Economic growth · Lesson 7 of 14

The golden rule: how much should an economy save?

In the Solow model a higher saving rate always raises output per worker. But what people live on is consumption, and saving is what they give up to get capital. So there must be a best saving rate, and there is: the one that makes steady-state consumption as large as it can be. This lesson derives it, shows why for Cobb–Douglas it equals capital’s share, and explains the surprising possibility that an economy can save too much.

← Economic growth moduleLesson 7 of 14 · about 40 minutes

The short answer

What is the golden rule saving rate in the Solow model?

Steady-state consumption is output minus break-even investment, c* = f(k*) − (n + g + δ)k*. It rises with capital at first (more output) and then falls (more of output is needed just to keep the larger stock intact). It has a peak, and the saving rate that puts the economy at the peak is the golden rule saving rate (Phelps, 1961).

f^{'} (k_{g o l d}) = n + g + δ s_{g o l d} = \frac{k _{g o l d} f ^{'} ( k _{g o l d} )}{f ( k _{g o l d} )} = α (Cobb-Douglas)

The golden rule picks the saving rate that makes steady-state consumption per worker as large as possible.
At it, the net marginal product of capital equals the growth rate: f′(k) − δ = n + g.
For Cobb–Douglas, s = α: save exactly what capital earns. With y = k^0.5 that is 50%.
Above the golden rule the economy is dynamically inefficient; below it, reaching it costs consumption now.

A question to hold on to

For the per capita production function y = k^0.5 with depreciation 0.1 and no population growth, what saving rate maximises consumption per capita in the steady state?

CUET-PG 2023Solved at the end ↓

New to this? The few things you need first

The Solow steady state: s f(k*) = (n + δ)k*: saving per worker equals break-even investment. Consumption per worker is (1 − s) f(k*).
Marginal product and factor shares: For Cobb–Douglas f(k) = k^α, f′(k) = αk^α−1 and capital’s share is k f′(k)/f(k) = α.
Effective labour: With technical progress, k̂ = K/(AL) and the break-even rate is n + g + δ. The golden rule is the same with g added.

The idea

How much should an economy save?

Saving buys capital, and capital makes output. But saving is also consumption not enjoyed. If an economy saves nothing, capital wears out and output goes to zero, so consumption is zero. If it saves everything, there is output but nothing to consume. Somewhere in between consumption is highest.

Make this exact. In the steady state, saving equals break-even investment, so consumption is what is left of output after break-even investment is paid for. Writing the break-even rate as n + g + δ (set g = 0 if there is no technical progress):

Steady-state consumption

c^{*} = f (k^{*}) - (n + g + δ) k^{*} = (1 - s) f (k^{*})

Output minus the investment that just keeps capital per effective worker constant. A hill-shaped function of k*, and therefore of the saving rate.

Both forms are the same number. The first shows why the shape is a hill: the first term rises with k* but flattens (diminishing returns), while the second rises in a straight line, so after some k* the line wins and consumption falls. The second form shows how the saving rate comes in: a higher s lowers the consumed share (1 − s) but raises f(k*).

The derivation

Deriving the golden rule

Pick k* to make c* as large as possible. Differentiate c* = f(k*) − (n + g + δ)k* with respect to k* and set it to zero:

f^{'} (k_{g o l d}) - (n + g + δ) = 0 ⟹ f^{'} (k_{g o l d}) - δ = n + g

In words, the golden-rule capital stock is the one at which the net marginal product of capital equals the growth rate. The second derivative is f″ < 0, so this is a maximum.

Now find the saving rate that gets the economy there. At any steady state s f(k) = (n + g + δ)k, so s = (n + g + δ)k/f(k). Substitute the golden-rule condition n + g + δ = f′(k_gold):

The golden-rule saving rate

s_{g o l d} = \frac{k _{g o l d} f ^{'} ( k _{g o l d} )}{f ( k _{g o l d} )} = α for f (k) = k^{α}

The saving rate equals the share of output that goes to capital. Often put as: save what capital earns, consume what labour earns.

For Cobb–Douglas, f′(k)k/f(k) = α at every k, so the golden-rule saving rate is simply α, whatever n, g and δ are. Those parameters only change how much capital that saving rate supports:

k_{g o l d} = (\frac{α}{n + g + δ})^{\frac{1}{1 - α}}

CUET-PG 2023 · hard

y = k^0.5, depreciation 0.1, no population growth. Find the golden-rule capital per capita and saving rate.

Golden-rule condition.
$f^{'} (k) = 0.5 k^{- 0.5} = δ = 0.1$
Solve for k.
k^0.5 = 5, so k_gold = 25, and y = 5.
Saving rate.
At the steady state s·5 = 0.1 × 25 = 2.5, so s = 0.5.
Shortcut.
For y = k^α the golden-rule saving rate is α = 0.5, with no calculation.

Answer

k_gold = 25 and s_gold = 50%.

Interactive

Find the peak

The top chart plots steady-state consumption against the saving rate. Move the saving rate and watch your position relative to the peak. Change α and the break-even rate and see the peak move. The lower chart answers a practical question: if this economy moved to the golden-rule saving rate tomorrow, what would happen to consumption on the way?

This economy saves less than the golden rule. Saving more would raise steady-state consumption, but consumption falls first, so there is a real trade-off between the present and the future.

Two sides of the hill are very different. If you start above the golden rule, moving down to it raises consumption at once and for ever: there is nothing to pay. If you start below it, moving up lowers consumption first, while capital is being built, and raises it later: there is a price to pay now for the gain later.

Pause and predict

An economy with y = k^0.4 and positive depreciation is saving 60% of income in the steady state. Which is correct?

What it means

Dynamic efficiency and the cost of getting there

An economy is dynamically efficient if no way of changing its saving rate can raise consumption at every date. The golden rule divides the cases:

Saving rate compared with α	Net return f′(k) − δ	Verdict	Moving to the golden rule
s < α (below)	above n + g	Efficient, but income is lower than it could be	Costs consumption now; gains it later
s = α	equal to n + g	The golden rule: maximum steady-state consumption	Nothing to do
s > α (above)	below n + g	Dynamically inefficient	Raises consumption at every date, at no cost

Dynamic inefficiency is surprising: it means people are saving, but the saving earns less than the economy’s growth rate, so it would have been better to eat the capital. In the Solow model it can happen because the saving rate is simply assumed. It cannot happen in a model where households choose their saving, as Lesson 9 shows, except in overlapping-generations economies.

Is it a real problem? There is a simple test: at the golden rule investment equals capital income, so an economy is efficient if capital income exceeds investment. Abel, Mankiw, Summers and Zeckhauser applied this to the United States and other major OECD economies in 1989 and concluded that they appear to be dynamically efficient: they were saving less than the golden rule, not more.

Why the golden rule is not a recommendation to save more

Maximising consumption in the steady state is one objective, and not necessarily the right one. It ignores the generation that must give up consumption to build the capital, and it ignores impatience: people prefer consumption now. When households discount the future, the best long-run capital stock lies below the golden rule. That is the modified golden rule of Lesson 9.

Solving the opening question

The opening question, solved

CUET-PG 2023 · hard

y = k^0.5, δ = 0.1, no population growth. Which saving rate maximises steady-state consumption per capita?

Golden rule.
f′(k) = δ gives 0.5k^−0.5 = 0.1, so k = 25.
Saving rate from the steady state.
s√25 = 0.1 × 25, so 5s = 2.5 and s = 0.5.
Check against the shortcut.
s_gold = α = 0.5.

Answer

50%. The same exam asks for the steady-state capital when s = 0.4 (it is 16, Lesson 5); the two are different questions and a common trap.

Watch out for

The mistakes examiners are fishing for

Saying the golden rule maximises growth. It maximises steady-state consumption per worker. Growth per worker in the steady state is g whatever s is.
Using n or δ in the golden-rule saving rate for Cobb–Douglas. It is α, with no parameter but the capital exponent.
Assuming more saving is always better. Above α, more saving lowers consumption at every date.
Forgetting that reaching the golden rule from below has a cost. Consumption falls first.
Confusing the golden rule with Robinson’s golden age. One is a saving rate, the other a growth path.

Revise

The lesson on one screen

Steady-state consumption: c* = f(k*) − (n + g + δ)k* = (1 − s)f(k*). A hill in s and in k*.
Golden-rule capital: f′(k) = n + g + δ, equivalently the net return f′ − δ equals n + g.
Golden-rule saving: s = k f′(k)/f(k): capital’s share. Cobb–Douglas: s = α.
Above the golden rule: Dynamically inefficient: cut saving, raise consumption at every date.
Below the golden rule: Efficient. Reaching it costs consumption now, and pays later.
Evidence: Abel et al. (1989): the United States and other major OECD economies appear efficient.
Not the final word: Ignores transition costs and impatience; with discounting the best capital stock is below the golden rule.

Practice

Check yourself

0 of 14 correct so far

A. Numbers

Problem 1

y = k^0.5, n = 0.02, δ = 0.06, no technical progress. Find the golden-rule capital per worker and the golden-rule consumption per worker.

k goldc gold

Problem 2

y = k^0.25, n + δ = 0.1. Find the golden-rule saving rate (per cent) and the golden-rule capital per worker.

s gold (%)k gold

Problem 3

y = k^0.5, δ = 0.1, n = 0. An economy saves 80%. Find its steady-state consumption per worker and the golden-rule consumption per worker.

c at s = 0.8c gold

Problem 4

y = k^0.5, δ = 0.1, n = 0. An economy saves 20%. It raises its saving rate to the golden-rule rate. Find consumption per worker just before the change, just after it (capital has not yet moved), and in the new steady state.

BeforeJust afterNew steady state

B. Concepts

1The golden rule saving rate refers to the saving rate at which

2In the Solow model, at the golden rule the net marginal product of capital, f′(k) − δ, equals

3With y = k^α, the golden-rule saving rate is

4An economy saves more than the golden rule. A fall in its saving rate to the golden rule would

5Which statement is true of the golden age and the golden rule?

Real questions

Past papers, by level

Questions on the golden rule, from CUET-PG and UGC NET. The Delhi School’s golden-rule problems, including the economy with a production function F = L^2/3K^1/3, are in Lesson 14 when it opens.

Standard

Growth accounting, steady-state algebra, statement questions and matching, up to JNU, IIT JAM and UGC NET level.

4 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 14: The golden rule.

9 questionsLesson 14 →

All 4 past-paper questions for this lesson, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

Quick answers

Frequently asked questions

What is the golden rule saving rate?

The golden rule saving rate is the saving rate that makes steady-state consumption per worker as large as possible. Saving too little leaves too little capital to produce with; saving too much uses up output on capital that is not needed. In a Solow economy with Cobb–Douglas production the golden-rule saving rate equals capital's share of income, α.

What is the golden rule level of capital?

It is the capital per worker k_gold at which the marginal product of capital net of depreciation equals the growth rate of effective labour: f′(k_gold) − δ = n + g. Equivalently f′(k_gold) = n + g + δ. For Cobb–Douglas f(k) = k^α this gives k_gold = (α/(n + g + δ))^(1/(1 − α)).

Why does the golden rule saving rate equal capital's share in Cobb–Douglas?

At the steady state saving equals break-even investment, s·f(k) = (n + g + δ)k, so s = (n + g + δ)k/f(k). At the golden rule f′(k) = n + g + δ, so s = k f′(k)/f(k), which is the share of output paid to capital. With Cobb–Douglas that share is α. In words: the golden-rule economy saves exactly the income that capital earns.

What is dynamic inefficiency?

An economy is dynamically inefficient if it has so much capital that the net return on capital, f′(k) − δ, is below the growth rate n + g. Saving rate above the golden rule is the cause. Everyone could consume more, today and for ever, by saving less and letting capital fall to the golden-rule level, so the extra saving is pure waste.

Is the golden rule saving rate the optimal saving rate?

Not necessarily. The golden rule maximises consumption in the steady state only. It ignores the cost of getting there, since a country that saves less than the golden rule must cut consumption now to reach it, and it ignores impatience. When households discount the future, the optimal long-run capital stock is below the golden-rule level (the modified golden rule, Lesson 9).

What is the difference between the golden rule and the golden age?

They are different ideas. The golden rule (Phelps, 1961) is the saving rate that maximises steady-state consumption per head. The golden age (Joan Robinson) is a path of steady growth with full employment and a constant profit rate, in which the economy grows at the natural rate with capital fully used. Exam questions test the attribution.

← Lesson 6Solow in motion: saving, population and technology Lesson 8 →Convergence: do poor countries catch up?

The golden rule: how much should an economy save?

How much should an economy save?

Deriving the golden rule

Find the peak

Dynamic efficiency and the cost of getting there

The opening question, solved

The mistakes examiners are fishing for

The lesson on one screen

Check yourself

A. Numbers

B. Concepts

Past papers, by level

Standard

Delhi School level

Frequently asked questions

Read next

The golden rule: how much should an economy save?

How much should an economy save?

Deriving the golden rule

Find the peak

Dynamic efficiency and the cost of getting there

The opening question, solved

The mistakes examiners are fishing for

The lesson on one screen

Check yourself

A. Numbers

B. Concepts

Past papers, by level

Standard

Delhi School level

Frequently asked questions

Read next