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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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