Romer Model

Macro

Economic Growth

1. Solow Model

• There is a steady state level of output (for a constant $A$)
• There is a balanced growth rate of output. (When $A$ grows over time)

1. Ramsey Model (for a constant $A$ or $N$)

• Centralized version (optimal allocation)
• Decentralized version (market allocation)
  • Perfect Competition (efficient)
  • Monopolistic Competition (Market Failure)

1. Romer Model

The Model

There are a few components:

1. Households (provide $K$, $L$)
2. Intermediate Good (provide $X_i$)
3. Final Good

Key part is that we introduce R&D sector here, which means the “Research and Development”. They are aiming to increase $N$

In the monopolistic Ramsey Model, $N$ is a constant. In the Romer Model, R&D entrepreneurs develop new product. So that $N$ increases over time ⇒ Economic Growth.

• Final-good production function: $Y=L_Y^{1-\alpha }{\sum }_{i=1}^N{X}_i^{\alpha }$
• Resource constraint on capital: $K={\sum }_{i=1}^N{X}_i=N{X}_i$
• Cobb-Douglas Production Function: $Y=L_Y^{1-\alpha }N{\left(\frac{K}{N}\right)}^{\alpha }$

By defining $A=N^{1-\alpha }$, we get that:

$$Y=N^{1-\alpha }{K}^{\alpha }{L}_Y^{1-\alpha }$$

==Growth rate of $Y$ ==

$$\frac{\dot{Y}}{Y}=(1-\alpha )\frac{\dot{N}}{N}+\alpha \frac{\dot{K}}{K}+(1-\alpha )\frac{\dot{L_Y}}{L_Y}$$

On the BGP,

$\frac{\dot{L_Y}}{L_Y}=0$, and $\frac{\dot{Y}}{Y}=\frac{\dot{K}}{K}$

In the long run, the growth rate of output is:

$$\frac{\dot{Y}}{Y}=(1-\alpha )\frac{\dot{N}}{N}+\alpha \frac{\dot{Y}}{Y}+(1-\alpha )\frac{\dot{L_Y}}{L_Y}$$

So that the growth rate of $Y$ is exactly the growth rate of $N$

$$\frac{\dot{Y}}{Y}=\frac{\dot{K}}{K}=\frac{\dot{N}}{N}$$

Try to claim: $\frac{\dot{Y}}{Y}=\frac{\dot{K}}{K}$ in the long-run.

starting from:

$$\begin{array}{r}\dot{K}=I-\delta K⟹\frac{\dot{K}}{K}=\frac{I}{K}-\delta =\frac{sY}{K}-\delta \end{array}$$

Recall that $s$ is exogenous in Solow Model, and endogenous in Ramsey Model and Romer Model. The endogenous saving rate $s$ is stationary/constant in the long-run.

On the BGP, $\{\dot{K},s\}$ are constant. So $\frac{K}{Y}$ must be constant ⇒ $\frac{\dot{Y}}{Y}=\frac{\dot{K}}{K}$ in the long-run.

Key implication: $\frac{\dot{Y}}{Y}=\frac{\dot{N}}{N}$ in the long-run.

Economic Growth is determined by technological progress, which is endogenously determined in a market economy.

Key question is: what determines $\frac{\dot{N}}{N}$? That is R&D!

R&D sector:

• $v$ is the value of an invention
• $\dot{N}$ is the number of new inventions

Revenue from R&D = $\dot{N}V$

Cost of R&D = $W{L}_R$

R and D Profit