Supplemental Material
Output Maximization

The textbook concentrates on cost minimization. If its technology has smooth isoquants, the firm can determine the best factor proportions to use by answering either of two questions:

  1. Cost minimization: What is the lowest cost, C*, at which the firm can produce output q*?

  2. Output maximization: What is the most output, q*, that can be produced at cost C*?

By answering each of these questions in turn, we show how the firm determines the optimal combination of inputs. [Economists (who apparently have nothing better to do with their time) have discovered a few strange production processes in which minimizing costs gives a different answer about factor proportions than maximizing output. For virtually any production function where output varies continuously with inputs, however, the answers to these two questions are the same.]

Cost minimization by a Norwegian printing firm is discussed in the text. Equivalently, the printing firm could ask: What is the most output, q*, that can be produced at cost C* = 2,000 kr? The analysis is virtually the same as was used to answer the cost-minimization question. Here, the firm finds the highest isoquant that touches the 2,000 kr-isocost line. The q = 100 isoquant is tangent at point x (L = 50, K = 100), as shown in the figure:

It is possible to produce 50 units of output for 2,000 kr (points y and z), but the firm does not do so because it would rather produce 100 units than 50 units if they cost the same. At a cost of 2,000 kr, the firm cannot produce q = 150, as shown.

Thus, at the output-maximizing solution, the firm operates where its 2,000 kr-isocost is tangent to its q = 100 isoquant. The output maximization combination of inputs is, again, determined by the tangency between an isocost line and an isoquant. Thus, we get the same condition for output maximization as we did for input minimization: Produce where the last dollar spent on labor gives you as much extra output as the last dollar spent on capital: MPL/w = MPK/r.

© 2003 Jeffrey M. Perloff. Reprinted by permission.

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