Economic growth · Lesson 9 of 14

Where saving comes from: Ramsey–Cass–Koopmans and Diamond

Solow gave the economy a saving rate and asked what follows. A fair objection: people choose how much to save. This lesson lets them. First an infinitely lived household that weighs consumption today against consumption tomorrow, which gives the Euler equation, the phase diagram and the saddle path. Then Diamond’s economy of overlapping generations, in which saving is for old age, and in which an economy can end up saving too much.

← Economic growth moduleLesson 9 of 14 · about 55 minutes

The short answer

What do the Ramsey–Cass–Koopmans and Diamond models add to the Solow model?

In the Ramsey–Cass–Koopmans model a representative, infinitely lived household chooses consumption to maximise discounted lifetime utility. The household’s optimal saving behaviour replaces Solow’s constant saving rate, and the long-run level of capital is set by preferences (patience ρ) and technology, not by an assumed s.

\frac{c ˙}{c} = \frac{f ^{'} ( k ) - δ - ρ}{θ} k ˙ = f (k) - c - (n + δ) k f^{'} (k^{*}) = δ + ρ

Saving is chosen, not assumed. A household with discount rate ρ weighs consumption now against later, and its choice is summarised by the Euler equation.
The steady state is the modified golden rule, f′(k*) = δ + ρ. Because ρ > n, capital is below the golden-rule level, and the saving rate is α(n + δ)/(δ + ρ) for Cobb–Douglas.
Only one path leads there: the saddle path. Consumption jumps to put the economy on it.
Diamond’s overlapping generations derive saving from the life cycle. Unlike Ramsey’s economy, theirs can have too much capital: dynamic inefficiency.

A question to hold on to

In the Ramsey–Cass–Koopmans model, y = k^0.5, depreciation is 5%, population growth 1% and households discount the future at 3%. Find steady-state capital per worker and the saving rate, and compare them with the golden rule.

Practice questionSolved at the end ↓

New to this? The few things you need first

The golden rule: The saving rate that maximises steady-state consumption: f′(k) = n + g + δ, with s = α for Cobb–Douglas.
The Solow model: k̇ = s f(k) − (n + δ)k, with the saving rate s given. Steady state when saving per worker equals break-even investment.
Discounting and utility: Utility u(c) rises with consumption at a diminishing rate. Utility next year counts for less than utility today by a factor e^−ρ, where ρ is the rate of time preference. Constant relative risk aversion: u(c) = c^1−θ/(1 − θ), with u = ln c when θ = 1.

The idea

From an assumed saving rate to a chosen one

Solow’s households save a fixed share of income, whatever the return on capital and whatever their wealth. Real households do not behave like that: they save more when the return is high, and they save to smooth consumption over time. The Ramsey–Cass–Koopmans model replaces the rule of thumb with a choice.

The ingredients are the Solow technology and capital accumulation, plus a household that picks the path of consumption to maximise

U = \int_{0}^{\infty} e^{- ρt} u (c_{t}) \frac{L _{t}}{L _{0}} d t u (c) = \frac{c ^{1 - θ}}{1 - θ}

where ρ is impatience and θ measures how strongly the household dislikes an uneven consumption path. It is subject to the same resource constraint as before, k̇ = f(k) − c − (n + δ)k: output not consumed becomes capital. The household has perfect foresight and takes the technology as given. Frank Ramsey solved a version of this in 1928, as a problem for a planner, and Cass and Koopmans returned to it in 1965 to study economic growth.

The key result

The Euler equation

The best way to see the household’s choice is a two-period trade. Give up one unit of consumption today and invest it. Next year you get back 1 + r units, where r = f′(k) − δ is the net return on capital. At the best plan you cannot gain by shifting: the utility lost today equals the discounted utility gained tomorrow.

u^{'} (c_{t}) = \frac{1 + r}{1 + ρ} u^{'} (c_{t + 1})

If the return r exceeds impatience ρ, the right side is bigger than the left unless c_t+1 is larger than c_t (marginal utility falls as consumption rises): saving is worthwhile, and consumption rises over time. If r < ρ, consumption falls. Taking the period to zero and using CRRA utility turns this into the Euler equation:

The Euler equation

\frac{c ˙}{c} = \frac{1}{θ} (f^{'} (k) - δ - ρ)

The growth rate of consumption is the gap between the net return and impatience, divided by θ. 1/θ is the elasticity of intertemporal substitution: how readily the household accepts a tilted consumption path in return for a higher return.

Check the logic. With θ = 1 (log utility) and a net return of 7% against impatience of 3%, consumption grows 4% a year. A household that finds uneven consumption more painful, θ = 2, lets it grow only 2%. Both are on the same two-sided logic: save if the return beats impatience. The formal derivation, by the Hamiltonian, is on its own page: the Euler equation, derived from two periods to continuous time.

Pause and predict

In the Ramsey model with CRRA utility, consumption per head is rising over time when

Putting it together

The two-equation system and the steady state

The economy is described by two equations in capital per worker k and consumption per worker c:

k ˙ = f (k) - c - (n + δ) k c ˙ = \frac{c}{θ} (f^{'} (k) - δ - ρ)

In the steady state both are zero. Setting ċ = 0 gives $f^{'} (k^{*}) = δ + ρ$ , which pins down capital, and setting k̇ = 0 gives consumption, c* = f(k*) − (n + δ)k*.

The modified golden rule

f^{'} (k^{*}) = δ + ρ ⟹ k^{*} = (\frac{α}{δ + ρ})^{\frac{1}{1 - α}}, s^{*} = \frac{( n + δ ) k ^{*}}{f ( k ^{*} )} = \frac{α ( n + δ )}{δ + ρ}

Compare the golden rule f′(k) = n + δ. Because impatience ρ exceeds n, the Ramsey economy's capital stock is smaller than the golden-rule stock, and steady-state consumption lower, but no one wants to change that.

Three things stand out. The steady-state saving rate comes out of the model, and it is below α, the golden-rule rate, because ρ > n: the household would never choose to oversave, so the Ramsey economy is always dynamically efficient. With technical progress at rate g the conditions become f′(k̂*) = δ + ρ + θg and s* = α(n + g + δ)/(δ + ρ + θg). And the intertemporal elasticity 1/θ affects how fast the economy moves but not where it ends up (without technical progress). Impatience ρ does the work: a more patient household saves more and ends up with more capital.

Practice

y = k^0.5, δ = 5%, n = 1%, ρ = 3%, no technical progress. Find k*, c* and the saving rate, and compare with the golden rule.

Modified golden rule.
$0.5 k^{- 0.5} = δ + ρ = 0.08 \Rightarrow k^{0.5} = 6.25, k^{*} = 39.06$
Output and consumption.
y* = 6.25; c* = 6.25 − (0.01 + 0.05)(39.06) = 6.25 − 2.34 = 3.91.
Saving rate.
s* = 2.34 / 6.25 = 0.375, which equals α(n + δ)/(δ + ρ) = 0.5 × 0.06 / 0.08.
Golden rule for comparison.
0.5k^−0.5 = 0.06 gives k = 69.4, c = 8.33 − 0.06(69.4) = 4.17, with saving rate 50%.

Answer

k* = 39.1, c* = 3.91, saving rate 37.5%. The golden rule would have k = 69.4, c = 4.17 and a saving rate of 50%. Impatience keeps the Ramsey economy at lower capital and a lower saving rate, and it is right to: the extra consumption is available only after waiting for the capital to accumulate.

Interactive

The phase diagram and the saddle path

Plot capital per worker k across and consumption c up. The k̇ = 0 curve is output minus break-even investment, f(k) − (n + δ)k, a hill. Below it, saving exceeds what is needed and capital rises; above it capital falls. The ċ = 0 line is vertical at k*: to its left the return is above δ + ρ and consumption rises, and to its right it falls. They cross once, at the steady state.

The arrows point toward the steady state only along one path, the saddle path. Capital is a state variable, fixed today, but consumption can jump, and the household chooses the jump that puts the economy on the saddle path. Use the last slider to move starting consumption off the path and see where the other choices go.

Households pick the one initial consumption that puts the economy on the saddle path, the only route that arrives at the steady state. Capital below k* means consumption is low and saving high; both then rise or fall toward the steady state.

Why are the other paths excluded? A household that starts with too much consumption runs capital down to zero in finite time, and cannot consume what it no longer produces. A household that starts with too little accumulates capital past the steady state and then cuts consumption for ever: it dies with unspent wealth, which is not optimal. The ruling-out of this second case is the transversality condition: the present value of capital must tend to zero, since wealth unspent at the end of time is wasted. The saddle path is the only one that satisfies both the Euler equation and the transversality condition.

Solow against Ramsey in one line

In Solow the saving rate is fixed and capital adjusts slowly. In Ramsey the household changes its saving rate along the path: when capital is scarce, saving is typically high and consumption low, and as capital builds up the saving rate moves toward its steady-state value. Both models share the same steady-state structure and the same convergence logic. What Ramsey adds is a reason for the saving rate.

A different story about saving

Diamond: overlapping generations

Ramsey’s household lives for ever, which is a convenient fiction. In Samuelson’s (1958) and Diamond’s (1965) models of overlapping generations (OLG), people live for two periods. The young work, earn a wage and save part of it. The old consume their savings and the return on them. In each period a young generation, N_t strong, coexists with an old one.

Keep it simple: log utility ln c₁ + β ln c₂, Cobb–Douglas production Y = K^αL^1−α, and capital that is fully used up in a period (δ = 1), with population growing at n per generation. Log utility makes the young save a fixed share of their wage, whatever the interest rate:

s_{w} = \frac{β}{1 + β} ⟹ (1 + n) k_{t + 1} = s_{w} (1 - α) k_{t}^{α} k^{*} = (\frac{s _{w} ( 1 - α )}{1 + n})^{\frac{1}{1 - α}}

The saving of the young this period is next period’s capital, so the path of capital is a curve in k_t, like Solow’s, and converges to a steady state k*. What is new is the question of whether k* is too high. The gross return on saving at the steady state is R* = αk*^α−1, and the golden rule requires R = 1 + n: the return should equal the growth of the population. Substituting k*:

When Diamond's economy has too much capital

R^{*} = \frac{α ( 1 + n )}{s _{w} ( 1 - α )} < 1 + n ⟺ s_{w} > \frac{α}{1 - α}

R* < 1 + n if and only if the young save more than α/(1 − α) of their wage. This can happen for patient generations (large β) when capital's share is small.

The steady state lies to the right of the golden rule: the return on saving is below the growth rate of the population, so the economy has more capital than is good for it. Each generation could be better off if some capital were eaten, a case the Ramsey model rules out and Diamond's allows.

With the default settings (α = 0.3, β = 0.9) the young save 47% of their wage, above α/(1 − α) = 43%, so the economy is dynamically inefficient: the return on saving, 1.09, is below the population growth factor, 1.2. If each generation could pass part of its saving back to the old, as a pay-as-you-go pension does, every generation could be better off. In a Ramsey economy this cannot happen: one infinitely lived household would never let it. In an OLG economy no individual can fix it, because the generations that would gain never meet.

	Solow	Ramsey–Cass–Koopmans	Diamond OLG
Saving	Constant rate s, assumed	Chosen by one infinitely lived household	Chosen by each young generation for old age
Steady state	s f(k) = (n + δ)k	f′(k) = δ + ρ (modified golden rule)	(1 + n)k = s_w(1 − α)k^α
Dynamic inefficiency	Possible, if s > α	Impossible, since ρ > n	Possible
Government debt and pensions	No role	Ricardian equivalence	Real effects: shifts burdens between generations

Solving the opening question

The opening question, solved

Practice

y = k^0.5, δ = 5%, n = 1%, ρ = 3%. Steady-state capital, saving rate, and the comparison with the golden rule?

Modified golden rule.
f′(k*) = δ + ρ = 0.08, so 0.5k^−0.5 = 0.08 and k* = 39.06.
Saving rate.
s* = α(n + δ)/(δ + ρ) = 0.5 × 0.06 / 0.08 = 37.5%.
Golden rule.
k_gold = 69.4 and s_gold = 50%.

Answer

k* = 39.1 and a saving rate of 37.5%, both below the golden rule’s 69.4 and 50%, because ρ (3%) exceeds n (1%).

Watch out for

The mistakes examiners are fishing for

Using f′(k) = n + δ in the Ramsey steady state. That is the golden rule. The Ramsey steady state is f′(k) = δ + ρ.
Thinking the saddle path is a policy. It is the household’s optimal choice. Consumption jumps at the start to land on it.
Saying a higher θ changes the steady state. Without technical progress, θ affects the speed of adjustment, not k*.
Mixing up the models on dynamic inefficiency. Possible in Solow and Diamond, impossible in Ramsey.
Forgetting ρ > n. It is needed for lifetime utility to be finite, and it is what puts k* below the golden rule.

Revise

The lesson on one screen

Ramsey–Cass–Koopmans: An infinitely lived household maximises ∫e^−ρtu(c)dt. Saving is chosen.
Euler equation: ċ/c = (f′(k) − δ − ρ)/θ. Consumption rises when the net return exceeds ρ.
System: k̇ = f(k) − c − (n + δ)k and the Euler equation. One saddle-point steady state.
Modified golden rule: f′(k*) = δ + ρ (+ θg). Capital below the golden rule; s* = α(n + δ)/(δ + ρ).
Saddle path: The one path to the steady state; consumption jumps onto it. Transversality rules out all others.
Diamond OLG: Two-period lives; k′ from saving of the young; with log utility s_w = β/(1 + β).
Dynamic inefficiency: Possible in OLG if s_w > α/(1 − α); impossible in Ramsey.

Practice

Check yourself

0 of 14 correct so far

A. Numbers

Problem 1

Ramsey model: y = k^0.5, δ = 5%, n = 1%, ρ = 3%, no technical progress. Find k*, c* (two decimals) and the saving rate (per cent).

k*c*Saving rate (%)

Problem 2

For the same economy, find the golden-rule capital per worker and the golden-rule consumption per worker.

k goldc gold

Problem 3

CRRA utility, net return on capital 7%, time preference 3%. Find the growth rate of consumption (per cent a year) when θ = 1, and when θ = 2.

θ = 1θ = 2

Problem 4

Diamond OLG with log utility: α = 0.3, β = 0.9. Find the share of the wage the young save (per cent), and the threshold α/(1 − α) above which the economy is dynamically inefficient (per cent).

Saving share (%)Threshold (%)

B. Concepts

1In the Ramsey–Cass–Koopmans model, the steady-state capital stock satisfies

2Compared with the golden rule, the Ramsey steady-state capital stock is

3Why is the saddle path the only optimal path in the Ramsey model?

4Which of these models can be dynamically inefficient?

5In the Ramsey model, a more patient household (lower ρ) in the long run has

Real questions

Past papers, by level

No past-paper question in our bank asks the Ramsey or Diamond model directly: the UGC NET and the Delhi School tend to ask about the Solow model and the golden rule, and refer to these models in the history of growth theory. The ideas of this lesson are the background for Lesson 7’s golden rule and the intertemporal choice questions in microeconomics and public finance, and they are the starting point of the research literature on growth.

All 0 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 Ramsey–Cass–Koopmans model?

It is the neoclassical growth model in which the saving rate is chosen, not assumed. A representative household that lives for ever picks its path of consumption to maximise lifetime utility, discounted at rate ρ, subject to the economy's capital accumulation equation. The result is two equations, one for capital per worker and one for consumption (the Euler equation), whose solution is a saddle path to a steady state.

What is the Euler equation in the Ramsey model?

With constant relative risk aversion utility, the Euler equation is ċ/c = (f′(k) − δ − ρ)/θ, where f′(k) − δ is the net return on capital, ρ is the rate of time preference and θ is the inverse of the elasticity of intertemporal substitution. Consumption grows when the return exceeds impatience, and the more the household dislikes uneven consumption (the larger θ), the more slowly consumption adjusts.

What is the modified golden rule?

It is the steady-state condition of the Ramsey model, f′(k*) = δ + ρ (plus θg with technical progress). It differs from the golden rule f′(k) = n + δ because households are impatient: ρ is greater than n, so the optimal capital stock is smaller than the one that maximises steady-state consumption. The Ramsey economy therefore cannot be dynamically inefficient.

What is the saddle path and why is it the only solution?

The phase diagram of capital per worker and consumption has one steady state, which is a saddle point: exactly one path leads to it. Capital is a state variable that starts at a given value, but consumption is a jump variable that the household chooses. It chooses the starting consumption that puts the economy on the stable arm. A higher starting consumption exhausts capital, a lower one accumulates capital for ever, and the transversality condition rules both out.

What is the Diamond overlapping generations model?

In Diamond's (1965) model, each generation lives for two periods, working and saving when young and consuming savings when old. Capital next period is what the young save. Unlike the Ramsey model it has many generations who never meet, and so a steady state can be dynamically inefficient: the economy can have too much capital, because no individual has the incentive to eat it.

What is the difference between the Solow, Ramsey and Diamond models?

All three have the same technology and capital accumulation. They differ in who saves and why. Solow assumes a constant saving rate. Ramsey derives saving from an infinitely lived household that maximises lifetime utility, so the saving rate can vary along the path and the long-run saving rate depends on preferences. Diamond derives saving from life-cycle choices by two-period lives, which allows dynamic inefficiency and a role for government debt and pensions.

← Lesson 8Convergence: do poor countries catch up?Lesson 10 →The AK model and learning by doing

Where saving comes from: Ramsey–Cass–Koopmans and Diamond

From an assumed saving rate to a chosen one

The Euler equation

The two-equation system and the steady state

The phase diagram and the saddle path

Diamond: overlapping generations

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

Frequently asked questions

Read next

Where saving comes from: Ramsey–Cass–Koopmans and Diamond

From an assumed saving rate to a chosen one

The Euler equation

The two-equation system and the steady state

The phase diagram and the saddle path

Diamond: overlapping generations

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

Frequently asked questions

Read next