ECON7074 Chap.7 Solow Steady States, Convergence and New Growth
Solow Steady States, Convergence and New Growth
This chapter solves the Solow-Swan model at its rest point and asks what the solution says about the world. Setting investment equal to depreciation gives closed-form steady-state capital and output, a capital-output ratio equal to the investment rate over the depreciation rate, and an exponent of 3/2 on productivity, larger than in the production model because better technology also attracts more capital.
A ratio of steady-state incomes then splits a rich-poor gap into an investment term and a productivity term, and the evidence points to productivity as the dominant source. The chapter compares parameter changes, explains why the economy settles and why capital alone cannot sustain growth, and weighs convergence evidence among OECD countries against the wider world.
It closes with endogenous growth theory, the constant-returns AK contrast and the policy levers of saving, research and development, trade and education. Each section ends with a short written routine you can reuse under exam conditions, so the algebra and the argument arrive together.
What this chapter covers
- 01
Closed-form steady-state capital and output
- 02
Why productivity carries an exponent of 3/2
- 03
Splitting income gaps into investment and productivity
- 04
Parameter experiments and the absence of long-run growth
- 05
Convergence among OECD countries and the world
- 06
Endogenous growth, AK and growth policy
Worked example · free
Splitting a predicted sixteen-fold income gap into investment and depreciation versus productivity
- +1Write the ratio of steady-state incomes: the productivity ratio to the power 3/2 times the ratio of s/d values to the power 1/2; because depreciation differs, it does not cancel.
- +1Investment and depreciation term: R has s/d = 0.24/0.05 = 4.8 and P has 0.12/0.10 = 1.2, a ratio of 4 whose square root is 2.
- +1Productivity term: 4 to the power 3/2 is 8, so the predicted income ratio is 2 times 8, which is 16.
- +1Sixteen is four doublings: one comes from the investment and depreciation term and three from productivity, so productivity supplies three quarters of the gap on a log scale.
Key terms
- Capital-output ratio
- Steady-state capital divided by output, which in this model equals the investment rate over the depreciation rate.
- Income-gap decomposition
- Splitting a rich-poor income ratio into the part explained by investment rates and the part attributed to productivity.
- Convergence
- Tendency of poorer economies to grow faster than richer ones and so narrow the gap in income per person.
- Endogenous growth model
- Growth model in which sustained growth arises from choices inside the model, such as investment in new ideas.
- Knowledge capital
- Stock of ideas and discoveries usable by many producers at once, whose returns do not diminish in new growth theory.
Solow Steady States, Convergence and New Growth FAQ
Why does productivity have an exponent of 3/2 in the Solow steady state?
A rise in productivity raises output directly, as in the production model, but it also raises investment, so the economy accumulates more capital before reaching its new rest point. That extra capital adds still more output. The combined effect shows up as an exponent of 3/2 on productivity in steady-state income per person, compared with an exponent of one when capital is held fixed.
Can a higher investment rate raise long-run growth in the Solow model?
No. A higher investment rate raises the steady-state levels of capital and output, and growth is positive while the economy moves to the new rest point. Once it arrives, growth returns to zero, because diminishing returns stop extra capital from adding enough investment to outrun depreciation. Only the level changes, not the long-run growth rate.
Does the evidence support convergence?
Partly. Among OECD economies, those that were poorer in 1960 grew faster afterwards, as transition dynamics predict if they share a steady state. Across the whole world, rich and poor economies have grown at similar average rates, which suggests many have different steady states set by weaker parameters rather than temporary shortfalls of capital.
What does new growth theory add to the Solow model?
It explains sustained growth from inside the model. Discoveries come from profit-seeking choices, can be used by many people at once and do not run into diminishing returns, so knowledge can keep raising productivity indefinitely. That gives a role to policies on research and development, saving, trade and education that the basic model cannot capture.
Why is productivity called a residual?
Productivity is not measured directly. It is whatever part of the income difference the measured investment rates, and in other exercises capital and labour, cannot explain once the model is assumed to hold. Any error in the measured inputs therefore ends up in the productivity estimate, which is why conclusions about it should be stated with care.
Exam move
Derive the steady state yourself from sY = dK until you can reach the capital and output formulas in a few lines, then check them with the capital-output ratio. Practise income-ratio problems in both directions: predict a gap from given investment and productivity ratios, and back out the productivity ratio from an observed gap.
For each parameter change, write one row of a table naming the moving curve, the new steady state and the sign of growth during the transition. Rehearse a short paragraph on convergence that separates OECD evidence from world evidence, and another contrasting diminishing returns to capital with non-diminishing returns to knowledge. Those two paragraphs answer most written questions on this topic.
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