Question

Given the following National Income Model

Y = C + I0 + G0 + X0 – Z
Where C = C0 + bYd
Yd = Y – T

T = T0 + Ty and Z = Z0 + ZYd
In this equation X and Z are export and import
• Identify endogenous and exogenous variables and parameters.
• Find out equilibrium level of Y and C
(full solution)

Answer 1

1) Identify endogenous and exogenous variables and parameters

The exogenous variables are $I_0, \;G_0, \; X_0, \; T_0, \; Z_0$

The endogenous variables are $Y, \; C, \; T, \; Z$

2) Find out equilibrium level of Y and C

At equilibrium,

$$Y = C + I + G + (X - Z)$$

Therefore:

$$Y = C_0 + b(Y - T_0 - tY) + I_0 + G_0 + X_0 - z(Y - T_0 - tY)\\[0.3cm] Y(1 - b + tb + z - tz) = C_0 - bT_0 + I_0 + G_0 + X_0 + zT_0\\[0.3cm] \color{red}{Y^* = \dfrac{C_0 - bT_0 + I_0 + G_0 + X_0 + zT_0}{1 - b + tb + z - tz}}$$

This gives an equilibrium consumption of:

$C = C_0 + bY_d\\[0.3cm] C = C_0 + b(Y - T_0 - tY)\\[0.3cm] C^* = C_0 + b(1 - t)Y^* - bT_0\\[0.3cm] C^* = C_0 + \dfrac{b(1 - t)(C_0 - bT_0 + I_0 + G_0 + X_0 + zT_0)}{1 - b + tb + z - tz} - bT_0\\[0.3cm] \color{red}{C^* = \dfrac{(C_0 - bT_0)(1 - b + tb + z - tz) + b(1 - t)(C_0 - bT_0 + I_0 + G_0 + X_0 + zT_0)}{1 - b + tb + z - tz}}$