3. Consider a model where agents maximize lifetime utility: $\sum_{t=0}^{\infty} \beta^t \ln c_t$, subject to the
constraint: $k_t = f(k_{t-1}) - c_t$ (note this capital accumulation constraint assumes a
depreciation rate equal to one). Here $f(k_t)$, the production technology, is expressed in
intensive form and $\beta$ is a discount rate satisfying $\beta \in (0, 1)$.
(a) Write down a dynamic equation for $c_t$ that describes necessary conditions which
must be satisfied on the optimal path.
(b) Using your answer to part (a), write an expression for $\frac{c_{t+1}}{c_t}$ when production is
described by the Cobb-Douglas technology: $y_t = k_{t-1}^{\alpha}$. [You do not have to redo
the dynamic optimization problem. Just sub out for $f_k$ in your Euler equation].
(c) Sketch a graph of optimal consumption growth condition you derived in part (c),
with $\frac{c_{t+1}}{c_t}$ on the vertical axis and $k$ on the horizontal axis. [Note: I said $\frac{c_{t+1}}{c_t}$ on
the vertical axis NOT $c_t$.
(d) Label the following points on the relevant axes with appropriate values: Steady-
state values of $\frac{c_{t+1}}{c_t}$ and $k$, as well as the (Phelps) golden capital-labor ratio, $k^*$.
