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Econ 5021W - Winter 2023

Assignment 3

Due at the beginning of the tutorial (6:05pm) on Wednesday, March 1, 2023.

Answer all of the following questions. The assignment is out of 40 marks total. Be sure to show your work, but also please visually emphasize your final answer (e.g., by putting a box around it, or writing it in a different colour).

1.  (20 marks total) Consider Hall’s “random walk” model of Section 4.2.1 of LN2, where utility is quadratic, and rt = r = ρ for all t. Suppose initial assets are a0 = 0, and household income Yt  is made up of two parts: a deterministic part Dt  that grows at some constant exogenous rate each period, and a stochastic part Zt  that follows an AR(1) stochastic process. That is, we can write

Yt = Dt + Zt ,                                                            (1)

with

Dt = (1 + γ) Dt-1                                                      (2)

Zt = δZt − 1 + ϵt .                                                          (3)

Here, γ ∈ [0, r) is the constant growth rate of D , δ ∈ (−1, 1) is the persistence parameter of the AR(1) process for Z , and {ϵt} is a white noise process, i.e., a sequence of i.i.d.  random variables with mean zero, so that Et[ϵt+τ] = 0 for τ ≥ 1.  We take the initial values D0  > 0 and Z0  as given.

(a)  (3 marks) Find an expression for Et[Zt+τ] for τ ≥ 0 as a function of Zt  and the model parameters only. (HINT: Start from the date- (t+τ ) version of (3), and then use repeated substitution of that same recursive relationship backwards through time to obtain an expression for Zt+τ  as a function of (i) Zt, and (ii) the values of the innovations ϵt+s  for s = 1, . . . , τ . Then take date-t expectations of this expression.)

(b)  (2 marks) Find an expression for Dt+τ  for τ  ≥ 0 as a function of Dt  and the model parameters only.

(c)  (5 marks) Solve for Et[Wt] (i.e., the expected discounted present value of lifetime income from date t onward) as a function of Dt , Zt, and the model parameters only.  (HINT: You should end up with an expression of the form Et[Wt] = ADt+ BZt, where A and B are functions of exogenous parameters only.)

(d)  (5 marks) Solve for Et[Wt+1] as a function of Dt , Zt, and the model parameters only. (HINT: You should again end up with an expression of the form Et[Wt+1] = ADt+ BZt , where A and B are functions of exogenous parameters only. Note that this A and B are not necessarily the same as the ones from part (c).)

(e)  (5 marks) Solve for the revision from t to t + 1 in the HH’s forecast about Wt+1, i.e., Et+1[Wt+1] − Et[Wt+1], as a function of Dt , Zt , ϵt+1, and the model parameters only.

2.  (5 marks total) Consider a finite-horizon version of the asset-pricing model discussed in Section

6.1 of LN2.   Let T  < ∞ be the last period that time exists in this model. Let Ij,t ≡ 对τ(T)1(t) βτ Et[dj,t+τ] denote the intrinsic value in this case  (where we adopt the summation convention that if b < a, then we always set 对j(b)=axj  = 0, so that in particular Ij,T  = 0). Under what condition(s) (if any) is it possible for the non-intrinsic value pˆj,t  ≡ pj,t − Ij,t  to be positive at some date t? Comparing your answer to the analogous result obtained in LN2 for the infinite-horizon case (where we haven’t assumed the no-bubbles condition necessarily holds), explain the economic intuition for any similarities/differences.

3.  (15 marks total) Consider the stochastic asset-pricing model with risk-averse HHs (i.e., Section

6.2 of LN2). Suppose there are two assets, indexed by j = 1, 2. Assume the actual (realized) gross return on asset j = 2 from t to t + 1 is related to the return on asset j = 1 by

G2,t+1 − µ2,t = α (G1,t+1 − µ1,t) ,                   (4)

where we have defined µj,t  ≡ Et[Gj,t+1], and α ∈ R is a parameter.1   That is, the difference between G2,t+1  and its mean is equal to α times the difference between G1,t+1  and its mean. In solving this question, it will be useful to recall that for two random variables X and Y , and two non-random variables δ, γ, we have Cov (δX + γ, Y) = δCov (X, Y).

(a)  (3 marks) Letting X be any random variable, using (4) obtain an expression for Covt(X, G2,t+1) in terms of Covt(X, G1,t+1).

(b)  (8 marks) Under what condition(s) will we have µ2,t  > µ 1,t?

(c)  (4 marks) Explain clearly the economic intuition for your answer to part (b).