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the rate of deprecation would be greater (less) than the return of the capital in the economy. Only at these points is there no incentive to change the capital stock.

In Figure 9.4 panel B a savings function is added. The golden rule level of capital (K GR) corresponds to the point where depreciation = savings and consumption is maximized. This outcome is far from certain for an economy. One unique savings function must apply, as shown in Figure 9.4 panel B. For any other savings function, the optimum level of consumption does not occur and the level of capital stock differs from K GR.

9.4.2THE SOLOW MODEL AND CHANGES IN LABOUR INPUT

Using the Solow model to examine the impact of changes in labour on output and economic growth is complicated. Why? Because using the production function approach we assumed T and L are fixed. Relaxing this assumption for a while, take the case where the labour input increases and we assume the economy is initially operating efficiently. With a rise in quantity of the labour input (L) only and no change in capital, each worker (on average) has less capital with which to work. The net effect on output depends on the relative effects of

1.lower average productivity due to the drop in capital per worker;

2.rising total output produced by the extra workers.

See the box for examples.

An economy produces output efficiently. It has 100 workers and average productivity is 10 baskets of output. Therefore, national output is 100 × 10 = 1 000.

Example 1

Labour rises by 5% to 105 workers. Capital does not change. Each worker has less capital and so their productivity declines by 10% to 9 baskets. New national output is 105 × 9 = 945. This represents a drop of 5.5%.

Example 2

Labour rises by 10% to 110 workers. Capital does not change. Each worker has less capital and so their productivity declines by 5% to 9.5 baskets. New national output is 110 × 9.5 = 1045. This represents a rise of 4.5%.

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Thus, the impact on output depends on the precise relationship between output, capital, labour and technology for the economy at that point in time, i.e. its production function!

There is a further possibility. It could be envisaged that an increase in the labour input could occur either if more workers become available or if the quality of the workforce improves (due to better education or skills, for example). In such a case, greater numbers of more efficient or productive workers might reduce or even cancel out the effect of having less capital with which to work. All of this implies that it is difficult to predict exactly what the effect of changes in the labour force will have on real output.

If the quantity of workers and capital are both increasing, however, the Solow model is useful for considering the effects of increasing labour inputs for output and economic growth. A rising labour input reduces capital per worker (as with depreciation). In order to maintain a constant amount of capital for each worker in the economy if the labour input is rising, investment must cover not only depreciation but also the capital required to supply all new workers with capital: capital widening is required.

In an economy where the capital stock increases at the same rate as the labour force and covers depreciation, capital widening occurs.

Capital deepening occurs when the capital stock increases faster than the labour force so that the amount of capital per worker rises.

At the level of the macroeconomy, if capital widening occurs, a given level of capital per worker (K1 for example) can be maintained, which is associated with a specific level of output (Y1). To maintain Y1 if the labour input is rising requires greater investment than if labour is unchanging.

Capital deepening would be funded through savings leading to higher investment. In an economy with a workforce growing by 5% each year, for example, investment would have to grow by 5% just to keep a constant amount of capital available to each worker. This is apart from the investment required to replace depreciating capital. If our economy above experiences 20% depreciation and 5% labour force growth then investment to the value of 25% of the capital stock is required to maintain a constant level of capital stock per worker.

A rise in labour coupled with capital deepening would lead to a further expansion of the production function. In an economy where constant returns to scale prevail, output grows at the same rate as input growth. In other words if both capital and labour increase by 5% then output grows by 5% also, reflected in an upward shift of the production function.

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In measuring the changes in labour input, economists usually use average annual hours worked in an economy, which is estimated based on broad surveys of companies usually carried out by national statistical agencies. This measure of labour input is preferred to a measure of the number of workers because of the changing patterns of work for many individuals. Focusing solely on the numbers of workers employed reveals that numbers employed internationally have risen (this is true using historical data going back to 1870) but since part-time work is quite commonplace and given labour policies and regulations, hours worked per person have actually fallen internationally.

9.4.3THE SOLOW MODEL AND CHANGES IN TECHNOLOGY

The Solow model assumes a given level of technology. Allowing technology to improve, ceteris paribus, allows more output to be produced from a given amount of inputs. This means the production function moves up as shown in Figure 9.5.

An economy with an amount K1 of capital can increase its output from Y2 to Y1 given better technology. If the economy begins with K2 capital its output increases from Y3 to Y4. The country with the higher initial capital stock experiences a greater absolute rise in output. We can extend this figure to illustrate that technology improvement is a more important source of growth than increased capital for more industrialized countries, ceteris paribus.

This case is shown in Figure 9.6. Imagine two countries with production functions like PF1 that both experience technology improvements allowing their production functions to move to PF2. One country is relatively poor with a low capital stock of K1; the second country has a higher capital stock of K2. Both countries experience the same absolute increase in capital stock.

Y4	PF2
Output
Y3	PF1
Y2
Y1
K1	Capital: K
	K2
F I G U R E 9 . 5 I M P R O V E D T E C H N O L O G Y

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		Output
		Y2*		PF2
		Y2		PF1
		Y1*
				Improved technology explains
		Y1		a portion of increased output
				Higher capital stock explains
				a portion of increased output
			K1 K1*	Capital: K
				K1 K2*
	F I G U R E 9 . 6 I M P R O V E D T E C H N O L O G Y : R I C H V S .
	P O O R C O U N T R I E S

K1 rises to K1 as shown in Figure 9.6 while K2 rises to K2 . Since the marginal product of capital is higher for the poorer country with the lower stock of capital, the expansion of its capital stock is responsible for approximately half of its increase in output from Y1 to Y1 . Compare this to the rise in output for the richer country. Because the production function is relatively flatter for the richer country since its capital stock is higher, most of the expansion in its output from Y2 to Y2 is caused by the shift in the production function.

This implies that developing or non-industrialized countries benefit more from increasing their capital stock than do more advanced economies. Advanced economies can expand their output but technology improvements are a relatively more important source of their growth.

9.4.4EXPLAINING GROWTH: LABOUR, CAPITAL AND TECHNOLOGY

Any change in national output can be explained by changes in capital or labour or technology. More specifically a technique known as growth accounting categorizes the separate sources of output changes.

Growth accounting is a method of identifying the proximate causes of changes in output and its rate of growth. The main causes are:

1.The marginal product of capital – linked to where on the production function an economy produces, ultimately explained by past investment decisions influencing the capital stock.