# Timing of an equation

Dear All,

suppose you have a capital accumulation equation

k(t)=p*k(t-1)+I(t)

and that k(t) (not k(t-1) as usual) is the stock of capital used in today’s production. In other words investment becomes new capital within the same period. This implies that K is not a state variable.

How would you write the equation above in dynare? Notice that If I write k as reported above it would be considered as a state variable.

what if I write it as

k(t+1)=p*k(t)+I(t)

and put k(t) in the production function?

thanks

k(t-1) is the state variable because you bring it in from last period. You should keep your former law of capital accumulation equation, but use k(t) in the current production function instead of k(t-1).

Using k(t) instead of k(t-1) in the production function for today’s y(t) makes little economic sense. The resource constraint for the economy would read as
y(t)=f(k(t))=c(t)+k(t)-(1-d)k(t-1)
So this would mean that the same capital that is used to produce today’s y(t) is then used to produce today’s k(t), which is kind of weird, because you would at the same produce produce y(t) using k(t) and then produce k(t) using whatever y(t) you do not use for consumption. It is conceptually ok if you think of k(t) as being intermediate goods, but then you should set d=1 (intermediate goods are used up in production within the period), but then you could write the entire model without reference to k(t) at all since k(t) becomes a purely static variable.
Bottom line: better think hard of why it is k(t) and not k(t-1) in the production function.

Thanks for the replies,

actually K is not the variable I’m considering it was just an example, obviously it was not good.

So briefly, consider the following. Each firm in each period can post vacancies V which with probability q becomes new matches in the same period and contribute to the firm’s existing workforce n

So n(t)=p*n(t-1)+Vq

n(t) workers are used to produce in each period t.
However if I write the equation in dynare as above then n would be condider as a state variable, but it is not since Vq contribute to the stock of empolyment in the same period.

How would you write equation and prod. fucntion assuming Y is linear in n?

Sorry, I am not sure, if I fully understand the problem, but n(t-1) definitely is a state in your problem as you have to keep track of it due to your “law of motion” n(t)=pn(t-1)+Vq. Hence, you should enter it this way to Dynare. The difference to capital is that capital is a predetermined state. Next period’s capital is determined by today’s investment, but yesterday’s capital is used in today’s production function.

```k=(1-delta)*k(-1)+I; y=A*k(-1)```
In your case, today’s n is determined according to your law of motion (in Dynare notation)

```n=p*n(-1)+V*q; y=A*n;```
and is hence not predetermined. Note the difference to the treatment of capital, where the production factor has a different timing.

Ok,
thanks for your reply. That is the way I wrote the equation since the beginning, but I have determinacy problems.

Maybe it is just the calibration.

thanks anyway