Kenji Sato

January 18, 2017

Today, we are going to study the Ramsey–Cass–Koopmans model (Ramsey model, hereafter), which is the simplest general equilibrium macroeconomic model.

As it will turn out, the structure of the dynamics is identical to the optimal growth model.

The Ramsey model incorporates the following considerations:

- Consumers are the owners of capital and they rent out capital (saving behavior) to firms.
- Saving decision is made according to consumers’ lifetime utility maximization.

- Firms borrow capital in the capital market.
- Firms maximize profit at each moment in time.
- The rental rate is determined by the productivity of capital.

- There are a large number of identical firms.
- Each firm has access to the production function \(Y = F(K, AL)\)
- \(F\) has CRS
- \(f(k) = F(k, 1)\) satisfies the standard conditions.

- The firms hire workers and rent capital in competitive factor markets.
- Firms take \(A\) as given. \(A\) grows exogenously at rate \(g\) (as in the Solow model)

- There are \(H\) identical households.
- The size of each household grows at rate \(n\).
- \(L(t)\) is the total population and \(L(t) / H\) is the number of members in each household.
- Each member of the household supplies 1 unit of labor at every point in time.

- The household has initial capital holdings of \(K(0) / H\), where – \(K(0) > 0\) is the initial amount of capital in the economy
- A household rents whatever capital it owns to firms.

- The household divides its income (from labor and capital) between consumption and saving so as to maximize its lifetime utility:

\[ U=\int_{0}^{\infty}e^{-\rho t}u(c(t))\frac{L(t)}{H}dt \]

- \(c(t)\): consumption of each member of the household at time t; i.e., per capita consumption.
- \(u(\cdot)\): instantaneous utility function.
- \(L(t)\): is the total population of the economy.
- \(L(t)/H\) is the number of members in each household.
- \(u(c(t))L(t)/H\) is the total instantaneous utility of each household.
- \(\rho\) is the discount rate

- CRRA Utility function \[
u(c)=\frac{c^{1-\theta}}{1-\theta},\quad\theta>0
\]
- Notice that the standard CRRA form \(\frac{c^{1-\theta} - 1}{1 - \theta}\) gives the same optimum but makes analysis a bit more verbose

- Assume

\[ \beta := \rho-n-(1-\theta)g>0 \]

under which \(U < \infty\).

The profit maximization. Profit is \(F(K, AL) - \bar r K - wL\), where \(\bar r\) is gross rate of return from the capital owner’s viewpoint. \(r(t) + \delta\), \(r(t)\) is net rate of return and \(\delta\) the depreciation rate.

\[ \max_{K, L} F(K, AL) - (r + \delta) K - w L \]

The first order conditions are given as below:

\[ \begin{aligned} \frac{\partial F}{\partial K} &= r + \delta, \\ \frac{\partial F}{\partial L} &= w \end{aligned} \]

We will use a different symbol for capital per unit of effective labor: \(\hat k = K/AL\). Small symbols are for per-labor variables; small symbols with hat for per-effective-labor variables.

Note that with \(f(\hat k) = F(\hat k, 1) = F(K, AL) / AL\), \(f'(\hat k) = \partial F / \partial K\). We thus have

\[ f'(\hat k) = r + \delta \]

and

\[ w = A\left[f(\hat{k})-\hat{k}f'(\hat{k})\right], \] namely,

\[ \hat w = \frac{w}{A} = f(\hat{k})-\hat{k}f'(\hat{k}) \]

The household divides income between consumption \(c(t)L(t)\) and saving \(S(t)\). Each household is subject to the budget constraint:

\[ \dot{S}(t) + c(t)\frac{L(t)}{H} \le r(t)S(t)+w(t)\frac{L(t)}{H} \]

- A household spends the left-hand side
- \(\dot S (t)\) is the increment/decrement in bank balance (saving/lending),
- \(c(t) L(t) / H\) the consumption of the household,

- A household earns the right-hand side
- \(r(t) S(t)\) the interest income,
- \(w(t) L(t) / H\) the labor income.

The capital market clearing condition is

\[ H\cdot S(t)=K(t),\qquad t\ge0, \]

from which we get

\[ \begin{aligned} \dot{K}(t) &\le r(t)K(t) + w(t)L(t) - c(t)L(t)\\ &= F(K(t),A(t)L(t)) - \delta K(t) - c(t)L(t). \end{aligned} \]

or

\[ \frac{\dot{K}(t)}{A(t)L(t)}\le f(\hat{k}(t))-\delta\hat{k}(t)-\hat{c}(t) \]

where \(\hat{c}(t) = c(t) / A(t)\) is consumption per unit of effective labor.

The inequality

\[ \frac{\dot{K}(t)}{A(t)L(t)}\le f(\hat{k}(t))-\delta\hat{k}(t)-\hat{c}(t) \]

gets us

\[ \begin{aligned} \dot{\hat{k}}(t) &= \frac{\dot{K}(t)}{A(t)L(t)}-(g+n)\hat{k}(t)\\ &\le f(\hat{k}(t))-(\delta+g+n)\hat{k}(t)-\hat{c}(t) \end{aligned} \]

Utility function

\[ \begin{aligned} u(c(t)) &= \frac{c(t)^{1-\theta}}{1-\theta} \\ &= \frac{[A(t)\hat{c}(t)]^{1-\theta}}{1-\theta} \\ &= \frac{[A(0)e^{gt}\hat{c}(t)]^{1-\theta}}{1-\theta} \\ &= A(0)^{1-\theta}e^{(1-\theta)gt}\frac{\hat{c}(t)^{1-\theta}}{1-\theta}. \end{aligned} \]

Since \(L(t)=L(0)e^{nt}\),

\[ \begin{aligned} U &=\int_{0}^{\infty}e^{-\rho t}\frac{c(t)^{1-\theta}}{1-\theta}\frac{L(t)}{H}dt\\ &=\int_{0}^{\infty} e^{-\rho t}\left[A(0)^{1-\theta}e^{(1-\theta)gt} \frac{\hat{c}(t)^{1-\theta}}{1-\theta}\right]\frac{L(0)e^{nt}}{H}dt\\ &=A(0)^{1-\theta}\frac{L(0)}{H}\int_{0}^{\infty}e^{-[\rho-n-(1-\theta)g]t} \frac{\hat{c}(t)^{1-\theta}}{1-\theta}dt\\ &=:B\int_{0}^{\infty}e^{-\beta t}\frac{\hat{c}(t)^{1-\theta}}{1-\theta}dt \end{aligned} \]

where \(\beta=\rho-n-(1-\theta)g > 0\) by assumption.

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 \(B\).

**Excercise 1**: Derive the differential equations that govern the 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.

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.