Real Business Cycle Model

Motivation. The real business cycle model asserts that all the cycles are caused by changes in $z$ or:

def. Solow Residual (=Total Factor Productivity, TFP). Simply defined as $\ln (z)$ from the Intertemporal Consumption-Leisure Optimization (Full General Equilibrium) model.

Improving Our General Equilibirum Model §

Before proceeding, however, we need to add a few more items into the variables that affect our general equilibrium. From General Equilibirum Functions we have:

• Labor market
  • $N^s(w^{\oplus },r^{\oplus },W^{\ominus })$
  • $N^d(w^{\ominus },z^{\oplus })$
• Goods market
  • $Y^s=z^{\oplus }F(K,\stackrel{\oplus \text{LR}}{N^{\star }}(w,r^{\oplus \text{LR}},W))$
  • $Y^d=C^d(r^{\ominus },\dots )+I(r^{\ominus },\dots )+G$ We will now consider a change in $z^{\prime }$ (not $z$) and its effects:

Investment §

On $Y^d$ via $I$, investment decisions. Expand the optimal investment decision condition:

$$\begin{array}{rl}{\text{MP}}_{K_{t+1}}-d& =r\\ z_{t+1}\alpha \underset{\text{ Law of Motion }}{\underbrace{K_{t+1}^{\alpha -1}}}{N}_{t+1}^{1-\alpha }& =r+d\\ \stackrel{\uparrow }{z_{t+1}}\alpha {\left(\frac{1}{(1-d)K_t+\stackrel{\uparrow }{I}}\right)}^{1-\alpha }& =r+d\end{array}$$

Observe that for the optimal investment decision, as $z_{t+1}$ increases, $I$ has to increase to too balance the equation, as $r,d,K_t$ is held the same. Also note that ${\text{MP}}_{K_{t+1}}$ also increases Intuition. Firms want to invest in more capital since productivity of capital in the future will increase.

Consumption §

On $Y^d$ via $C$, current consumption decisions. The firm prices the optimal future wage as:

$$\stackrel{\uparrow }{w_{t+1}}=\stackrel{{\text{MP}}_{N_{t+1}}}{\overbrace{\frac{\partial Y_{t+1}}{\partial N_{t+1}}}}=(1-\alpha )\stackrel{\uparrow }{z_{t+1}}{K}_{t+1}^{\alpha }{N}_{t+1}^{-\alpha }$$

Profits (which go to HHs eventually) also increase:

$$\stackrel{\uparrow }{{\pi }_{t+1}}=\stackrel{\uparrow }{z_{t+1}}F(K_{t+1})-\stackrel{\uparrow }{w_{t+1}}{N}_{t+1}+(1-d)K_{t+1}$$

Intuition. As lifetime wealth increases, consumption increases too, with the same reasoning as shown here.

Laborx §

On $N_t^s$. As lifetime wealth increases, consumers reduce their working hours, with the same reasoning as shown here again.

Effects of Current and Future $z$ Increase §

1. Direct Effects ($z_t$ increases)
   • Production function increases
   • $Y_t^s$ increases as firms take advantage of more efficiency
   • $N_t^d$ increases as firms produce more
2. Secondary Effects ($z_{t+1},w_{t+1}$ increases)
   • $Y_t^d$ increases as explained above (via $C,I$)
   • $N_t^s$ decreases as consumers’ lifetime wealth increases
3. Equilibirum condition achieved.
   1. Wages are higher
   2. Assume direct effects are stronger than indirect effects then
   3. … $r_1>r_2$ interest rates fall
   4. … $N_1<N_2$ employment rises

Utility Function Change §