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# The production function

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Tall Paul Stewart Brewery produces beer. The most popular style is Tall Paul Stewart Stout. The MBA's in marketing estimate TPS Stoutâ??s production function to be Q = 400L2B, where Q is the number of bottles of TPS Stout produced each week, L is the weekly labor hours, and B is weekly total hours running the bottling line. Each bottling line can operate 120 hours in a week. The cost of labor is \$15/hour, and each bottling line costs \$1200 per week.

Suppose TPS has 20 lines and is producing at an optimal B/L ratio. How many people does TPS employ to make Stout? Assume each person works 40 hours a week.

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#### Solution Preview

The production function is given to be

Q = 400(L^2)*B.

The first step is to find the marginal product with respect to the two inputs: labor, and bottling line. The marginal product with respect to labor is the partial derivative of Q with respect to L, and this means

MPL = (partial 400 L^2 B)/(partial L) = ...

#### Solution Summary

This solution provides a step by step explanation of how to calculate the number of employees that a business needs to employ to produce a given product.

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