January 20, 2017

# Optimal Growth and the Ramsey Model

The Ramsey model boils down to the optimal growth problem:

\begin{aligned} &\max \int_{0}^{\infty} Be^{-\beta t}\frac{\hat{c}(t)^{1-\theta}}{1-\theta}dt\\ &\text{subject to }\\ &\quad\dot{\hat{k}}(t)=f(\hat{k}(t))-(\delta+g+n)\hat{k}(t)-\hat{c}(t). \end{aligned}

Note that $$B$$ before the total utility doesn’t change the optimality conditions and so you can safely remove it. Check this fact by solving the problem without removing $$B$$.

# First-order Dynamic Equations

The dynamics of the Ramsey model is governed by the following system of equations:

\begin{aligned} \dot{\hat{k}}(t)&=f\left(\hat{k}(t)\right)-(\delta+g+n)\hat{k}(t)-\hat{c}(t)\\ \dot{\hat{c}}(t)&=\left(\frac{r(t)-\rho-\theta g}{\theta}\right)\hat{c}(t), \end{aligned} where $$r(t) = f'(k(t)) - \delta$$.

# Long-run Dynamics of the model

For a given $$\hat k (0) = K(0)/A(0)L(0)$$, there is only one initial $$\hat c (0) = C(0) / A(0) L(0)$$ such that the optimal path from $$(\hat k(0), \hat c(0))$$ converges to the steady state $$(\hat k^\star, \hat c^\star)$$.

In the steady state (if the economy starts from it)

\begin{aligned} C(t) &= A(t) L(t) \hat c^\star \\ K(t) &= A(t) L(t) \hat k^\star \end{aligned}

The growth rates:

• $$C$$ and $$K$$ grow at rate $$g + n$$
• $$c$$ and $$k$$ grow at rate $$g$$.

Implications of the Solow model are maintained.

# Treatment in Romer 4e

Romer’s textbook employs a Lagrangian method after introducing the lifetime budget constraint.

# Households’ budget constraint (in present terms)

The flow budget constraint written in present terms:

$e^{-R(t)}\dot{S}(t) \le r(t)e^{-R(t)}S(t)+e^{-R(t)}w(t)\frac{L(t)}{H}-e^{-R(t)}c(t)\frac{L(t)}{H},$

where

$R(t)=\int_{0}^{t}r(s)ds,\qquad\dot{R}(t)=r(t)$

Integrate the flow budget constraint:

$e^{-R(t)}\dot{S}(t) \le r(t)e^{-R(t)}S(t)+e^{-R(t)}w(t) \frac{L(t)}{H}-e^{-R(t)}c(t)\frac{L(t)}{H},$

The left-hand side becomes:

\begin{aligned} \int_{0}^{\infty}e^{-R(t)}\dot{S}(t)dt &= \left[e^{-R(t)}S(t)\right]_{0}^{\infty}-\int_{0}^{\infty}-r(t)e^{-R(t)}S(t)dt\\ &= \lim_{t\to\infty}e^{-R(t)}S(t)- S(0)+\int_{0}^{\infty}r(t)e^{-R(t)}S(t)dt. \end{aligned}

The right-hand side becomes:

$\int_{0}^{\infty}\left[r(t)e^{-R(t)}S(t)+e^{-R(t)}w(t) \frac{L(t)}{H}-e^{-R(t)}c(t)\frac{L(t)}{H}\right]dt$

# Life-time budget constraint (cont’d)

We get

$\lim_{t\to\infty}e^{-R(t)}S(t)-S(0) \le \int_{0}^{\infty}e^{-R(t)}w(t)\frac{L(t)}{H}dt - \int_{0}^{\infty}e^{-R(t)}c(t)\frac{L(t)}{H}dt$

By assuming $$\lim_{t\to\infty}e^{-R(t)}S(t)\ge 0$$ we get

$\int_{0}^{\infty}e^{-R(t)}c(t)\frac{L(t)}{H}dt \le S(0)+\int_{0}^{\infty}e^{-R(t)}w(t)\frac{L(t)}{H}dt$

This inequality states that the present value of the consumption of a household must not exceed initial saving plus the present value of their lifetime income.

# No-Ponzi Game condition

The condition

$\lim_{t\to\infty}e^{-R(t)}S(t) \ge 0$

is called the No-Ponzi Game condition. This condition is violated if the household always borrows money to repay their borrowings; the interest accrue and present value of borrowing in the infinite future becomes nonzero.

Such a financial scheme is excluded.

NB: NPG condition is different from the transversality condition (a few professional authors are confused!). NPG is an assumption for the model, while transversality is for the model’s optimal path.

# Optimization problem

The consumer’s problem is thus

\begin{aligned} &\max \int_{0}^{\infty} B e^{-\beta t}\frac{\hat{c}(t)^{1-\theta}}{1-\theta}dt\\ &\text{subject to }\\&\quad\int_{0}^{\infty}e^{-R(t)+(n+g)t}\hat{c}(t)dt \le\hat{k}(0)+\int_{0}^{\infty}e^{-R(t)+(n+g)t}\hat{w}(t)dt. \end{aligned}

Romer defines the Lagrangian function and bypasses (sacrifycing mathematical rigor) the technical stuff.

# Lagrangian

$\begin{multline} \mathcal{L}=B\int_{0}^{\infty} e^{-\beta t}\frac{\hat{c}(t)^{1-\theta}}{1-\theta}dt+ \lambda\bigg[\hat{k}(0) \\ +\int_{0}^{\infty} e^{-R(t)+(n+g)t}\hat{w}(t)dt-\int_{0}^{\infty}e^{-R(t)+(n+g)t}\hat{c}(t)dt\bigg] \end{multline}$

# Comparative Dynamics

Suppose that the economy is on the balanced growth path. Describe the dynamic of the economy after the following events:

1. An unanticipated permanent increase in $$n$$.
2. An unanticipated permanent fall in $$\rho$$.
3. An unanticipated permanent increase in $$g$$.

# Effect of government purchases

Let’s introduce the government purchases (per unit of effective labor) to the capital accumulation equation. (Disposable income is reduced by $$\hat G$$)

$\dot{\hat{k}}(t) = f(\hat{k})-(\delta+g+n)\hat{k}-\hat{c}-\hat{G}(t)$

This does not chage the Euler equation (verify) and only brings down the $$\dot k = 0$$ locus.

Although reduction in the disposable income reduces consumption per unit of effective labor, whereby reducing the level of lifetime income, it doesn’t induce substitution to investment (unlike the Solow model).

$\int_{0}^{\infty}e^{-R(t)+(n+g)t}\hat{c}(t)dt\le\hat{k}(0) + \int_{0}^{\infty}e^{-R(t)+(n+g)t}\left[\hat{w}(t)-\hat G(t)\right]dt$

Thus, the steady state capital level is unchanged.

# Unanticipated permanent hike in $$\hat G$$

Suppose that the economy is on the balanced growth path and that there is an unanticipated permanent increase in $$\hat G$$. At $$t = 0$$,

$\hat G:0 \to \hat G_H > 0.$

Since the existence of government purchases doesn’t alter the $$\dot{\hat{c}}=0$$ locus, the steady state level of $$\hat{k}$$ does not change. The economy on the old balanced growth path jumps to new balanced growth path.

# Unanticipated hike in $$\hat G$$, anticipated to be temporary

Suppose that the economy is on the balanced growth path and that there is an unanticipated increase in $$\hat G$$, which is known to be temporary. At $$t = 0$$,

$\hat G: 0 \to \hat G_H,$ and it is announced that the policy ends at $$t = t_0$$,

$\hat G: \hat G_H \to 0.$

Describe the dynamics of the economy.

# Anticipated permanent hike in $$\hat G$$

Suppose that the economy is on the balanced growth path and that there is an anticipated permanent increase in $$\hat G$$. At $$t = t_0$$,

$\hat G: 0 \to \hat G_H,$

where the information is announced at $$t = 0$$.

At the time of announcement, the households (if they are patient) start to reduce their consumption, toward the new optimal path, on which they get just at $$t = t_0$$.