Asymmetric theoretical mean in symmetric model


model_stickyPricesComplete.mod (9.2 KB)

In the attached 2-country model, you notice that the theoretical means (oo_.mean) are not symmetric across the 2 countries even though the countries are symmetric. For instance, for consumption, there is a slight difference of 0.8081 vs. 0.8076.
This seems to be related to the way I implement the market clearing condition for population. Since the model features migration, one equilibrium condition is that world population is constant and this condition replaces one of the first-order conditions for optimal migration (similar to the bond market clearing condition replacing one of the UIP conditions in an incomplete market model). But depending on which country’s FOC I replace (that is either country H1 or country H2 below), I get that consumption of H1 is 0.8081 or consumption of H2 is 0.8081.
Also: If I remove the market clearing condition for population and keep both countries’ FOCs in, I get NaNs in the theoretical means.

Any thoughts on this?

[name='24. Stock of workers']
@#if i == "H1"    
@#for j in countries
@#if i != "H1"
    Popw@{i} = 
@#for j in countries
    +exp(gam*((cw@{i}^(1-1/sigma)-1)/(1-1/sigma) - tau@{i}@{j} + beta*V@{i}(+1)-V@{j})) * Popw@{j}(-1)

It seems you are looking at order=2. Here, the shocks affecting the respective economies will alter the mean. While the steady state is perfectly symmetric, that is not true for the ergodic mean. It seems it matters for the dynamics which country you consider as the baseline.

Thanks for your reply, Johannes. Yes, you’re right, I refer to the ergodic mean with a second order approximation.
Isn’t it concerning that such an arbitrary choice affects the ergodic mean and hence any conclusions for welfare?
The same thing seems to happen in a 2-country incomplete market model where the nth country’s budget constraint is replaced by the world resource constraint. Depending on which country I choose to be the nth country (and this choice is quite arbitrary), I get different ergodic means.

It seems the model is not perfectly symmetric. When inspecting the decision rules at order=2, you get

                                 CH2             CH1
Constant                    0.814353        0.814542
(correction)               -0.003338       -0.003149
PopwH1(-1),PopwH1(-1)        0.003503       -0.004399
PopwH2(-1),PopwH2(-1)       -0.000723       -0.071846
PopwH1(-1),sh_H1           -0.003503       -0.064199
PopwH1(-1),sh_H2            0.012774        0.054928
PopwH2(-1),sh_H1           -0.237229       -0.372093
PopwH2(-1),sh_H2            0.320667        0.288654

model.mod (7.3 KB)

Yes, I think this observation is another way of describing the problem.
What I do not understand is why the model is not symmetric.

Maybe it is easier to talk about the Baxter-Crucini 1995 case. I have rewritten the code, removed the migration component, but added incomplete financial markets. The same problem appears. Up to a first order both the steady state and the dynamics are symmetric. But with a second-order approximation, the policy functions become asymmetric although the code is mathematically symmetric. With complete financial markets, things are symmetric even with a second-order approximation.
The one place that I suspect to cause asymmetries is the budget constraint:

[name=‘21. Budget constraint’]
@#if i == nth_country
@#for j in countries
= 0;
@#if i != nth_country
@#if complete==1
B@{i} = 0;
B@{i}/(1+int@{nth_country})/s@{i} - B@{i}(-1)/s@{i} + iota/2*B@{i}^2=
pM@{i}*M@{i} - Y@{i};

where pM*M - Y is just net exports and s is the real exchange rate relative to the nth country.
But my setup seems rather standard: I’ve got 2 countries’ budget constraints + the bond market clearing condition. By Walras’ law, if two of them are satisfied, the third is satisfied, too. Mathematically, it makes no difference which country’s budget constraint I use. So why does it make a difference to dynare?

Could it be that you stationarized the model using one of the “tricks” of Schmitt Grohe and Uribe. For instance a debt elastic interest rate? I mean if its percetly symetric with complete financial markets it has to be the introduction of incomplete financial markets (i.e. UIP condition etc).

model_simple.mod (4.5 KB)

I agree that it has to be related to the part with incomplete financial markets. I’ve set up the simplest (endowment) NK model of a currency union with Armington trade. The issue persists…
I find it interesting that the bond market does not seem to clear when I enter the values from oo_.mean. Both countries borrow.
Yes, I add bond adjustment costs as in SGU. Without them, I get NaNs in the oo_.mean. Does anyone know of an incomplete market 2-country model implemented in dynare?

I made a bit of progress. Now, things look more symmetric except for the correction term and constant in the policy function. For C in cty 1 it’s 1.000032 and 0.000032 correction, and for cty 2 it’s 0.999968 and -0.000032 correction. The policy function of C with respect to the various variables in the system is symmetric. But the means of the stochastic steady state are still different.
Does anyone know what this correction term is and why it is different across the two countries?
model_simple.mod (4.7 KB)

The only thing I can think of is an approximation error at order=2. Here, it would matter which country is the baseline due to ratios. Approximating the variable in the numerator will be different from the denominator. That would explain