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Households

Representative household": all households are identical, both ex ante and ex post. The household maximizes

\[ max \quad E[\sum_{t=0}^{\infty} \beta^t (log(c) +\eta\cdot log(1-L)) ] \\s.t. \quad K_t=(1+r_t)K_{t-1} + w_tL_t-C_t\\lim_{k\rightarrow \infty} = 0 \]

We rewrite the household’s problem into Bellman equation form

\[V(z,k) = max_{ l,c} \quad [u(c,l) + \beta EV(z', k')]\]

With Euler equation:

\[ U_c (c_t, L_t) = \beta E_t(1+ r_{t+1}) Uc(c_{t+1}, L_{t+1}) \]

Which equals to:

\[ \frac{1}{c_t}= \beta E_t[ \frac{1}{c_{t+1}}(1+\alpha k_t^{\alpha-1}z_tl_t^{1-\alpha}-\delta)] \]

Firms

Firms rent capital and labor from households. In each period t, they maximize profits. Notice: the firm solves a sequence of static optimization problems, not a dynamic problem. Where \(r^K_t\) is the rental rate of capital. \[ max_{K_{t-1}, L_t} \quad z_tK_t^\alpha L_t^\alpha-w_tL\_t - r^K_tK_{t-1}\ \]

Investment

We assume that physical capital suffers from a constant rate of depreciation, \(\delta\). Then aggregate capital evolves according to:

\[ K_t = (1 -\delta)K_{t-1}+I_t \]

Where \(K_t\) is is capital at the beginning of period t. The rental rate of capital \(r^K_t\) is then related to households’ net return on capital rt by

\[r^K_t = r_t +\delta\]

Aggregate resource constrain

We assume that the production function Yt = F(Kt−1, Lt, zt) is constant returns to scale in K and L. This implies that:

\[Y_t =r^KK_t +w_tL_t\]Which allows us to obtain the aggregate resource constrain:

\[Y_t = C_t +I_t\]

Model equations

Definitions Equations Number
TFP \(log(z_t) = \rho log(z_{t-1}) + \epsilon_t\) 1
HH Euler Equation \(\frac{1}{c_t}= \beta E_t[ \frac{1}{c_{t+1}}(1+r_t)]\) 2
Labor Supply \(w_t =\eta\frac{c_t}{1-l_t}\) 3
Aggregate resource constrain \(Y_t = C_t +I_t\) 4
Production function \(Y_t = z_tk_t^\alpha l_t^\alpha\) 5
Labour Demand (MPL) \(w_t = (1-\alpha) k_t^{\alpha}z_tl_t^{-1}\) 6
MPK \(\alpha k_t^{\alpha-1}z_tl_t^{1-\alpha}=r_t+\delta\) 7
Capital dynamics \(k_t = (1 -\delta)k_{t-1}+I_t\) 8

If we follow this process we reduce the above table to only four equations:

  • Substitute (7) in (2).

  • Substitute (6) in (3)

  • Substitute (5) in (4) and latter (4) in (8)

\[\begin{align} \frac{1}{c_t}&= \beta E_t[ \frac{1}{c_{t+1}}(1+\alpha k_t^{\alpha-1}z_tl_t^{1-\alpha}-\delta)] \\ \eta\frac{c_t}{1-l_t} &= (1-\alpha) k_t^{\alpha}z_tl_t^{-1}\\ c_t + k_{t+1} &= \alpha k_t^{\alpha-1}z_tl_t^{1-\alpha} + (1-\delta)k_t \\ log(z_t) &= \rho log(z_{t-1}) + \epsilon_t \end{align} \]

If there are no shocks to the economy, e.g. a deterministic scenario, we would get rid of two things. First, the dynamics of TFP which would be constant with no growth and second the expectation operator in the Euler equations. Moreover, in a steady state, we also get rid of the time index because the value of a variable today is the same as the value of that variable tomorrow. Therefore we have three equations:

\[\begin{align} \frac{1}{c}&= \beta \frac{1}{c_{}}(1+\alpha k^{\alpha-1}z^{1-\alpha}-\delta) \\ \eta\frac{c}{1-l} &= (1-\alpha) k_t^{\alpha}l^{-1}\\ c + \delta k &= \alpha k^{\alpha-1}l^{1-\alpha} -\delta \\ \end{align} \]

Where the first equation can be written as:

\[ \frac{1}{\beta}= 1+\alpha k^{\alpha-1}z^{1-\alpha}-\delta \]

Know we can isolate the variables of interest of the Steady State:

\[\begin{align} k = \frac{\color{blue} {1/\eta (1-\alpha)\varphi^{-\alpha}}}{\color{green} {\varphi^{1-\alpha}-\delta+ } \color{red}{(1/\alpha(1/\beta-1+\delta))^{\frac{1}{1-\alpha}}}\color{blue} {1/\eta (1-\alpha)\varphi^{-\alpha}}} \end{align} \]

\[ l = \color{red}{(1/\alpha(1/\beta-1+\delta))^{\frac{1}{1-\alpha}}}\cdot k = \color{red} \varphi \cdot k \]

\[ c =\color{green} {\varphi^{1-\alpha}-\delta+ } \cdot k \]

\[ y = k^\alpha l^{1-\alpha} \]