Steady-State in NK Model - How to treat gains

Hi there,

I work with a standard NK model including final and intermediate firms. Intermediate firms face monopolistic competition. There is investment as well as consumption. I have implemented habit persistence preferences in consumption and capital adjustment costs.

In this model, marginal costs in steady-state are equal the inverse mark-up (markup > 1), therefore, they are smaller than 1. This means that labor effort is paid lower than the additional value of output it produces (wedge). Because households own the intermediate firm (which pays less wage than under no mark-up), there must be dividend payments for the household.

How I solve for the steady-state: I solve for labor market equilibrium (I isolate steady-state consumption depending only on output) and then I solve for capital market equilibrium (I isolate steady-state investment depending only on output). Then I take the resource constraint, plug in consumption and investment and solve for steady-state output. The rest is straightforward.

I’m unable to find the steady-state. I don’t know how to deal with gains because they enter the household’s budget constraint exogenously. I would expect the household not incorporating lower wage payments as he owns the firm and gets dividends instead. But actually, he does, in terms of marginal cost-depending steady-state wage payments.

At the end, I don’t know how to correct the wedge between actually produced output and the corresponding too low wage payments (which should not be incorporated… :woozy_face: …).

Should I introduce steady-state fix costs? How does it work? Or fixing labor effort in steady- state? Any consequences?

Here are some steady-state equations:

b_habit = 0.87;        % Habit persistence in consumption
xi = 0.23;                  % Capital adjustment costs parameter
alpha  = 0.20;		 % Share of capital in technology
beta = 0.997;		  %  Discount factor
delta = 0.025;		  % Depreciation of capital
sigmaC = 2;	          % Risk aversion in individual's consumption
sigmaL = 1.5;		  % Labor disutility
theta_p	= 0.65;       % Proportion of price setting constrained firms 
rho_r = 0.8;		   % Monetary policy parameter
phi_r = 2;		  % Monetary policy parameter for inflation weight
mu = 1.1;		  % Mark-up on intermediate goods
psi = 1.5;		          % Substitutability between intermediate goods 
rho_a = 0.95; 		  % Productivity coefficient
chi_X = 1;                  % Labor disutility parameter

Infl_x_H = 1;
Infl_H = 1;
price_dispersion_h = 1;
A_h = 0;
q_h = 1;
mc_h = (1/mu);
r_h = 1/beta;
a_3 = (1/xi);  
a_1 = ((delta)^(a_3));
a_2 = (delta - ((delta)/(1-a_3)));

z_h =  (1/beta) - (1-delta) - (((a_1)/(1-a_3))*(delta^(1-a_3))) - a_2 + a_1*((delta)^(1-a_3)) ; % real remuneration of capital
w_h = (1-alpha)* ((mc_h)^(1/(1-alpha)))  *  ( ( alpha/z_h )^(alpha/(1-alpha)) )      ; % real wage

y_h =  (   (   (1-b_habit)*(    (1-delta*alpha*(1/z_h))/   (      ((chi_X/w_h)*((1-alpha)^(sigmaL))*((1/w_h)^(sigmaL))*(1/(1-b_habit*beta)))^(-1/sigmaC)  )   )     )    ^(-1/ (  (sigmaL/sigmaC)+1) )  )       ;                        % output

d_h = (1 - mc_h)*y_h;    % dividends

y_h = d_h + y_h;  % Updating y_h in terms of dividend payments (?)

i_h = delta*alpha*(y_h /z_h);    % investment
k_h = i_h/delta;     % capital
c_h = y_h   - i_h ;   % consumption
lambda_h = ((c_h - b_habit*c_h)^(-sigmaC))*(1-beta*b_habit);
h_h =  (((lambda_h*w_h)*(1/chi_X))^(1/sigmaL))   ;     % labor-effort


Thank you very much for your inputs! :four_leaf_clover:

You should have a look at
In case of a distortionary steady state, there are profits.

Of course output needs to take dividends into account, because people will consume and save out of those profits.

Many thanks for your answer, I will check the file.

I saw a lot of similar models fixing labor effort in steady-state. My model, of course, works with that assumption. Can I also use this feature to make the model stationary? I use my model (levels, 2nd order approx.) to observe theoretical moments (e.g. Does such a fixed steady-state labor effort distort the results?

It would be very helpful for me if somebody would give me a hint why the steady-state code above is not correct. There is a small deviation: the production function’s output (implied by steady state capital and labor) is not exactly the same as the output implied by the resource constraint (steady-state consumption and investment).

Thank you for your help

Did you have a look at the NK_baseline.mod in the Dynare examples folder and the reference therein?
Also, I do not understand the line

y_h = d_h + y_h;  % Updating y_h in terms of dividend payments (?)

I added a fixed cost block into the production function. The fixed cost is simply the difference between the implied steady-state production function’s output (too high because wage payments are lower in terms of lower marginal costs) and the steady-state output implied by labor market and capital market equilibria. The model works now.