I am trying to figure out how to run a basic NK model with trend inflation and I managed to get the steady state of nominal interest rate to be greater than the real interest rate. However, what I find is that a problem emerges when I analyse the trend inflation case. The usual cost minimisation problem of monopolistically competitive firms gives the following:
Y=(A*(K(-1)^kappa)/THETA_P)*((N/THETA_W)^(1-kappa)); // AGGREGATE PRODUCTION FUNCTION
w=MC*(1-kappa)*(Y/N)*(THETA_W*THETA_P); // LABOUR DEMAND
r=MC*kappa*((THETA_P*Y)/K(-1)); // CAPITAL DEMAND
MC=(1/(A))*((r/kappa)^kappa)*((w/(1-kappa))^(1-kappa)); // REAL MARGINAL COSTS
Where the notation is standard except THETA_P is the price dispersion and THETA_W denotes wage dispersion.
Whenever I run my model, however, all of the 4 equations above turn out to be collinear. In a sense, it is not surprising, because the MC is derived by substituting w and r into Y and rearranging. But how do I then identify aggregate output, if the standard resource constraint of Y=C+I+G identifies the aggregate consumption? If it is not the problem of identification, could this be a problem of the timing? I never really understood this well, but I did not encounter this problem before working with zero trend inflation models.
Another collinear relationship is found in the Taylor rule:
(R/STEADY_STATE(R))=((R(-1)/STEADY_STATE(R))^(rho_r))*((((PI/STEADY_STATE(PI))^nu_pi)*((Y/STEADY_STATE(Y))^nu_y))^(1-rho_r))*M; // TAYLOR RULE
Where e_r is just an exogenous shock. I don’t really understand how the Taylor rule can be collinear with itself? Note that this holds even if there is no smoothing or output response.
I had a look at the NK baseline model of fernandez-villaverde and they use not separate unconditional factor demand schedules, but rather the optimal ratio of the factors of production, which gets rid of the singularity problem so long as one of the factors of production has an additional equation. I still don’t get what the additional equation is even though the NK baseline code is right in front of my eyes, but this isn’t a dynare problem, so I’ll figure it out eventually.
Why is your production function
i.e. why the division by THETA_P?
Apologies for the long silence, I was on a holiday. Both wage and price dispersion have a negative effect on aggregate employment and output in a Calvo framework because of staggered adjustment (THETA_P being price dispersion). I think this is a fairly standard way of modelling it. Intuition being that although with constant returns to scale all firms have identical reset price, adjustment of prices is non-synchronous and consumers demand more of the goods that are under-priced at any given period. However, output of such firms does not alter, which implies that aggregate demand (=aggregate supply in equilibrium) must fall for the goods market to clear.
In any case, the identification issue was totally unrelated to the set of equations describing cost-minimisation displayed above. I just missed another equation in the nominal part of the model that identified the stochastic discount factor. Since then, I took out the stochastic discount factor as one of the endogenous variables as it is simply the inverse of the gross nominal interest rate.
The collinearity of the Taylor rule still persists, however. And interestingly, it is only the Taylor rule collinear with itself when it is represented as above, but when it is represented as below, it becomes collinear with the Euler equation:
PI(+1)=beta*R*((PSI(+1)/PSI)*(UC(+1)/UC)); //(4) EULER EQUATION
R=D*PI(+1); //(5) FISCHER EQUATION
((R*beta)/STEADY_STATE(PI))=(((R(-1)*beta)/STEADY_STATE(PI))^(rho_r))*((((PI/STEADY_STATE(PI))^nu_pi)*((Y/STEADY_STATE(Y))^nu_y))^(1-rho_r))*exp(-sigma_r*e_r); //(38) TAYLOR RULE
PSI being a preference shock. The two representations are in a sense identical given the Fischer equation, but give rise to completely different results. Any idea what I might be missing?
Ok. I did not see that this was the price dispersion term. That explains it.
Regarding collinearity: you have the issue described in: [An infinity of steady states with Taylor rules)
You cannot simultaneously determine the nominal interest rate and inflation in steady state in the model due to the Fisher equation.
This was quite careless of me. I am identifying steady state inflation with a parameter, but I completely forgot to write the Taylor rule by explicitly taking it into the account. Now that I have re-written the Taylor rule in the following way:
code=(((R(-1)beta)/(1+tau))^(rho_r))((((PI/(1+tau))^nu_pi)*((Y/STEADY_STATE(Y))^nu_y))^(1-rho_r))exp(-sigma_re_r); //(38) TAYLOR RULE
the model works without any warnings (tau being a parameter =0.02). Interestingly, the impulse response functions did not seem to change. But I guess its always good practice to try and eliminate all the collinearity (especially when it is non-trivial).