# Timing of returns in NK model with government

Hi all,

I have a silly question. The timing of the interest rate on government debt ® and the rental rate of capital ® in NK_baseline.mod (the replication of Fernandez-Villaverde (2009)), while both government debt and capital are predetermined variables, are different; R(-1) and r in the consumer budget constraint. Thus the FOC,

lambda=betta*lambda(+1)mu_z(+1)^(-1)/PI(+1)R;
q=betta
lambda(+1)/lambda
mu_z(+1)^(-1)mu_I(+1)^(-1)((1-delta)*q(+1)+**r(+1)*u(+1)-(gammma1(u(+1)-1)+gammma2/2(u(+1)-1)^2));

The timing difference, is this common to use? This, I think, is problematic when I have fiscal policy in the model. If I add a government budget constraint using the government debt timing on NK_baseline.mod and the original paper, R(-1)*b(-1), this won’t work; the rank condition isn’t verified. When I declared the debt as predetermined variable and use the same timing as the capital, for example,

g + Rb = tax(ldw+r*k) + b(+1)*PI(+1) + lumpsum;

the Euler equation is now

lambda=betta*lambda(+1)*mu_z(+1)^(-1)/PI(+1)*R(+1);

This, however, is working when the Taylor rule is defined as

R(+1)/Rbar=(R/Rbar)^gammmaR*((PI/PIbar)^gammmaPI*((yd/yd(-1)*mu_z)/exp(LambdaYd))^gammmay)^(1-gammmaR)*exp(epsm);

But, really, the Taylor rule looks really weird with (+1) in it. Do you have any thoughts on this? Did I make any mistake on something? Can you suggest how can I fix this problem? Any comments will be very much appreciated. Thank you!

Note: I let g, tax, and lumpsum as completely exogenous, so the government budget constraint determines debt.

The first part of the answer, why the setup is not stable, is here: [RBC with Government Sector).

The second part is: the timing difference comes from the fact that nominal bonds are risk-free for agents. Bonds holdings decided today, b, were promised to return tomorrow a nominal interest rate R, that was decided already today. This makes them nominally risk-free in the Euler equation. But the inflation part makes them risky in real terms.
This differs from capital decided upon today, k. It will return a stochastic return whose realization is only known tomorrow, r(+1).

In what you wrote down, the government budget constraint is wrong. It should be

See e.g. equation 22 of Born/Peter/Pfeifer (2013): Fiscal News and Macroeconomic Volatility, sciencedirect.com/science/ar … 8913001437

Dr. Pfeifer,

Thank you very much for taking the time to answer to my questions. I followed your suggestions but I still had a problem with the Blanchard-Kahn conditions. So I, (1) changed the government bond timing to r(-1)*d, (2) switched on the responses of government debt-to-GDP ratio to government consumption,

g/gss - (g(-1)/gss)^GAMMA_G * ((d/y(-1))/dyss)^(GAMMA_D*(1-GAMMA_G)) * exp(eps_g);

where GAMMA_D = 1.5, and (3) changed the timing on the Taylor rule. I attached the two codes. The “funny” one is fiscal5.mod and the one following your suggestion is fiscal6.mod. Did I make any other mistake? Thank you.
fiscal6.mod (4.05 KB)
fiscal5.mod (4.05 KB)

Why is there a price level showing up in your government budget constraint? Everything should be in real terms. If anything, there should be an inflation term. Also, for

```GAMMA_D = -0.5; ```
it runs. Check the signs of your definitions for B and whether this makes sense.

Dr. Pfeifer,

The p in the government budget constraint, actually, is my notation for inflation. So, I divided a nominal bond D(+1) by price at time t, thus the inflation term showed up. Also, I made a mistake on the sign of the response. Thank you very much for the correction.