EF5472 Chap.4 Ramsey Equilibrium and Transition Paths
Ramsey Equilibrium and Transition Paths
Select one admissible infinite path
The Ramsey model combines feasibility, Euler optimality and transversality. Local first-order conditions alone allow excessive saving paths, so the boundary condition rules out valuable capital left unused in the limit. Competitive factor pricing and household optimisation reproduce the planner path when markets are undistorted and clear.
In the phase diagram, capital is a predetermined state while consumption is a jumping control. The capital and consumption zero-motion loci divide the plane, and only the stable saddle path converges to the modified-golden-rule steady state from a given initial stock.
Read dynamics in three passes
First derive the zero-motion loci from the laws of motion.
On the capital locus, consumption uses exactly net output; above it capital falls and below it capital rises. On the vertical consumption locus, the net marginal product equals impatience; returns on either side determine whether consumption grows or falls. Second mark the inherited capital stock.
Because installed capital cannot jump, any immediate response to news occurs through consumption, which moves vertically onto the new stable arm. Third separate impact, transition and destination. Impact identifies the jump variable, transition follows the arrows and destination solves both stationary equations. A direct diagonal leap to the new steady state violates the state-control distinction.
In computations, use the stationary Euler condition to locate capital and the resource equation to recover consumption, then verify positivity and the transversality requirement. The ordinary golden rule and modified golden rule use different objectives, so name the relevant condition rather than treating their capital stocks as interchangeable.
When decentralising the allocation, state factor pricing, household optimality, market clearing and the boundary condition; matching a planner Euler equation alone leaves implementation and admissibility unproved.
What this chapter covers
- 01
Infinite-horizon feasibility
- 02
Transversality
- 03
Competitive decentralisation
- 04
Modified golden rule
- 05
Phase-diagram signs
- 06
Saddle-path dynamics
Worked example · free
Find a Ramsey steady state
- 4Apply the stationary Euler condition.
- 3Use resources to recover consumption.
- 3Verify marginal return and positivity.
Key terms
- Transversality Condition
- The limiting condition excluding paths that accumulate valuable resources without sufficient utility benefit.
- Modified Golden Rule
- The Ramsey steady-state condition equating the net marginal product of capital with impatience.
- Predetermined State
- A variable such as installed capital whose initial value is inherited and cannot jump when news arrives.
- Jumping Control
- A choice variable such as consumption that can change immediately to place the economy on an optimal path.
- Stable Arm
- The unique local trajectory converging to the saddle-point steady state.
- No-Ponzi Condition
- A financing restriction preventing debt from being rolled over forever without backing resources.
Ramsey Equilibrium and Transition Paths FAQ
Why is a transversality condition needed after the Euler equation?
Euler conditions compare neighbouring dates and therefore enforce local optimality. They do not eliminate paths that save too much forever and leave valuable assets unused. Transversality supplies the boundary discipline, selecting a globally admissible path when combined with feasibility and the maintained concavity assumptions.
Why can consumption jump while capital cannot?
Installed capital is inherited from earlier investment and takes time to change, making it a state variable. Consumption is the current allocation decision and can respond immediately to new information. After a permanent shock, consumption jumps to the new stable arm while capital adjusts gradually along it.
How does the market allocation match the planner?
Competitive firms pay marginal products, households face the corresponding gross return, asset and goods markets clear, and the household boundary condition holds. These elements make the household Euler equation and aggregate resource constraint coincide with the planner system. Matching only the Euler equations is not sufficient.
Exam move
Draw the phase diagram from equations rather than memory. Derive the capital zero-motion curve from net output and the consumption zero-motion line from the Keynes-Ramsey rule, then mark the sign of each derivative in every region. Pick an initial capital stock and test consumption guesses above and below the stable arm.
For steady-state calculations, solve the Euler condition first and the resource constraint second, then compare the modified and ordinary golden rules. Practise shocks in three sentences: what jumps on impact, which arrows govern transition and which equation determines the destination. Finally, explain transversality in value terms so you do not incorrectly claim that physical capital itself must converge to zero.
Audit every drawn arrow against the law of motion in that region before tracing the convergent path.
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