Final answer:
The equation for the production possibilities frontier can be derived by determining the opportunity cost of producing one good over another in a trade-off. The slope of the PPF is constant in linear models and reflects this opportunity cost. The slope is calculated by determining the rate at which one good must be given up to produce an additional unit of another good.
Step-by-step explanation:
To find the equation for the production possibilities frontier (PPF), we consider the trade-off between the two goods being produced, refrigerators and shoes, in this case. The slope of the PPF represents the opportunity cost of producing one additional unit of a good at the expense of the other good. In linear PPF models, the slope is constant, hence the opportunity cost remains the same along the entire frontier. For example, in the production scenario at Plant 1 between snowboards and skis, the slope is calculated to be −2 pairs of skis/snowboard, indicating that for one additional snowboard produced, two pairs of skis must be forgone.
If the production function in the question above were similar, and assuming the PPF is linear, we would calculate the slope by determining the change in shoe production when one more refrigerator is produced. The negative slope would reflect the number of shoes that must be given up to produce an additional refrigerator. For instance, if producing one fridge required giving up five pairs of shoes, the slope of the PPF would be −5 shoes/refrigerator, and the equation would be based on the specific production capabilities of the economy.