ECON307 Problem Set 2
Question 1
A consumer chooses consumption and savings to maximize their welfare subject to budget constraints.
\[ \begin{aligned} \max_{\left\{{c_t, F_{t+1}}\right\}_{t=0}^{\infty}} & \sum_{t=0}^{\infty}\beta^t u(c_t) \\ \,\text{s.t.}\,& F_{t+1} = R_t\left(F_t + y_t - c_t\right), \text{ for all } t \geq 0 \\ & c_t \geq 0, \text{ for all } t \geq 0 \\ & \lim_{T\to\infty}\beta^T u'(c_T)F_{T+1} = 0 \quad \text{(Transversality Condition)} \end{aligned} \]
where \(u(c_t) = \frac{c_t^{1-\gamma}}{1-\gamma}\), \(\beta \in (0,1)\), \(\gamma > 0\), \(F_0 = 0\), and \(\left\{{y_t}\right\}_{t=0}^{\infty}\) is an exogenous, deterministic sequence of labor income with at least some positive \(y_t\). \(\left\{{R_t}\right\}_{t=0}^{\infty}\) are the gross interest rate on financial assets, and are an exogenous, deterministic sequence known to the consumer.
Question 1.1
Setup the Lagrangian for this problem, being clear on Lagrange Multipliers and equality/inequality constraints.
Hint: You can be sloppy and skip the multiplier on the Transversality condition, as we have done in class. As always, I strongly suggesting using present-value Lagrange multipliers to simplify algebra.
Question 1.2
Show that the \(c_t \geq 0\) constraint can never bind, then find the Euler equation for the consumer at all \(t\geq 0\).
Hint: The only change from our standard problem is the time varying interest rate. You will need to be careful with the timing when taking first order conditions. To prove that \(c_t > 0\), you will need to use the marginal utility and use the fact that there is at least some positive income.
Question 1.3
For some \(\delta \geq 0\) and \(\phi \geq 0\), let the labor income process be
\[ y_t = \begin{cases} y_0 \delta^t & t = 0,\ldots T \\ y_0 \delta^t \phi^{t-T} & t = T+1,\ldots \infty \end{cases} \]
It just happens that \(\left\{{R_t}\right\}_{t=0}^{\infty}\) is such that the consumer optimally sets \(c_t = y_t\) and \(F_{t+1} = 0\) for all \(t\) (i.e., this is a particular sequence of \(R_t\) which rationalizes this behavior). Find a formula for \(\left\{{R_t}\right\}_{t=0}^{\infty}\) and justify your formula.
Hint: What does optimality mean? Also, be a little careful around \(T\) for the calculation of \(R_t\).
Question 1.4
Interpret your formula for \(R_t\) in terms of (i) the consumer’s impatience, and (ii) the consumer’s income growth.
Question 2
There are two consumers (\(i=1,2\)) with potentially different consumption and income processes (\(c^i_t\) and \(y^i_t\)), initial financial wealth \(F_0^i = 0\), and identical preferences subject to an intertemporal budget constraint,
\[ \begin{aligned} \max_{\left\{{c_t}\right\}_{t=0}^{\infty}} & \sum_{t=0}^{\infty}\beta^t u(c_t^i) \\ \,\text{s.t.}\,& \sum_{t=0}^{\infty} \beta^t c_t^i = \sum_{t=0}^{\infty} \beta^t y_t^i \end{aligned} \]
where \(u'(c) > 0, u''(c) < 0, \beta \in (0,1),\) and \(\beta R = 1\). Assume that the two income processes are
\[ \begin{aligned} y_t^1 &= \left\{{0,1,0,1,\ldots}\right\} = \begin{cases} 0 & \text{ if } t \text{ even} \\ 1 & \text{ if } t \text{ odd} \end{cases} \\ y_t^2 &= \left\{{1,0,1,0,\ldots}\right\} = \begin{cases} 1 & \text{ if } t \text{ even} \\ 0 & \text{ if } t \text{ odd} \end{cases} \end{aligned} \]
Question 2.1
Apply the permanent income result to find \(c_t^i\) for both agents.
Hint: Note that if \(a_t = \left\{{1,0,1,0\ldots}\right\}\) then \(\sum_{t=0}^{\infty}\beta^t a_t = 1 + \beta^2 + \beta^4 + \ldots = \sum_{t=0}^{\infty}\left(\beta^2\right)^t\).
Question 2.2
For every \(t\), compare \(c^1_t + c^2_t\) vs. \(y^1_t + y^2_t\). Would this comparison change if \(\beta R \neq 1\)? (no need to solve for the exact \(c^i_t\) in that case)
Question 2.3
Assuming that both agents start with no financial wealth, i.e. \(F_0^1 = F_0^2 = 0\), compute the asset trades between consumer 1 and 2 to support the \(c^i_t\) where the period-by-period budget constraint for \(i=1,2\) is
\[ F_{t+1}^i = R(F_t^i + y_t^i - c_t^i) \]
Question 3
A consumer chooses consumption and savings to maximize their welfare subject to budget constraints.
\[ \begin{aligned} \max_{\left\{{c_t, F_{t+1}}\right\}_{t=0}^{T}} & \sum_{t=0}^{T}\beta^t c_t \\ \,\text{s.t.}\,& F_{t+1} = R\left(F_t - c_t\right), \text{ for } t=0, \ldots T \\ & c_t \geq 0, \text{ for } t = 0, \ldots T \\ & F_{T+1} = 0 \end{aligned} \]
where \(T < \infty, F_0 > 0, R > 0,\) and \(\beta \in (0,1)\). Note that there is positive initial wealth, but no labor income. They are choosing how to spend their wealth.
Hint: This has not assumed any relationship between \(\beta\) and \(R\). Consider that \(\beta R \lesseqqgtr 1\) as potentially having different behaviors, which might require analyzing different cases since you don’t have the luxury to pick a single \(R\) which is convenient. The \(\beta R = 1\) case will be very familiar.
Question 3.1
Check if the utility function is concave.
Question 3.2
Setup the Lagrangian and find the first-order necessary conditions.
Hint: You will have to be very careful with Lagrange multipliers here and cannot just directly use our formulas. The complementarity conditions will be important for the \(c_t\) constraints. And remember that linear objectives and linear constraints usually means corners, so the \(c_t \geq 0\) constraint may actually be important.
Question 3.3
Using the first-order necessary conditions, find the optimal path of \(\left\{{c_t, F_{t+1}}\right\}_{t=0}^{T}\). Is the solution always unique, and if not, why?
Question 3.4
Interpret how does the optimal allocation depends on the relationship between \(R\) and \(\beta\).