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Lecture 8
• Solow growth model • Growth accounting
Solow Growth Model
• This is a key model which is the basis for the modern theory of economic growth.
• A key prediction is that technological progress is necessary for sustained increases in standards of living.
Population growth
• In the Solow growth model, population is assumed to grow at a constant rate n.
• Discussion: How is this different from the Malthusian view of population growth? What’s the significance?
Representative Consumer
• Consumers are assumed to save a constant fraction s of their income, consuming the rest.
• The consumer has one unit of time available, inelastically supplying one unit of time as labor
Representative Firm
The firm produces using as inputs capital and labor:
Representative Firm
Constant returns to scale implies: Output per worker depends on capital per worker!
( , )Y zF K NN N
The Per-Worker Production Function
( )y zf k
Evolution of the capital stock
Future capital equals the capital remaining after depreciation, plus current investment.
Income-Expenditure Identity
The income expenditure identity holds as an equilibrium condition.
Future capital equals total savings plus what remains of
current K.
1 ' ? 1Y s Y K d K
Equilibrium capital
Substitute for output from the production function.
Capital per worker
Rewrite in per-worker form. ' ,1 1K K KszF d
N N N
' ' ,1 1'
K N K KszF dN N N N
Capital per worker (cont’d)
Re-arrange, to get: We can now use this condition to determine the
steady state of the model, where k’=k=k*
Steady State Capital per Worker
• k* is the steady state population, determined by the intersection of the curve and the 45 degree line.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-19
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-20
Now let’s explore how the steady state changes with changes in the savings rate
Steady state analysis
Equation determining the steady state quantity of capital per worker, k*:
set k’ = k = k*
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-24
Determination of the Steady State Quantity of Capital per Worker
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-25
An increase in the savings rate, s
• In the steady state, this increases capital per worker and real output per capita.
• In the steady state, there is no effect on the growth rates of aggregate variables.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-26
Effect of an Increase in the Savings Rate on the Steady State Quantity of Capital per Worker
Let’s study the transition the to new steady state as the savings rate increases
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-31
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-32
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-33
Effect of an Increase in the Savings Rate at Time T
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-35
Steady State Consumption per Worker
• Consumption per worker in the steady state is AB (output minus savings), given capital per worker k*
The Golden Rule Quantity of Capital per Worker
• The golden rule savings rate sgr is where
• Note that at the golden rule allocation, consumption is maximized
* )'(zf k n d= +
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-38
An increase in the population growth rate, n
• Capital per worker and output per worker decrease.
• There is no effect on the growth rates of aggregate variables.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-39
Steady State Effects of an Increase in the Labor Force Growth Rate
• An increase in the labor force growth rate from n1 to n2 decreases the steady state capital per worker
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-40
Increases in Total Factor Productivity, z
Sustained increases in z cause sustained increases in per capita income.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-41
Increases in Total Factor Productivity in the Solow Growth Model
• Increases in total factor productivity cause increase in the quantity of capital per worker, and thus output per worker
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