The Romer Model

Martin Ellison, Hilary Term 2017

2Introduction

The Romer model• Focusses on the distinction between ideas and objects• Stipulates that output requires knowledge and labour

The production function of the Romer model• Constant returns to scale in objects alone• Increasing returns to scale in objects and ideas

New ideas depend on• The existence of ideas in the previous period• The number of workers producing ideas• Worker productivity

Unregulated markets traditionally do not provide enough resources to produce ideas

• Hence they are underprovided

The population• Workers producing ideas and workers producing output

3Labour in the Romer model

Labour

• 𝐿𝐿𝑦𝑦𝑦𝑦 workers producing output

• 𝐿𝐿𝑎𝑎𝑦𝑦 workers producing ideas

• is an object so subject to a resource constraint 𝐿𝐿𝑦𝑦𝑦𝑦 + 𝐿𝐿𝑎𝑎𝑦𝑦 = �𝐿𝐿

Allocation of labour

• proportion 1 − ̅𝑙𝑙 of workers produce output

• proportion ̅𝑙𝑙 of workers produce ideas

• decided by markets in original Romer model, with patents and monopoly power

𝐿𝐿𝑦𝑦𝑦𝑦 = (1 − ̅𝑙𝑙)�𝐿𝐿𝐿𝐿𝑎𝑎𝑦𝑦= ̅𝑙𝑙 �𝐿𝐿

Endogenous variables Parameters ̅𝑙𝑙, �𝐿𝐿

4Production in the Romer model

Output

• Produced using stock of existing knowledge 𝐴𝐴𝑦𝑦 and workers 𝐿𝐿𝑦𝑦𝑦𝑦

• Constant returns to objects

• Increasing returns to objects and ideas

Ideas

• Produced using stock of existing knowledge 𝐴𝐴𝑦𝑦 and workers 𝐿𝐿𝑎𝑎𝑦𝑦

• ̅𝑧𝑧 is a productivity parameter for workers producing ideas

Standing on the shoulders of giants – Isaac Newton*

Stock of knowledge 𝐴𝐴𝑦𝑦 appears in both production of output and ideas - due to non-rivalry

𝑌𝑌𝑦𝑦 = 𝐴𝐴𝑦𝑦𝐿𝐿𝑦𝑦𝑦𝑦

Δ𝐴𝐴𝑦𝑦+1 = ̅𝑧𝑧𝐴𝐴𝑦𝑦𝐿𝐿𝑎𝑎𝑦𝑦

* Quoting Bernard de Chartres - nanos gigantum humeris insidentes

Unknowns / endogenous variables: 𝑌𝑌𝑦𝑦,𝐴𝐴𝑦𝑦, 𝐿𝐿𝑦𝑦𝑦𝑦 , 𝐿𝐿𝑎𝑎𝑦𝑦, parameters ̅𝑧𝑧, �𝐿𝐿, ̅𝑙𝑙,𝐴𝐴0

Differences between Romer and Solow models:

• Added production of ideas

• Left out capital

5Summary of the Romer model

Relationship Equation

Output production function 𝑌𝑌𝑦𝑦 = 𝐴𝐴𝑦𝑦𝐿𝐿𝑦𝑦𝑦𝑦Idea production function Δ𝐴𝐴𝑦𝑦+1 = ̅𝑧𝑧𝐴𝐴𝑦𝑦𝐿𝐿𝑎𝑎𝑦𝑦Resource constraint 𝐿𝐿𝑦𝑦𝑦𝑦 + 𝐿𝐿𝑎𝑎𝑦𝑦 = �𝐿𝐿

Allocation of labour 𝐿𝐿𝑎𝑎𝑦𝑦 = ̅𝑙𝑙 �𝐿𝐿

Solving the Romer model6

Output per person depends on the total stock of knowledge*

The growth rate of knowledge is constant

The stock of knowledge depends on its initial value and its growth rate

𝑦𝑦𝑦𝑦 ≡𝑌𝑌𝑦𝑦�𝐿𝐿

=𝐴𝐴𝑦𝑦𝐿𝐿𝑦𝑦𝑦𝑦�𝐿𝐿

= 𝐴𝐴𝑦𝑦(1 − ̅𝑙𝑙)

* Compare this to the Solow model, where output per person depends on capital per person

∆𝐴𝐴𝑦𝑦+1𝐴𝐴𝑦𝑦

= ̅𝑧𝑧𝐿𝐿𝑎𝑎𝑦𝑦 = ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿Growth rate of knowledge

All constants

Stock of knowledge

𝐴𝐴𝑦𝑦 = �̅�𝐴0 (1 + �̅�𝑔)𝑦𝑦

Growth rate of knowledge

Initial amount of knowledge

�̅�𝑔 ≡ ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿

Growth in output per person7

Output per person grows at a constant rate and is a straight line on a ratio scale

Combine 𝑦𝑦𝑦𝑦 ≡𝑌𝑌𝑡𝑡�𝐿𝐿

= 𝐴𝐴𝑦𝑦(1 − ̅𝑙𝑙) and 𝐴𝐴𝑦𝑦 = �̅�𝐴0 (1 + �̅�𝑔)𝑦𝑦

The level of output per person is now written entirely as a function of the parameters of the model

𝑦𝑦𝑦𝑦 = �̅�𝐴0 (1 − ̅𝑙𝑙)(1 + �̅�𝑔)𝑦𝑦

8Why is there growth in the Romer model?

The model produces the desired long-run growth that Solow did not

• In the Solow model, capital has diminishing returns so capital and income stop growing

The model does not have diminishing returns to ideas because they are non-rivalrous

Look at the exponents on the endogenous terms on the right hand side

• Labour and ideas have increasing returns together

• Returns to ideas are unrestricted

Balanced growth• The Romer model does not exhibit tranisition dynamics• Instead, has balanced growth path• The growth rate of all endogenous variables are constant at �̅�𝑔 = ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿

Δ𝐴𝐴𝑦𝑦+1 = ̅𝑧𝑧𝐴𝐴𝑦𝑦𝐿𝐿𝑎𝑎𝑦𝑦

9Experiments in the Romer model – an increase in �𝐿𝐿

An increase in population changes the growth rate of knowledge

𝑦𝑦𝑦𝑦 = �̅�𝐴0 (1 − ̅𝑙𝑙)(1 + �̅�𝑔)𝑦𝑦

�̅�𝑔 = ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿

An increase in population will immediately and permanently raise the growth rate of per capita output

10Experiments in the Romer model – an increase in ̅𝑙𝑙

An increase in the fraction of labour producing ideas, holding all other parameters equal, will increase the growth rate of knowledge

𝑦𝑦𝑦𝑦 = �̅�𝐴0 (1 − ̅𝑙𝑙)(1 + �̅�𝑔)𝑦𝑦

�̅�𝑔 = ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿

If more people work to produce ideas, less people produce output

• The level of output jumps down initially

• But the growth rate has increased for all future years so output per person will be higher in the long run

11Growth versus level effects

Growth effects are changes to the rate of growth of per capita output

Level effects are changes to the level of per capital output

The exponent on ideas in the production function determines the returns to ideas alone

If the exponent on ideas is not equal to 1

• The Romer model will still generate sustained growth

• Growth effects are eliminated if the exponent on ideas is less than 1, due to diminishing returns

12Combining Solow and Romer

The combined model adds capital into the Romer model production function

• The production function has constant returns to scale in objects but increasing returns in ideas and objects together 𝑌𝑌𝑦𝑦 = 𝐴𝐴𝑦𝑦𝐾𝐾𝑦𝑦

1/3𝐿𝐿𝑦𝑦𝑦𝑦2/3

• The change in the capital stock is investment minus depreciation ∆𝐾𝐾𝑦𝑦+1 = �̅�𝑠𝑌𝑌𝑦𝑦 − �̅�𝑑𝐾𝐾𝑦𝑦

• Researchers are used to produce new ideas Δ𝐴𝐴𝑦𝑦+1 = ̅𝑧𝑧𝐴𝐴𝑦𝑦𝐿𝐿𝑎𝑎𝑦𝑦

• The number of workers and researchers sum to equal the total population 𝐿𝐿𝑦𝑦𝑦𝑦 + 𝐿𝐿𝑎𝑎𝑦𝑦 = �𝐿𝐿

• A constant fraction of the population is assumed to work as researchers 𝐿𝐿𝑎𝑎𝑦𝑦= ̅𝑙𝑙 �𝐿𝐿

Combining the insights from Solow and Romer leads to a rich theory of economic growth

• The growth of world knowledge explains the underlying upward trend in incomes

• Countries may grow faster or slower than this world trend because of transition dynamics

Unknowns / endogenous variables: 𝑌𝑌𝑦𝑦,𝐾𝐾𝑦𝑦 ,𝐴𝐴𝑦𝑦, 𝐿𝐿𝑦𝑦𝑦𝑦 , 𝐿𝐿𝑎𝑎𝑦𝑦, parameters �̅�𝑠, �̅�𝑑, �𝐿𝐿, ̅𝑙𝑙,𝐴𝐴0,𝐾𝐾0

The combined model will result in

• a balanced growth path - since 𝐴𝐴𝑦𝑦 increases continually over time

• transition dynamics - due to capital accumulation

13Summary of the combined Solow-Romer model

Relationship Equation

Output production function 𝑌𝑌𝑦𝑦 = 𝐴𝐴𝑦𝑦𝐾𝐾𝑦𝑦1/3𝐿𝐿𝑦𝑦𝑦𝑦

2/3

Capital accumulation ∆𝐾𝐾𝑦𝑦+1 = �̅�𝑠𝑌𝑌𝑦𝑦 − �̅�𝑑𝐾𝐾𝑦𝑦Idea production function Δ𝐴𝐴𝑦𝑦+1 = ̅𝑧𝑧𝐴𝐴𝑦𝑦𝐿𝐿𝑎𝑎𝑦𝑦Resource constraint 𝐿𝐿𝑦𝑦𝑦𝑦 + 𝐿𝐿𝑎𝑎𝑦𝑦 = �𝐿𝐿

Allocation of labour 𝐿𝐿𝑎𝑎𝑦𝑦 = ̅𝑙𝑙 �𝐿𝐿

14Solving the combined model

Start from the output production function 𝑌𝑌𝑦𝑦 = 𝐴𝐴𝑦𝑦𝐾𝐾𝑦𝑦1/3𝐿𝐿𝑦𝑦𝑦𝑦

2/3

Apply logarithms and differentiate to give growth rates

𝑔𝑔𝑦𝑦𝑦𝑦 ≡∆𝑌𝑌𝑦𝑦+1𝑌𝑌𝑦𝑦

= 𝑔𝑔𝐴𝐴𝑦𝑦 +13𝑔𝑔𝐾𝐾𝑦𝑦 +

23𝑔𝑔𝐿𝐿𝑦𝑦𝑦𝑦

Growth rate of knowledge

Divide production function for ideas by 𝐴𝐴𝑦𝑦

𝑔𝑔𝐴𝐴𝑦𝑦 ≡∆𝐴𝐴𝑦𝑦+1𝐴𝐴𝑦𝑦

= ̅𝑧𝑧𝐿𝐿𝑎𝑎𝑦𝑦 = ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿

Growth rate of labour in output production

The number of workers in output production is a constant fraction of the

constant population

𝑔𝑔𝐿𝐿𝑦𝑦𝑦𝑦 = 0

15Growth rate of capital along the balanced growth path

Divide capital accumulation equation by 𝐾𝐾𝑦𝑦

If growth is constant then ratio 𝑌𝑌𝑡𝑡𝐾𝐾𝑡𝑡

must be constant as well, therefore

Capital and output grow at the same rate*

𝑔𝑔𝐾𝐾𝑦𝑦 ≡∆𝐾𝐾𝑦𝑦+1𝐾𝐾𝑦𝑦

= �𝑠𝑠𝑌𝑌𝑦𝑦𝐾𝐾𝑦𝑦− �𝑑𝑑

Constant along a balanced growth path

𝑌𝑌𝑦𝑦𝐾𝐾𝑦𝑦

= 𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 log 𝑌𝑌𝑦𝑦 − log(𝐾𝐾𝑦𝑦) = log(𝑐𝑐𝑐𝑐𝑐𝑐𝑠𝑠𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)1𝑌𝑌𝑦𝑦𝑑𝑑 𝑌𝑌𝑦𝑦𝑑𝑑𝑐𝑐

−1𝐾𝐾𝑦𝑦𝑑𝑑 𝐾𝐾𝑦𝑦𝑑𝑑𝑐𝑐

= 0

𝑔𝑔𝐾𝐾∗ = 𝑔𝑔𝑌𝑌∗

* The asterisk means these variables are evaluated along a balanced growth path

Put it all together

Solve for the growth rate of output

For the long-run combined model, this equation pins down the growth rate of output and the growth rate of output per person

The growth rate of output is even larger in the combined model than in the Romer model

• Output is higher in this model because ideas have a direct and indirect effect

• Increasing productivity raises output because productivity has increased and higher productivity results in a higher capital stock

16Growth rate of output

𝑔𝑔𝑦𝑦𝑦𝑦 = 𝑔𝑔𝐴𝐴𝑦𝑦 +13𝑔𝑔𝐾𝐾𝑦𝑦 +

23𝑔𝑔𝐿𝐿𝑦𝑦𝑦𝑦

𝑔𝑔𝐴𝐴𝑦𝑦 = ̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿 ≡ �̅�𝑔𝑔𝑔𝐾𝐾∗ = 𝑔𝑔𝑌𝑌∗𝑔𝑔𝐿𝐿𝑦𝑦𝑦𝑦 = 0

𝑔𝑔𝑌𝑌∗ = �̅�𝑔 +13𝑔𝑔𝑌𝑌∗

𝑔𝑔𝑌𝑌∗ =32�̅�𝑔 =

32̅𝑧𝑧 ̅𝑙𝑙 �𝐿𝐿

17Output per capita

The equation for the capital stock can be solved for the capital-output ratio along a balanced growth path

Substitute back into the production function for output and solve

• Growth in 𝐴𝐴𝑦𝑦 leads to sustained growth in output per person along a balanced growth path

• Output per capita 𝑦𝑦𝑦𝑦 depends on the square root of the investment rate

• A higher investment rate raises the level of output per person along the balanced growth path

∆𝐾𝐾𝑦𝑦+1𝐾𝐾𝑦𝑦

= �̅�𝑠𝑌𝑌𝑦𝑦𝐾𝐾𝑦𝑦− �̅�𝑑 𝑔𝑔𝐾𝐾∗ = 𝑔𝑔𝑌𝑌∗ = �̅�𝑠

𝑌𝑌𝑦𝑦∗

𝐾𝐾𝑦𝑦∗− �̅�𝑑

𝑌𝑌𝑦𝑦∗

𝐾𝐾𝑦𝑦∗=𝑔𝑔𝑌𝑌∗ + �̅�𝑑

�̅�𝑠𝐾𝐾𝑦𝑦∗

𝑌𝑌𝑦𝑦∗=

�̅�𝑠𝑔𝑔𝑌𝑌∗ + �̅�𝑑

𝑦𝑦𝑦𝑦∗ ≡𝑌𝑌𝑦𝑦∗�𝐿𝐿

= 𝐴𝐴𝑦𝑦∗𝐾𝐾𝑦𝑦∗�𝐿𝐿

1/3 𝐿𝐿𝑦𝑦𝑦𝑦∗

�𝐿𝐿

2/3

= 𝐴𝐴𝑦𝑦∗�̅�𝑠𝑦𝑦𝑦𝑦∗

𝑔𝑔𝑌𝑌∗ + �̅�𝑑

1/3

1 − ̅𝑙𝑙 2/3 𝑦𝑦𝑦𝑦∗ =�̅�𝑠

𝑔𝑔𝑌𝑌∗ + �̅�𝑑

1/2

𝐴𝐴𝑦𝑦∗3/2 1 − ̅𝑙𝑙

18Transition dynamics

The Solow model and combined model both have diminishing returns to capital

• transition dynamics applies in both models

The principle of transition dynamics for the combined model

• The father below its balanced growth path an economy is, the faster it will grow

• The father above its balanced growth path an economy is, the slower it will grow

A permanent increase in the investment rate in the combined model:

• The balanced growth path is higher (parallel shift)

• Current income is unchanged – the economy is now below the new balanced growth path

• The growth rate of income per capita is immediately higher – the slope of the output path is steeper than the balanced growth path

19Experiments in the combined model – an increase in �̅�𝑠

Next lecture20