Ramsey Model: Complete Description

# Ramsey Model: Complete Description
## Macroeconomics (2017Q4)
### Kenji Sato
### 2018-01-15

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## Exogenous variables

Population growth

`$$\dot{L} = n L$$`

Technical growth

`$$\dot{A} = g A$$`

---

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## Consumer Behavior: Objective

Consumers act altruistically. They care not only about 
their own consumption but also about offspring's.

Each individual maximizes the total utility:

`$$\max \int_0^\infty e^{-\rho t} u(c(t)) e^{nt} dt$$`

Since there are `$L(0)$` consumers at `$t = 0$`, the economy 
as a whole miximizes the following objective.

`$$\max \int_0^\infty e^{-\rho t} u(c(t)) L(t) dt$$`

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## Consumer Behavior: Budget constraint

Each consumer is subject to the following flow budget constraint:

`$$c(t) + \dot{s}(t) = \left(r(t) - n\right) s(t)  + w(t)$$`

We have `$-n s(t)$` in the interest term because of the population growth. Per-person 
share of the interest income is reduced by the factor of growth rate.

---

## Consumer Behavior: First-order condition

The representative consumer solves the following maximization problem:

`$$\max \int_0^\infty e^{-(\rho - n) t} u(c(t)) dt$$`
subject to

`$$c(t) + \dot{s}(t) = \left(r(t) - n\right) s(t)  + w(t)$$`

**Problem: Set up the Hamiltonian. (3min)**

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## Consumer Behavior: Canonical Equations

`$$\mathcal H =
e^{-(\rho - n) t} u(c) + \lambda [(r - n) s + w - c]$$`

**Problem: Compute the canonical equations. (5min)**

1. `$\partial \mathcal H / \partial c = 0$`
2. `$\dot{\lambda} = - \partial \mathcal H / \partial s$`
3. `$\dot{s} = \partial \mathcal H / \partial \lambda$`

---

## Consumer Behavior: Euler equation

**Problem: Derive the Euler equation. (5min)**

`$$\begin{aligned}
  \frac{\dot{c}(t)}{c(t)} = 
  \frac{r(t) - \rho}{\Theta(c)} 
\end{aligned}$$`

Under the assumption of CRRA utility

`$$u(c) = \frac{c^{1 - \theta} - 1}{1 - \theta},\quad \theta > 0$$`

the Euler equation is expressed as

`$$\begin{aligned}
  \frac{\dot{c}(t)}{c(t)} = 
  \frac{r(t) - \rho}{\theta} 
\end{aligned}$$`

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## Euler equation for `$C/(AL)$`

Define the consumption per unit of effective labor:

`$$\hat{c} = \frac{c}{A} = \frac{C}{AL}$$`

Since `$A$` grows exogenously with a constant growth rate `$g$`,

`$$\frac{\dot{\hat{c}}}{\hat{c}} = \frac{\dot{c}(t)}{c(t)} - g$$`

We therefore have

`$$\frac{\dot{\hat{c}}}{\hat{c}} =
\frac{r(t) - \rho - \theta g}{\theta}$$`

---

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## Firm behavior: Production function

Let's describe the behavior of the firms.

This is mostly equivalent to the setup of the Solow model.

The firms have access to the CRS production technology

`$$Y(t) = F(K(t), A(t) L(t)).$$`

In per-`$AL$` term,

`$$\hat{y}(t) = f\left(\hat{k} (t)\right) = 
\frac{F(K, AL)}{AL}$$`

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## Firm behavior: Profit maximization

Rent

`$$r(t) + \delta = f'\left( \hat{k}(t) \right)$$`

Wage

`$$w(t) = A(t) \left[
  f\left( \hat{k}(t) \right) - \hat{k}(t) f'\left( \hat{k}(t) \right)
\right]$$`

Wage per effective labor

`$$\hat{w}(t) = f\left( \hat{k}(t) \right) - \hat{k}(t) f'\left( \hat{k}(t) \right)$$`

---

## Equilibrium: Capital Accumulation

The capital market clearing condition:

`$$s(t) L(t) = K(t)$$`

The flow budget constraint

`$$c(t) + \dot{s}(t) = \left(r(t) - n\right) s(t)  + w(t)$$`

is transformed into

`$$\frac{\dot{\hat{k}}}{\hat{k}} =
\frac{\dot s}{s} - g = 
\frac{f\left(\hat{k}\right) - \hat{c} - (\delta + g + n) \hat{k}}{\hat{k}}$$`

---

## Equilibrium: First-order Dynamics

We have the following capital accumulation equation:

`$$\dot{\hat{k}} = f\left(\hat{k}\right) - \hat{c} - (\delta + g + n) \hat{k}$$`

The dynamics of the Ramsey model is determined by this capital 
accumulation equation together with 
the Euler equation:

`$$\frac{\dot{\hat{c}}}{\hat{c}} =
\frac{f'\left(\hat{k}\right) - \delta - \rho - \theta g}{\theta}$$`

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## Locus for `$\dot{\hat{k}} = 0$`

Let's derive the condition under which `$\dot{\hat{k}} = 0$`.

From the capital accumulation equation,

`$$\dot{\hat{k}} = f\left(\hat{k}\right) - \hat{c} - (\delta + g + n) \hat{k},$$`
we obtain

`$$\begin{aligned}
  \dot{\hat{k}} = 0 
  \Leftrightarrow
  \hat{c} = f\left(\hat{k}\right) - (\delta + g + n) \hat{k}
\end{aligned}$$`

---

## Regions for `$\dot{\hat{k}} > 0$` and `$\dot{\hat{k}} < 0$`

Moreover,

`$$\begin{aligned}
  \dot{\hat{k}} > 0 
  \Leftrightarrow
  \hat{c} < f\left(\hat{k}\right) - (\delta + g + n) \hat{k}
\end{aligned}$$`
and

`$$\begin{aligned}
  \dot{\hat{k}} < 0 
  \Leftrightarrow
  \hat{c} > f\left(\hat{k}\right) - (\delta + g + n) \hat{k}
\end{aligned}$$`

`$\dot{\hat{k}}$` is increasing below the `$\dot{\hat{k}} = 0$` locus
and decreasing above the locus.

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## Regions for `$\dot{\hat{k}} > 0$` and `$\dot{\hat{k}} < 0$` (cont'd)

You can rewrite these conditions as below:

`$$\begin{aligned}
  \dot{\hat{k}} \gtreqless 0 
  \Longleftrightarrow
  f\left(\hat{k}\right) - \hat{c} \gtreqless (\delta + g + n) \hat{k}
\end{aligned}$$`

Capital increases if and only if the investment, 
`$f(\hat{k}) - \hat{c}$`, is greater than the break-even level,
`$(\delta + g + n)\hat{k}$`.

This condition is exactly what you have already encountered in the 
analysis of the Solow model.

---

## Locus for `$\dot{\hat{c}} = 0$`

From

`$$\frac{\dot{\hat{c}}}{\hat{c}} =
\frac{f'\left(\hat{k}\right) - \delta - \rho - \theta g}{\theta}$$`

we can derive the locus on which `$\dot{\hat{c}} = 0$`.

`$$\dot{\hat{c}} = 0
\Longleftrightarrow
f'\left( \hat{k} \right) = \delta + \rho + \theta g$$`

---

## Regions for `$\dot{\hat{c}} > 0$` and `$\dot{\hat{c}} < 0$`

Obviously,

`$$\dot{\hat{c}} \gtreqless 0
\Longleftrightarrow
f'\left( \hat{k} \right) \gtreqless \delta + \rho + \theta g$$`

Since `$f'$` is a decreasing function of `$\hat{k}$`,

`$$\dot{\hat{c}} \gtreqless 0
\Longleftrightarrow
\hat{k} \lesseqgtr (f')^{-1}\left(\delta + \rho + \theta g\right)$$`

Let's define

`$$\hat{k}^* =  (f')^{-1}\left(\delta + \rho + \theta g\right)$$`

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## Regions for `$\dot{\hat{c}} > 0$` and `$\dot{\hat{c}} < 0$` (cont'd)

Recall `$f'(\hat{k}) - \delta = r$`. The first-order condition for the 
consumption is easy to grasp when `$g = 0$`.

`$$\dot{\hat{c}} \gtreqless 0
\Longleftrightarrow
r \gtreqless \rho + \theta\cdot 0$$`

When benefit from waiting `$r$` is greater than utility cost of waiting `$\rho$`, it's best to defer consuming. Thus, increasing consumption (`$\dot{\hat c} > 0$`)

When `$g \neq 0$`, an additional term is needed to adjust
for technology change.