Question

1. Write down the equation of motion for k, which shows Dk as saving minus break-even investment. Use this equation to draw a graph showing the determination of steady-state k. (Hint: This graph will look much like those we used to analyze the Solow model.)

please write it in your own words

Answer 1

Solow growth model has been used to explain how savings, technological advancement/progress and growth in population affect the output in an economy along with output growth overtime. The model is used to estimate how capital growth, labor force growth & technological advancement interact in an economy.

In the below model we will analyze output is determined solely by capital growth, keeping technology fixed and assuming no labor force growth.

Supply Side of the Model:

· We assume the production function is given by Y = F(K,L) which is homogeneous in nature, i.e. it exhibits constant returns to scale such that:

zY = f(zK,zL)

zY = zf(K,L)   

· We also assume the size of the economy does not matter, thus we use per-worker analysis. We can write the per-worker production function as:

Y/L = f(K/L, L/L)

y = f(k,1)

y = f(k) , where output is a function of capital per worker k

· Figure I, shows the per worker production function with capital per worker (k) on the X-axis and output per worker(y) on the Y-axis. The slope of the production function is given by MPK (marginal productivity of capital) which shows the change in output per worker when an additional unit of capital is used. The slope increases but at a decreasing rate, so as more and more capital is employed the production function becomes flatter.

Demand side of the model:

· Every year people use their income for consumption and investment. Thus the output per worker can be written using the generic aggregate demand equation :

y = c + i   , where c = consumption per worker and i = investment per worker

· The above statement can also means that ever year people save a fraction of their income and consumes the rest of it. Let’s say, the fraction by which they save is given by “s” and the fraction by which they consume is given by c = 1-s. These are the marginal propensity to save and consume respectively.

Fraction of income saved = sy

Fraction of income consumed = (1-s)y

Putting these values in the above demand equation we get :

y = c + i   

y = (1-s)y + i

y = y – sy + i

y – y + sy = I

sy = i

The above relation shows the equilibrium in the economy where savings per worker is equal to investment per worker. It also states that the saving rate shows the proportion of output/income devoted towards investment.

· Refer to figure 2: For a given amount of capital per worker (k’) and per worker output, the savings rate allocates the output (y’) between consumption and investment.

Growth of capital and motion equation (refer figure 2 and 3)

Change in capital is affected by :

1. Investment ( i ) that raises capital stock : A rise in investment induces more capital growth and vice as versa.

Given per worker production function y = f(k) and saving investment equilibrium “i = sy” investment can be written as “i = sf(k) "

2. Depreciation Rate (d) that reduces capital stock: This is wear and tear of capital stock over time. Given “d” as the Depreciation rate, depreciation can be written as a fraction of capital stock that wears out. Each year depreciation rises with an increase in capital stock. Refer to figure 3 which shows the relation between depreciation and capital stock with capital per worker on the x-axis and depreciation on the Y-axis.

Thus the motion equation of capital stock or growth in capital can be given by :

Dk = sf(k) - dk

Steady state level of capital

The level of capital where investment equal to the depreciation of capital is knows nas the steady stae level capital given by “k*”. Using the motion equation, the steady state level can be described as :

Dk = sf(k) – dk

0 = sf(k) – dk

sf(k*) = dk* , here capital growth is equal to zero.

· Refer to the below figure 4, with capital per worker on X-axis and depreciation and investment on the Y-axis. The depreciation curve and the investment curve intersect at equilibrium point E. Here steady state level out capital is k*, steady state level investment is i*.

· For any point below the steady state (k1<k*): Investment is more than depreciation which means more capital is replaced with new capital than it is weared out, thus capital stock keeps on increasing till it reaches the steady state. In diagram here i > d.

· For any point above the steady state (k2<k*): Investment is less than depreciation which means more capital weared out than it is added/replaced, thus capital stock keeps on falling till it reaches the steady state. In diagram here i < d.