Given the following national income model:

Y = C + I_0 + G_0C = C_0 + b(Y − T), and T = T_0 + tY

where Y = income, C = consumption, T = taxation, C_0, b, t, I_0, G_0 and T_0 (autonomous taxation) are constants and C_0 > 0; 0 < b < 1; 0 < t < 1:

(a) Write this model as three equations in terms of the variables Y, C and T.

(b) Use Cramer’s rule to derive expressions for the equilibrium level of income, consumption

and taxation.

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(a) The variable terms containing Y, C and T are arranged on the LHS of each equation, the

constants on the RHS:

Y − C = I_0 + G_0

−bY + C + bT = C_0

−tY + T = T_0

(b) Using Cramer’s rule, solve for Y, C, T:

Y={\frac{\Delta_{Y}}{\Delta}}={\frac{I_{0}+\mathrm{G}_{0}+\mathrm{C}_{0} – b\,T_{0}}{1 – b + b t}} = the equilibrium level of income

C=\frac{\Delta_{\mathrm{C}}}{\Delta}=\frac{C_{0}+b\,T_{0} + b(1 – t)(I_{0}+G_{0})}{1 – b + b t} = equilibrium level of consumption

T=\frac{\Delta_{T}}{\Delta}=\frac{T_{0}(1 – b) + t(\mathrm{C}_{0}+I_{0}+\mathrm{G}_{0})}{1 – b + b t} equilibrium level of taxation

Since:

Δ = \begin{vmatrix} 1 & -1 & 0 \\ -b & 1 & b \\ -t & 0 & 1 \end{vmatrix} = (1) \begin{vmatrix}\\ 1 & b \\ 0 & 1 \end{vmatrix} − (−1) \begin{vmatrix} -b & b \\ -t & 1 \end{vmatrix} + (0) = 1 + (−b − (−bt))= 1 − b + bt

Δ_Y = \begin{vmatrix} I_0 + G_0 & -1 & 0 \\ C_0 & 1 & b \\ T_0 & 0 & 1 \end{vmatrix} = (I_0 + G_0) \begin{vmatrix} 1 & b \\ 0 & 1 \end{vmatrix} − (−1) \begin{vmatrix} C_0 & b \\ T_0 & 1 \end{vmatrix} + (0)= (I_0 + G_0) + (C_0 − bT_0)

Δ_C = \begin{vmatrix} 1 & I_0 + G_0 & 0 \\ -b & C_0 & b \\ -t & T_0 & 1 \end{vmatrix} = (1) \begin{vmatrix} C_0 & b \\ T_0 & 1 \end{vmatrix} − (I_0 + G_0) \begin{vmatrix} -b & b \\ -t & 1 \end{vmatrix} + (0)= (C_0 − bT_0) − (I_0 + G_0)(−b − (−bt))

= C_0 − bT_0 + b(I_0 + G_0)(1 − t)

Δ_T = \begin{vmatrix} 1 & -1 & I_0 + G_0 \\ -b & 1 & C_0 \\ -t & 0 & T_0 \end{vmatrix} = (1) \begin{vmatrix} 1 & C_0 \\ 0 & T_0 \end{vmatrix} − (−1) \begin{vmatrix} -b & C_0 \\ -t & T_0 \end{vmatrix} + (I_0 + G_0) \begin{vmatrix} -b & 1 \\ -t & 0 \end{vmatrix}= T_0 − bT_0 + tC_0 + (I_0 + G_0)(0 − (−t))

= T_0(1 − b) + t(C_0 + I_0 + G_0)

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