Macroeconomics:Day 11

Kenji Sato

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:

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:

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


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

Lifetime budget constraint

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.


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

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 \(t = t_0\).