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Royal Economic Society

Frank Ramsey’s A Mathematical Theory of Saving

Orazio Attanasio University College London, EDePo@IFS, NBER Royal Economic Society Manchester - April 2015

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Outline

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Introduction

In 1928, Ramsey published, at 25, his second paper in economics, on the theory

• f optimal saving.

Introduction

In 1928, Ramsey published, at 25, his second paper in economics, on the theory

• f optimal saving.

At the time he was the head of mathematics at Cambridge and had written important contribution in Mathematics and Logics.

Introduction

In 1928, Ramsey published, at 25, his second paper in economics, on the theory

• f optimal saving.

At the time he was the head of mathematics at Cambridge and had written important contribution in Mathematics and Logics. He was close to Keynes, Sraffa, Wittengestein.

Introduction

In 1928, Ramsey published, at 25, his second paper in economics, on the theory

• f optimal saving.

At the time he was the head of mathematics at Cambridge and had written important contribution in Mathematics and Logics. He was close to Keynes, Sraffa, Wittengestein. As a student he had worked for Pigou.

(relevant for the taxation paper)

Introduction

In 1928, Ramsey published, at 25, his second paper in economics, on the theory

• f optimal saving.

At the time he was the head of mathematics at Cambridge and had written important contribution in Mathematics and Logics. He was close to Keynes, Sraffa, Wittengestein. As a student he had worked for Pigou.

(relevant for the taxation paper)

His contributions anticipated many subsequent developments:

Optimal growth. Ramsey pricing and optimal taxation. Truth and probability: expected utility and choice under uncertainty.

Introduction

A Mathematical Theory of Saving The paper is astonishingly modern.

Introduction

A Mathematical Theory of Saving The paper is astonishingly modern. It contains the main results of optimal growth theory.

Introduction

A Mathematical Theory of Saving The paper is astonishingly modern. It contains the main results of optimal growth theory. Cass and Koopmans re-derived the same results in the 1960s.

• David Cass: “In fact I always have been kind of embarrassed because that paper is

always cited although now I think of it as an exercise, almost re-creating and going a little beyond the Ramsey model.”

Introduction

A Mathematical Theory of Saving The paper is astonishingly modern. It contains the main results of optimal growth theory. Cass and Koopmans re-derived the same results in the 1960s.

• David Cass: “In fact I always have been kind of embarrassed because that paper is

always cited although now I think of it as an exercise, almost re-creating and going a little beyond the Ramsey model.”

It anticipates many further developments:

Life cycle theory of consumption. OLG models. Theory of inequality.

Outline

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

A Mathematical Theory of Saving

The paper sets to answer the following question: How much of its income should a nation save?

A MATHEMATICAL THEORY OF SAVING

its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows. The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment. In order to justify this rule it is, of course, necessary to make various simplifying assumptions: we have to suppose that our community goes on for ever without changing either in numbers

• r in its capacity for enjoyment or in its aversion to labour; that

enjoyments and sacrifices at different times can be calculated independently and added; and that no new inventions or improve- ments in organisation are introduced save such as can be regarded as conditioned solely by the accumulation of wealth.' One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in com- parison with earlier

• nes, a practice which is ethically indefensible

and arises merely from the weakness of the imagination; we shall, however, in Section II include such a rate of discount in some of our investigations. We also ignore altogether distributional considerations, assuming, in fact, that the way in which consumption and labour are distributed between the members of the community depends solely on their total amounts, so that total satisfaction is a function of these total amounts only. Besides this, we neglect the differences between different kinds of goods and different kinds of labour, and suppose them to be expressed in terms of fixed standards, so that we can speak simply of quantities of capital, consumption and labour without discussing their particular forms. Foreign trade, borrowing and lending need not be excluded, provided we assume that foreign nations are in a stable state, so

• No. 152.-VOL. XXXVIII.

0 0

A Mathematical Theory of Saving

The paper sets to answer the following question: How much of its income should a nation save? The answer, of course, is:

A MATHEMATICAL THEORY OF SAVING

its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows. The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment. In order to justify this rule it is, of course, necessary to make various simplifying assumptions: we have to suppose that our community goes on for ever without changing either in numbers

• r in its capacity for enjoyment or in its aversion to labour; that

enjoyments and sacrifices at different times can be calculated independently and added; and that no new inventions or improve- ments in organisation are introduced save such as can be regarded as conditioned solely by the accumulation of wealth.' One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in com- parison with earlier

• nes, a practice which is ethically indefensible

and arises merely from the weakness of the imagination; we shall, however, in Section II include such a rate of discount in some of our investigations. We also ignore altogether distributional considerations, assuming, in fact, that the way in which consumption and labour are distributed between the members of the community depends solely on their total amounts, so that total satisfaction is a function of these total amounts only. Besides this, we neglect the differences between different kinds of goods and different kinds of labour, and suppose them to be expressed in terms of fixed standards, so that we can speak simply of quantities of capital, consumption and labour without discussing their particular forms. Foreign trade, borrowing and lending need not be excluded, provided we assume that foreign nations are in a stable state, so

• No. 152.-VOL. XXXVIII.

0 0

A Mathematical Theory of Saving

The paper sets to answer the following question: How much of its income should a nation save? The answer, of course, is:

A MATHEMATICAL THEORY OF SAVING

its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows. The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment. In order to justify this rule it is, of course, necessary to make various simplifying assumptions: we have to suppose that our community goes on for ever without changing either in numbers

• r in its capacity for enjoyment or in its aversion to labour; that

enjoyments and sacrifices at different times can be calculated independently and added; and that no new inventions or improve- ments in organisation are introduced save such as can be regarded as conditioned solely by the accumulation of wealth.' One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in com- parison with earlier

• nes, a practice which is ethically indefensible

and arises merely from the weakness of the imagination; we shall, however, in Section II include such a rate of discount in some of our investigations. We also ignore altogether distributional considerations, assuming, in fact, that the way in which consumption and labour are distributed between the members of the community depends solely on their total amounts, so that total satisfaction is a function of these total amounts only. Besides this, we neglect the differences between different kinds of goods and different kinds of labour, and suppose them to be expressed in terms of fixed standards, so that we can speak simply of quantities of capital, consumption and labour without discussing their particular forms. Foreign trade, borrowing and lending need not be excluded, provided we assume that foreign nations are in a stable state, so

• No. 152.-VOL. XXXVIII.

0 0

The problem is posed in a normative fashion. In the first section of the paper, Ramsey derives the result without referring to interest rates and wages.

A Mathematical Theory of Saving

The paper sets to answer the following question: How much of its income should a nation save? The answer, of course, is:

A MATHEMATICAL THEORY OF SAVING

its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows. The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment. In order to justify this rule it is, of course, necessary to make various simplifying assumptions: we have to suppose that our community goes on for ever without changing either in numbers

• r in its capacity for enjoyment or in its aversion to labour; that

enjoyments and sacrifices at different times can be calculated independently and added; and that no new inventions or improve- ments in organisation are introduced save such as can be regarded as conditioned solely by the accumulation of wealth.' One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in com- parison with earlier

• nes, a practice which is ethically indefensible

and arises merely from the weakness of the imagination; we shall, however, in Section II include such a rate of discount in some of our investigations. We also ignore altogether distributional considerations, assuming, in fact, that the way in which consumption and labour are distributed between the members of the community depends solely on their total amounts, so that total satisfaction is a function of these total amounts only. Besides this, we neglect the differences between different kinds of goods and different kinds of labour, and suppose them to be expressed in terms of fixed standards, so that we can speak simply of quantities of capital, consumption and labour without discussing their particular forms. Foreign trade, borrowing and lending need not be excluded, provided we assume that foreign nations are in a stable state, so

• No. 152.-VOL. XXXVIII.

0 0

The problem is posed in a normative fashion. In the first section of the paper, Ramsey derives the result without referring to interest rates and wages. In the third section, however, he derives the equilibrium interest rate.

Table of Contents

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Individual behaviour

Ramsey first assumes a representative consumer with an infinite lives

Both assumptions are relaxed later in the paper.

Individual behaviour

Ramsey first assumes a representative consumer with an infinite lives

Both assumptions are relaxed later in the paper.

He considers an closed economy without government, so that the budget constraint is given by: dk dt + c = f(h, k) (1)

Individual behaviour

Ramsey first assumes a representative consumer with an infinite lives

Both assumptions are relaxed later in the paper.

He considers an closed economy without government, so that the budget constraint is given by: dk dt + c = f(h, k) (1) The instantaneous felicity function is given by: U(c) − V (h)

Individual behaviour

Ramsey first assumes a representative consumer with an infinite lives

Both assumptions are relaxed later in the paper.

He considers an closed economy without government, so that the budget constraint is given by: dk dt + c = f(h, k) (1) The instantaneous felicity function is given by: U(c) − V (h) Within period optimality implies: Vh(h) Uc(c) = ∂f ∂h (2)

Individual behaviour

Intertemporal optimality is given by an Euler equation d dtUc(c(t)) = −∂f ∂k Uc(c(t)) (3)

Individual behaviour

Intertemporal optimality is given by an Euler equation d dtUc(c(t)) = −∂f ∂k Uc(c(t)) (3) As Ramsey puts it: “This equation means that Uc(c), the marginal utility of consumption, falls at a proportionate rate given by the rate of interest.”.

Individual behaviour

Intertemporal optimality is given by an Euler equation d dtUc(c(t)) = −∂f ∂k Uc(c(t)) (3) As Ramsey puts it: “This equation means that Uc(c), the marginal utility of consumption, falls at a proportionate rate given by the rate of interest.”. Given equations (1) - (3), Ramsey derives by simple calculus the main result in the paper: dk dt = f(h, k) − c = B − (U(c) − V (h)) Uc(c) (4)

Individual behaviour

Intertemporal optimality is given by an Euler equation d dtUc(c(t)) = −∂f ∂k Uc(c(t)) (3) As Ramsey puts it: “This equation means that Uc(c), the marginal utility of consumption, falls at a proportionate rate given by the rate of interest.”. Given equations (1) - (3), Ramsey derives by simple calculus the main result in the paper: dk dt = f(h, k) − c = B − (U(c) − V (h)) Uc(c) (4)

• r, in words: [The] “rate of saving multiplied by marginal utility of consumption

should always equal bliss minus actual rate of utility enjoyed ”.

Table of Contents

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Factor Prices and Extensions

In the second section, Ramsey starts with the assumption that factor prices, are constants w and r: Then: f(h, k) = wh + rk (5)

Factor Prices and Extensions

In the second section, Ramsey starts with the assumption that factor prices, are constants w and r: Then: f(h, k) = wh + rk (5) Given this, he:

develops a useful graphical analysis;

Factor Prices and Extensions

In the second section, Ramsey starts with the assumption that factor prices, are constants w and r: Then: f(h, k) = wh + rk (5) Given this, he:

develops a useful graphical analysis;

studies finite lives: dk dt = f(h, k) − c = Q − (U(c) − V (h)) Uc(c) (6) where Q ≤ B depends on the parameters of the model, including the importance

• f the bequest motive.

Factor Prices and Extensions

In the second section, Ramsey starts with the assumption that factor prices, are constants w and r: Then: f(h, k) = wh + rk (5) Given this, he:

develops a useful graphical analysis;

studies finite lives: dk dt = f(h, k) − c = Q − (U(c) − V (h)) Uc(c) (6) where Q ≤ B depends on the parameters of the model, including the importance

• f the bequest motive.

studies discounting: d dt Uc(c(t)) = − ∂f ∂k − ρ

• Uc(c(t)) = −{r − ρ}Uc(c(t))

(7) dk dt = rk − y = Q − y

ω(y)

(8) where y = c − wh and ω(y) = Uc(c) − Vh(h)

Table of Contents

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Equilibrium Factor Prices

In Section 3, Ramsey derives equilibrium interest rates.

Equilibrium Factor Prices

In Section 3, Ramsey derives equilibrium interest rates. He first considers steady states.

The three equations that determine the equilibrium levels of c, h and k are: c = f(h, k) Vh(h) = ∂f ∂hUc(c) ∂f ∂k = ρ

Equilibrium Factor Prices

In Section 3, Ramsey derives equilibrium interest rates. He first considers steady states.

The three equations that determine the equilibrium levels of c, h and k are: c = f(h, k) Vh(h) = ∂f ∂hUc(c) ∂f ∂k = ρ He stresses that the third equation is a long run relationship that links the discount factor to the interest rate and might never be reached. In the short run the interest rate is determined by the marginal product of capital. that interest rate will then generate a supply of capital, which in turn will determine a new level for the interest rate.

Equilibrium Factor Prices

Ramsey then discusses the determination of equilibrium interest rates in a model with finite lives. He uses a sophisticated combination of life cycle analysis and an equilibrium framework that anticipates the OLG model of Samuelson (1958) and Diamond (1965).

• f the rate of interest would be to ensure

• f income which should be saved.

• f

• f such children as are necessary to maintain the population, but

• wn lifetime he has a constant utility schedule for consumption

• way. Suppose that the rate of interest is constant and equal to

• f the rate of interest would be to ensure

• f income which should be saved.

• f

• f such children as are necessary to maintain the population, but

• wn lifetime he has a constant utility schedule for consumption

• way. Suppose that the rate of interest is constant and equal to

• now. If we neglect variations in his earning power, his action

• price. This supply price depends on people's rates of discount

Equilibrium Factor Prices

Ramsey then discusses the determination of equilibrium interest rates in a model with finite lives. He uses a sophisticated combination of life cycle analysis and an equilibrium framework that anticipates the OLG model of Samuelson (1958) and Diamond (1965).

• f the rate of interest would be to ensure

• f income which should be saved.

• f

• f such children as are necessary to maintain the population, but

• wn lifetime he has a constant utility schedule for consumption

• way. Suppose that the rate of interest is constant and equal to

• f the rate of interest would be to ensure

• f income which should be saved.

• f

• f such children as are necessary to maintain the population, but

• wn lifetime he has a constant utility schedule for consumption

• way. Suppose that the rate of interest is constant and equal to

• now. If we neglect variations in his earning power, his action

• price. This supply price depends on people's rates of discount

Table of Contents

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Inequality

In the last section of the paper, Ramsey attempts to generate inequality in his model. 1928]

A MATHEMATICAL THEORY OF SAVING

559

Now suppose equilibrium 1 is obtained with capital c, income f(c) and rate of interest f'(c) or r. Then those families, say m(r) in number, whose rate of discount is less than r must have attained bliss or they would still be increasing their expenditure according to equation (9a). Consequently they have between them an income m(r) . x1. The other families, n -

m(r) in number (where

n is the total number

• f families), must be down

to the subsistence level, or they would still be decreasing their expenditure. Con- sequently they have between them a total income {n -M(r)}X2 whence f(c) = m(r)x1 + n - m(r)}x2

n.

n. + m(r){x,

X

which, together with r = f'(c), determines r an;d

• c. m(r)

being an increasing function of r, it is easy to see, by drawing gaphs of r against f(c), that the two equations have in general a unique

solution.2

In such a case, therefore, equilibri wvould be attained by a division of society into two classes, the thrifty enjoying bliss and the improvident at the subsistence level.

• F. P. RAMSEY

King'8 College, Cambridge.

1 We suppose

eaoh family in equflibrium, which is the only way in which that state could be maintained, since otherwise, although the savings of some might at any moment balnce the borrog

• f others, they would not continue to do

so except by an extraordinary accident.

2 We have neglected in this the negligible number

• f families for which p is

exactly equal to r.

• No. 152.-VOL.

x2XVnI. PP

Inequality

In the last section of the paper, Ramsey attempts to generate inequality in his model. He does so by introducing heterogeneity in discount factors. 1928]

A MATHEMATICAL THEORY OF SAVING

559

Now suppose equilibrium 1 is obtained with capital c, income f(c) and rate of interest f'(c) or r. Then those families, say m(r) in number, whose rate of discount is less than r must have attained bliss or they would still be increasing their expenditure according to equation (9a). Consequently they have between them an income m(r) . x1. The other families, n -

m(r) in number (where

n is the total number

• f families), must be down

to the subsistence level, or they would still be decreasing their expenditure. Con- sequently they have between them a total income {n -M(r)}X2 whence f(c) = m(r)x1 + n - m(r)}x2

n.

n. + m(r){x,

X

which, together with r = f'(c), determines r an;d

• c. m(r)

being an increasing function of r, it is easy to see, by drawing gaphs of r against f(c), that the two equations have in general a unique

solution.2

In such a case, therefore, equilibri wvould be attained by a division of society into two classes, the thrifty enjoying bliss and the improvident at the subsistence level.

• F. P. RAMSEY

King'8 College, Cambridge.

1 We suppose

eaoh family in equflibrium, which is the only way in which that state could be maintained, since otherwise, although the savings of some might at any moment balnce the borrog

• f others, they would not continue to do

so except by an extraordinary accident.

2 We have neglected in this the negligible number

• f families for which p is

exactly equal to r.

• No. 152.-VOL.

x2XVnI. PP

Inequality

In the last section of the paper, Ramsey attempts to generate inequality in his model. He does so by introducing heterogeneity in discount factors. 1928]

A MATHEMATICAL THEORY OF SAVING

559

Now suppose equilibrium 1 is obtained with capital c, income f(c) and rate of interest f'(c) or r. Then those families, say m(r) in number, whose rate of discount is less than r must have attained bliss or they would still be increasing their expenditure according to equation (9a). Consequently they have between them an income m(r) . x1. The other families, n -

m(r) in number (where

n is the total number

• f families), must be down

to the subsistence level, or they would still be decreasing their expenditure. Con- sequently they have between them a total income {n -M(r)}X2 whence f(c) = m(r)x1 + n - m(r)}x2

n.

n. + m(r){x,

X

which, together with r = f'(c), determines r an;d

• c. m(r)

being an increasing function of r, it is easy to see, by drawing gaphs of r against f(c), that the two equations have in general a unique

solution.2

In such a case, therefore, equilibri wvould be attained by a division of society into two classes, the thrifty enjoying bliss and the improvident at the subsistence level.

• F. P. RAMSEY

King'8 College, Cambridge.

1 We suppose

eaoh family in equflibrium, which is the only way in which that state could be maintained, since otherwise, although the savings of some might at any moment balnce the borrog

• f others, they would not continue to do

so except by an extraordinary accident.

2 We have neglected in this the negligible number

• f families for which p is

exactly equal to r.

• No. 152.-VOL.

x2XVnI. PP

Outline

• 1. Introduction
• 2. A Mathematical Theory of Saving

2.1 Individual behaviour 2.2 Factor Prices and Extensions 2.3 Equilibrium Factor Prices 2.4 Inequality

• 3. Influences and Anticipations

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature.

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature. Cass (1965) Koopmans (1966).

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature. Cass (1965) Koopmans (1966). Consumption theory: life cycle and permanent income model.

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature. Cass (1965) Koopmans (1966). Consumption theory: life cycle and permanent income model. OLG models

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature. Cass (1965) Koopmans (1966). Consumption theory: life cycle and permanent income model. OLG models Inequality:

Accumulation is similar to Solow (1956) or Piketty (2014). However: saving is endogenous in Ramsey’s model.

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature. Cass (1965) Koopmans (1966). Consumption theory: life cycle and permanent income model. OLG models Inequality:

Accumulation is similar to Solow (1956) or Piketty (2014). However: saving is endogenous in Ramsey’s model. Inequality is introduced through differences in patience.

Individual behaviour

As mentioned the paper anticipates many strands of the subsequent literature. Cass (1965) Koopmans (1966). Consumption theory: life cycle and permanent income model. OLG models Inequality:

Accumulation is similar to Solow (1956) or Piketty (2014). However: saving is endogenous in Ramsey’s model. Inequality is introduced through differences in patience. The same considerations were make 70 years later in Krusell and Smith (1998)

Royal Economic Society

The Directions of Growth Theory

Timo Boppart IIES, Stockholm University

RES Manchester, special session on Ramsey and Harrod, March 30, 2015

Ramsey and Harrod on growth

• F. P. Ramsey,“A Mathematical Theory of Saving,” Economic

Journal, 38(152).

• R. F. Harrod, “An Essay in Dynamic Theory,” Economic

Journal, 49(193). I briefly talk about the Ramsey (1928) and the Harrod (1939) paper and the impressions I got while rereading them. The aim of the presentation is to show that both Ramsey and Harrod were true pioneers and their contributions still live on.

Ramsey: “A Mathematical Theory of Saving,” Economic Journal, 1928.

This paper is really modern in style! Ramsey states a concrete research question: “how much of its income should a nation save?” Important assumptions are: (i) no population growth, (ii) no time discounting, and (iii) a given “maximum obtainable rate

• f enjoyment” called bliss B (because capital or consumption

saturates). The optimization problem Ramsey proposes is: min ∞ B − [U(C(t)) − V (L(t))] dt (1) subject to ˙ K(t) = F [K(t), L(t)] − C(t). (2)

Solution of the Ramsey problem

An economic argument that marginal products should equalize, leads to the Euler equation ˙ UC(C(t)) UC(C(t)) = FK [K(t), L(t)] , (3) and the labor/leisure trade-off FL [K(t), L(t)] UC(C(t)) = VL(L(t)) (4) Then, the solution is given by those optimality conditions, the resource constraint, a given K(0), and a terminal condition as t → ∞.

Optimal saving rule

Moreover, by substituting dt =

dK(t) F[K(t),L(t)]−C(t) in the

• bjective, the problem can be rewritten as

min

{C(t),L(t)}∞

∞

K(0)

B − [U(C(t)) − V (L(t))] F [K(t), L(t)] − C(t) dK(t). (5) The first-order condition gives the Keynes-Ramsey rule: [F [K(t), L(t)] − C(t)]·UC(C(t)) = B −[U(C(t)) − V (L(t))] , (6)

• r in words: “saving multiplied by marginal utility of

consumption should always equal bliss minus actual rate of utility enjoyed.”

Discussion: Ramsey paper

The paper concludes with a remarkable discussion of the assumptions made and further extensions like:

Back-of-the-envelope calculation (leading to the quantitative statement that the saving rate should be 60 percent). Life-cycle saving, Time discounting and time consistency, Heterogeneity in discount rates and its effect on wealth distribution, Assumption of a single production sector.

Harrod: “An Essay in Dynamic Theory,” Economic Journal, 1939.

The paper constitutes one of the starting points of a dynamic macroeconomic theory. The main new element is the warranted rate of growth, Gw, given by Cr · Gw · Y (t) = s · Y (t), (7) where s is the saving rate and Cr is the capital coefficient. The warranted rate is “that rate of growth which, if it occurs, will leave all parties satisfied that they have produced neither more nor less than the right amount.” The warranted growth path is unstable because of the acceleration principle.

Discussion: Harrod paper

An additional element is the natural rate of growth determined by population growth and technical improvements. Contrasting the warranted and the natural rate of growth leads to the knife-edge condition for stable growth as in the Harrod-Domar model. The (1939) paper is just one of the starting points and Harrod continued to contribute to growth theory until the 60s. The definition of labor augmenting technical change (1948) “which, at a constant rate of interest, does not disturb the value of the capital coefficient”, serves as a good example.

Conclusions

The work of Ramsey and Harrod were both pathbreaking! The gist of their contributions lives on and still shapes the current research on growth (far from just my own research).

Conclusions

The work of Ramsey and Harrod were both pathbreaking! The gist of their contributions lives on and still shapes the current research on growth (far from just my own research). “It is almost certainly because of Harrod’s rediscovery of growth theory in the 1930s and his notable contributions to it that Assar Lindbeck, the Chairman of the Nobel Prize Committee, chose to state that he was among those who would have been awarded a Nobel Prize in economics if he had lived a little longer.” The New Palgrave Dictionary of Economics.

Royal Economic Society

Growth and Ideas

Chad Jones Stanford GSB

Royal Economic Society Meetings – March 30, 2015

Growth and Ideas, RES 2015 – p. 1

Overview

• Reflections on growth theory
• Lessons inspired by Harrod / Domar?
• Double meaning of the title “growth and ideas”
• The ideas of growth theorists
• The crucial role of the nonrivalry of ideas
• Are ideas getting harder to find?

Growth and Ideas, RES 2015 – p. 2

Harrod / Domar / Von Neumann

• Conclusion of any growth theory:

˙ yt = gyt and a story about g

• Key to this result is (essentially) a linear differential equation

somewhere in the model: ˙ Xt = Xt

• Growth models differ according to what they call the Xt

variable and how they fill in the blank.

Growth and Ideas, RES 2015 – p. 3

Catalog of Growth Models: What is Xt?

Solow

˙ kt = skα

t

Solow

˙ At = ¯ gAt

AK model

˙ Kt = sAKt

Lucas

˙ ht = (1 − u)ht

Romer/AH/GH

˙ At = νLatAt

Jones/Kortum/S

˙ Lt = nLt

Growth and Ideas, RES 2015 – p. 4

The Linearity Critique

˙ Xt = sXφ

t

• To explain the U.S. 20th century, φ ≈ 1 is required
• φ < 1: Growth slows to zero
• φ > 1: Growth will explode
• Solow (1994 JEP) criticizes new growth theory for this: “You

would have to believe in the tooth fairy to expect that kind of luck.”

• But the same criticism applies to ˙

At = gAt

• Facts ⇒ we need linearity somewhere. Where?

Growth and Ideas, RES 2015 – p. 5

• Any successful growth model requires a knife-edge linearity
• One way of distinguishing among growth theories
• Why is the differential equation for Xt precisely linear?
• In nearly all cases, there is no explanation provided
• Rather, linearity is assumed because it is needed.
• Next, I describe an exception!

Growth and Ideas, RES 2015 – p. 6

Nonrivalry, Increasing Returns, and Population Growth

• Romer (1990):
• Ideas are nonrivalrous
• That gives rise to increasing returns to scale
• IRS means that “bigger = more productive”
• Population growth makes the economy bigger and more

productive

• Can sustain growth

The nonrivalry of ideas and population growth together explain exponential growth in incomes

Growth and Ideas, RES 2015 – p. 7

The Simplest Model

Production of final good

Yt = Aσ

t LY t Each researcher makes α ideas

˙ At = αLAt

Resource constraint

LY t + LAt = Lt = L0ent

Allocation of labor

LAt = ¯ sLt, 0 < ¯ s < 1

Growth and Ideas, RES 2015 – p. 8

Solving

Income depends on stock of ideas

yt ≡ Yt

Lt = Aσ t (1 − ¯

s)

Growth of income

gy = σgA

Growth of ideas

gA = n

gy = σn

• Growth rate is the product of (1) degree of increasing

returns σ and (2) rate at which scale is rising n

• More people ⇒ more ideas ⇒ more income per capita.

Growth and Ideas, RES 2015 – p. 9

Response to Linearity Critique

• No linearity beyond population growth is needed
• Population growth is a historical fact.
• If we take it as given, then growth in per capita income is

not surprising

• Pushing: why is the population growth equation linear?
• Example: economy of N identical agents
• Each has n kids
• Then the law of motion for population is ˙

Nt = nNt People reproduce in proportion to their number. Linearity comes directly from aggregation!

Growth and Ideas, RES 2015 – p. 10

Are ideas getting harder to find?

Growth and Ideas, RES 2015 – p. 11

Setup

• Let’s define ideas to be proportional improvements (e.g.

Kortum, Aghion-Howitt)

Romer/AH/GH benchmark ˙ At At = αSt Jones/Kortum/Segerstrom ˙ At At = αStA−β t

β > 0 means ideas are getting harder to find

• Benchmark case of β = 0 is assumed in many recent

models: Acemoglu, Akcigit, Bloom, and Kerr (2014), Aghion, Akcigit, Howitt (2013), Acemoglu, Aghion, Bursztyn, Hemous (2012)

Growth and Ideas, RES 2015 – p. 12

Measuring Productivity in the Idea Production Function

• Idea productivity is

Idea TFP= ˙ At/At St = Proportional growth

# of Researchers

Null hypothesis: Idea TFP = α (constant) Alternative: if ideas are getting harder to find, then

Idea TFP will be declining...

Growth and Ideas, RES 2015 – p. 13

Micro Data Possibilities (in progress)

• Moore’s Law vs Intel’s research
• Pharmaceutical innovations vs pharma research
• Price of telephone call vs AT&T research, 1930–1980
• Apple profits, sales, productivity vs R&D since 2000
• Patents versus firm research with micro data
• TFP growth versus firm research with micro data
• As many other things as possible
• Sequencing DNA in the Human Genome Project?
• Hard disks, memory chips, battery technology
• Specific micro ideas where output and input are readily

measured

Growth and Ideas, RES 2015 – p. 14

Moore’s Law – Steady exponential growth (Wikipedia)

Growth and Ideas, RES 2015 – p. 15

Idea TFP for Moore’s Law

1990 1995 2000 2005 2010 2015 1/8 1/4 1/2 1

YEAR PRODUCTIVITY INDEX FOR MOORES LAW

Growth and Ideas, RES 2015 – p. 16

Annual Flow of New Molecular Entities

1980 1985 1990 1995 2000 2005 2010 10 15 20 25 30 35 40 45 50 55

YEAR NEW MOLECULAR ENTITIES (NUMBER)

Growth and Ideas, RES 2015 – p. 17

Idea TFP for NMEs (pharmaceuticals)

1980 1985 1990 1995 2000 2005 2010 1/8 1/4 1/2 1

YEAR PRODUCTIVITY INDEX FOR NMES

Growth and Ideas, RES 2015 – p. 18

How this supports Romer, not detracts...

• This might wrongly be seen as a criticism of Romer (1990)
• Instead — highlights Romer’s key insight: nonrivalry
• Why? Consider the R&D model used by Akcigit, Celik, and

Greenwood (2014) “Buy, Keep, or Sell...” Y = AσKθL1−σ−θ

constant returns

˙ At At = αS

• Ideas are fully rivalrous here, just like capital!
• Innovations occur in a perfectly competitive model

Growth and Ideas, RES 2015 – p. 19

• The ACG model generates endogenous growth even

though ideas are just another rivalrous input!

• If you are willing to put linearity in the idea PF

⇒ no need for nonrivalry, increasing returns, imperfect competition, patents, and so on!

• Rejecting this idea production function

⇒ nonrivalry has a key role to play

Growth and Ideas, RES 2015 – p. 20

Why growth?

• If (proportional) ideas are getting harder and harder to find,

then the idea production function essentially looks like: ˙ At At = TFPt · St

• If TFP is falling, the only way to sustain constant

exponential growth is with exponential growth in scientists (Jones 95, Kortum 97, and Segerstrom 98) Growing resources devoted to R&D offsets rising difficulty of discovering new ideas

Growth and Ideas, RES 2015 – p. 21

Conclusion

Many good ideas ⇒ growing field of economic growth!

Growth and Ideas, RES 2015 – p. 22

Royal Economic Society