# Why do investment adjustment and capital adjustment costs yield different dynamics?

Both the Business Cycle and DSGE literature find that Investment adjustment costs and capital adjustment costs give rise to different dynamics for investment, output, consumption etc. Investment adjustment cost fit the data considerably better, leading to the familiar hump-shaped responses to a shock.

In general, capital accumulation in the presence of adjustment costs is given by: $$K_{t+1}=\left(1-\kappa \phi(.) \right)I_t - \delta K_t$$ In the case of capital adjustment costs, $$\phi(.)$$ becomes a function of $$\frac{I_t}{K_t}$$, whereas under investment adjustment cost, we have $$\phi(.)$$ as a function of $$\frac{I_t}{I_{t-1}}$$ (with certain conditions imposed upon the functional form of $$\phi(.)$$ depending on the case we're dealing with but which I believe are largely unimportant for the difference in dynamics).

Since capital itself is just an accumulation of investment, I would have expected similiar effects in both types of adjustment costs - the smoothing of investment over time incl. anticipatory investment, potentially a buildup etc.

Why is it that they yield so different results? Can in principle similiar responses be reached in both models but it is a case of realistic calibration that causes these different responses, or do different mechanisms govern the two models of adjustment cost? Sadly I was not able to find either literature or lecture notes covering the reason for the difference; any help would be appreciated for the intuition behind the result.