Answer to Question #187854 in Macroeconomics for umair

Question

Derive the Equation of Motion for Solow growth Model and discuss the break-even level of investment.

Derive the general case of the Endogenous Growth model and discuss the dynamics for both K and A given β + θ > 1 and β +θ = 1.

Answer 1

In the Solow model, L in turn grows at rate being n and A grows at rate being g. The growth of K is being determined by saving.

Since Y depends on K, AL, it seems highly unlikely that output would be going to be unchanging in steady state.

Characterizing Solow steady state is as a situation where y and k are held constant over period of time. 

since,

$k=\frac{K}{AL}=\frac{k}{k}=\frac{K}{K}-\frac{A}{A}-\frac{L}{L}=\frac{K}{K}-g-n,$

so if k is unchanging then k=0 and k must be growing at rate g+n.

By using the equation above and substituting for yields of K,

$\frac{k}{k} =\frac{K'}{K}-g-n=\frac{sY−δK}{K}-g-n=\frac{sY}{K}-δ-g-n=\frac{sy}{k}-δ-g-n=\frac{sf(k)}{k}-δ-g-n$

$k=sf(k)−(g+n+δ)k$

This is considered to be central equation of motion for the slow model

Break-even level of investment :

At k1 the amount of new investment per effective worker (on curve) would exceed the amount required for breakeven (on the line) by the gap between the line and the curve ,thus k is increasing ( k >0 ).

At k2 the amount of new investment per effective worker would tend to fall short of the amount required for breakeven, thus k is decreasing ( k < 0 ).

At k* the amount of new investment per effect worker would exactly tend to balance the need for breakeven investment, thus k is stable: k =0

k =0.At this level of k the economy has in turn settled into a steady state in which k would not tend to change.

The AK model of economic growth is an endogenous growth model used in the theory of economic growth

The AK model production function is a cobb-Douglas function with constant returns to scale.

$Y=AK^aL^{1-a}$

Where,

Y-Total production in an economy

A-Total factor productivity

K- capital

L- Labor

a- measures output elasticity of capital

Given $g_t=1$ and $\beta + \theta=1$

$Y=AK$

The production function becomes linear in capital and does not have the property of decreasing returns to scale in the capital stock which would prevail for any value of the capital intensity between 0 and 1.