2 equations with 2 unknows Y & C

Hi this is part of a matrix question. I am ok with matrix but am having trouble with getting the equations -

Here it is -

Y = C + I + G

C =C* + yYD C*>0: 0<y<1 (The D is supposed to be a small D at the bottom of the Y

I=I*

G=G*

YD = Y -T (Again the D is supposed to be a small D down below the line

T = tY 0<t<1

Reduce the model to 2 equations in 2 unknowns Y and C.

thanks again - Helen

Re: 2 equations with 2 unknows Y & C

$\displaystyle \text{We have the following system of equations:}$

$\displaystyle \text{(1) } Y = C + I + G$

$\displaystyle \text{(2) } C = C^{*} + yY_{d}$

$\displaystyle \text{(3) } I = I^{*}$

$\displaystyle \text{(4) } G = G^{*}$

$\displaystyle \text{(5) } Y_{d} = Y - T$

$\displaystyle \text{(6) } T = tY$

$\displaystyle \text{From (3), (4)}\rightarrow \text{(1), we have:}$

$\displaystyle \text{(7) } Y = C + I^{*} + G^{*}$

$\displaystyle \text{From (6)}\rightarrow \text{(5), we have:}$

$\displaystyle \text{(8) } Y_{d} = (1 - t)Y$

$\displaystyle \text{From (8)}\rightarrow \text{(2), we have:}$

$\displaystyle \text{(9) } C = C^{*} + y(1 - t)Y$

$\displaystyle \text{Hence, from (7) and (9), we have the final system of equations in the unknowns } Y \text{ and } C \text{:}$

$\displaystyle Y = C + I^{*} + G^{*}$

$\displaystyle C = C^{*} + y(1 - t)Y$