Multisector DSGE Model

Hello everyone, I am new to dynare and currently trying to run a multisector model. Typically in the single sector model, you have interest rates determined in two equations - the bond euler and monetary rule this creates allows the model to be identified. However, when i add another identical sector to the model, it becomes under-identified by one equation. Please does anyone know how to these multisector models are closed?

Why does it become underdetermined? The central bank sets the interest rate. Given this, the Euler equation determines savings by the respective agents. If you have two sectors, you have to Euler equations telling you the savings in the two sectors.

So am I meant to have sectoral bonds for both sectors? This allows for two bond euler equations.In this case i would have to use the sectoral inflation otherwise it would be perfectly colinear with the first euler.

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No, you don’t need sectoral bonds, just that both sectors have access to a common bond.

I am highly confused. So both sectors have access to a common bond, how are the eulers different since the common bonds is growing at the same rate of interest. I am really sorry for disturbing.

Say you have log utility. Then a typical Euler equation will be
So you have the same interest rate and inflation, but consumption differs according to sector i. That consumption is what this equation determines.

Berholt and Lansen 2016-compressed.pdf (2.5 MB)
Ohhh! I was assuming aggregate consumption. I was going by this paper. Thank you very much.

Dear Prof. Johannes, It turns out that the typical euler would not work, as consumption is aggregated in a CES basket. So the FOC for consumption is at the aggregate level meaning there is only one euler equation for the aggregate level of consumption. Sectoral consumption is then obtained using the CES aggregator.

Then I don’t understand why you would have two Euler equations if there are not two separate choices. Or are you confusing aggregation and optimality conditions? Symmetry is only imposed afterwards.

I do not have two euler equations, but one. I tried the two euler equations you suggested but it resulted in colinearity. So, I returned it to the one euler equation for aggregate consumption, but still short one equation.

The euler is given as:
1=\beta E\left[\frac{(C_{t+1} -hC_t)^{-\sigma}}{(C_{t} -hC_{t-1})^{-\sigma}} \frac{R_t}{\Pi_{t+1}}\right].

Then aggregate consumption is aggregated as:
C_{t} = \left[a_m^{\frac{1}{\vartheta_c}} C_{m,t}^{\frac{\vartheta_c-1}{\vartheta_c}} + a_s^{\frac{1}{\vartheta_c}} C_{s,t}^{d\frac{\vartheta_c-1}{\vartheta_c}} \right]^{\frac{\vartheta_c}{\vartheta_c-1}}

I may be missing something here, but why can you not check which variable is not determined in your model? If there were two identical sectors in autarky, you could determine everything. Now you link those two sectors. What changes?

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Perhaps it is the equation linking both sectors i am missing, as i have market clearing for each individual sector, with every other variable determined

That sound about right.

Dear Prof. Pfeifer
I am constructing an economy with formal and informal sectors.
Formal households consume formal and informal goods, similarly informal households consume formal and informal goods.

I specify a cobb-douglas utility function for both the households.

Now when I write the Euler equations in terms of consumption goods, I have 4 euler equations. Is this correct?
Or, should I have a consumption index for the formal households, and another consumption index for informal households. And then obtain two euler equations?

Thank you,

From your description, you have two households maximize with respect to two consumption goods. Thus, you need four first order conditions. Even if you use consumption indices, you will have two Euler equations, but also two more equations governing the consumption index evolution.

Thank you for the reply Prof. Pfeifer