Consider the Keynesian cross model in a closed economy where the private consumption not only depends on tax rate but also depends on the real interest rate, decided by the Central Bank of the country.

Specifically, the planned expenditure is given by the following equation:

^D=_o+₁(1)₂++

where, := ₀+₁(1)₂

a) Derive the slope of the planned expenditure function and explain the economic implication of it. Show both mathematically and graphically how a fall in the tax by would shift the planned expenditure and change the equilibrium output level of the country in good's market (Make sure to mention the axes and the direction of the shift of the curves). How the change in the consumption pattern from the conventional Keynesian cross model would alter the tax multiplier? (2 + 2 + 1 = 5 points)

Now, for this economy the IS-curve is given by

=₀+₁(1)₂+₀₁+

where, := ₀₁

b) Derive the slope of IS- curve, and compare that with the slope of conventional IS curve. Show the comparison both mathematically and graphically with proper explanations. (2 + 2 = 4 points)