ECON2000 Chap.2 Solow Capital Dynamics
Solow Capital Dynamics
The Solow model turns production, saving and depreciation into a dynamic path for capital. This chapter emphasises the within-period order: capital is inherited, output is produced, output is divided between consumption and investment, and investment determines next period's capital. That timing explains which variables can jump after a surprise and which are predetermined.
A steady state occurs when saving-financed investment equals depreciation or other break-even investment. Below it, capital rises; above it, capital falls. The chapter derives the steady state, interprets comparative statics, distinguishes a new long-run level from the transition path and explains why a higher saving rate does not produce permanent per-capita growth in the baseline model.
What this chapter covers
- 01
Production and resource constraints
- 02
Capital law of motion
- 03
Within-period timing
- 04
Steady-state condition
- 05
Comparative statics
- 06
Transition dynamics
- 07
Golden Rule consumption
Worked example · free
Solving capital one period ahead
- +1Output is the square root of 16, so Y = 4.
- +1Investment is 0.36 times 4, so I = 1.44.
- +1Consumption is 0.64 times 4, so C = 2.56.
- +1Depreciation is 0.09 times 16, so lost capital is 1.44.
- +2Next-period capital is 16 minus 1.44 plus 1.44, which remains 16.
Key terms
- Capital law
- The equation carrying undepreciated capital and current investment into next period's capital stock.
- Steady state
- A position where the relevant per-worker variables remain constant because investment covers break-even needs.
- Predetermined capital
- Capital inherited at the start of a period and therefore unable to jump after a current surprise.
- Transition dynamics
- The period-by-period path followed between an initial condition and a steady state.
- Golden Rule
- The steady-state capital position that maximises steady-state consumption rather than output.
Solow Capital Dynamics FAQ
Why can investment jump while capital cannot?
Current capital was created by earlier investment and is inherited when the period begins. A new saving decision can immediately change how current output is divided between consumption and investment. That additional investment affects capital only in the next period through the law of motion.
What makes the Solow steady state stable?
Below the intersection, saving-financed investment exceeds depreciation, so capital rises. Above it, depreciation exceeds investment, so capital falls. Diminishing returns flatten the production and saving curve, allowing the linear break-even requirement to pull the economy back toward the intersection.
Does a higher saving rate cause permanent growth?
In the baseline model it raises the steady-state level of capital and output and creates temporary transition growth. Once the higher steady state is reached, per-capita growth returns to zero unless exogenous technology or another sustained growth mechanism is introduced.
Why can higher saving reduce consumption at first?
The saving rate divides current output. When it rises unexpectedly, a larger share becomes investment and a smaller share remains for consumption, while current capital and output are fixed. Later output increases may lift consumption, but the impact response is a decline.
Exam move
Memorise the timing sequence rather than a finished transition graph. For every shock, write capital, production, allocation and next-period capital in order, then mark where the changed parameter first appears. Re-derive the steady state by setting investment equal to break-even investment and use the diagram to verify the direction of motion on both sides.
Practise saving, depreciation, productivity and population shocks separately. For each one, distinguish the impact response, the transition and the final steady-state comparison. End with a consumption statement so a higher output level is not automatically treated as higher welfare. Redraw the investment and depreciation schedules from blank axes until their curvature and intersection follow from the equations.
Create a transition ledger with one row per period and columns for inherited capital, output, consumption, investment, depreciation and next-period capital. Fill three rows by hand before using a spreadsheet. This reveals that current capital is predetermined and that current saving affects the next stock through investment.
After solving a steady state algebraically, substitute it back into both investment and depreciation; the values must match. Use the diagram for a second check by selecting a capital stock on either side of the intersection and comparing the two schedules. For a saving shock, write the impact fall in consumption before describing later output gains.
For a productivity shock, note that current output can jump even though current capital cannot. Add population growth only after the baseline mechanics are secure, because it changes break-even investment rather than the production timing. Finally, compare steady-state output with Golden Rule consumption so a larger capital stock is not automatically labelled a welfare improvement.
Accurate transition answers describe impact, direction, speed and destination in that order. Run a verbal transition without numbers after every calculation. Say what is inherited, what changes on impact, what moves next period and why convergence slows. If the shock is to saving, contrast the immediate consumption loss with the later output path. If it is to productivity, explain why output can jump while capital cannot.
This narration is the quickest way to detect a graph that shows the right endpoints but an impossible route between them.
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