Question

Given: C = 500 + .6Yd I = 200 + .3Y - 2000i G = 300 T = 200




(M/P)d = 4Y -12000i M/P = 3000




Show your solution:


a. Derive the IS relation or equation.


b. Derive the LM relation or equation.


c. Solve for the equilibrium interest rate.


d. Solve for the equilibrium output.

Answer 1

a. Derive the IS relation or equation.

The IS equation shows the equilibrium in the goods market.

We know that:

$$Y = C + I + G$$

substituting the given equations in the formula above:

$$Y = 500 + 0.6Yd + 200 + 0.3Y - 2000i + 300$$

We know that:

$$Yd = Y - T$$

But T = 200. Therefore:

$$Y = 500 + 0.6(Y - 200) + 200 + 0.3Y - 2000i + 300$$

$$Y = 500 + 0.6Y - 120 + 0.3Y - 2000i + 300$$

$$Y = 0.9Y + 80 - 2000i$$

$$0.1Y = 80 - 2000i$$

$$\color{red}{Y^* = 800 - 20000i}.....\text{Eqn 1}$$

The above equation is the IS Equation

b. Derive the LM relation or equation.

The LM curve provides equilibrium in the money market. This happens when the money demand is equal to the money supply.

$$M^d = M^s$$

Equating the given real money demand to the real money supply, we get:

$$4Y - 12000i = 3000$$

$$Y - 3000i = 750$$

$$\color{red}{Y ^* = 3000i + 750}..........\text{Eqn 2}$$

The above equation is the LM curve.

c. Solve for the equilibrium interest rate.

Equilibrium in the economy is at the point where the IS curve cuts the LM curve.

$$800 - 20000i = 3000i + 750$$

$$23000i =50$$

$$i^* =\dfrac{50}{23000} = \color{red}{0.00217}$$

d. Solve for the equilibrium output.

$$Y ^* = 3000(0.00217) + 750$$

$$Y ^* = \color{red}{756.52}$$