A bottling company uses two inputs to produce bottles of the soft drink Sludge: bottling machines (K) and workers (L). The isoquants have the usual smooth shape. The machines cost $1,000 per day to run (r), and the workers earn $200 per day (w). At the current level of production, the marginal product of machines (MP Subscript Upper K) is an additional 316 bottles per day, and the marginal product of labor (MP Subscript Upper L) is 39 more bottles per day. Is this firm producing at minimum cost? If it is minimizing cost, explain why. If it is not minimizing cost, explain how the firm should change the ratio of inputs it uses to lower its cost.

In order to minimizing the cost for a given level of output, the firm should equate the weighted marginal product of capital with the weighted marginal product of labor.

[tex]\frac{MP_{K} }{r}= \frac{MP_{L} }{w}[/tex]

Put the value in the above equation, we get

[tex]\frac{316}{1,000}= \frac{39}{200}[/tex]

0.316 > 0.195

Now, [tex]\frac{MP_{K} }{r}>\frac{MP_{L} }{w}[/tex], so the firm is not minimizing its cost in producing the bottles of the soft drink Sludge.

Hence, in order to minimize cost the firm should substitute labor with more of capital, so that MP 'K' falls and become equal to MP 'L'.