11 Solow-Swann Growth Model W8 L14,15

Who developed the Solow-Swan model and what are its two key relationships?

Robert Solow and Trevor Swan (an Australian) in 1956.

1. The amount of capital determines output, and

2. the amount of output determines new investment,

which together determine capital accumulation over time.

What is the key behavioural assumption of the Solow-Swan model?

A constant fraction s of output is saved each period, so S = I = sY and consumption C = (1−s)Y.

What is the capital accumulation equation?

K(t+1) − K(t) = sAF(K, L) − δK, which is new investment minus depreciation.

What is δ?

The depreciation rate, the fraction of the capital stock that wears out each period.

What do the investment curve and depreciation line look like in the Solow diagram?

Investment sAF(K, L) curves upward and flattens because of diminishing returns to capital. Depreciation δK is a straight line.

What happens to capital if investment is above, below or equal to depreciation?

Above means capital rises, below means capital falls, and equal means capital is constant (the steady state).

capital stock K (the total machines, buildings and tools the economy has) depending on how two flows compare:

• Investment is new capital being added each period: sAF(K, L).

• Depreciation is old capital wearing out each period: δK.

The Solow-Swan equation puts them together:

K(t+1) − K(t) = investment − depreciation

What is the steady state?

The situation where there is no further capital accumulation, so sAF(K, L) = δK. On the diagram it is where the investment curve crosses the depreciation line.

What is steady-state capital K* for Cobb-Douglas and what raises it?

K* = (sA/δ)^(1/(1−α)) × L. It rises with a higher saving rate s, higher productivity A and more labour L, and falls with a higher depreciation rate δ.

Why can’t capital accumulation sustain long-run growth on its own?

Diminishing returns to capital. Below K* the marginal product of capital is high so the economy grows, but as capital rises the marginal product falls and growth slows to zero as K approaches K*.

What happens after a permanent increase in the saving rate from s to s′?

The investment curve shifts up, so K* and Y* rise. Investment exceeds depreciation, so capital accumulates and the economy grows for a while, but diminishing returns slow growth back down.

What is the difference between a level effect and a growth effect?

A higher saving rate raises the level of capital and output in the long run (level effect) but has no effect on long-run growth (not a growth effect).

What can generate sustained long-run growth?

Increases in total factor productivity A, which shift the whole production function up. Countries cannot save their way to permanent growth.

How do productivity and the labour force grow in the extended model?

A(t+1)/A(t) = 1 + g_A and L(t+1)/L(t) = 1 + g_L.

What is labour-augmenting productivity and effective labour?

Technology makes labour more productive, so output is F(K, AL). The product A × L is effective labour, workers adjusted for productivity.

What are output and capital per effective worker?

y = Y/(AL) and k = K/(AL).

What is the intensive form of the production function?

Using constant returns to scale, y = f(k) where f(k) = F(k, 1). For Cobb-Douglas this is y = k^α.

What is effective depreciation and why is the line steeper?

(δ + g_A + g_L) × k. Capital per effective worker is diluted by wear and tear, by labour force growth and by productivity growth.

What happens to capital per effective worker when effective investment is above, below or equal to effective depreciation?

Above means k rises, below means k falls, and equal means k is constant (the steady state k*).

At what rate does output per worker grow in the steady state?

At g_A, the rate of productivity growth. Total output grows at about g_A + g_L.

What is the effect of a permanent increase in the saving rate in the extended model?

k* and y* rise. Growth is faster than trend on impact, but diminishing returns slow it back down to trend (g_A) in the long run.

What does the convergence hypothesis predict?

An economy grows faster when it is far below its steady state and slower when it is close to it.

What is the difference between conditional and unconditional convergence?

Conditional convergence means economies with the same parameters (s, δ, g_A, g_L) reach the same long-run output per worker whatever their starting capital. Unconditional convergence would mean all economies reach the same level regardless of parameters, which the model does not predict.

What is growth accounting and the Solow residual?

A tool for splitting observed growth into growth in inputs and a productivity residual. g_Y = g_A + α g_K + (1−α) g_L, so g_A = (g_Y − g_L) − α(g_K − g_L).