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Optimal mix of input factors of production for long run cost

Explain why a firm's long-run costs are minimized when it employs a mix of resources such that the ratio of all of the resources' marginal products to their wage rates are equalized. Use a graph to illustrate.

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See the attached file. The text here may not print correctly for tables and symbols. Thanks

Long Run Costs
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Explain why a firm's long-run costs are minimized when it employs a mix of resources such that the ratio of all of the resources' marginal products to their wage rates is equalized. Use a graph to illustrate.
Let us assume that there are two types of resources used for production.
X= Quantity of input X
Y= Quantity of input Y
Px=Wage rate for resource X
Py=Wage rate for resource Y
I= Budget available with the firm to spent on resource X and Y.
The amount of money spent on resource X= X*Px
The amount of money spent on resource Y= Y*Py
Total amount spent on both resources = X*Px + Y*Py
Total Cost (TC) = X*Px + Y*Py
Since the total ...

Solution Summary

This post proves mathematically with the help of equations for budget line, isocost curves and isoquant curves that the long run costs are minimized when the firm employs a mix of resources such that the ratio of all of the resources' marginal products to their wage rates are equalized. All assumptions and symbols used in the equations are explained and the post also includes the graphical representation for easier understanding.

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