{I}{K})$$

The first term is decreasing in $K$ for fixed $I$ but the second need not be.