Blanchard Kahn conditions are not satisfied: no stable equilibrium

Hello, I have the problem about the stability conditions of Blanchard and Khan, I can not solve if the problem is by the initial parameters or maybe some error in the temporary variables.

I hope you can help me greetings

AB.mod (10.2 KB)

It seems you have bonds in the model. What assures the solvency of the issuer, i.e. what stabilizes debt?

Dear professor, I find when there is distortianary tax in the model, like 图片 , then bonds will be stabilized and can be included in the modeldistortion_tax_logs.mod (3.7 KB) . But when there is no distortianary tax, like 图片 , bonds will not be stabilized. Can you tell me why distortianary taxs will have such effect? (in my opinion, it seems that those exogenous distortianary taxs would not influence the BK condition).

What exactly stabilizes debt in your model without distortionary taxation? There is only G left and I presume that it is exogenous.

Professor, thanks for answering me. My point is: in the RBC model without distortionary taxation and lump sum tax, the debt can not be stabilized. Besides, Debt also won’t get into the equibrilium equations and there is only exogenous G. But I find when distortionary taxation is introduced into the model (still without lump sum tax), we can stabilizes Debt. I just wonder why distortionary taxation have such power.

My guess is because the tax revenue now varies with the state of the economy.

Dear professor, I try to understand your point with this: in the RBC model like 图片 , the debt can not be stabilized because this equation only depends on Gt and is expolisive for debt Dt (which means the eigenvalue>1), while Dt is a state variable in the model (not a control variable), therefore BK condition is violated. However, when we use distortianary tax 图片 , the equation of debt will vary with the state of the economy and becomes convergent (which means the eigenvalue<1), so there will be a solution and debt will be stabilized
Am I right?

Hello, you need to consider that equation:

implies that G_t is negative in steady state, unless r_{t-1}^{} equals zero in steady state as well, in that case G_t = 0. Then the rule of government spending as a share of output can’t hold unless output is negative too.

Yes, you are completely right, and I think this is why lump sum tax Tt is included in the RBC model.
My quesiton arises from this: When I see lump sum tax Tt does’t get into the final equibrilium equations, I wonder why Tt is needed in the household constrain and goverment spengding equation. And I find when Tt is eliminated (which is 图片 ), the only chage with the RBC model is that it seems that I can solve Dt with the goverment spending equation. But actually I can not solve it. And the reason I think is what you say or what I mention above (you give your answer from the perspective of economic reality, and mine is from whether the equations are solvable). So really thanks, and I look forward to your next answer or suggestion.

As far as I know, the lump sum tax has to be included into the agents budget constraints and the governement, and following your example, the size of T_t has to balance/assure the Gov spending constraint (including the payment on bonds) and the amount of G_t that you want to calibrate as a fraction of Y_t.
In case you want to include government spending only (so as its shock), I recommend you to follow the mod file in: https://www3.nd.edu/~esims1/rbc.mod, there you can notice no tax or bonds are necessary (or even included) to close the model economy. I hope it helps.

Note that in models with lump sum taxation, there is no government debt. It balances the budget each period. That’s the reason only G shows up and not T. Government spending perfectly determines the tax rate, so the government budget constraint reads G_t=T_t. But you now have spending but no way to pay for that spending.

Thanks ! your answer really helps me to understand it better. :smiley:

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I see, so this is why lump sum taxation is needed in the model: to pay for goverment spending. Thank you professor :grin: