# Question on capital shock?

Dear Professor Pfeifer,

I want to add a shock on capital stock. I search the forum and find your code of RBC_capitalstock_shock.mod which makes a typical example. I just want to make sure that:

(1) In the notation of the code, you said that “the timing for k in all other equations changes to being contemporaneous” and the the law of motion for capital is written as
exp(k) = exp(-eps_cap)*(exp(invest(-1))+(1-delta)*exp(k(-1)));

So in other equations which include variable “invest”, I sould substitue it by
exp(invest)=exp(k(+1))/exp(-eps_cap(+1))-(1-delta)*exp(k);
and calculate derivative with respect to k(+1). Is this correct ?

(2) In your code, eps_cap is exogenous variable. I wonder if I could use the code as follows:
exp(k) = exp(eps_k)*(exp(invest(-1))+(1-delta)*exp(k(-1)));
and
exp(eps_k) = (1 - rhokUU) * 1 + rhokUU * exp(eps_k(-1)) + e_k;
in which eps_k is endogenous variable and e_k is the exogenous one. Is this correct ?

The bayesian estimation results on the two types of introducing capital shock above are quite different and I have no idea about the reason.

Thank you for your time, Professor.
Best regards

Sorry, but I don’t get what you are after.

1. The constraint I wrote down is the correct one and you take the derivative with respect to k and invest at time t, i.e. (0).
2. I assumed an iid shock (autocorrelation 0). Yours is autocorrrelated. Of course results will be different.

Dear Professor Pfeifer,

I’m very sorry that I did not make the first question clear.

I know the entrepreneur’s equations without capital shock is as:

exp(k) = exp(invest)+(1-delta)*exp(k(-1));
exp(y) = exp(z)*exp(k(-1))^alpha * exp(l)^(1-alpha);
exp(consume)+exp(invest)+ exp(w)*exp(l)+exp(rate(-1))*exp(b(-1))/exp(pi) = exp(y)+ exp(b);
exp(rate) * exp(b)=m * ( exp(qk(+1)) * (1-deltaUU)*exp(k)*exp(pi(+1)) ); %credit constraint

While when capital shock came in, the equations become as:

exp(k) = exp(e_k)*( exp(invest(-1))+(1-delta)*exp(k(-1)) );
exp(y) = exp(z)exp(k)^alphaexp(l)^(1-alpha);
exp(consume) +exp(invest)+ exp(w)*exp(l)+exp(rate(-1))*exp(b(-1))/exp(pi) = exp(y)+ exp(b);
exp(rate) * exp(b)=m * ( exp(qk(+1)) * (1-deltaUU)*exp(k(+1))*exp(pi(+1)) ); //credit constraint

In this case, I feel a little confused about the first-order condition on variable “invest” and “k”.
How should I calculate derivative with respect to invest and k ?

Thanks again for your time, Professor.
Best regards

What do you mean with

It’s a standard derivative…

Dear Professor Pfeifer,

When taking the derivative with respect to k at time t, how should I deal with k(+1) in the budget constraint and credit constraint ? It seems that I have never encountered this kind of situation before. And sorry if I asked a stupid question.

Best regards

But capital does not show up in the budget constraint. With respect to the credit constraint: what is the logic here behind the timing, i.e. is the timing correct?

Dear Professor Pfeifer,

As the notation in your code of RBC_capitalstock_shock.mod says:
“Similarly, the timing for k in all other equations changes to being contemporaneous.”
I thought it means that I should go forward one period for k in other equation, so I changed the k(t) in credit constraint into k(t+1). Did I misunderstand this ?

With respect to variable invest, I thought it could be substituted by
exp(invest)=exp(k(+1))/exp(e_k(+1))-(1-delta) * exp(k);
according to the law of motion for capital
exp(k) = exp(e_k)*(exp(invest(-1))+(1-delta) * exp(k(-1)));
Then there is k(t+1) in the budget constraint. Did I also misunderstand something in this ?

Best regards

1. You need to think about the economic logic of the model, not blindly shift the timing. Why should the constraint depend on expected capital?
2. Given the new timing, you cannot substitute out investment.

Dear Professor Pfeifer,

1. So the timing of k in credit constraint will retain k(t) as it still represents capital stock at the end of time t. Is it correct?

2. Sorry but I really feel confused about the timing here, and I cannot find this capital shock in any paper. As you said, I should think about the economic logic of the model. I used to treat k(t-1) as the capital stock at the beginning of t and k(t) as which at the end of t. However, when capital shock comes in, the law of motion for capital changes to
exp(k) = exp(e_k)*(exp(invest(-1))+(1-delta) * exp(k(-1))).
k(t) seems not like the meaning I considered before ? While you said I cannot substitute out the investment, how should I calculate derivative with respect to investment in this case ? The equation may become even more complex when investment adjustment cost come in.