Euler equation with capital adjustment costs (a la Gali et. al (2007)

Dear all,
I have a question regarding the Euler equation w.r.t. physical capital in Gali, Lopez-Salido and Valles (2007) paper.

If I try to derive it using the capital stock (like it’s normally done), it’s all fine. However…

If I try to use investment only (essentially the stock variable doesn’t exist, there is no K at all, but rather only I_1, I_2, I_3 and so on), then I don’t quite get the same result.

The attachment must shed more light on the problem I am currently facing.

Any ideas would be highly appreciated.

capital_adj_derivation.pdf (136.6 KB)

Why are you trying to do it that way?

Hi Johannes,
Yes, this is a logical question. I am working with long-term loans and I stumbled upon this issue. Then my supervisor mentioned that it would be interesting to explore this problem and see whether the issue here is the same. And it is.

So in the frictionless case, it doesn’t matter whether one differentiates w.r.t. the flow (investment) or the stock(capital) variable. The results are identical. But with adjustment costs, or long-term loan contracts…there seems to be a discrepancy, which has been puzzling me. From a maths perspective, shouldn’t the result be the same, as we are not changing the problem?


I am not sure I understand that point. Unless you can completely substitute out one variable, you need to take the FOCs with respect to both capital and investment.

Yes, but the stock of capital variable only exist in time t-1. Afterwards, simply because K_t=(f+1-delta)K_t-1, we can represent K_t in each period as a function of all past investment decisions and the initial stock of capital. That means, K_t as a decision variable simply doesn’t exist.

I mean, if you see my attachment, I do exactly that at the beginning each period.

Then you should get identical results, but I don’t have the time right now to check your full computations.

Of course not, I wouldn’t ask for this. And thanks for your input.