econ.studio
Ramsey-Cass-Koopmans Model
Exercises

Ramsey-Cass-Koopmans Model

Exercises

Work through the prompts first, then compare against the solutions once you are ready.

Exercise 1
**Derive the Euler equation from scratch.** Set up the current-value Hamiltonian for the representative household with CRRA preferences, take first-order conditions, and obtain the Keynes-Ramsey rule.
  1. (a)
    Write the current-value Hamiltonian H(k,c,μ)\mathcal{H}(k, c, \mu) for the per-worker problem with effective discount rate ρ−n\rho - n and per-period utility u(c)=(c1−θ−1)/(1−θ)u(c) = (c^{1-\theta} - 1)/(1-\theta).
  2. (b)
    Take the FOC with respect to cc, the co-state equation for μ˙\dot \mu, and combine them to eliminate μ\mu.
  3. (c)
    Show the resulting equation reduces to c˙/c=(1/θ)(f′(k)−δ−ρ)\dot c / c = (1/\theta)(f'(k) - \delta - \rho) and identify the sign of c˙\dot c in each of the three regimes (MPK above, at, or below ρ+δ\rho + \delta).
Exercise 2
**Steady state with Cobb-Douglas.** Let f(k)=Akαf(k) = A k^\alpha. Take α=0.33\alpha = 0.33, A=1A = 1, ρ=0.04\rho = 0.04, δ=0.05\delta = 0.05, n=0.01n = 0.01.
  1. (a)
    Compute k∗k^*, y∗y^*, c∗c^*, the implied savings rate s∗s^*, and the steady-state interest rate r∗r^*.
  2. (b)
    Compute the golden-rule kGRk_{GR} and cGRc_{GR}. By what percentage does steady-state consumption fall short of the golden rule?
  3. (c)
    Show analytically that k∗<kGRk^* < k_{GR} iff ρ>n\rho > n.
Exercise 3
**Linearize the dynamics and compute the convergence rate.** Use the steady state from Exercise 2 and θ=2\theta = 2.
  1. (a)
    Write down the Jacobian JJ of the RHS of (k˙,c˙)(\dot k, \dot c) at (k∗,c∗)(k^*, c^*) for Cobb-Douglas.
  2. (b)
    Compute the trace, determinant, and the two eigenvalues numerically.
  3. (c)
    Report the half-life of convergence (in 'years' if tt is annual). How does the half-life change if θ\theta is halved?
Exercise 4
**Transversality condition.** Show that on the saddle path the TVC lim⁡t→∞e−(ρ−n)tc(t)−θk(t)=0\lim_{t\to\infty} e^{-(\rho - n)t} c(t)^{-\theta} k(t) = 0 is satisfied automatically.
  1. (a)
    On the saddle path, k(t)−k∗=(k0−k∗)eλ−tk(t) - k^* = (k_0 - k^*) e^{\lambda^- t}. Use this to argue k(t)k(t) converges to k∗k^* at rate ∣λ−∣|\lambda^-|.
  2. (b)
    Compute the asymptotic behaviour of e−(ρ−n)tc(t)−θk(t)e^{-(\rho-n)t} c(t)^{-\theta} k(t). Argue that (ρ−n)>0(\rho - n) > 0 guarantees the limit is zero.
  3. (c)
    Sketch what would go wrong if the household *over-saved* (chose c0c_0 below the saddle path). Which boundary condition would be violated?
Exercise 5
**A permanent productivity boom.** The economy is at its steady state when AA unexpectedly jumps from A0A_0 to A1>A0A_1 > A_0 at t=0t = 0.
  1. (a)
    Compute the new steady state (k1∗,c1∗)(k^*_1, c^*_1). Is k1∗>k0∗k^*_1 > k^*_0? Is c1∗>c0∗c^*_1 > c^*_0?
  2. (b)
    On a phase diagram, show the *old* and *new* zero-loci. Where does the economy jump on impact at t=0t = 0?
  3. (c)
    Describe in words what happens to consumption on impact, and during the transition.
Exercise 6
**Capital-income tax.** Suppose the government taxes capital income at rate τ∈(0,1)\tau \in (0, 1) and rebates the revenue lump-sum to households.
  1. (a)
    Show that the after-tax return on capital is (1−τ)(f′(k)−δ)(1-\tau)(f'(k) - \delta).
  2. (b)
    Derive the new Euler equation and steady-state condition.
  3. (c)
    Compute the long-run effect on k∗k^* as a function of τ\tau. What is the welfare cost (in terms of c∗c^*) of a 10\% tax?
Exercise 7
**Comparing RCK and Solow.** Suppose the planner could set Solow's savings rate to anything she likes. What value would replicate the RCK steady state, and would she choose it?
  1. (a)
    Compute the savings rate s^\hat s that makes Solow's steady-state equal k∗k^*.
  2. (b)
    Is s^\hat s unique?
  3. (c)
    If the planner maximises c∗c^*, what ss would she pick instead?
Exercise 8
**Heterogeneity preview.** Consider two RCK economies, identical except that economy A has ρA=0.03\rho_A = 0.03 and economy B has ρB=0.05\rho_B = 0.05.
  1. (a)
    Which economy has higher steady-state capital? Higher consumption? Higher interest rate?
  2. (b)
    Suppose households from A and B can trade freely with each other. In a long-run common-asset equilibrium, who ends up owning all the capital? Why?
  3. (c)
    What does this say about the legitimacy of the representative-agent assumption when households differ in ρ\rho?