Answer to Question #144957 in Macroeconomics for Donald

Question

1. Consider an economy that is described by the the following behavioural equations:
C = c0+c1(YT) T = b0+b1Y
G = G0
I = I0 NX = 0
0 < b1<1
a. Solve for the quilibrium output, and equilibrium consumption.

Answer 1

$Solution$

From the National income Accounting, we have.

$Y=C+I+G+NX\\ Therefore, Y=C_0+G(Y-T)+I_O+G_O+0$

Introduce $T$, to the equation, we get

$Y=C_o+I_O+G(Y-(b_o+b_1Y))\\ =(C_o+I_o+G_o)+G(Y-b_o-b_1Y)\\ Let, C_o+I_o+G_o=A,$

Where $A$ will be the autonomous expenditure, which does not depend on $Y$

$\therefore\ Y=A+GY-Gb_o-Gb_1Y\\ \implies\ Y-GY+Gb_1Y=A-Gb_0\\ \implies\ Y(1-G+Gb_1)=A-Gb_o\\ \implies\ Y=\frac{A-Gb_O}{1-G+Gb_1}=Y^* (Is\ the\ Equilibrium\ Output)$

$C=C_o+G(Y^*-T)=C_o+G(Y-(b_o+b_1Y))\\ C=C_o+G(Y^-b_o-b_1Y^*)\\ =C_o+GY^-Gb_o-Gb_1Y^* \\ =C_o-Gb_o+Y^*(G-Gb_1)\\ =\frac{(C_o-Gbo)(1-G+Gb_1)+(A-Gb_o)(G-Gb_1)}{(1-G+Gb_1)}\\ =C=\frac{C_o(1-G+Gb_1)+G(A(1-b_1)-b_o}{(1-G+Gb_1)} \ is\ the\ equilibrium\ consumption.$