ECON7074 Chap.6 Solow-Swan Capital Accumulation and Dynamics
Solow-Swan Capital Accumulation and Dynamics
This chapter turns the static production model into a growth model by explaining where capital comes from. Following the Solow-Swan set-up, output is produced with Cobb-Douglas technology, divided between consumption and investment, and a fixed share is invested each period while a constant fraction of the capital stock wears out.
Combining the five equations gives one law of motion for capital, which the Solow diagram shows as an investment curve against a straight depreciation line. Their crossing is the steady state, the rest point where capital, output and consumption stop changing.
The chapter solves a steady state in levels, advances an economy one period at a time, and plots its capital stock to show transition dynamics: growth is fastest when the economy starts far from its rest point and fades as the gap closes. It ends with how output and consumption follow capital, and with written practice in the style of the course papers.
What this chapter covers
- 01
Why the model makes capital endogenous
- 02
The five equations of the Solow-Swan system
- 03
Saving, investment and the real interest rate
- 04
Reading the Solow diagram
- 05
Solving the steady state in levels
- 06
Transition dynamics and the time path of capital
Worked example · free
Capital stock growth slows period by period on the way to the steady state
- +1Period 1: production is 25 times the cube root of 216, which is 150; 0.3 of it is 45 and wear is 0.075 times 216, or 16.2, so the stock rises by 28.8 to 244.8.
- +1Growth in period 1 is 28.8 divided by 216, about 13.3%.
- +1Period 2: production is about 156.4, so 46.9 is added and 18.4 wears out; the stock rises by about 28.6 to about 273.4, growth of about 11.7%.
- +1Additions rise more slowly than wear because of diminishing returns to capital, so net growth shrinks relative to the stock and the growth rate slows toward zero near 1000.
Key terms
- Capital accumulation
- Growth of the capital stock when gross investment exceeds the capital that wears out in the same period.
- Depreciation rate
- Constant fraction of the existing capital stock lost to wear and obsolescence each period, written d in the model.
- Investment rate
- Fixed share of output devoted to new capital each period in the Solow-Swan model, so investment equals sY.
- Law of motion
- Equation that maps this period's capital stock into the next period's, here net investment added to current capital.
- Solow diagram
- Graph plotting investment and depreciation against capital, whose crossing marks the rest point of the economy.
Solow-Swan Capital Accumulation and Dynamics FAQ
Why does the Solow-Swan model need a steady state?
Investment grows with capital but by less and less because of diminishing returns, while depreciation grows in proportion to the capital stock. Sooner or later the two are equal, and at that point net investment is zero, so capital, output and consumption all stop changing. That rest point is what the model calls the steady state.
What does the vertical gap in the Solow diagram mean?
At any capital level, the distance between the investment curve and the depreciation line is net investment, the change in capital over the period. Divided by the capital stock it gives the growth rate of capital, which is fastest when capital is far below the steady state. A negative gap, where depreciation sits above investment, means the capital stock is shrinking.
How do I describe transition dynamics in an exam answer?
State where the economy starts relative to its steady state, say whether investment exceeds or falls short of depreciation, and conclude whether capital and output rise or fall. Then add the speed: movement is fastest when the economy is far from the rest point and slows as it gets close. A labelled diagram plus a sketch of output against time completes the answer.
Why is the real interest rate equal to the marginal product of capital?
In the model saving equals investment, and the return savers receive must match what capital earns when rented to firms. Competitive firms rent capital until its marginal product equals the rental price, so the real interest rate equals the marginal product of capital, which is high when capital is scarce and falls as it accumulates.
Does the model say saving more is always good?
Saving more raises the steady-state capital stock and output, but the course flags that the effect on steady-state consumption is not automatic. The slides name the golden rule level of capital in this context, and a practice-paper solution notes the answer depends on which side of it the economy sits, something the course did not develop further.
Exam move
Learn the five equations by writing them from memory, then collapse them into the single law of motion until the substitution is automatic. Draw the Solow diagram many times with labelled axes, the investment curve, the depreciation line and the crossing, and practise shading the gap that represents net investment.
For every numeric problem, solve the steady state, check that capital divided by output equals the investment rate over the depreciation rate, then advance one or two periods by hand. Finish each practice answer with a sketch of capital or output against time, marking the starting point, the rest point and the slowing slope.
Practice papers reward exactly this combination of diagram, transition and steady state, so rehearse writing it in under half a page.
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