# Blanchard Kahn in Human Capital Model

Hi Johannes,

In addition to the energy/pollution paper which I have posted about in the past, the other that I am working on features human capital. I wanted to test a naive setup of the model, yet it violates the Blanchard Kahn conditions. I have a simple log-log preference and production given by y = K ^ theta * (human capital * labor)^1-theta.

I attached the equations as they appear in Dynare and the Dynare code. I have made many alterations to this to see what might be causing the problem, ranging from changing parameters to changing wage income from “hours worked * human capital * wage* (1- taxes)” to “human capital * wage * (1- taxes)”. I wanted to get the model working in Dynare with perturbation solutions before I code it up with a global projection algorithm in Matlab.

Thank you for your advice – everyone I’ve asked thus far hasn’t been able to figure out the problem!
naive_dynamic.pdf (71.7 KB)
naive.mod (1.37 KB)

The labor FOC looks strange. Shouldn’t a contemporaneous h show up somewhere? And why is there a h(-2)?

That’s a valid concern and I have tried many different variations of the simple model – one with the FOC you’re describing is the type of model that I expected to work in the first place. The budget constraint you’re describing would have wage income equal to

hours worked * human capital * wage * (1 - taxes)

In that case, the FOC on the attached .mod is the right one. The algebra just involves taking the derivative of the budget constraint with respect to “n”, where “h=(1-xi)h(-1) + (h(-1)*n)^gamma”

The reason I put / tried h(-2) before was because of the timing protocol for Dynare. Since human capital is predetermined, like physical capital, then what I should include if I want to write “h” is really “h(-1)”; but since what I wanted was not “h”, but actually “h(-1)”, I put “h(-2)”. In the current FOC that’s attached, I have “h” and “h(-1)”.

The value for gamma is important; if gamma<1, it will not converge; if it is gamma>1, it violates BK.
naive.mod (1.37 KB)

It seems you are totally confusing the timing conventions here. Don’t try all combinations, but sit down and think about the correct timing implied by economic theory.

It looks as if your LOM for human capital is

where h is the stock of human capital at the end of the current period. But then production today involves h(-1) and wage income is
n*h(-1)*wage * (1 - taxes)
The derivative needs to be taken w.r.t the variable that can be chosen today and that would be h. This would already be the timing Dynare requests. No additional switching is involved.

That’s a good point – thank you for emphasizing that, and replying so quickly! My intuition is that production should then include end of period human capital, i.e.

y=k(-1)^theta*(h*n)^(1-theta);

rather than: y=k(-1)^theta*(h(-1)*n)^(1-theta);

Peculiarly, even after making this change (and not changing it, but only changing for wage income as you said), and trying for gamma>1 and gamma<1, the same problem persists. Note that the choice variable is not human capital, but rather labor supply; the purpose of this naive model is in fact to show that the mechanical “learning” setup between human capital and labor supply is not accurate.

Again, it depends on your model. If human capital is a predetermined stock, then you are producing with h(-1). The treatment is exactly like the one for capital. If you are producing with h, then it is not predetermined and everything must be shifted into the future by one period. You cannot mix those two.

Regarding choice variables: I am not sure I am following. Consider the standard RBC model. If you don’t substitute out for investment from the LOM, you have to take the derivative w.r.t to capital and investment. You cannot simply decide that people only choose investment rather than capital. In that case, you end up with an additional equation that determines the Langrange multiplier on the LOM (Tobin’s marginal q). This equation might be so trivial that it basicall drops out (e.g. q=1). But a priori you have to consider that households take being able to choose (human) capital into account.

Your point seems obvious now, but surprisingly no one mentioned that to me yet (nor did I realize the ramifications of the structure I was imposing by not including it as a choice variable). Thank you again for thinking about my problem – since it is no longer a Dynare issue, I will end the question!