---
tags: macro, public
---
# Lucas Tree
## Basic Model

In this economy, a tree produces fruit at every period. The identical household can and only can buy a share of the tree and eat a portion of the fruit. There is no other technology and labor in this economy.

- $p_t$: the price of the tree/stock at time $t$.
- $s_t$: the share of the tree/company the household holds at time $t$.
- $S_t$: the summation of $s_t$, which equals to $1$ when market cleans.
- $d_t$: the fruit/dividend of the tree/stock at time $t$.
- $c_t$: the household consumption at time $t$.
- $C_t$: the summation of $c_t$, which equals to $d_t$ when market cleans.
- $q_t$: the price of the bond at time $t$.
- $b_t$: the amount of bond the household holds at time $t$.

The household needs to solve the following problem:

\begin{gather*}
\max_{\{c_t, s_{t+1}\}_{t=0}^\infty} E\left[ \sum_{t=0}^\infty \beta^t u(c_t) \right]\\
\text{s.t. } c_t + p_t s_{t+1} \le (p_t + d_t) s_t
\end{gather*}

### Solution

The household's first-order condition is

$$
p_t u'(c_t) = \beta E_t [u'(c_{t+1}) (p_{t+1} + d_{t+1})]
$$

The transversality condition is

$$
\lim_{k \to \infty} \beta^k E_t [u'(c_{t+k}) (p_{t+k}) s_{t+k+1} ]=0
$$

Market cleaning condition,

\begin{align*}
s_{t+1} = S_{t+1} = 1\\
c_t = C_t = d_t
\end{align*}

Hence,

$$
p_t = \beta E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} (p_{t+1} + d_{t+1}) \right].
$$

Equivalently,

$$
\beta E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} \frac{p_{t+1} + d_{t+1}}{p_t} \right] =1,
$$

where we can define the realized return as,

$$
R_{t+1} = \frac{p_{t+1} + d_{t+1}}{p_t}.
$$

We can also define the intertemporal marginal rate of substitution between $t$ and $t+k$ as

$$
M_t^{t+k} = \beta^k \frac{u'(d_{t+k})}{u'(d_t)}.
$$

By iteration,

\begin{align*}
p_t &= E_t \left [ M_t^{t+1}(p_{t+1} + d_{t+1}) \right]\\
&= E_t \left [ M_t^{t+1}(E_{t+1} \left [ M_{t+1}^{t+2}(p_{t+2} + d_{t+2}) \right] + d_{t+1}) \right]\\
&= E_t \left [ M_t^{t+1}( \left [ M_{t+1}^{t+2}(p_{t+2} + d_{t+2}) \right] + d_{t+1}) \right]\\
&= E_t \left [ M_{t}^{t+2}(p_{t+2} + d_{t+2}) + M_t^{t+1} d_{t+1}) \right],\\
\end{align*}

where the third equality is based on the law of iterated expectation, and the fourth equality is by definition. Therefore,

By the transversality condition/no-bubble condition,

$$
\lim_{k \to \infty} E_t [M_t^{t+k} p_{t+k} ]=0
$$

We have,

$$
p_t = E_t \left \{ \sum_{k=1}^\infty M_t^{t+k} d_{t+k} \right\},
$$

In other words, the current price is the weighted average of future dividends. The weight is based on the customer's discounting marginal utility.

### Bond

Let's introduce a risk-free asset into the economy. If a household buy $b_t$ unit of bond with bond price $q_t$ at time $t$, they will receive $1$ dollar at $t+1$. Hence, the budget constraint becomes,

$$
(p_t + d_t) s_t + b_t = c_t + p_{t} s_{t+1} + q_t b_{t+1}
$$

Under market cleaning, the supply and demand of bonds are equal, and the holding amount is net zero. Therefore, we have the following Euler equation,

\begin{align*}
p_t &= \beta E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} (p_{t+1} + d_{t+1}) \right]\\
q_t &=\beta E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} \right]
\end{align*}

### Returns

Define the return of equity and bond as

\begin{align*}
R^e_{t+1} = \frac{d_{t+1} + p_{t+1} - p_t}{p_t},\\
R^b_{t+1} = \frac{1-q_t}{q_t} = \frac{1}{q_t} -1.\\
\end{align*}

The ex-post equity premium is

$$
R^e_{t+1} - R^b_{t+1}.
$$

At time $t$, $R^e_{t+1}$ is uncertain but $R^b_{t+1}$ is certain. In the data, we can estimate the average equity premium,

$$
E_t[R^e_{t+1}] - R^b_{t+1}.
$$

## CAPM

We can rewrite the Euler equations in terms of returns,

\begin{align*}
\beta E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} (R^e_{t+1} +1) \right] =1,\\
\beta E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} (R^b_{t+1} +1) \right] =1.
\end{align*}

Hence,

$$
E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} (R^e_{t+1} -R^b_{t+1}) \right] =0
$$

We can apply the formula of covariance,

$$
E[XY] = E[X] E[Y] + cov(X, Y).
$$

That is,

$$
E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} (R^e_{t+1} -R^b_{t+1}) \right] = E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} \right] E_t \left[ R^e_{t+1} -R^b_{t+1} \right] + cov(\frac{u'(d_{t+1})}{u'(d_t)},R^e_{t+1} -R^b_{t+1} )=0.
$$

However, $R^b_{t+1}$ is certain; we can subtract it from the covariance. Therefore,

$$
E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} \right] E_t \left[ R^e_{t+1} -R^b_{t+1} \right] + cov(\frac{u'(d_{t+1})}{u'(d_t)},R^e_{t+1} )=0,
$$

equivalently,

$$
E_t \left[ R^e_{t+1} -R^b_{t+1} \right] = -\frac{cov(\frac{u'(d_{t+1})}{u'(d_t)},R^e_{t+1} )}{E_t \left[ \frac{u'(d_{t+1})}{u'(d_t)} \right]},
$$

which is the consumption-CAPM formula. CAPM stands for capital assets pricing model. If the marginal utility is decreasing, higher $d_{t+1}$ induce higher return of stock and lower $\frac{u'(d_{t+1})}{u'(d_t)}$. Hence, the equity premium is positive.

## Modified Version

Consider a Lucas tree model with a single tree that produces dividends. Households value consumption of dividends as well as their wealth, defined as the following,

$$
w_t = (p_t +d_t)s_t +b_t.
$$

The utility is $u_t(c_t, w_t)$, and the houshold needs to solve the following problem:

\begin{gather*}
\max_{\{c_t, s_{t+1}\}_{t=0}^\infty} E\left[ \sum_{t=0}^\infty \beta^t u(c_t, w_t) \right]\\
\text{s.t. } c_t + p_t s_{t+1} + q_t b_{t+1} \le (p_t + d_t) s_t + b_t
\end{gather*}

### Solution

The household's first-order condition w.r.t equity is,

$$
p_t u_1(c_t, w_t) = \beta E_t [u_1(c_{t+1}, w_{t+1}) (p_{t+1} + d_{t+1}) + u_2(c_{t+1}, w_{t+1}) (p_{t+1} + d_{t+1})]
$$

The household's first-order condition w.r.t bond is,

$$
q_t u_1(c_t, w_t) = \beta E_t [u_1(c_{t+1}, w_{t+1}) \cdot 1 + u_2(c_{t+1}, w_{t+1}) ],
$$

where $u_1(\cdot)$ and $u_2(\cdot)$ are the partial derivative with respect to the first and second arguments. In this context, $u_1(\cdot) = u_c(\cdot)$ and $u_2(\cdot)= u_w(\cdot)$.

Under market cleaning, $c_t =d_t, b_t=0, s_t = S_t=1$. Hence, the prices are,

\begin{align*}
p_t &= \beta E_t \left [ \left (\frac{u_1(d_{t+1}, w_{t+1})}{u_1(d_t, w_t)} + \frac{u_2(d_{t+1}, w_{t+1})}{u_1(d_t, w_t)} \right ) (p_{t+1} + d_{t+1}) \right]\\
q_t &= \beta E_t \left [ \frac{u_1(d_{t+1}, w_{t+1})}{u_1(d_t, w_t)} + \frac{u_2(d_{t+1}, w_{t+1})}{u_1(d_t, w_t)} \right]\\
\end{align*}

Let $u_{1,t} = u_1(d_t, w_t), u_{1,t+1} = u_1(d_{t+1}, w_{t+1})$, we can have a neat expression,

\begin{align*}
p_t &= \beta E_t \left [ \left (\frac{u_{1, t+1}}{u_{1, t}} + \frac{u_{2, t+1}}{u_{1, t}} \right ) (p_{t+1} + d_{t+1}) \right],\\
q_t &= \beta E_t \left [ \frac{u_{1, t+1}}{u_{1, t}} + \frac{u_{2, t+1}}{u_{1, t}} \right].\\
\end{align*}