IS Equation and the IS-LM Model

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To derive the IS equation, we need to combine the consumption, investment, and output equations. The consumption equation is:
C=C+mpcY-T-br
The investment equation is:
I=I-dr
The output equation is:
Y=C+I+G 
Substituting the consumption and investment equations into the output equation,
we get: Y=C+mpcY-T-br+I-dr+G
Simplifying, we get: Y=C+I+G-br-dr+mpcY-T
Moving the Y term to the left-hand side and collecting like terms, we get:
Y-mpcY=C+I+G-br-dr-mpcT
Solving for Y, The IS equation is,
Y=C+I+G-br-dr-mpcT/1-mpc
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Y=C+I+G-br-dr-mpcT/1-mpc
where T is the sum of autonomous transfer payments and taxes. If autonomous transfer payments and taxes are increased by $600 billion each, then T will increase by $1.2 trillion. Substituting the given values into the equation, we get:
∆Y = (0.6 / (1 - 0.6)) * $1.2 trillion
∆Y = (0.6 / 0.4) * $1.2 trillion
∆Y = 1.5 * $1.2 trillion
∆Y = $1.8 trillion
Therefore, if autonomous transfer of payments and autonomous taxes were increased by $600 billion each, real GDP (∆Y) would increase by $1.8 trillion.
C.
Slope of the curve is given by,
∆Y/∆r=-1-mpc/1-mpc1-t
This slope is different from the standard IS slope, which is typically given by: ∆Y/∆r=-1/1-mpc1-t
The difference between the two slopes is the presence of the (1-t) term in the numerator of the first equation. This term represents the effect of taxes on the responsiveness of output to changes in interest rates. When taxes are present, the responsiveness of output to interest rates is reduce...