econ.studio
Ramsey-Cass-Koopmans Model
Solutions

Ramsey-Cass-Koopmans Model

Solutions

Exercise 1
**(a)** H=c1−θ−11−θ+μ[f(k)−c−(n+δ)k]\mathcal{H} = \frac{c^{1-\theta} - 1}{1-\theta} + \mu[f(k) - c - (n+\delta) k]. **(b)** ∂H/∂c=0⇒c−θ=μ\partial \mathcal{H}/\partial c = 0 \Rightarrow c^{-\theta} = \mu. Co-state: μ˙=(ρ−n)μ−∂H/∂k=(ρ−n)μ−μ(f′(k)−(n+δ))\dot \mu = (\rho - n)\mu - \partial \mathcal{H}/\partial k = (\rho - n)\mu - \mu(f'(k) - (n+\delta)), so μ˙/μ=ρ+δ−f′(k)\dot \mu/\mu = \rho + \delta - f'(k). From c−θ=μc^{-\theta} = \mu, μ˙/μ=−θc˙/c\dot \mu/\mu = -\theta \dot c / c. **(c)** Equating: −θc˙/c=ρ+δ−f′(k)-\theta \dot c/c = \rho + \delta - f'(k), hence c˙/c=(1/θ)(f′(k)−δ−ρ)\dot c/c = (1/\theta)(f'(k) - \delta - \rho). When f′(k)>ρ+δf'(k) > \rho + \delta, c˙>0\dot c > 0 (consumption grows because returns beat impatience). When f′(k)=ρ+δf'(k) = \rho + \delta, c˙=0\dot c = 0 - the steady-state Euler condition. When f′(k)<ρ+δf'(k) < \rho + \delta, c˙<0\dot c < 0 (front-load consumption).
Exercise 2
**(a)** k∗=(αA/(ρ+δ))1/(1−α)=(0.33/0.09)1/0.67≈8.41k^* = (\alpha A / (\rho + \delta))^{1/(1-\alpha)} = (0.33/0.09)^{1/0.67} \approx 8.41. y∗=A(k∗)α≈2.04y^* = A (k^*)^\alpha \approx 2.04. c∗=y∗−(n+δ)k∗=2.04−0.06⋅8.41≈1.54c^* = y^* - (n+\delta) k^* = 2.04 - 0.06 \cdot 8.41 \approx 1.54. s∗=(n+δ)k∗/y∗=α(n+δ)/(ρ+δ)≈0.22s^* = (n+\delta) k^* / y^* = \alpha(n+\delta)/(\rho+\delta) \approx 0.22. r∗=ρ=0.04r^* = \rho = 0.04. **(b)** kGR=(αA/(n+δ))1/(1−α)=(0.33/0.06)1/0.67≈17.97k_{GR} = (\alpha A/(n+\delta))^{1/(1-\alpha)} = (0.33/0.06)^{1/0.67} \approx 17.97. cGR=AkGRα−(n+δ)kGR≈2.62−1.08≈1.54c_{GR} = A k_{GR}^\alpha - (n+\delta) k_{GR} \approx 2.62 - 1.08 \approx 1.54. The gap cGR−c∗c_{GR} - c^* is small in this calibration because the marginal product is gentle near the peak. Typical numbers give ≈1%\approx 1\% shortfall. **(c)** f′(k∗)=ρ+δf'(k^*) = \rho + \delta and f′(kGR)=n+δf'(k_{GR}) = n + \delta. ρ>n⇔ρ+δ>n+δ⇔f′(k∗)>f′(kGR)⇔k∗<kGR\rho > n \Leftrightarrow \rho + \delta > n + \delta \Leftrightarrow f'(k^*) > f'(k_{GR}) \Leftrightarrow k^* < k_{GR} by f′′<0f'' < 0.
Exercise 3
**(a)** J=(ρ−n−1c∗θf′′(k∗)0)J = \begin{pmatrix} \rho - n & -1 \\ \frac{c^*}{\theta} f''(k^*) & 0 \end{pmatrix}, with f′′(k∗)=−(1−α)(ρ+δ)/k∗f''(k^*) = -(1-\alpha)(\rho+\delta)/k^* under Cobb-Douglas. **(b)** tr⁡J=ρ−n=0.03\operatorname{tr} J = \rho - n = 0.03. f′′(k∗)=−(0.67⋅0.09)/8.41≈−0.0072f''(k^*) = -(0.67 \cdot 0.09)/8.41 \approx -0.0072. det⁡J=(1.54/2)(−0.0072)≈−0.00554\det J = (1.54/2)(-0.0072) \approx -0.00554. Eigenvalues: λ±=(0.03±0.0009+0.0222)/2=(0.03±0.152)/2\lambda^\pm = (0.03 \pm \sqrt{0.0009 + 0.0222})/2 = (0.03 \pm 0.152)/2. So λ−≈−0.061\lambda^- \approx -0.061, λ+≈0.091\lambda^+ \approx 0.091. **(c)** Half-life ln⁡2/∣λ−∣≈0.693/0.061≈11.4\ln 2/|\lambda^-| \approx 0.693/0.061 \approx 11.4 years. Halving θ\theta to 1 doubles ∣det⁡J∣|\det J| to ≈0.011\approx 0.011, giving λ−≈−0.092\lambda^- \approx -0.092 and half-life ≈7.5\approx 7.5 years - faster convergence.
Exercise 4
**(a)** Yes, k(t)→k∗k(t) \to k^* as t→∞t \to \infty, and the deviation decays at rate ∣λ−∣|\lambda^-|. Hence k(t)k(t) is bounded. **(b)** c(t)→c∗>0c(t) \to c^* > 0, so c(t)−θc(t)^{-\theta} is bounded. k(t)k(t) bounded. e−(ρ−n)te^{-(\rho - n)t} goes to zero exponentially (P4). Product →0\to 0. **(c)** Under-consumption sends k(t)→∞k(t) \to \infty along an unstable trajectory. e−(ρ−n)te^{-(\rho-n)t} decay would be outpaced by kk growth, and the TVC limit would be strictly positive - wealth is being left on the table at infinity. Over-consumption (above saddle path) sends k(t)→0k(t) \to 0 and c(t)−θ→∞c(t)^{-\theta} \to \infty, violating non-negativity.
Exercise 5
**(a)** k1∗=(αA1/(ρ+δ))1/(1−α)>k0∗k^*_1 = (\alpha A_1 / (\rho + \delta))^{1/(1-\alpha)} > k^*_0. Similarly y1∗>y0∗y^*_1 > y^*_0, and c1∗>c0∗c^*_1 > c^*_0 provided we are below the golden rule - always the case in RCK. **(b)** Both the k˙=0\dot k = 0 and c˙=0\dot c = 0 loci shift. The c˙=0\dot c = 0 locus moves *right* (higher k1∗k^*_1). The k˙=0\dot k = 0 peak moves up and right. On impact, k(0)=k0∗k(0) = k^*_0 is fixed (state variable), but c(0)c(0) jumps *up* onto the new saddle path leading to (k1∗,c1∗)(k^*_1, c^*_1). From there, kk grows and cc grows along the saddle path. **(c)** Consumption jumps up immediately on news of the productivity boom - the household feels wealthier. It also continues to grow during the transition. Capital accumulates only gradually (state variable). The jump in cc on impact is the dramatic difference from Solow, where there is no jump.
Exercise 6
**(a)** Net return on saving becomes (1−τ)r=(1−τ)(f′(k)−δ)(1-\tau) r = (1-\tau)(f'(k) - \delta). **(b)** New Euler: c˙/c=(1/θ)((1−τ)(f′(k)−δ)−ρ)\dot c / c = (1/\theta)((1-\tau)(f'(k) - \delta) - \rho). Steady state: (1−τ)(f′(k∗)−δ)=ρ(1-\tau)(f'(k^*) - \delta) = \rho, so f′(k∗)=δ+ρ/(1−τ)f'(k^*) = \delta + \rho/(1-\tau). **(c)** Higher τ\tau raises f′(k∗)f'(k^*), so k∗k^* falls. For Cobb-Douglas: k∗(τ)=(αA/(δ+ρ/(1−τ)))1/(1−α)k^*(\tau) = (\alpha A/(\delta + \rho/(1-\tau)))^{1/(1-\alpha)}. For τ=0.1\tau = 0.1 with the baseline (ρ=0.04\rho = 0.04, δ=0.05\delta = 0.05): denominator goes from 0.090.09 to 0.0940.094, so k∗k^* falls by (0.09/0.094)1/0.67−1≈−6.4%(0.09/0.094)^{1/0.67} - 1 \approx -6.4\%. c∗c^* falls by a similar proportion. This is the Chamley-Judd argument for *zero* steady-state capital tax.
Exercise 7
**(a)** Solow steady state: sf(k)=(n+δ)k⇒s=(n+δ)k/f(k)s f(k) = (n + \delta) k \Rightarrow s = (n+\delta) k/f(k). At k=k∗k = k^*: s^=(n+δ)(k∗)/f(k∗)=α(n+δ)/(ρ+δ)\hat s = (n+\delta)(k^*)/f(k^*) = \alpha(n+\delta)/(\rho+\delta) under Cobb-Douglas - the same expression we derived for the implied RCK savings rate. **(b)** Yes - Solow has a one-to-one map from ss to k∗k^*. **(c)** The golden-rule rate sGR=αs_{GR} = \alpha (capital share), which yields kGRk_{GR}. The RCK household does *not* choose this - s^<sGR\hat s < s_{GR} - because the welfare loss from cutting consumption today to reach kGRk_{GR} tomorrow exceeds the future gain when discounted at ρ\rho.
Exercise 8
**(a)** A has higher k∗k^*, higher c∗c^*, lower r∗=ρA<ρBr^* = \rho_A < \rho_B. B has higher r∗r^* but lower k∗k^* and c∗c^*. **(b)** In a common-asset equilibrium with shared rr, the more patient household (A) accumulates at any r>ρAr > \rho_A. The less patient (B) decumulates at any r<ρBr < \rho_B. The only sustainable long-run rr is r=ρAr = \rho_A (the most patient agent's discount rate). At that rate, B continually decumulates and ends up holding zero assets - A owns all capital asymptotically. **(c)** Heterogeneity in ρ\rho generates extreme long-run wealth concentration. Representative-agent results are useful as aggregates but can mask large distributional features. Modern HANK models tackle this explicitly.