Module 2 | Real

Set up

Production Function：

\[ Y_t = F(K_t, A_t L_t) \]

Where：

\[ K_{t+1} = (1-\delta)K_t + I_t, \quad I_t = sY_t, \] \[ A_{t+1} = (1+g)A_t, \quad L_{t+1} = (1+n)L_t \]

Intensive Form：

\[ y_t = \frac{Y_t}{A_t L_t} \equiv f(k_t), \quad k_t \equiv \frac{K_t}{A_t L_t} \]

Assumption & Condition

Assumption

$F_K > 0, F_{KK} < 0$ (Concave)
Constant return to scale (CRS): \[ F(\lambda K, \lambda AL) = \lambda F(K,AL), \quad \forall \lambda > 0 \]

Inada Condition

\[ F_K(0,AL) = \infty, \quad F_K(\infty,AL) = 0 \] or in intensive form: \[ f'(0) = \infty, \quad f'(\infty) = 0 \]

Continuous Time

Define the derivative of its continuous time to describe its rate of change: t = 0, $\Delta$, 2$\Delta$, ...

For $K$

\[ K_{t + \Delta} - K_t = I_t \Delta - \delta K_t \Delta \] \[ \dot{K}(t) = \lim_{\Delta \to 0} \frac{K(t+\Delta) - K(t)}{\Delta} = sF(K) - \delta K \]

For $A,L$

\[ A_{t+ \Delta} = (1 + g) A_t = e^{\ln(1+g)} A_t \approx e^{g \Delta} A_t where g \rightarrow 0, ln(1+g) \approx g \] \[ \dot{A}(t) = gA(t) = \lim_{\Delta \to 0} \frac{A_{t+\Delta} - A_{t}}{\Delta} = \frac{e^{g \Delta} - 1}{\Delta}A(t) = \frac{dA(t)}{dt} \] \[ \frac{d\ln A(t)}{dt} = \frac{1}{A(t)} \frac{dA(t)}{dt} = g \]

So as the L:

\[ \dot{L}(t) = nL(t), \quad \frac{d\ln L(t)}{dt} = n \]

Capital Accumulation

Discrete:

\[ k_{t+1} = \frac{(1-\delta)k_t + sf(k_t)}{(1+g)(1+n)} \]

Continous:

\[ \dot{K}(t) = sF(K(t), A(t)L(t)) - \delta K(t) \] \[ \frac{d ln(k(t))}{dt} = \frac{d ln(K(t))}{dt} - \frac{d ln(A(t))}{dt} - \frac{d ln(L(t))}{dt} = \frac{\dot{K}(t)}{K(t)} - g - n \] \[ \frac{\dot{k}(t)}{k(t)} = \frac{\dot{K}(t)}{K(t)} - \frac{\dot{A}(t)}{A(t)} - \frac{\dot{L}(t)}{L(t)} \] \[ \dot{k}(t) =[ \frac{\dot{K}(t)}{A(t)L(t)k(t)} - g - n ] k(t) = sf(k(t)) - (\delta + g + n)k(t) = \psi k(t) \]

Steady State

Definition：$k_{t+1} = k_t$ and $\dot{k} = 0$

\[ sf(k^*) = (\delta+g+n+gn)k^*, \quad sf(k^*) = (\delta+g+n)k^* \]

Comparative Dynamic

For saving rate：$sf(k(s))= (\delta+g+n)k(s)$

\[ k'(s) = - \frac{f(k^*)}{sf'(k^*) - (\delta+g+n)} > 0 \]

where molecule(slope) is always positive under s-s

Golden Rule

Definition：What should the savings rate be in order to maximize the level of consumption?

1、Consumption maximization：

\[ c^* = f(k^*) - (\delta+g+n)k^* = 0 \] \[ c'(s) = f'(k^*) - (\delta+g+n) = 0 \]

Hence, if $f'(k^*) > \delta + g + n$, then $c'(s) > 0$, which means that increasing the saving rate will increase consumption. Vice versa.

2、Optimal Saving Rate：

\[ f'(k)k^* = sf(k^*) = (\delta+g+n)k^* \Rightarrow f'(k^*)k^* = f(k^*) \] \[ s = \frac{f'(k^*)k^*}{f(k^*)} \]

Speed of Convergence

Definition：How fast does the economy converge to the steady state?

\[ \psi'(k^*) = \frac{sf'(k^*)k^*}{k^*} - (\delta + g + n) = \frac{sf'(k^*)k^*}{f(k^*)}(\delta + g + n) - (\delta + g + n) \] \[ \psi'(k^*) = (f'(k^*)k^*/f(k^*) - 1)(\delta + g + n) < 0 \]

where $\epsilon \equiv f'(k^*)k^*/f(k^*)$ is the degree of concavity. $\delta + g + n$ is effective depreciation rate.

If $\epsilon$ is close to 1(higher), the speed of convergence is slow. If $\epsilon = 1$, never converge.

Real Model Overview

Solow Model