In the Ramsey-Cass-Koopmans model, an economy has a capital stock below its steady-state level, and the current marginal product of capital exceeds the household's discount rate. What does the Euler equation predict about household behavior?
AHouseholds reduce consumption now to accumulate capital, since the return to saving exceeds their impatience
BHouseholds immediately maximize consumption because they are currently below the steady state and therefore poor
CHousehold consumption remains constant because the Euler equation only applies at the steady state
DHouseholds increase consumption to stimulate demand and move the economy toward the steady state
The Euler equation (Keynes-Ramsey rule) says consumption growth is positive when the marginal product of capital exceeds the discount rate — when the return to saving outweighs impatience. Below the steady state, capital is scarce and productive, so the return to saving is high. Rational households defer consumption to exploit this arbitrage. This is the key difference from the Solow model: savings behavior here responds to the return on capital, not a fixed exogenous rate.
Question 2 Multiple Choice
What ensures that the Ramsey-Cass-Koopmans model produces a unique equilibrium path rather than many possible trajectories?
AThe government enforces the optimal consumption path through fiscal policy
BThere are multiple saddle paths, and households randomize among them
CThe saddle path is the unique trajectory converging to the steady state; rational forward-looking households select it because all other paths diverge
DAll trajectories eventually converge to the steady state, so the starting consumption level is irrelevant
In the phase diagram, most trajectories either overshoot (consumption collapses, capital explodes) or undershoot (capital collapses) and never reach the steady state. Only the saddle path converges. Rational households, knowing the long-run outcome of each trajectory, select the saddle-path consumption level for their current capital stock. This saddle-path selection is what makes the model determinate — there is exactly one optimal consumption level given any initial capital stock.
Question 3 True / False
In the Ramsey-Cass-Koopmans model, a permanent increase in government spending raises the long-run capital stock by stimulating aggregate investment.
TTrue
FFalse
Answer: False
A permanent increase in government spending reduces household lifetime wealth (households anticipate higher future taxes). To smooth consumption optimally, they reduce saving — which crowds out capital accumulation. The long-run capital stock falls, not rises. This is the opposite of what a naive Keynesian multiplier intuition might suggest. The RCK model produces this result because savings behavior responds endogenously to wealth and the return on capital.
Question 4 True / False
Unlike the Solow model, the Ramsey-Cass-Koopmans model rules out dynamic inefficiency — capital over-accumulation beyond the golden rule — because optimizing households would never save so much that the return on capital falls below their discount rate.
TTrue
FFalse
Answer: True
In the Solow model, a fixed savings rate could push capital past the golden rule, where the net marginal product of capital falls below zero — the economy saves more than needed just to maintain the capital stock. In the RCK model, if the return on capital fell below the discount rate, rational households would consume more and save less, preventing over-accumulation. Dynamic inefficiency is impossible by construction, which is one of the model's key results.
Question 5 Short Answer
What does the Euler equation in the Ramsey-Cass-Koopmans model tell households about the timing of their consumption, and what economic intuition underlies it?
Think about your answer, then reveal below.
Model answer: The Euler equation says that consumption grows at a rate proportional to the gap between the marginal product of capital (net of depreciation) and the household's discount rate. If saving yields more than the household discounts the future, it is optimal to defer consumption now and consume more later. If the return falls below the discount rate, consume more now. The intuition is intertemporal arbitrage: households equate the marginal utility cost of sacrificing consumption today with the marginal utility benefit of the extra consumption tomorrow that saving enables.
This is the continuous-time analogue of the standard consumer-theory result that a forward-looking agent equates marginal utility across time periods, adjusted for the discount rate and return on saving. In the Solow model this optimization is bypassed by assuming a fixed saving rate; the RCK model makes it explicit, allowing savings to respond to policy changes in a way the Solow model cannot capture.