Constraint optimization problem

Question

Suppose pigs $\left(Q\right)$ can be fed corn-based feed $\left(C\right)$ or soybean-based feed $\left(S\right)$ such that the production function is $Q = 2C + 5S$. If the price of corn is feed is $\$4$ and the price of soybean feed is $\$5$, what is the cost minimizing combination of producing $p=200$?

Answer 1

The first thing to do is to look at the marginal utilities of each of your inputs, i.e. the corn and the soybean feed. From the production function, we see that per unit of corn, we get $2$, and per unit of soybean, we get $5$. Thus, the ratio of marginal utilities of corn to soybean is $\frac{2}{5}$. Then, we can look at the price ratio of corn to soybean, which is $\frac{4}{5}$. What does this tell us about which feed we're going to prefer? How much of that feed are we going to use? (Hint: look at the difference in price ratio and marginal utility.)

An alternative method: for every four dollars I invest in corn, I get two units of "pig" in return. For every five dollars I invest in soybean, I get five units of "pig" in return. Which of these two has a better cost/return ratio? Would I use the other input if these ratios are not equal?