ECON307 Problem Set 1
Question 1
A dividend process follows the deterministic process:
\[ x_{t+1} = A \cdot x_t \]
where \(x_t\) is an \(n \times 1\) vector and \(A\) is an \(n \times n\) matrix, and
\[ y_t = G \cdot x_t \]
where \(y_t\) is the dividend (a scalar), and \(G\) is a \(1 \times n\) vector. Assume future profits are discounted by \(\beta \in (0,1)\), and that \(I - \beta A\) is invertible.
Question 1.1
What is the stock price of the firm \(p_t\), in terms of \(A\) and \(G\) if there was no bubble?
Question 1.2
What is the stock price of the firm today, \(p_t\), in terms of the price tomorrow, \(p_{t+1}\), and the state today, \(x_t\)?
Question 1.3
A friend guesses that the stock price should be
\[ p_t = H \cdot x_t + c \cdot \lambda^t \]
for some vector \(H \in {\mathbb{R}}^n\) and scalars \(c\), \(\lambda\).
Get as far as you can in finding formulas for \(H\), \(c\), \(\lambda\). (Hint: use the guess and verify to find the undetermined constants, with the recursive definition of the price from Section 1.2).
Question 1.4
Is \(H\) unique? How about \(c\) and \(\lambda\)?
Question 2
A dividend obeys:
\[ y_{t+1} = \lambda_0 + \lambda_1 y_t + \lambda_2 y_{t-1} \]
where \(y_t\) is scalar.
The stock price obeys:
\[ p_t = \sum_{j=0}^{\infty} \beta^j y_{t+j} \]
Question 2.1
Find a solution for the price \(p_t\) of the form:
\[ p_t = a_0 + a_1 y_t + a_2 y_{t-1} \]
for some \(a_0\), \(a_1\), and \(a_2\) in terms of model parameters. (Hint: Set it up as a linear state space.)
(No need to actually invert matrices, etc. to find the solution to the particular \(a_0\), \(a_1\), \(a_2\))
Question 3
Take an asset which owns claims to a single claim on two streams of dividends (both paying out to the owner of the asset at time \(t\)):
- \(d_{A,t+1} = (1 + \delta_A) d_{A,t}\) for \(\delta_A \geq 0\)
- \(d_{B,t+1} = (1 + \delta_B) d_{B,t}\) for \(\delta_B \geq 0\)
where \(d_{A,0} = d_{B,0} = 1\). Let the price of this asset, using discount rate \(\rho > 0\) (i.e., \(\frac{1}{1+\rho}\) is the discount factor) be \(p_t^{AB}\).
That is, if I own 1 unit of the asset at time \(t\), I get \(y_t = d_{A,t} + d_{B,t}\) in payoffs.
Question 3.1
Write this problem in our linear state space model.
Question 3.2
Find an expression for the price, \(p_0^{AB}\), of the underlying asset at time 0 using the tools from our linear state space models. (Hint: to take the inverse of a diagonal matrix, just take the reciprocal along the diagonals. i.e., \(\begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}^{-1} = \begin{bmatrix}1/a & 0 \\ 0 & 1/b\end{bmatrix}\))
Question 3.3
Roughly describe the conditions on \(\delta_A\), \(\delta_B\) and \(\rho\) required for this to be a well defined problem.
Question 3.4
Now assume that instead of a joint asset, consider an asset, priced at \(p_t^A\) which only has claims to the \(d_{A,t}\) sequence, and another \(p_t^B\) with claims to the \(d_{B,t}\) sequence. Calculate \(p_0^A\) and \(p_0^B\).
Question 3.5
Describe the intuition for how \(p_0^{AB}\), \(p_0^A\), \(p_0^B\) relate, and how agents would behave differently if the relationship was broken.