Answer to Question #187178 in Macroeconomics for Pragya Sood

Question

Given the following national income model:

and

𝑇=𝑇 +𝑡𝑌 0

𝑌 = 𝐶 + 𝐼0 + 𝐺0

𝐶 = 𝐶0 + 𝑏(𝑌 − 𝑇) 

where 𝑌 = 𝑖𝑛𝑐𝑜𝑚𝑒, 𝐶 = 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛, 𝑇 = 𝑡𝑎𝑥𝑎𝑡𝑖𝑜𝑛;

𝑤h𝑒𝑟𝑒 𝐶0 > 0; 0 < 𝑏 < 1; 0 < 𝑡 < 1

Using Cramer’s rule to derive an expression for the equilibrium level of income, consumption, and taxation. 

Answer 1

Solution:

First re-arrange the equations so that the endogenous variables are on the LHS and the exogenous variables are on the RHS:

Y − C = I + G0

C − bY + bT = C0

T – tY0 = T

 

Then rewrite the equations in matrix form:

The equilibrium national income (Y) = C + I0 + G0

Y = C0 + b(Y – T) + I0 + G0

$=\frac{C0 − bT + I0 + G0 }{1 − b(1 − t)}$

Consumption:

$=\frac{C0 − bT + b (1 − t) (I0 + G0) }{1 − b(1 − t)}$

Taxation:

$==\frac{C0 − bT + at + t(I0 + G0)) }{1 − b(1 − t) }$