Ramsey Model

Ramsey growth model endogenise $s$ in the Solow Model. The question is what determines the saving rate?

The key difference between Ramsey Model and Solow Model is that in Ramsey Model, household chooses $C$ to maximize $U$.

We introduce an optimizing household to the Solow Model.

Household

$$\begin{array}{rl}Ma{x}_{C}U& =\sum _{t=0}\frac{\ln C_t}{(1+\rho {)}^t}\\ s.t.& \\ \dot{K}& =I-\delta K\\ \dot{K}& =Y-C-\delta K\end{array}$$

Recall that $s=\frac{I}{Y}=1-\frac{C}{Y}$

Hamiltonian

$$H=\ln C+\lambda \dot{K}$$

F.O.C:

$$\begin{array}{rl}\frac{\delta H}{\delta C}& =\frac{1}{C}-\lambda =0\\ \frac{\delta H}{\delta K}& =\lambda \left(\underset{MPK}{\underbrace{\alpha A{\left(\frac{L}{K}\right)}^{1-\alpha }}}-\delta \right)=\rho \lambda -\dot{\lambda }\end{array}$$

We could derive that $-\frac{\dot{\lambda }}{\lambda }=MPK-\delta -\rho$

and $\frac{\dot{C}}{C}=MPK-\delta -\rho$

The optimal consumption path in the Ramsey Model

• If $MPK-\delta >\rho$, then $\frac{\dot{C}}{C}>0$ because the household wants to invest in capital and reduce the consumption today in order to enjoy a higher level of consumption in the future.

Two differential equations

$$\begin{array}{rl}\frac{\dot{C}}{C}& =\alpha A{\left(\frac{L}{K}\right)}^{1-\alpha }-\delta -\rho \\ \dot{K}& =A{K}^{\alpha }{L}^{1-\alpha }-C-\delta K\end{array}$$

There are 2 endogenous variables $\{C,K\}$

This dynamic system converges to the steady state (We won’t do it here since it is an undergrad course)

Steady state: $\dot{C}=\dot{K}=0$

We have a steady state because $A$ is constant.

$\frac{\dot{C}}{C}=MPK-\delta -\rho =\alpha A{\left(\frac{L}{K}\right)}^{1-\alpha }-\delta -\rho$

If $\frac{\dot{C}}{C}=0$, then $\alpha A{\left(\frac{L}{K}\right)}^{1-\alpha }=\delta +\rho$

Steady-state level of capital: $K^{\ast }=(\frac{\alpha A}{\delta +\rho }{)}^{1/1-\alpha }L$

How does the household’s discount rate $\rho$ affect $K^{\ast }$

$$\rho \uparrow ⟹K\ast \downarrow$$

As the household becomes more important by discounting future utility at a higher rate, it accumulates less capital.

Culture (People’s preference) affects Economic Growth.

• Solow: The saving rate $s$ affects economic growth
• Ramsey: The household’s discount rate $\rho$ affects saving and Economic Growth: $\rho \uparrow ⟹K\ast \downarrow$

But what about income? it decreases. ($Y\ast =A(\frac{\alpha A}{\delta +\rho }{)}^{\alpha /1-\alpha }L$)

income = $\frac{Y^{\ast }}{L}$

Deriving the steady-state saving rate.

记得补充笔记

$$s^{\ast }=\frac{\alpha \delta }{\delta +\rho }$$

---

• In the Solow Model, $s$ is exogenous.
• In the Ramsey Model, $s$ is endogenous and chosen by the household.

$$s^{\ast }({\rho }_{-},{\alpha }_{+},{\delta }_{+})$$

Ramsey Model with a Perfect Competition Market

The setting of the Ramsey Model in the market economy is unrealistic. In a market economy, firms are the ones who produce output.

So, we will introduce a market economy and firms to the Ramsey Model. We will refer to this extended version as a decentralized Ramsey Model.

Firms

$$\pi =Y-wL-rK=A{K}^{\alpha }{L}^{1-\alpha }-wL-rK$$

and we could derive the $w$ and $r$ by F.O.C…

Household

• Decentralized Ramsey Model: $\frac{\dot{C}}{C}=r-\delta -\rho$
• Centralized Ramsey Model: $\frac{\dot{C}}{C}=MPK-\delta -\rho$

It is surprised that by firms, $MPK=r$, so that the outcome of decentralized and centralized are the same.

Note that the $r=MPK$ doesn’t always hold.

The centralized Ramsey Model is not realistic, but important. because it gives us an optimal allocation. Decentralized Ramsey Model gives us a market allocation. If market allocation is not the same as the optimal allocation, we could consider there could be a Market Failure, which is not efficient.

Decentralized Ramsey Model.

Hint: the firms do not care about the households’ fortune. They only care about their own profit.

However, by maximizing profit, the firms chooses $MPK=r$,which ends up maximizing the household’s utility.

The reason is that a market economy with perfect competition (and without externality) yields an efficient allocation of resources.

$$Y=wL+rK$$

From the firms’ profit maximization problem,

$$\begin{array}{rl}r& =\frac{\delta Y}{K}⟹rK=\alpha Y\\ w& =(1-\alpha )\frac{Y}{L}⟹wL=(1-\alpha )Y\end{array}$$

---

Exercise: Derive the steady-state saving rate in the decentralized Ramsey Model.

---

Monopolistic

Competitive Ramsey Model	Monopolistic Ramsey Model
$R=MPK⟹K_c^{\ast }={\left(\frac{\alpha A}{\delta +\rho }\right)}^{1/1-\alpha L}$	$p_i=\frac{r}{\alpha },\alpha <1$
	$r=\alpha p_i=\alpha MP{X}_i=\alpha MPK,r<MPK$
	$K_m^{\ast }={\left(\frac{{\alpha }^{2}A}{\delta +\rho }\right)}^{1/1-\alpha }L$

Monopolistic distortion reduces the rental price of capital (relative to $MPK$), such that the household accumulates a lower level of capital than the optimum.

• $K_c^{\ast }:r=MPK=\alpha A{\left(\frac{L}{K}\right)}^{1-\alpha }$
• $K_m^{\ast }:r=\alpha MPK={\alpha }^{2}A{\left(\frac{L}{K}\right)}^{1-\alpha }$ so that $K_c^{\ast }>K_m^{\ast }$

Final Good

$$Y=L^{1-\alpha }\sum _{i=1}^N{x}_i^{\alpha }$$

We take the FOC:

$$\begin{array}{rl}w& =MPL=(1-\alpha )\frac{Y}{L}⟹wL=(1-\alpha )Y\\ p_i& =MP{X}_i=\alpha L^{1-\alpha }{X}_i^{\alpha -1}\end{array}$$

This is the demand function for $X_i$

Intermediate Goods

• Production function: $X_i=K_i$.
• Profit function: ${\pi }_i=p_i{X}_i-r{K}_i=\alpha L^{1-\alpha }{X}_i^{\alpha }-r{X}_i$

Recall that we define: $A=N^{1-\alpha }$

$$N{\pi }_i=(1-\alpha )\alpha Y$$

In the competitive Ramsey Model, the government, the government should not subsidize capital investment because the levels of capital and output are already at the optimal levels, without government intervention.