Question

Production Function: ?=2?^2?^1/4. ??????????????????????=12?^2?^―3/4. ????????????????????????=4??^1/4. Wages = $20 Rental Rate = $150 Output = 1000

Solve for the cost-minimizing input bundle.

Graph the isocost line, isoquant, and cost-minimizing bundle. Make sure to label the slope and intercepts.

Answer 1

Given:

Production function:

The marginal product of labor: MP_L =

The marginal product of capital: MP_K =

Wage = $20

Rental rate = $150

Output q = 1000

The cost minimization input bundle is where the marginal rate of technical substitution is equal to the ratio of the price of inputs.

That is, where the ratio of marginal product of labor and capital is equal to the ratio of wage and rental rate.

This is a cost minimization condition.

Now we have to find the optimal bundle for the production level of q = 1000

Therefore putting this value in the production function we get the equation as:

Now putting from the cost minimization condition:

into the production function

There the number of labor required is:

Putting this value of L into the cost minimization condition we found above,

Here for the sake of clarity, taking the round off.

Therefore the cost minimization optimal input bundle is:

L = 15 and K = 16

b.)

Isocost line:

The isocost line shows all the combinations of two inputs that the firm could hire with the given cost.

here, the cost is:

Putting the optimal quantity of inputs and prices.

Therefore now forming the equation of the isocost line:

Putting the values of the price of inputs and total cost.

Therefore the isocost line is:

Putting labor units on the x-axis and capital on the y-axis.

The x-intercept is when K = 0

therefore x-intercept is 135.

The Y-intercept is when L = 0

therefore y-intercept is 18

The slope of the isocost is the ratio of the price of inputs = 20/150 = 0.1333

Therefore the isocost cost line:

Isoquant curve for output level 1000:

Optimal bundle:

Labor units = 15 and capital units = 16

The IC is the isocost line. the cost minimization bundle is (15,16) at point o where the isocost is tangent to the isoquant curve.