Study sheet: Ramsey-Cass-Koopmans Growth Model

📋 Course Outline

1. RCK Model Motivation and Assumptions
2. Households, Population, and Utility
3. CRRA Preferences and Risk Aversion
4. Two-Period Consumption Choice
5. Continuous-Time Euler Condition
6. Assets, Firms, and Factor Incomes
7. Intertemporal Budgets and Solvency
8. Steady State and Capital Dynamics
9. Phase Diagram and Saddle Path
10. Golden Rule and Patience
11. Government and Ricardian Equivalence
12. Government Spending Shocks

📖 1. RCK Model Motivation and Assumptions

🔑 Key Concepts & Definitions

• RCK model : endogenizes saving by allowing individuals to choose consumption and saving according to their preferences and incentives.

★ Must-know

📌 The Solow model cannot analyze the efficiency of different growth paths or compare welfare between steady states because its saving rate is exogenous.

• In the RCK model, the saving rate varies over time and depends on patience, interest rates, preferences, and new information such as expected tax changes.

• The main assumptions are:

  • perfect competition
  • infinitely lived households
  • a finite but large number of households
  • price flexibility
  • no market imperfections
  • minor risk
  • rational expectations

Further detail

📌 Intergenerational altruism means that households internalize the well-being of future generations, unlike overlapping-generations models in which agents live for two periods and do not necessarily care about descendants.

💡 Memory Hook

Solow fixes saving; RCK makes saving an individual choice.

📖 2. Households, Population, and Utility

★ Must-know

• Population grows at rate $n$, and every household member supplies one unit of labor at each point in time.

📐 Formula — Total labor supply satisfies $L(t)=L(0)e^{nt}$.

• Household utility in continuous time is the discounted integral of individual utility multiplied by household population, with discount factor $e^{-\rho t}$.

📌 Population growth changes the effective discount rate from $\rho$ to $\rho-n$ because future consumption benefits more descendants.

Further detail

• The capital stock is distributed equally among households, and the model assumes no depreciation of capital.

💡 Memory Hook

Population growth → more descendants benefit → future consumption is discounted less.

📖 3. CRRA Preferences and Risk Aversion

★ Must-know

📐 Formula — The CRRA utility function is $u(C(t))=\frac{C(t)^{1-\theta}}{1-\theta}$ for $\theta>0$.

• A low value of $\theta$ indicates low risk aversion and greater willingness to accept consumption risk, whereas a high value indicates strong risk aversion and a preference for smoother consumption.

📐 Formula — The marginal utility of consumption is $u'(C(t))=C(t)^{-\theta}$.

Further detail

📐 Formula — When $\theta=1$, CRRA utility becomes logarithmic utility, $u(C(t))=\ln(C(t))$.

💡 Memory Hook

Low θ accepts consumption risk; high θ prefers smoother consumption.

📖 4. Two-Period Consumption Choice

📝 Essential Points

📐 Formula — In the two-period model, the household’s budget constraints are $C_1=E_1-B_1$ and $C_2=(1+r)B_1+E_2$.

📌 A household borrows when first-period consumption exceeds first-period endowment, $C_1>E_1$, and saves when $C_1<E_1$.

📌 The intertemporal budget constraint requires the present value of lifetime consumption to equal or remain below the present value of lifetime endowments.

📐 Formula — Utility maximization gives the optimal consumption ratio $\frac{C_1}{C_2}=\left(\frac{1+r}{1+\rho}\right)^{-1/\theta}$.

💡 Memory Hook

Endowment → consumption or bonds → future consumption.

📖 5. Continuous-Time Euler Condition

🔑 Key Concepts & Definitions

• Elasticity of intertemporal substitution : measures how willing households are to shift consumption between periods when the interest rate changes.

📝 Essential Points

📐 Formula — The continuous-time Euler condition is $\frac{C_t}{C_{t+1}}=\left(\frac{1+r_t}{1+\rho}\right)^{-1/\theta}$.

📌 A higher interest rate makes saving more attractive, reducing current consumption and increasing future consumption.

• The elasticity of intertemporal substitution is inversely related to $\theta$, so higher $\theta$ makes households more resistant to changing consumption over time.

💡 Memory Hook

Optimize utility → impose the budget constraint → obtain the Euler equation.

📖 6. Assets, Firms, and Factor Incomes

★ Must-know

📐 Formula — Household wealth consists of physical capital and bonds: $W(t)=K(t)+B(t)$.

• The aggregate stock of bonds equals zero because one household’s positive bond holdings are offset by another household’s negative position.

📌 The no-arbitrage condition requires capital and bonds to offer identical returns in equilibrium; otherwise households shift investment toward the higher-yielding asset.

📐 Formula — Firms demand capital until its marginal product equals the rental rate: $r(t)=f'(k_t)$.

Further detail

📐 Formula — The intensive-form wage is $w(t)=f(k_t)-k_tf'(k_t)$, and the real wage per unit of labor is $W(t)=A(t)w(t)$.

💡 Memory Hook

Capital is a household asset and firm input; bonds are private assets but public liabilities.

📖 7. Intertemporal Budgets and Solvency

★ Must-know

📐 Formula — The household period budget constraint is $\frac{C(t)}{H}+\frac{\dot K(t)}{H}+\frac{\dot B(t)}{H}=\frac{W(t)L(t)}{H}+r(t)\frac{K(t)+B(t)}{H}$.

📌 The intertemporal budget constraint states that the present value of consumption cannot exceed initial wealth plus the present value of lifetime labor income.

📌 The No-Ponzi-game condition requires discounted terminal wealth to remain non-negative: $\lim_{T\to\infty}e^{-R(T)}V(T)\ge0$.

Further detail

📐 Formula — Accumulated interest is $R(t)=\int_0^t r(\tau)d\tau$.

💡 Memory Hook

Unlimited borrowing → postponed repayment → No-Ponzi solvency condition.

📖 8. Steady State and Capital Dynamics

📝 Essential Points

📐 Formula — Individual consumption growth follows the Euler equation $\frac{\dot c(t)}{c(t)}=\frac{r(t)-\rho-\theta g}{\theta}$.

📐 Formula — Capital accumulation in efficiency units satisfies $\dot k(t)=f(k(t))-c(t)-(n+g)k(t)$.

📐 Formula — At the steady state, the marginal product of capital satisfies $f'(k^*)=\rho+\theta g$.

• A high marginal product of capital encourages saving and faster capital accumulation, while diminishing returns make MPK fall as capital grows.

💡 Memory Hook

High MPK → saving and capital accumulation → falling MPK → steady state.

📖 9. Phase Diagram and Saddle Path

🔑 Key Concepts & Definitions

• Phase diagram : represents the dynamics of the two endogenous variables, consumption and capital, in the $(k,c)$ plane.

📝 Essential Points

📌 Consumption grows to the left of the $\dot c=0$ locus, where $f'(k)>\rho+\theta g$, and falls to its right.

📐 Formula — The zero-capital-dynamics locus is $c=f(k)-(n+g)k$.

• The steady state requires both consumption and capital growth to be zero: $\dot c=0$ and $\dot k=0$.

💡 Memory Hook

On the saddle path the economy converges; outside it, consumption or capital eventually collapses.

📖 10. Golden Rule and Patience

🔑 Key Concepts & Definitions

• Saddle path : the unique trajectory from a given initial capital stock that satisfies household optimization and converges to the steady state.

★ Must-know

📌 If initial consumption is above the critical saddle-path level, consumption eventually rises while capital is depleted until both collapse.

• A decentralized equilibrium is Pareto efficient when markets are competitive, there are no externalities, and information is perfect.

Further detail

📌 If initial consumption is below the critical saddle-path level, capital accumulates while consumption eventually falls, producing another divergent path.

💡 Memory Hook

RCK’s intertemporal choice generally differs from Solow’s Golden Rule.

📖 11. Government and Ricardian Equivalence

★ Must-know

📐 Formula — The RCK steady state satisfies $f'(k^*)=\rho+\theta g$, while the Solow Golden Rule satisfies $f'(k_{GR})=n+g$ when depreciation is zero.

📌 Higher technology growth creates a substitution effect that encourages saving for higher future returns and a wealth effect that encourages present consumption because future income is expected to be higher.

Further detail

📐 Formula — The difference between the RCK and Golden Rule marginal products is $f'(k^*)-f'(k_{GR})=\rho-n-(1-\theta)g$.

• A higher population growth rate can increase saving because households care about the consumption of future generations.

💡 Memory Hook

Public spending reduces resources, while government bonds do not create net social wealth.

📖 12. Government Spending Shocks

🔑 Key Concepts & Definitions

• Ricardian equivalence : states that households anticipate the future taxes required to repay government debt, so government bonds do not constitute net wealth for society.

★ Must-know

📐 Formula — With government purchases, capital dynamics become $\dot k(t)=f(k(t))-c(t)-G(t)-(n+g)k(t)$.

• Government purchases are financed through lump-sum taxes and government bond issuance, and they are assumed to provide public consumption rather than public investment.

Further detail

📐 Formula — The government’s bond dynamics satisfy $\dot B(t)=G(t)+r(t)B(t)-T_{LS}(t)$ after normalization by the number of households.

💡 Memory Hook

Permanent spending shifts the capital locus permanently; temporary spending creates transitional dynamics.

📊 Synthesis Tables

RCK and Solow Comparisons

Dimension	Solow model	RCK model
Saving	Exogenous saving rate	Endogenous, time-varying saving
Welfare	Cannot compare welfare across steady states	Can evaluate welfare across growth paths
Steady-state condition	$f'(k_{GR})=n+g$	$f'(k^*)=\rho+\theta g$

Government Financing Effects

Shock	Capital locus	Consumption dynamics
Permanent government spending increase	Shifts $\dot k=0$ downward	Private consumption falls immediately
Temporary government spending increase	Transitional shift	Consumption falls, then recovers when spending ends