Question

Mimic the deviation of Solow's simple model (k y, per capita) to derive Solow's extended model (k* y*, per effective labor) (Simple instructions is ok]

Answer 1

In the diagram;

Savings Deprecation antok -savings TK Steady-state Per capita The Solow capital, K Model

sy = skα: represents saving (and investment) per capita

(n + d)k: represents the amount of investment needed to keep per capita capital constant given:

n: population growth, d: depreciation

The level of per capita capital at which, ˙k = skα − (n + d)k = 0 (25)

is called the “steady-state capital stock per capita”, and denoted k∗. In the steady-state, aggregate capital, K ( t ), is not constant: K˙/K = ˙k /k + L˙ /L = n > 0.

Similarly, total (not per capita) output, Y ( t ) grows as well: Y = K ^α L^ 1 − α

ln Y = α ln K + (1 − α) ln L

So, Y˙/ Y = α K˙/ K + (1 − α ) L˙/ L

= αn + (1 − α ) n = n

The economy grows at the rate of population growth. This means that in the Solow model, growth of per capita income is not sustained. In the steady-state, per capita income is constant.

Outside of the steady-state, there will be growth, positive or negative:

1. Suppose k ( t ) < k ∗ : sk ^α − ( n + d ) k = ˙k > 0, Capita per capita grows over time.

2. Suppose k ( t ) > k ∗ : sk^ α − ( n + d ) k = ˙k < 0, Capital per capita falls over time.

These forces cause the economy to tend toward the steady-state over time.

EXTENSION OF SOLOW MODEL

The two Extension of the Solow Model are: 1. Population Growth 2. Technological Progress.

Population Growth:

We now assume that population does not remain fixed. Instead, the population and the size of labour force grow at a constant rate n.

The Steady State with Population Growth

We may now discuss how population growth, along with investment and depreciation, influences the accumulation of capital per worker. In the basic Solow model, while investment increases capital stock, depreciation reduces it. In this extended model, another factor changes the amount of capital per worker: the growth in the number of workers causes capital per worker to fall.

We assume that the number of workers is growing over time at the rate ‘n’ per period. So the change in capital stock per worker is

Δk = i – (δ + n)k …

Equation shows that new investment (i) increases k, while depreciation (δ k) and population growth (n) decrease k. These factors thus, together, determine capital stock per worker.The term (δ + n)k may be treated as the break-even level of investment which is necessary to keep per capita capital stock constant. Break-even investment has two components: replacement investment nk which is a measure of depreciation of existing capital and new investment — the amount of investment necessary to provide new workers with capital. The required amount of investment is nk because there are n workers for each existing worker arid k is the amount of capital per worker. Equation shows that population growth, has a negative effect on the accumulation of capital per worker. The equation can then be expressed as

Δk = sf(k) – (δ + n)k

Actual investment & break then inst. (Sn)K Capital peu Wouken, K

Figure shows what determines the steady-state level of capital per worker. We know that an economy is in steady state if capital per worker k remains constant at k*.

Technological Progress

The next source of economic growth is technological progress. It is called the residual factor of economic growth. We may now incorporate this factor into the Solow model. Technological progress alters the relationship between inputs (capital and labour) and the output of goods and services and thus leads to exogenous increases in society’s capacity to produce. It leads to intensive growth by shifting the production function upward.

Efficiency of Labour:

Since technological progress improves the efficiency of labour the production function may now be expressed as

Y = F(K, L x E)

where E is efficiency of the labour force, which is essentially a reflection of a society’s knowledge about the methods of production. With an improvement in technology, the efficiency of labour rises.

The Steady State with Technological Progress:

The effect of technological progress is the same as that of population growth because it is labour-augmenting in nature. Now we assume the number of effective workers to rise and describe the growth of the economy in terms of quantities (output levels) per effective worker.

Let k = K/(L x E) denote capital per effective worker, and y = Y/(L x E) denote output per effective worker. If the efficiency of labour is growing, then k and y refer to quantities per effective worker. In the presence of technological progress we have the following relation which shows how k grows over time:

Δk = sf(k) – (δ + n + g)k

Thus the change in the capital stock Δk equals actual investment sf(k) minus the break­even investment (δ + n + g)k. Now, since k = K/LxE, break-even investment includes a new term, viz., gk, which is needed to provide capital for the new ‘effective workers’ created by tech­nological progress.

Actual inust. $ berak even inust (dontg)k Tastlk) Capital per effective, worken (K)

As shown in Fig. even in the presence of technological progress there is one level of k, viz., k*, which ensures the existence of steady state. See at this level of k, both capital per effective worker and output per effective worker remain constant. As in the basic Solow model, this steady state represents the long-run equilibrium of the macro-economy.