Model Estimation with Complete Markets

Hi all,

I am setting a LOE model (3 countries indexed as c: center, i: EME1, e:EME2) with complete markets. I need to include two Risk Sharing Conditions. I am departing from a financial autarky model that works and the only feature I am changing is that I allow for bonds to not be zero. This removes 3 equations from my system. But now I have a zero net supply equation for the bonds and the aforementioned two Risk sharing conditions to close the model.

If I set the following condition it works:

Cc^(-sigmma)/Cc(-1)^(-sigmma) = Ci^(-sigmma)/Ci(-1)^(-sigmma) ;
Ci^(-sigmma)/Ci(-1)^(-sigmma) = Ce^(-sigmma)/Ce(-1)^(-sigmma) ;

However, if I use the following equations the B-Kahn conditions won’t hold:

(Cc(+1))^(-sigmma)/(Cc)^(-sigmma) = (Ci(+1))^(-sigmma)/(Ci)^(-sigmma) ;
(Ci(+1))^(-sigmma)/(Ci)^(-sigmma) = (Ce(+1))^(-sigmma)/(Ce)^(-sigmma) ;

Why one works and the other doesn’t, aren’t them analog?

Additionally the one working shows me the following prompt:

All endogenous are constant or non stationary, not displaying correlations and auto-correlations

why is that?, Am I making the system non-stationary by adding Risk Sharing?. This didn’t occur with the Financial Autarky model, the only changes made were the mentioned above.

Finally, in order to see whether such formulation led to non-stationarity I wanted to change my formulation and include the conditions not as equality of MRS but as equality of the marginal utilities (following Schmitt-G and Uribe (2003, JIE) among others). So I set the conditions as:

Cc^(-sigmma) = Ci^(-sigmma) ;
Ci^(-sigmma) = Ce^(-sigmma) ;

But then, the model can’t compute the Steady State. The model is simple in which it only has one final good so that the RER is 1 and it’s not missing from the equations. But I don’t now how to deal with the constant of proportionality between center countries and emergent. That constant in fact is the cause of errors and is not an issue if I set the ratios of MRS as above. In SG-U2003 this constant is calibrated in the steady state but different levels won’t cause an error (however their model is an SOE model).

How can I deal with this source of errors in the steady state so that the model can compute?

Is there any way to get the constant I should consider? Maybe for the steady state of the model that’s working in terms of the MRS so that I can get it as \xi = u'(C_0)/u'(C_0^*)=u'(C_{ss})/u'(C_{ss}^*)?

Thank you so much for any reply I can get.

CG (10.0 KB)

  1. The risk sharing conditions is about equalizing actual marginal utilities, not their expected values. That is why the first one works, but the second one with the future timing does not.
  2. Regarding the non-stationarity: you need to find out where this comes from. Your model has two unit roots, but the IRFs nevertheless seem stationary (you can see this if you use a large number of periods).
  3. You can see in Chari et al (2002): “Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates” that going from the MRS to marginal utilities involves backward iteration and normalization of the initial condition. It may be related to 2., because your current formulation may not uniquely pin down this initial condition.
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Thank you, I understand better now.

To prevent messing with the dynamics of the model I finally backed down the appropriate constant in terms of the initial conditions as you suggest:

Cc^(-sigmma) = Ci^(-sigmma)*(steady_state(Cc)/steady_state(Ci))^(-sigmma) ;
Ci^(-sigmma) = Ce^(-sigmma)*(steady_state(Ci)/steady_state(Ce))^(-sigmma) ;

the prompt about unit roots is gone now. The IRFs are stationary as before too.


Camilo G.