Find the real interest rate that clears the goods market

Question

Question 1

Consider the following model of the economy.

Production function: Y = AKN - (N²/2)

Marginal product of labor: MPN = AK - N

where the initial values of A = 8, and K = 10.

The initial labor supply curve is given as: N^s = 20 + 9w

Also, the desired consumption, desired investment, government purchases, and taxes are given as:

C^d = 401 + 0.50 (Y - T) - 500r
I^d = 800 - 500r
G = 500
T = 100

and, the real demand for money is given as: (M^d/P) = 469 + 0.5Y - 1000r

and nominal supply is M^s = 4000.

We assume that expected inflation rate is zero (π^e = 0) so that money demand depends directly on the real interest rate (so i = r).

(a) Solve for the labor market clearing real wage (w*), the prot maximizing level of labor input (N*), and the full employment level of output (Y*). Please show your work.

(b) Draw two diagrams vertically with the labor market on the bottom graph and the production function on the top graph. Be sure to label everything including this initial equilibrium point as point A.

(c) Derive an expression for the IS curve (r in terms of Y). Please show your work.

(d) Find the real interest rate that clears the goods market. Please show your work.

(e) Find the price level needed to clear the money market. Please show your work.

(f) Find the expression for the LM curve (r in terms of Y). Please show your work.

(g) Now draw four separate diagrams.

Top left: a desired savings equals desired investment
Top right: a FE-IS-LM diagram
Bottom left: a money market diagram
Bottom right: An AD-AS diagram, locating this initial equilibrium point as point A.

Be sure to LABEL all diagrams .

Question 2

How does each of the following factors aect the LM curve? Start your argument from the diagram of MD-MS (money demand-money supply) and show how the associated change will aect LM curve.

(a) An increase in nominal money supply.

(b) An increase in the risk of non-monetary assets relative to the risk of holding money.

Question 3

How does each of the following factors aect the IS curve? Start your argument from the diagram of I^d - S^d and show how the associated change will aect IS curve.

(a) An increase in government purchases
(b) An increase in marginal productivity of capital.