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… 
…).
Should I introduce steady-state fix costs? How does it work? Or fixing labor effort in steady- state? Any consequences?
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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
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Thank you very much for your inputs! 