Chapter 6.4, Problem 9E

Interpretation Introduction

Interpretation:

To show that there is a fixed point for the model and equilibrium state of economy if the rate of the government spending is constant and classify the fixed points a function of $\alpha$ and $\beta$. To show that if $\beta =1$ economy is predicted to oscillate. To determine the condition when there is an economically sensible equilibrium, if government spending increases linearly with national income. Show that this kind of equilibrium will stop existing if the parameter exceeds a critical value. To determine how economy is predicted to behave under this condition. To show that the system can have two, one or no fixed points in first quadrant if government expenditures increase quadratic ally. Discuss the implications for the economy in the above cases by interpreting the phase portraits.

Concept Introduction:

The fixed point is the point at which the first derivative of the system equals to zero.

To check the stability of fixed point use Jacobian matrix

$\left(\begin{array}{cc}\frac{\partial \dot{x}}{\partial x}& \frac{\partial \dot{x}}{\partial y}\\ \frac{\partial \dot{y}}{\partial x}& \frac{\partial \dot{y}}{\partial y}\end{array}\right)$

The point $(x^{*},y^{*})$ is said to be a stable fixed point if eigenvalues of Jacobian matrix evaluated at this point having negative real parts and point is said to be unstable if one of its eigenvalue has positive real part. If the both eigenvalues are purely real, then the fixed point is saddle point.