Science:Math Exam Resources/Courses/MATH105/April 2010/Question 04
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The production function for a firm is
where and are the number of units of labour and capital utilized respectively. Suppose that labour costs $108 per unit and capital costs $2 per unit and that the firm decides to produce 600 units of goods. Use Lagrange multipliers to determine the amounts of labour and capital that should be utilized in order to minimize cost. You need not show that your solution minimizes the cost.
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What are the objective function and the constraint function?
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We are given that labour costs $108 per unit and capital costs $2 per unit, and we want to minimize cost. Thus the objective function is
The firm wants to produce 600 units, so using the production function we want . Thus, after dividing both sides by 5, the constraint function is
and the constraint is g(x, y) = 0.
The Lagrange multiplier method says that the gradients of the objective function and the constraint function should be proportional, or
The gradient of the objective function is
and the gradient of the constraint function is
Then we have three equations that must be satisfied
Solving the second equation for gives
and substituting into the first equation to eliminate gives
Simplifying, we have
and then we solve for y to get
Now substituting into the third equation, the constraint equation, gives
Simplifying, we have
We then have that
Therefore the cost is minimized when using 40 units of labour and 1080 units of capital.