Question

d 21. Consider the following IS-LM model: C = co +61 (Y – T) I = bo + b Y – bai M d¡Y – dzi Р M P Р a. where (b+c) <1 b. Derive IS equation. Derive and determine its sign. [5 points]- di di c. b. Derive LM equation. Derive and determine its sign. [5 points]- d. c. Assume that LM curve is ** dY t dY () M P =dY Solve for the equilibrium output and the equilibrium interest rate. How does output change with changes in tax? (5 points]-

Answer 1

a) IS equation gives the Good market equilibrium represented by identity:  Total Income = Aggregate Expenditure = Consumption + Investment (for two sector model)

=>Y = C +I = c₀ + c₁(Y-T) + b₀ + b₁Y - b₂i

rearranging the Y terms on one side, we get: (1-c₁ - b₁)Y = c₀ - c₁T + b₀ - b₂i

=> Y = (c₀ - c₁T + b₀ )/(1-c₁ - b₁)- [b₂/(1-c₁ - b₁)] i

Let: (c₀ - c₁T + b₀ )/(1-c₁ - b₁) = A and [b₂/(1-c₁ - b₁)] = m

then IS equation :

>Y = A-m *

$\frac{\partial Y}{\partial i} = -m < 0 : b2\geq 0 , b1+c1<1$

Therefore, IS curve has a negative slope. A rise in interest rate leads to a fall in investment and hence a fall in equilibrium output.

b) LM equation gives the Money market equilibrium represented by identity:  Money Demand = Money Supply

=> (M/P)^d = (M/P)^s = > d₁Y - d₂i = M/P

or, i = (M/P)/d₂ + (d₁/d₂)Y

LM equation :

$\frac{\partial i}{\partial Y} = d1/d2 > 0 : d1\geq 0 , d2\geq 0$

Therefore, LM curve has a positive slope. A rise in output/income leads to a rise in money demand and therefore, for given money supply increase in money demand leads to increase in interest rate.

c) If (M/P)^d = d₁Y => Y = (M/P)/d₁, it implies that money demand is a function of only income and interest rate has no effect on money demand. therefore, we have a vertical LM curve implying that money market equilibrium is attained at a given level of output Y* = (M/P)/d₁

At this level of equilibrium output, good market equilibrium will gives us the equilibrium interest, i*: Y = A - mi*

=> (M/P)/d₁ = A - mi*

=> i* = [A - (M/P)/d₁]/m

Equilibrium output will not change if tax changes because the LM curve is vertical. Any change in tax will only lead to a change in equilibrium interest rate by shifting the IS curve and hence, we can say that fiscal policy is ineffective in case of Vertical LM curve.