Questions: Capital Accumulation and the Golden Rule

Question 1

An economy is operating above the Golden Rule capital stock. A policymaker proposes reducing the saving rate. What happens to consumption in both the short run and the long run?

• A. Consumption falls in the short run and falls further in the long run — reducing saving always hurts
• B. Consumption rises in the short run but falls in the long run as the capital stock erodes
• C. Consumption rises in both the short run and the long run — above the Golden Rule, less investment means more for consumption now and later
• D. Consumption is unchanged in the short run but rises in the long run once the new steady state is reached

Answer: (C)

Above the Golden Rule, the economy is dynamically inefficient — it has over-accumulated capital. Reducing the saving rate immediately frees up output for consumption (since less is diverted to investment), raising consumption in the short run. As the capital stock depreciates to its new (lower) steady-state level, output falls, but the gain from consuming a higher share of that output means steady-state consumption also rises — to a level closer to the Golden Rule maximum. This is a 'free lunch': both present and future consumption increase. It is the hallmark of dynamic inefficiency — a Pareto-improving reallocation from investment to consumption.

Question 2

The Golden Rule condition states that at the optimal capital stock, f'(k_gold) = δ. What is the economic interpretation of this condition?

• A. The marginal product of capital equals the depreciation rate, meaning one additional unit of capital produces just enough output to replace itself — all remaining output is available for consumption
• B. The savings rate equals the depreciation rate, ensuring the capital stock neither grows nor shrinks
• C. Output per worker equals the depreciation of capital per worker, meaning all output goes to replacing worn-out machines
• D. The interest rate equals the depreciation rate, satisfying the Fisher equation for capital market equilibrium

Answer: (A)

The Golden Rule maximizes steady-state consumption. Steady-state consumption per worker is c* = f(k*) − δk* (output minus depreciation investment). To maximize this with respect to k*, take the derivative and set it to zero: f'(k*) − δ = 0, so f'(k_gold) = δ. The marginal product of capital equals the depreciation rate. Intuitively: if the marginal unit of capital produces more than δ (its depreciation cost), adding more capital increases the surplus available for consumption — so you should accumulate more. If it produces less than δ, that capital costs more to maintain than it produces — so you should have less. Equality means we're at the peak.

Question 3

An economy operating below the Golden Rule capital stock is dynamically inefficient because it consumes too little and invests too much.

• T. True
• F. False

Answer: False

Dynamic inefficiency specifically describes the case of *too much* capital — operating *above* the Golden Rule. An economy below the Golden Rule is dynamically *efficient* (no waste, no Pareto-improving reallocation possible) but not at optimal consumption. Below the Golden Rule, increasing saving would raise long-run consumption — but this comes at the cost of reduced consumption during the transition. That transition cost means the policy change is not a free lunch: you sacrifice present consumption to gain future consumption, and whether this is worthwhile depends on how much society discounts the future. Above the Golden Rule, by contrast, reducing investment raises consumption immediately *and* in the long run — a genuine free lunch.

Question 4

In the Solow model, the Golden Rule capital stock is the steady state that forward-looking households will naturally achieve when they optimize their own utility.

• T. True
• F. False

Answer: False

This is the key distinction between the Solow model and optimizing models like Ramsey-Cass-Koopmans. In the Solow model, the saving rate s is an exogenous parameter — households don't optimize; they just save a fixed fraction of income. The Golden Rule requires choosing the specific s such that f'(k*) = δ, but there is no mechanism guaranteeing markets select this s. In the Ramsey model, households maximize lifetime utility with discount rate ρ, and the steady state satisfies f'(k**) = ρ + δ. If households are impatient (ρ > 0), they save less than the Golden Rule level. The Golden Rule is a welfare benchmark — the best possible steady state — not a market equilibrium.

Question 5

Why is maximizing steady-state capital per worker not the same as maximizing steady-state consumption per worker? Explain the tradeoff.

Model answer: Steady-state consumption equals output minus the investment needed to sustain the capital stock: c* = f(k*) − δk*. As capital increases, output f(k*) rises (due to diminishing returns, at a decreasing rate) but depreciation δk* rises linearly. Consumption is the gap between these two curves, which first grows then shrinks. At maximum capital (saving rate = 1), all output goes to investment and consumption is zero. The Golden Rule picks the capital stock where this gap is maximized — where the slope of the production function equals the slope of the depreciation line (f'(k) = δ). Beyond this point, additional capital raises maintenance costs faster than output, shrinking the consumption gap.

The intuition is that capital is a means to an end (consumption), not the end itself. A policy of maximizing capital accumulation — saving everything — is internally inconsistent with welfare: people accumulate capital in order to consume, so consuming nothing to maximize capital produces zero welfare. The Golden Rule identifies the sweet spot where productive capacity and consumable output are optimally balanced.