Consider an economy with the production function L^(1/3)
a) Derive the Labor demand Curve
b) If Labor supply curve is L^s= (w/p), calculate the equilibrium levels of real wage, labor and output.
c) Assume M^s= 45 and k=3 . What is the equation of the aggregate demand curve for this example? What is the price level? What is the wage?
a) To derive the labor demand curve, we need to determine the quantity of labor demanded at various wage rates. The labor demand is derived from the marginal product of labor (MPL) in the production function.
To find MPL, we take the derivative of the production function with respect to labor (L):
MPL = d/dL (L^(1/3))
Taking the derivative, we obtain:
MPL = (1/3) * L^(-2/3)
This represents the labor productivity in the economy.
b) To determine the equilibrium levels of real wage, labor, and output, we need to set labor supply equal to labor demand.
Labor supply, L^s, is given by:
L^s = (w/p)
where w is the real wage rate and p is the price level.
Setting labor supply equal to labor demand:
(w/p) = (1/3) * L^(-2/3)
Simplifying, we can solve for L:
L^(-2/3) = 3 * (w/p)
Taking the reciprocal of both sides:
L^(2/3) = 1 / (3 * (w/p))
Cubing both sides:
L = (1 / (3 * (w/p)))^3
Now we can substitute this value of L into the production function to find the equilibrium output:
Y = L^(1/3) = (1 / (3 * (w/p)))^1
c) To derive the aggregate demand curve, we need to express output as a function of the price level.
Using the equation for output from part (b):
Y = (1 / (3 * (w/p)))^1
Substituting the known values of M^s and k:
Y = (1 / (3 * (w/p)))^1 = (1 / (3 * (45/3)))^1 = (1 / 45)^1 = 1 / 45
So, the equation of the aggregate demand curve is:
Y = 1 / 45
For this example, the price level is 1 and the wage rate can be determined by substituting the equilibrium level of labor (L) into the labor supply equation:
L^s = (w/p) = (w/1) = w
Since L^s = L, we can substitute 1 / (3 * (w/1)) = 1 for L in the labor supply equation:
1 = (w/1) -> w = 1
Therefore, the wage rate is 1.
a) To derive the labor demand curve, we need to determine the quantity of labor demanded at different wage rates. In this case, the production function is given as L^(1/3), where L represents the quantity of labor.
The marginal product of labor (MPL) is given by the derivative of the production function with respect to labor:
MPL = (∂L^(1/3))/ (∂L)
To find the labor demand curve, we equate the MPL to the real wage rate (w) divided by the price level (p):
MPL = w/p
Now, let's solve for the labor demand curve:
L^(1/3) = w/p
Cubing both sides of the equation:
L = (w/p)^3
Therefore, the labor demand curve is L = (w/p)^3.
b) To find the equilibrium levels of real wage, labor, and output, we need to equate labor supply and labor demand in the market.
The labor supply curve is given as L^s = (w/p), where L^s represents the quantity of labor supplied.
Setting the labor supply equal to the labor demand:
(w/p) = (w/p)^3
Multiplying both sides by (w/p)^2:
1 = (w/p)^2
Taking the square root of both sides:
(w/p) = 1 (equilibrium condition)
From the labor supply equation L^s = (w/p), we substitute (w/p) with 1:
L^s = 1
Thus, the equilibrium level of labor is L = 1.
To find the equilibrium level of output, substitute the equilibrium value of labor (L = 1) into the production function:
Y = L^(1/3)
Y = (1)^(1/3) = 1
Therefore, the equilibrium level of output is Y = 1.
c) In this example, we are given the money supply (M^s = 45) and the value of k (3). To find the equation of the aggregate demand curve, we use the quantity theory of money:
M^s * V = P * Y
where M^s is the money supply, V is the velocity of money, P is the price level, and Y is the level of output.
Given M^s = 45 and k = 3 (the velocity of money equals 1/k), we substitute these values into the equation:
45 * (1/3) = P * 1 (since V = 1/k)
15 = P
Therefore, the equation of the aggregate demand curve is P = 15.
The price level is P = 15.
The wage rate is not directly provided in the given information, so we cannot determine it.