Real Business Cycle (RBC) model

Model description

Consider the basic Real Business Cycle (RBC) model with leisure. The representative household maximizes present as well as expected future utility \begin{align*} \underset{{C_{t},I_{t},L_t,K_{t}}}{\max} E_t \sum_{j=0}^{\infty} \beta^{j} U_{t+j} \end{align*} with $\beta <1$ denoting the discount factor and $E_t$ is expectation given information at time $t$. We will consider two functional specifications for the contemporaneous utility function. \begin{align*} U_t &= \gamma \log(C_t) + \psi \log{(1-L_t)} & ~~[\text{log-utility}] \\ U_t &= \gamma \frac{C_{t}^{1-\eta_C}-1}{1-\eta_C} +\psi \frac{(1-L_{t})^{1-\eta_L}-1}{1-\eta_L} &~~[\text{CES}] \end{align*} Note that due to L’Hopital’s rule $\eta_C=\eta_L=1$ implies the log-utility specification. In both cases, contemporenous utility is additively separable and has two arguments: consumption $C_t$ and labor $L_t$. The marginal utility of consumption is positive, whereas more labor reduces utility. Accordingly, $\gamma$ is the consumption utility parameter and $\psi$ the labor disutility parameter. In each period the household takes the real wage $W_t$ as given and supplies perfectly elastic labor service to the representative firm. In return, she receives real labor income in the amount of $W_t L_t$ and, additionally, profits $\Pi_t$ from the firm as well as revenue from lending capital $K_t$ at interest rate $R_t$ to the firms, as it is assumed that the firm and capital stock are owned by the household. Income and wealth are used to finance consumption $C_t$ and investment $I_t$. In total, this defines the (real) budget constraint of the household: \begin{align*} C_t + I_t = W_t L_t + R_t K_t + \Pi_t \end{align*}

The law of motion for capital $K_{t-1}$ at the end of period $t$ is given by \begin{align*} K_{t} = (1-\delta)K_{t-1} + I_t \end{align*} and $\delta$ is controlling depreciations. Assume that the transversality condition¹ is full-filled.

Productivity $A_t$ is the driving force of the economy and evolves according to \begin{align*} \log{A_{t}} &= \rho_A \log{A_{t-1}} + \varepsilon_t^A \end{align*} where $\rho_A$ denotes the persistence parameter and $\varepsilon_t^A$ is assumed to be normally distributed with mean zero and variance $\sigma_A^2$.

Real profits $\Pi_t$ of the representative firm are revenues from selling output $Y_t$ minus costs from labor $W_t L_t$ and renting capital $R_t K_{t-1}$: \begin{align*} \Pi_t = Y_{t} - W_{t} L_{t} - R_{t} K_{t-1} \end{align*} The representative firm maximizes expected profits \begin{align*} \underset{{L_{t},K_{t-1}}}{\max} E_t \sum_{j=0}^{\infty} \beta^j Q_{t+j}\Pi_{t+j} \end{align*} subject to a Cobb-Douglas production function \begin{align*} f(K_{t-1}, L_t) = Y_t = A_t K_{t-1}^\alpha L_t^{1-\alpha} \end{align*} The discount factor takes into account that firms are owned by the household, i.e. $\beta^j Q_{t+j}$ is the present value of a unit of consumption in period $t+j$ or, respectively, the marginal utility of an additional unit of profit; therefore $Q_{t+j}=\frac{\partial U_{t+j}/\partial C_{t+j}}{\partial U_{t}/\partial C_{t}}$.

Finally, we have the non-negativity constraints $K_t \geq0$, $C_t \geq 0$ and $0\leq L_t \leq 1$ and clearing of the labor as well as goods market in equilibrium, i.e.: \begin{align*} Y_t = C_t + I_t \end{align*}

First-order conditions of the representative household

Due to our assumptions, we will not have corner solutions and can neglect the non-negativity constraints. Due to the transversality condition and the concave optimization problem, we only need to focus on the first-order conditions. Therefore, we will now show that the first-order conditions of the representative household are given by \begin{align*} U_t^C &= \beta E_t\left[U_{t+1}^C\left(1-\delta + R_{t+1}\right)\right] \\\ W_t &= -\frac{U_t^L}{U_t^C} \end{align*} where $U_t^C=\frac{\partial U_t}{\partial C_t}$ and $U_t^L = \frac{\partial U_t}{\partial L_t}$. In case of log-utility, we have the following functional specifications: \begin{align*} U_t^C &= \gamma C_t^{-1}\\\ U_t^L &= - \psi (1-L_t)^{-1} \end{align*} whereas for CES utility we have: \begin{align*} U_t^C &= \gamma C_t^{-\eta_C}\\\ U_t^L &= - \psi (1-L_t)^{-\eta_L} \end{align*}

The Lagrangian for the household problem is: \begin{align*} L = E_t\sum_{j=0}^{\infty}&\beta^j U_{t+j}\left(C_{t+j},L_{t+j}\right) \\\ &+\beta^j\lambda_{t+j}\left[W_{t+j} L_{t+j} + R_{t+j} K_{t-1+j} - C_{t+j} - I_{t+j}\right] \\\ &+\beta^j \mu_{t+j} \left[(1-\delta)K_{t-1+j} + I_{t+j} - K_{t+j}\right] \end{align*} Note that the problem is not to choose $[C_t,I_t,L_t,K_{t}]{t=0}^\infty$ all at once in an open-loop policy, but to choose these variables sequentially given the information at time $t$ in a closed-loop policy, i.e. at period $t$ decision rules for ${C_t,I_t,L_t,K{t}}$ given the information set at period $t$; at period $t+1$ decision rules for $[C_{t+1},I_{t+1},L_{t+1},K_{t+1}]$ given the information set at period $t+1$. The first-order condition w.r.t. $C_t$ is given by \begin{align*} \frac{\partial L}{\partial C_{t}} &= E_t \left(U_t^{C}-\lambda_{t}\right) = 0 \\\ \Leftrightarrow \lambda_{t} &= U_t^{C} & (I) \end{align*} The first-order condition w.r.t. $L_t$ is given by \begin{align*} \frac{\partial L}{\partial L_{t}} &= E_t \left(U_t^{L} + \lambda_{t} W_{t}\right) = 0 \\\ \Leftrightarrow \lambda_{t} W_{t} &= - U_t^{L} &(II) \end{align*} The first-order condition w.r.t. $I_{t}$ is given by \begin{align*} \frac{\partial L}{\partial I_{t}} &= E_t \beta^j \left(-\lambda_{t} + \mu_{t}\right) = 0 \\\ \Leftrightarrow \lambda_{t} &= \mu_{t} & (III) \end{align*} The first-order condition w.r.t. $K_{t}$ is given by \begin{align*} \frac{\partial L}{\partial K_{t}} &= E_t (-\mu_{t}) + E_t \beta \left(\lambda_{t+1}R_{t+1}+\mu_{t+1}(1-\delta)\right) = 0 \\\ \Leftrightarrow \mu_{t} &= E_t \beta(\mu_{t+1}(1-\delta)+\lambda_{t+1}R_{t+1}) & (IV) \end{align*}

(I) and (III) in (IV) yields \begin{align*} U_t^c &= \beta E_t U_{t+1}^c\left(1-\delta + R_{t+1}\right) \end{align*} This is the Euler equation of intertemporal optimality. It reflects the trade-off between consumption and savings. If the household saves a (marginal) unit of consumption, she can consume the gross rate of return on capital, i.e. $(1-\delta+R_{t+1})$ units, in the following period. The marginal utility of consuming today is equal to $U_t^c$, whereas consuming tomorrow has expected utility $E_t(U_{t+1}^c)$. Discounting expected marginal utility with $\beta$ the household must be indifferent between both choices in the optimum.

(I) in (II) yields: \begin{align*} W_t = - \frac{U_t^L}{U_t^c} \end{align*} This equation reflects intratemporal optimality, particularly, the optimal choice for labor supply: the real wage must be equal to the marginal rate of substitution between labor and consumption.

First-order conditions of the representative firm

First, we note that even though firms maximize expected profits here we actually have a simple static problem. That is, the objective is to maximize profits \begin{align*} \Pi_t = A_t K_{t-1}^\alpha L_t^{1-\alpha} - W_t L_t - R_t K_{t-1} \end{align*} We will now show that the first-order conditions are given by: \begin{align*} W_t &= f_L\\\ R_t &= f_K \end{align*} where $f_L = (1-\alpha) A_t \left(\frac{K_{t-1}}{L_t}\right)^\alpha$ and $f_K = \alpha A_t \left(\frac{K_{t-1}}{L_t}\right)^{1-\alpha}$.

Let’s start with the first-order condition w.r.t. $L_{t}$ which is given by \begin{align*} \frac{\partial \Pi_t}{\partial L_{t}} &= (1-\alpha) A_t K_{t-1}^\alpha L_t^{-\alpha} - W_t = 0 \\\ \Leftrightarrow W_t &= (1-\alpha) A_t K_{t-1}^\alpha L_t^{-\alpha} = f_L = (1-\alpha) \frac{Y_t}{L_t} \end{align*} Intuitively, the real wage must be equal to the marginal product of labor. Due to the Cobb-Douglas production function it is a constant proportion $(1-\alpha)$ of the ratio of total output and labor. This is the labor demand function.

Second, the first-order condition w.r.t. $K_{t-1}$ is given by \begin{align*} \frac{\partial \Pi_t}{\partial K_{t-1}} &= \alpha A_t K_{t-1}^{\alpha-1} L_t^{1-\alpha} - R_t = 0 \\\ \Leftrightarrow R_t &= \alpha A_t K_{t-1}^{\alpha-1} L_t^{1-\alpha} = f_K = \alpha \frac{Y_t}{K_{t-1}} \end{align*} Intuitively, the real interest rate must be equal to the marginal product of capital. Due to the Cobb-Douglas production function it is a constant proportion $\alpha$ of the ratio of total output and capital. This is the capital demand function.

Steady state

The steady state of this model is a fixed point, in the sense that there is a set of values for the endogenous variables that in equilibrium and in the absence of shocks remain constant over time. Usually, we want to provide a recipe how to compute these values sequentially.

First, consider the steady state value of technology: \begin{align*} \log\bar{A}=\rho_A \log\bar{A} + 0 \Leftrightarrow \log\bar{A} = 0 \Leftrightarrow \bar{A} = 1 \end{align*} The Euler equation in steady state becomes: \begin{align*} \bar{U}^C &= \beta \bar{U}^C(1-\delta+\bar{R})\\\ \Leftrightarrow \bar{R} &= \frac{1}{\beta} + \delta - 1 \end{align*}

Now the trick is to provide closed-form expressions for all variables recursively, but in relation to steady state labor.

• The firms demand for capital in steady state becomes \begin{align*} \bar{R} &= \alpha \bar{A} \bar{K}^{\alpha-1}\bar{L}^{1-\alpha}\\\ \Leftrightarrow \frac{\bar{K}}{\bar{L}} &= \left(\frac{\alpha \bar{A}}{\bar{R}}\right)^{\frac{1}{1-\alpha}} \end{align*}
• The firms demand for labor in steady state becomes: \begin{align*} W =(1-\alpha) \bar{A}\bar{K}^\alpha \bar{L}^{-\alpha} = (1-\alpha)\bar{A} \left(\frac{\bar{K}}{\bar{L}}\right)^\alpha \end{align*}
• The law of motion for capital in steady state implies \begin{align*} \frac{\bar{I}}{\bar{L}} &= \delta\frac{\bar{K}}{\bar{L}} \end{align*}
• The production function in steady state becomes \begin{align*} \frac{\bar{Y}}{\bar{L}} = \bar{A} \left(\frac{\bar{K}}{\bar{L}}\right)^\alpha \end{align*}
• The clearing of the goods market in steady state implies \begin{align*} \frac{\bar{C}}{\bar{L}} = \frac{\bar{Y}}{\bar{L}} - \frac{\bar{I}}{\bar{L}} \end{align*}

Now, we need to derive steady state labor from the equilibrium on the labor market.

Log-utility

Due to the log-utility, we can derive a closed-form expression: \begin{align*} \psi \frac{1}{1-\bar{L}} &= \gamma \bar{C}^{-1} W \\\ \Leftrightarrow \psi \frac{\bar{L}}{1-\bar{L}} &= \gamma \left(\frac{\bar{C}}{\bar{L}}\right)^{-1} W \\\ \Leftrightarrow \bar{L} &= (1-\bar{L})\frac{\gamma}{\psi} \left(\frac{\bar{C}}{\bar{L}}\right)^{-1} W \\\ \Leftrightarrow \bar{L} &= \frac{\frac{\gamma}{\psi} \left(\frac{\bar{C}}{\bar{L}}\right)^{-1} W}{1+\frac{\gamma}{\psi} \left(\frac{\bar{C}}{\bar{L}}\right)^{-1} W} \end{align*}

Then, it is straigforward to compute the remaining steady state values, i.e. \begin{align*} \bar{C} = \frac{\bar{C}}{\bar{L}}\bar{L},\qquad \bar{I} = \frac{\bar{I}}{\bar{L}}\bar{L},\qquad \bar{K} = \frac{\bar{K}}{\bar{L}}\bar{L}, \qquad \bar{Y} = \frac{\bar{Y}}{\bar{L}}\bar{L} \end{align*}

CES-utility

The steady state for labor changes to \begin{align*} W \gamma C^{-\eta_C} &= \psi(1-L)^{-\eta_L}\\\ W \left(\frac{C}{L}\right)^{-\eta_C} &= \frac{\psi}{\gamma}(1-L)^{-\eta_L}L^{\eta_C} \end{align*} This cannot be solved for $L$ in closed-form. Rather, we need to condition on the values of the parameters and use an numerical optimizer to solve for $L$.

Dynare implementation

Note that I define local variables for marginal utility of current consumption #UC, marginal utility of future consumption #UCp and marginal utility of current labor #UL. These are text-substituted by the macro preprocessor in the equations of the model block only.

Log-utility

var Y C K L A R W I;
varexo eps_A;
parameters alph betta delt gam pssi rhoA;

alph = 0.35; betta = 0.99; delt = 0.025; gam = 1; pssi = 1.6; rhoA = 0.9;

model;
    #UC  = gam*C^(-1);
    #UCp = gam*C(+1)^(-1);
    #UL  = -pssi*(1-L)^(-1);
    UC = betta*UCp*(1-delt+R(+1));
    W = -UL/UC;
    K = (1-delt)*K(-1)+I;
    Y = I+C;
    Y = A*K(-1)^alph*L^(1-alph);
    W = (1-alph)*Y/L;
    R = alph*Y/K(-1);
    log(A) = rhoA*log(A(-1))+eps_A;
end;

CES-utility

var Y C K L A R W I;
varexo eps_A;
parameters alph betta delt gam pssi rhoA etaC etaL;

alph = 0.35; betta = 0.99; delt = 0.025; gam = 1; pssi = 1.6; rhoA = 0.9;
etaC  = 2;   etaL  = 1.5;

model;
    #UC  = gam*C^(-etaC);
    #UCp = gam*C(+1)^(-etaC);
    #UL  = -pssi*(1-L)^(-etaL);
    UC = betta*UCp*(1-delt+R(+1));
    W = -UL/UC;
    K = (1-delt)*K(-1)+I;
    Y = I+C;
    Y = A*K(-1)^alph*L^(1-alph);
    W = (1-alph)*Y/L;
    R = alph*Y/K(-1);
    log(A) = rhoA*log(A(-1))+eps_A;
end;