# Duality in the forestry industry

Loggers often face a dual problem regarding the means used to carry timber out of the harvesting place. They can either use rented trucks (A) or order the transport to someone (B). In the first case, the logger ends up paying a fixed cost plus an additional cost per ton, while in the second choice he ends up paying a cost per ton that is higher than just the cost per ton in the former case.

This choice problem can be modeled as giving rise to two different budget constraints. Let us plot the number of tons of timber carried in a truck in the horizontal axis and Swedish kronor (SEK) of other consumption of goods in the vertical axis. If a logger is of type B, the budget constraint will have a higher intercept, since type B loggers do not face the costs perceived as fixed, which are the same regardless of how many tons they carry. At the same time, the slope of the budget constraint would be steeper for type B loggers, because each additional ton carried has a higher opportunity cost.

These two loggers face the choice between two budget constraints, as follows:

Both consumers are referred as having the same preferences and facing the same choice problem. Hence, they are indifferent between these two modes of transportation (and thus between the two budget constraints). The loggers end up in the same indifference curve and they have the same map of indifference curves, as well as the same level of exogenous income.

Figure 2 shows both optimal choices:

Each budget constraint is tangent to the single indifference curve, therefore the loggers reach the same utility level. Logger B’s optimal choice is at point 2 (indicating less tons carried out than logger A) and logger A’s optimal choice is at point 1, that is, they ended up making different choices. Different opportunity costs impact differently on behavior.

Intuitively, once logger A rents a truck, he has to pay the fixed costs, no matter what else he did. So, the opportunity cost of transporting a ton of timber (once logger A has decided to rent a truck) was only the cost per ton. Conversely, logger B faces a higher opportunity cost since he had to pay a higher cost per ton. Even though their optimal choices rely on two different budget constraints, they reach the same utility level. However, it is clear that the lower opportunity cost of transporting the timber made logger A to transport more tons and consume less of other goods than logger B.

Thus, under the implementation of a reforestation plan by a landowner where loggers face a term to cut the old trees and carry the timber out of the harvesting place, i.e. loggers only cut the forest (win the harvesting proposal) if they sign a contract with a landowner that states clearly a term, type A loggers will most likely perform this operation faster than type B loggers, since they transport more tons of timber. Nevertheless, they consume less of other goods and still reach the same utility level.  Therefore, a logger will rent trucks instead of ordering the transport to someone, since it gives him a higher chance of fulfilling the contract.

Nuno Lourenço #85

Source: Thomas Nechyba – Microeconomics: An Intuitive Approach, adapted example