Hello,
I am trying to run a perfect foresight simulation to see the effect of labor income taxes on household utility. Unfortunately DSGE modeling and I are not on good terms 
and I struggle to understand which equations I can (and should) use for my model. I ran a similar model setup with a productivity and monetary shock, but I’m not able to apply this to this fiscal shock. This is the model I am trying to set up:
Households have an instantaneous utility function that depends on C, consumption; N, hours worked; (1/theta), the intertemporal elasticity of substitution; and (1/phi), the Frisch elasticity of labor supply.
They are under the following budget constraint:
P(t)*C(t) + 1/(1+i(t)) * B(t) + T(t) = (1 - theta(t)) * W(t)*N(t) + B(t-1) + PI(m)
where i is the nominal interest rate, B are government bonds, T are lump-sum transfers and PI are the profits of m firms (as households are the owners). There is no capital in the model, but saving is somewhat implemented using the bonds.
The shock I want to simulate is a 5 % decrease in the labor income tax rate for some periods, like this:
theta(t) = theta(bar) - epsilon(theta)
where theta is the labor income tax rate (the bar denotes steady state) and epsilon is the fiscal shock.
I want the income tax reduction to be financed by debt using this government budget restriction:
where gamma is the constant share of spending on GDP, b is the real public debt share, w is the gross real wage and t is the real share of lump sum taxes on GDP.
The government reacts to this debt by adjusting the lump sum taxes according to:
t(t) = t(bar) + omega(b) * (b(t-1) - b(bar))
So this is what I would like to do: I want to plot labor income tax rate, net and gross real wage, working hours, HH utility, output, consumption, nominal and real interest rate, public debt, lump-sum taxes and inflation. Can you help me finding the formulas I can use to set this up? Sorry for such a broad question, I have tackled this several times but found myself stuck somewhere again and again…
I hope I described the setup thoroughly enough, if you would need more information, please let me know! Unfortunately I couldn’t show more formulas, like the HH Utility function or others. I hope it is still enough info.