R and D Profit

Macro

${\pi }_R=\dot{N}V-W{L}_R$

Recall the Innovation Equation:

$${\pi }_R=\theta N{L}_{R}V-W{L}_R$$ $$\frac{\partial {\pi }_R}{\partial L_R}=\theta NV-W=0⟹\theta NV=W$$

Exactly same.

The zero-profit condition is equivalent.

The value of an invention: $V=\frac{{\pi }_i}{r}$

We could finally conclude that:

$$\theta N\frac{{\pi }_i}{r}=W$$

• We have $\{W,N{\pi }_i,r\}$
• $W$ is from final good sector.
• $N{\pi }_i$ is from intermediate goods sector.
• $r$ is from households sector.

In Final Goods

$$Y=L_Y^{1-\alpha }\sum _{i=1}{X}_i^{\alpha }$$ $${\pi }_Y=Y-W{L}_Y-\sum P_i{X}_i$$

FOC:

$$W=MPL=\frac{(1-\alpha )Y}{L_Y}$$ $$P_i=MP{X}_i=\alpha L_Y^{1-\alpha }{X}_i^{\alpha -1}$$

Intermediate Goods

• Production function: $X_i=K_i$
• Profit function: ${\pi }_i=P_i{X}_i-R{K}_i=(P_i-R)X_i$

FOC: $P_i=\frac{R}{\alpha }⟹R=\alpha P_i=\alpha MP{X}_i=\alpha MPK$

$$\begin{array}{rl}{\pi }_i& =\left(\frac{R}{\alpha }-R\right)X_i=\frac{1-\alpha }{\alpha }R{X}_i>0\\ {\pi }_i& =\frac{1-\alpha }{\alpha }\frac{RK}{N}⟹N{\pi }_i=\frac{1-\alpha }{\alpha }RK\\ R& =\alpha \frac{\partial Y}{\partial K}=\alpha \alpha \frac{Y}{K}\\ ⟹& N{\pi }_i=\frac{1-\alpha }{\alpha }{\alpha }^{2}Y=(1-\alpha )\alpha Y\end{array}$$

Households

Where the interest rate $r$ happens

In order to determine the interest rate $r$, we introduce financial asset, which is the ownership of monopolistic firms. Monopolistic firms generate positive profits, so their shares are financial assets

$$F=NV$$

We could also do the Hamiltonian in this case.

The non-arbitrage condition.

If $\frac{C}{Y}$ and $\frac{I}{Y}$ are not constant in the Long-run, then either $\frac{C}{Y}$ or $\frac{I}{Y}$ converges to zero:

• $\frac{C}{Y}\to 0⟹U\to -\infty$
• $\frac{I}{Y}\to 0⟹K\to 0⟹Y\to 0$

So $\frac{C}{Y}$ must be constant ⇒ $\frac{\dot{C}}{C}=\frac{\dot{Y}}{Y}$ is the long -run