Question

Tiffany's company has the production function Q=2K^0.5L^0.5, where Q measures output, K measures machine hours, and L measures labor hours. Let the wage rate be W, and suppose that the rental rate of capital is R=$20 and the firm wants to produce 400 units of output. Use the Lagrange method to find the demand curve for labor as a function of the wage rate. Your answer will have L on the left hand side of the equation. On the right hand side, you will have a constant multiplied by W raised to an exponent

-What is the exponent on W? (Note that the exponent is a negative number, so make sure to enter the exponent including the negative sign.)

-What is the constant number on the right hand side of the equation?

Tiffany's company has the production function Q = 2K0.5 10.5, where Q measures output, K measures machine hours, and L measures labor hours. Let the wage rate be W, and suppose that the rental rate of capital is R= $20 and the firm wants to produce 400 units of output. Use the Lagrange method to find the demand curve for labor as a function of the wage rate. Your answer will have L on the left hand side of the equation. On the right hand side, you will have a constant multiplied by W raised to an exponent.

Answer 1

Total Cost is given by

TC=WL+20R

Production function is given by

Q=2K^0.5L^0.5

Set Q=400

400=2K^0.5L^0.5

We need to minimize Cost i.e.

Minimize TC=WL+20*L

subject to

400=2K^0.5L^0.5

Lagrange is given by

Ł=WL + 20K + X(400 – 2K0.5 0.5)

First order conditions give us

dL/dL = W - XK05-0.5 = 0

or

W = XK0.5 -0.5 ---------(1)

dL/0K = 20 - 4K-0.5 0.5 = 0

20 = XK-0.50.5 ---------(2)

dL/dx = 400 – 260.50.5 = 0

or

2K0.50.5 = 400 ----------(3)

Divide equation (1) by equation (2) we get

W/20=K/L

K=WL/20

Set K=WL/20 in equation (3) we get

2*(WL/20)^0.5L^0.5=400

(W/20)^0.5*L=200

L=200*20^0.5/W^0.5

L=894.4272*W^-0.5

-What is the exponent on W?

Exponent on W=-0.5

-What is the constant number on the right hand of equation

894.4272