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The Basic Solow Model Notes

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The Basic Solow Model
20 October 2010

Topics Reading Key Points
• The basic Solow model: • Williamson (Chapter 6, p207-224) • Basic Solow model
• The Solow model with technological progress: • Weil (Appendix of Chapter 8)
• Cobb-Douglas production function
• Competitive equilibrium in the Solow model
The Basic Solow Model • Plotting the steady state
• The Solow model is the basis for the modern theory of economic growth • Summary of steady state
• It is based around capital accumulation • Multiple steady states
• A key prediction is that technological advancement is necessary for sustained • Stability of steady state
increases in living standards
• Comparing countries
• Population is assumed to grow at a constant rate :
○ • Conditional convergence and absolute convergence
• Consumers are assumed to save a constant fraction of their income, consuming • The golden rule
the rest: • Effects of changes in savings rate

• Effects of changes in population growth rate
• Representative firm's production function is neoclassical:
g • Effects of technological progress

• We drop land from the model, so that output depends on capital and labour • Solow model with technological progress
• Capital accumulation equation: • Labour augmenting technological progress

• Dynamics of capital per effective worker
• Key assumption in the Solow model
• Investment becomes future capital which becomes future investment etc. • Steady state with technological progress
• The constant-returns-to-scale (CRS) property allows us to focus on output per • Summary of Solow model with technological progress
worker and capital per worker: • Kaldor stylized facts

 Where Formulae
○ This then produces a 2-dimensional graph rather than a 3-dimensional •

The Cobb-Douglas Production Function •
○ The Cobb-Douglas production function is the most widely used production •
function i.e. f in the Solow model
○ It has constant returns to scale: •
• We assume that firms are competitive

○ The coefficient is the capital share (the share of income that capital

receives)
○ •

• Using the Cobb-Douglas production function: •
○ •




Competitive Equilibrium •
○ The income-expenditure identity holds as an equilibrium condition: •

 We've simplified by saying there is no G or NX •
○ Consumer's budget constraint: •

○ Because of this, in equilibrium:

○ The capital accumulation equation becomes:

 Which is the key of the Solow model
○ We then transform the capital accumulation equation into per-worker form
by putting in the production function:


• This is a dynamic equation showing that future capital per worker is
related to current future capital per worker and saving

Plotting the Steady State for Capital Per Worker
○ We plot current capital per worker against future capital per worker
○ Graph showing steady state for capital per worker
○ The blue line tells us how k' depends on k
○ Again the steady state is when the graph meets the 45 degree line
○ We can then solve for k* because of the facts of steady state:

 →
 →
○ Graph demonstrating solving for steady state as the intersection of the
'savings curve' and the 'effective depreciation curve'

Summary of Steady State
Growth rates are as follows:

Course Notes Page 11

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