The Basic Solow Model
The Basic Solow Model 20 October 2010
Topics • The basic Solow model: • The Solow model with technological progress:
Reading • Williamson (Chapter 6, p207-224) • Weil (Appendix of Chapter 8)
Key Points • Basic Solow model
• Cobb-Douglas production function • Competitive equilibrium in the Solow model
The Basic Solow Model • The Solow model is the basis for the modern theory of economic growth • It is based around capital accumulation • A key prediction is that technological advancement is necessary for sustained increases in living standards • Population is assumed to grow at a constant rate : ○ • Consumers are assumed to save a constant fraction of their income, consuming the rest: ○ • Representative firm's production function is neoclassical: ○ • We drop land from the model, so that output depends on capital and labour • Capital accumulation equation: ○ • Key assumption in the Solow model • Investment becomes future capital which becomes future investment etc. • The constant-returns-to-scale (CRS) property allows us to focus on output per worker and capital per worker:
• Plotting the steady state • Summary of steady state • Multiple steady states • Stability of steady state • Comparing countries
• Conditional convergence and absolute convergence • The golden rule
• Effects of changes in savings rate g
• Effects of changes in population growth rate • Effects of technological progress • Solow model with technological progress • Labour augmenting technological progress
• Dynamics of capital per effective worker • Steady state with technological progress
• Summary of Solow model with technological progress • Kaldor stylized facts
○ Where ○ 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) ○
Formulae •
• • • • • • • •
• 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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Topics • The basic Solow model: • The Solow model with technological progress:
Reading • Williamson (Chapter 6, p207-224) • Weil (Appendix of Chapter 8)
Key Points • Basic Solow model
• Cobb-Douglas production function • Competitive equilibrium in the Solow model
The Basic Solow Model • The Solow model is the basis for the modern theory of economic growth • It is based around capital accumulation • A key prediction is that technological advancement is necessary for sustained increases in living standards • Population is assumed to grow at a constant rate : ○ • Consumers are assumed to save a constant fraction of their income, consuming the rest: ○ • Representative firm's production function is neoclassical: ○ • We drop land from the model, so that output depends on capital and labour • Capital accumulation equation: ○ • Key assumption in the Solow model • Investment becomes future capital which becomes future investment etc. • The constant-returns-to-scale (CRS) property allows us to focus on output per worker and capital per worker:
• Plotting the steady state • Summary of steady state • Multiple steady states • Stability of steady state • Comparing countries
• Conditional convergence and absolute convergence • The golden rule
• Effects of changes in savings rate g
• Effects of changes in population growth rate • Effects of technological progress • Solow model with technological progress • Labour augmenting technological progress
• Dynamics of capital per effective worker • Steady state with technological progress
• Summary of Solow model with technological progress • Kaldor stylized facts
○ Where ○ 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) ○
Formulae •
• • • • • • • •
• 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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