The production of a unit of good \(\mathrm{Y}\) requires the employment of 3 workers and 7 units of capital. The going wage is \(\$ 4 .\) The rent on a unit of capital is \(\$ 1\). What should be the marginal physical product of capital in order for the production to be carried out at the least cost and what is this cost if the marginal physical product of labor is \(\$ 2 ?\)

The concept of the Marginal Rate of Technical Substitution (MRTS) is a crucial piece of machinery in the workshop of economics. It's essentially a way to gauge how much of one input, say labor, a producer can replace with another input, like capital, while maintaining the same level of output. Think of it as a trade-off ratio.
MRTS is calculated by dividing the Marginal Physical Product (MPP) of one production factor by the MPP of another. If we're talking about labor and capital, the formula might look something like \( MRTS = \frac{MPPL}{MPPK} \), where MPPL is the marginal physical product of labor, and MPPK is that of capital. A vital understanding here is that MRTS is connected to cost minimization because it helps businesses decide the most cost-effective combination of inputs.
For example, imagine you want to make a pizza and you can use either more workers or more ovens. If hiring another worker can replace the need for an extra oven and if workers are cheaper, the MRTS tells you how many ovens one worker can replace. This way, you can minimize costs while still churning out the same tasty number of pizzas.

Diving further into the cauldron of economics, we find 'cost minimization', a goal for any profit-seeking sorcerer out there. The magic spell here is to produce a specified level of output at the lowest possible cost.
It’s like going shopping with a list; a business needs to tick off all its production needs without splurging unnecessarily. By using MRTS, companies can figure out the exact blend of inputs to minimize costs without sacrificing product quality. It's not about just buying cheaper materials or paying less for labor, but about finding that sweet spot where each penny is perfectly allocated for efficiency.
Returning to our pizza parlor, cost minimization would be about deciding how many extra hours to have your current ovens running vs. hiring more workers or buying more efficient machinery. You want the best pizzas (output) for the fewest dimes spent, and analyzing MRTS can help you achieve that, by showing the most cost-effective way to substitute one resource for another.

Production factors are the bread and butter of any manufacturing or service industry. They refer to the resources employed in the creation of goods and services—namely labor, capital, and land. These components are the heroes in the narrative of production; think of them as the main characters of a story, each playing a distinct role.
In our given problem, labor and capital are the two production factors under the spotlight. Labor represents the human effort—workers rolling out dough in our metaphorical pizza kitchen—while capital is any manufactured aid used in production, like the ovens baking the pies. Each factor comes with a cost: labor demands wages, and capital requires rent or interest.

Understanding the Interplay of Factors

Analyzing how these factors interact is vital for a business. They must consider the laws of diminishing returns, where after a point, adding more of one factor, say labor, brings in less and less extra output. Businesses have to juggle these factors, constantly calculating the optimal combination that leads to cost-effective yet bountiful production. And with the concepts of MRTS and cost minimization in hand, they're well-equipped to make these crucial decisions.