ECON307 Problem Set 3

Author
Affiliation

Jesse Perla

University of British Columbia

Question 1

Consider two scenarios for a consumer planning consumption with income process \(y_t = y_0 \delta^t\) for all \(t \geq 0\) and \(F_0 = 0\).

Scenario 1 for Consumer: The consumer maximizes the following welfare

\[ \begin{aligned} U \equiv \max_{\left\{{c_t, F_{t+1}}\right\}_{t=0}^{\infty}} & \sum_{t=0}^{\infty} \beta^t \log(c_t) \\ \,\text{s.t.}\,& F_{t+1} = R(F_t + y_t - c_t), \quad \forall t \geq 0 \\ & \text{(transversality condition)} \end{aligned} \]

Scenario 2 for Consumer: The consumer faces the same problem as Scenario 1, except with no borrowing: \(F_{t+1} \geq 0\) for all \(t \geq 0\), and the initial level of \(y_0\) is potentially different (define it as \(y_0^{NB}\)).

Define the PDV of utility (i.e., the welfare) of this as \(U^{NB}\).

Question 1.1

Let \(\beta = 0.95\), \(R = 1.04\), \(\delta = 1.02\), \(y_0 = 1\), and \(y^{NB}_0 = 1\). Calculate \(U\) and \(U^{NB}\).

Question 1.2

Let \(y_0 = 1\). Now find a \(y_0^{NB}\) such that \(U = U^{NB}\). The difference between \(y_0\) and \(y^{NB}_0\) is the amount of sacrifice in terms a consumer with a borrowing constraint would pay to be free to borrow. A measure of the welfare loss of the no borrowing constraint.

Question 1.3

Maintain \(y_0 = 1\). Now, let \(\beta = 0.99\), \(R = 1.04\), and \(\delta = 1.01\). What is \(c_0\) and \(F_1\) here under Scenario 1? Repeat part (b) to find \(y_0^{NB}\) such that \(U = U^{NB}\) with these new parameters. What can you conclude about the welfare cost of no borrowing in this case?

Question 2

Let the consumer have power utility,

\[ u(c) = \frac{c^{1-\gamma}}{1 - \gamma}, \quad \gamma > 1 \]

Given \(F_0 = 0\), \(B \geq 0\), \(\beta R = 1\), and the deterministic income stream \(y_t = \delta^t\), the consumer maximizes

\[ \begin{aligned} \max_{\left\{{c_t, F_{t+1}}\right\}_{t=0}^{\infty}} & \sum_{t=0}^{\infty} \beta^t u(c_t) \\ \,\text{s.t.}\,& F_{t+1} = R(F_t + y_t - c_t), \quad \forall t \geq 0 \\ & F_{t+1} \geq -B \\ & F_0 = 0 \\ & \text{(transversality condition)} \end{aligned} \]

Question 2.1

Derive the Euler equation as an inequality, and the condition for it holding with equality.

Question 2.2

Let \(\delta > 1\) and \(B = \infty\). What is \(\left\{{c_t}\right\}_{t=0}^{\infty}\)?

Question 2.3

Let \(\delta > 1\) and \(B = 0\). What is \(\left\{{c_t}\right\}_{t=0}^{\infty}\)?

Question 2.4

Let \(\delta < 1\) and \(B = 0\). What is \(\left\{{c_t}\right\}_{t=0}^{\infty}\)?

Question 2.5

Assume that the consumer optimally eats their entire income each period, i.e., \(c_t = y_t = \delta^t\) which implies \(c_{t+1} = \delta c_t\). Setup, using dynamic programming, an equation to find the value \(V(c)\) recursively.

Question 2.6

Guess that \(V(c) = k_0 + k_1 c^{1-\gamma}\) for some undetermined \(k_0\) and \(k_1\). (Note: this equation deliberately avoids any \(t\) subscripts, making it a truly recursive expression.) Solve for \(k_0\) and \(k_1\) and evaluate \(V(1)\) (i.e., the value of starting with \(c_0 = 1\)).

Question 3

Consider a Markov chain with two states: \(U\) for unemployment and \(E\) for employment.

  • With probability \(\lambda \in (0,1)\), a person unemployed today becomes employed tomorrow.
  • With probability \(\alpha \in (0,1)\), a person employed today becomes unemployed tomorrow.

Question 3.1

Let \(N \geq 1\) be the number of periods until a currently unemployed person becomes employed. Calculate \({\mathbb{E}_{{}}\left[ {N} \right]}\).

Question 3.2

Let \(M \geq 1\) be the number of periods until a currently employed person becomes unemployed. Calculate \({\mathbb{E}_{{}}\left[ {M} \right]}\).

Question 3.3

Please compute the fraction of time an infinitely lived person can expect to be unemployed and the fraction of time they can expect to be employed.

Question 4

An economy has 3 states for workers:

  • \(U\): unemployment
  • \(V\): verification (found a potential employer and being verified for fit)
  • \(E\): employed (verified and working)

The probabilities of jumping between states each period are shown below:

G U U U->U 1-λ V V U->V λ V->U 1-γ E E V->E γ E->U α E->E 1-α

i.e., probability \(\gamma\) they are a good fit, and the verification takes 1 period.

Question 4.1

Write a Markov transition matrix for this process, \(P\).

Question 4.2

Write an expression for the stationary distribution across states in the economy, \(\pi \in {\mathbb{R}}^3\) (You can leave in terms of \(P\)).

Question 4.3

If a worker is \(U\) today, write an expression for the probability they will be employed exactly \(j\) periods in the future (considering any possible transitions which end in employment at \(j\) periods).

Note: This is only looking at \(j\) periods into the future, i.e., this is not the probability that they become employed at least once during the \(j\) periods, which is a much more difficult calculation.

Question 4.4

Assume that \(\alpha = 0\), \(\lambda = 0\). Is the stationary distribution unique? If not, describe the sorts of distributions that could exist and the intuition from the perspective of the Markov chain.