1. Consider the following version of the Lucas’s tree economy. There are two kinds of trees. The first kind is ugly and gives no direct utility to consumers, but yields a stream of fruit {d_{1t}}, where d_{1t} denotes a positive random process obeying a first-order Markov process. The second tree is beautiful and yields utility on itself. This tree also yields a stream of the same kind of fruit d_{2t}, where it happens that d_{2t} = d_{1t} = (\frac{1}{2}) d_{t} ∀ t, so that the physical yields of the two kinds of trees are equal. There is one of each tree for each N individuals in the economy. Trees last forever, but the fruit is not storable. Trees are the only source of fruit. Each of the N individuals in the economy has preferences described by

E_{0} ∑_{t=0}^{∞} β^{t} u(c_{t}, s_{2t})  (1)

where u(c_{t}, s_{2t}) = ln c_{t} + γ ln(s_{2t}) where γ ≥ 0, c_{t} denotes consumption of the fruit in period t and s2t is the stock of beautiful trees owned at the beginning of the period t. The owner of a tree of either kind i at the start of the period receives the fruit d_{it} produced by the tree during that period. Let p_{it} be the price of a tree of type i (where i = 1, 2) during period t. Let R_{it} be the gross rate of returns of tree i during that period held from period t to t + 1.

(a) Write down the consumer optimization problem in sequential and recursive form.

Consumer optimization in a recursive form

The Bellman’s equation is given by

v(d_{t}, s_{1t}, s_{2t}) = max_{{c_{t},s_{1t+1},s_{2t+1}}} (ln(c_{t}) + γ ln(s_{2t}) + E_{t}βv(d_{t}, s_{1t+1}, s_{2t+1}))

where c_{t} + p_{1t}s_{1t+1} + p_{2t}s_{2t+1} ≤ (d_{1t} + p_{1t})s_{1t} + (d_{2t} + p_{2t})s_{2t}.

Consumer optimization in a sequential form The sequential form is given by

max_{{c_{t},s_{1t+1},s_{2t+1}}} E_{0} ∑_{t=0}^{∞} β^{t} (ln(c_{t}) + γ ln(s_{2t}))

where c_{t} + p_{1t}s_{1t+1} + p_{2t}s_{2t+1} ≤ (d_{1t} + p_{1t})s_{1t} + (d_{2t} + p_{2t})s_{2t}.

(b) Define a rational expectations equilibrium.

Definition The following is called the market clear condition

∑_{i=1}^{I} c_{t}^{i} = ∑_{i=1}^{I} d_{t}^{i}  (2)
∑_{i=1}^{I} s_{1t}^{i} = ∑_{i=1}^{I} s_{10}^{i} = I
∑_{i=1}^{I} s_{2t}^{i} = ∑_{i=1}^{I} s_{20}^{i} = I

where s_{10}^{i} and s_{20}^{i} are each agent’s number of trees at initial time. Definition A sequential household problem is defined by each agent’s utility optimization problem:

max_{{c_{t},s_{1t+1},s_{2t+1}}} E_{0} ∑_{t=0}^{∞} β^{t} (ln(c_{t}) + γ ln(s_{2t}))

where c_{t} + p_{1t}s_{1t+1} + p_{2t}s_{2t+1} ≤ (d_{1t} + p_{1t})s_{1t} + (d_{2t} + p_{2t})s_{2t}. Definition A rational competitive equilibrium is an allocation, (c) Find the pricing functions mapping the state of the economy at t

unto p_{1t} and p_{2t} (give precise formulas). [Hint: You should be able to directly derive p_{1t} from the example seen in class, then since pricing function have to be linear you can guess a pricing function p_{2t} = kd_{t} and solve for k parameter using Euler equation of the second stock.]

I am going to use the sequential form to find a solution. Using the Lagrange Multiplier,

L = E_{0} ∑_{t=0}^{∞} β^{t} ((ln(c_{t}) + γ ln(s_{2t}))
+ λ_{t}((d_{1t} + p_{1t})s_{1t} + (d_{2t} + p_{2t})s_{2t} − c_{t} − p_{1t}s_{1t+1} − p_{2t}s_{2t+1}))

By taking differentiations with respect to {c_{t}, s_{1t+1}, s_{2t+1}}.

c_{t} : E_{0}β^{t}(\frac{1}{c_{t}} − λ_{t}) = 0
s_{1t+1} : E_{0}β^{t}(βλ_{t+1}(d_{1t+1} + p_{1t+1}) − λ_{t}p_{1t}) = 0
s_{2t+1} : E_{0}β^{t} (βλ_{t+1}(d_{2t+1} + p_{2t+1}) − λ_{t}p_{2t} + β \frac{γ}{s_{2t+1}} ) = 0

Using the Markov property E_{0} = E_{t}.

c_{t} : \frac{1}{c_{t}} = λ_{t}
s_{1t+1} : p_{1t} = E_{t}β \begin{matrix}λ & t+1 \\ λ & t \end{matrix} (d_{1t+1} + p_{1t+1})
s_{2t+1} : p_{2t} = E_{t}β \begin{matrix}λ & t+1 \\ λ & t \end{matrix} (d_{2t+1} + p_{2t+1}) + β \frac{γ}{s_{2t+1}λ_{t}}

Combining the first-order conditions, we can obtain the pricing formula.

p_{1t} = E_{t}β \frac{c_{t}}{c_{t+1}} (d_{1t+1} + p_{1t+1})
p_{2t} = E_{t}β \frac{c_{t}}{c_{t+1}} (d_{2t+1} + p_{2t+1}) + β \frac{γc_{t}}{s_{2t+1}}

Recall c_{t} = d_{t} in a general equilibrium.

p_{1t} = E_{t}β \frac{d_{t}}{d_{t+1}} (d_{1t+1} + p_{1t+1})
p_{2t} = E_{t}β \frac{d_{t}}{d_{t+1}} (d_{2t+1} + p_{2t+1}) + β \frac{γd_{t}}{s_{2t+1}}

Let us assume the linear form of the first tree’s pricing function.

p_{1t} = k_{1t}d_{t}

Next apply the above linear form to the Euler equation.

k_{1t}d_{t} = E_{t}β \frac{d_{t}}{d_{t+1}} (d_{1t+1} + k_{1t+1}d_{t+1})
k_{1t} = E_{t}βk_{1t+1} + E_{t}β \frac{d_{1t+1}}{d_{t+1}}

Recursively we obtain k_{1t}.

k_{1t} = E_{t} ∑_{j=1}^{∞} β^{j} \frac{d_{1t+j}}{d_{t+j}}

Since we are given d_{1t} = \frac{1}{2}d_{t}.

k_{1t} = \frac{1}{2}E_{t} ∑_{j=1}^{∞} β^{j}

= \frac{β}{2(1 − β)}

Let us assume the linear form of the second tree’s pricing function.

p_{2t} = k_{2t}d_{t}

Next apply the above linear form to the Euler equation.

k_{2t}d_{t} = E_{t}β \frac{d_{t}}{d_{t+1}} (d_{2t+1} + k_{2t+1}d_{t+1}) + β \frac{γd_{t}}{s_{2t+1}}
k_{2t} = E_{t}βk_{2t+1} + E_{t}β (\frac{d_{2t+1}}{d_{t+1}} + \frac{γ}{s_{2t+1}} )

Recursively we obtain k_{2t}.

k_{2t} = E_{t} ∑_{j=1}^{∞} β^{j} (\frac{d_{2t+j}}{d_{t+j}} + \frac{γ}{s_{2t+1}} )

Since we are given d_{2t} = \frac{1}{2}d_{t} and s_{2t+1} = 1 in equilibrium.

k_{2t} = E_{t} ∑_{j=1}^{∞} β^{j} (\frac{1}{2} + γ)

= (\frac{1}{2} + γ) \frac{β}{1 − β}

Finally, we have got the pricing equations.

p_{1t} = (\frac{β}{2(1 − β)}) d_{t}
p_{2t} = ((\frac{1}{2} + γ) \frac{β}{1 − β} ) d_{t}

(d) Prove that if γ > 0, then R_{1t} > R_{2t} ∀ t

The returns R_{1t}, R_{2t} are defined as:

R_{1t} = \frac{p_{1t+1} + d_{1t+1}}{p_{1t}}

R_{2t} = \frac{p_{2t+1} + d_{2t+1}}{p_{2t}}

From the derived pricing equations,

R_{1t} = \frac{\frac{1}{2} (\frac{β}{1 − β} ) d_{t+1} + \frac{1}{2}d_{t+1}}{\frac{1}{2} (\frac{β}{1 − β} ) d_{t}}

R_{2t} = \frac{(\frac{1}{2} + γ) (\frac{β}{1 − β} ) d_{t+1} + \frac{1}{2}d_{t+1}}{(\frac{1}{2} + γ) (\frac{β}{1 − β} ) d_{t}}

By rearranging the equations,

R_{1t} − R_{2t} = (1 − \frac{1}{1 + 2γ} ) \frac{1 − β}{β} \frac{d_{t+1}}{d_{t}}

If γ > 0, then R_{1t} > R_{2t}