Kenji Sato

January 20, 2017

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 \(B\).

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 \(r(t) = f'(k(t)) - \delta\).

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.

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

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:

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

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**.

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.

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.

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

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

- An unanticipated permanent increase in \(n\).
- An unanticipated permanent fall in \(\rho\).
- An unanticipated permanent increase in \(g\).

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.

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.

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.

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 \(t = t_0\).