Household Dynamic Problem

[matdehaven]

Hi Pascal,

Here's what I've written out for the household's problem this time. Does this look right?

[pmichaillat]

This looks much better.

There is only one possible issue. In the budget constraint it should be $-n(t)$ instead of $-c(t)$ since all butler must be paid, not just those who provide consumption. Shouldn't it?

It would also be good to specify that $I(t)$ is real wage income.

[matdehaven]

I see, that makes sense. I think it should be $-a n(t)$. Making that change I was able to get to a single Euler equation by solving out for all of the shadow prices.

It feels like the Euler equation is off, however. When I try to find the steady state for it, it is still a big mess. I think that this model should collapse down to the same steady state as the EBC model, right? Since in steady steady state we are on the Beveridge curve, no dynamics, etc.

I'm also attaching a pdf with my Lagrangian -> Euler derivations, and the set of equations I have defining the equilibrium.
DeHaven_Euler.pdf

[pmichaillat]

That does seem too complicated. The Euler equation should just link c(t) to c(t+1)—it should be a first-order difference equation, not a second-order difference equation.

One thing you could try is to have w(t) in the utility function instead of w(t+1). Sometimes such small timing differences simplify things. (As an aside, this is why continuous time is very good for analytical work: you do need to make such arbitrary timing decisions.)

And indeed, the steady-state version of the Euler equation should give you the same aggregate demand curve as in the EBC model—at least when the discount factor $\beta = 1$. With $\beta<1$, you might have some small extra terms involving $\beta$.

[matdehaven]

Can I solve this first order difference equation for the shadow price on real wealth directly?

Right now I'm plugging it into the other first order conditions, which gets me the second-order difference equation as a a result.

[pmichaillat]

Absolutely. Lagrange multipliers are valid variables of the model, and it is perfectly acceptable to formulate the Euler equation in terms of the Lagrange multiplier on the budget constraint. (This is actually how we solve the EBC model.)

If you do that, however, you have to make sure to keep the first-order condition that links the Lagrange multiplier to observable variables (such as wealth & consumption).

But in the simulations, it is perfectly fine to have the Lagrange multiplier as an endogenous variable.