A bottling company uses two inputs to produce bottles of the soft drink Sludge: bottling machines $)$ 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 ${?}_K)$ is an additional $432$ bottles per day, and the marginal product of labor $(M{P}_L)$ is $25$ 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.

The bottling company

A$.$ is not minimizing the cost of production because $M\frac{P_K}{r}>M\frac{P_L}{w}$ and should use less labor and more machines.

B$.$ is not minimizing the cost of production because $M\frac{P_K}{r}>M\frac{P_L}{w}$ and should use more labor and fewer machines.

C$.$ is not minimizing the cost of production because $M\frac{P_K}{M}{P}_L=\frac{r}{w}$ and should use more labor and fewer machines.

D$.$ is minimizing the cost of production because $M\frac{P_K}{r}=M\frac{P_L}{w}.$

E$.$ is not minimizing the cost of production because $M\frac{P_K}{r}$ and should use more labor and fewer machines.