## 1 – Marginal cost and average cost: what is the difference?

The notion of average cost corresponds well to the intuitive meaning that we can have of it, that is to say the total cost of production divided by the quantity produced.

Marginal cost, on the other hand, corresponds to the cost of producing the last unit produced (or one additional unit), which is different from average cost.

In an optimization context, we talk about marginal cost, but in reality, it would be more appropriate to talk about marginal valuation. Indeed, the rigorous definition of this quantity is the variation of the economic function if we vary the quantity produced by one unit.

We immediately see, from this definition, that it is not necessarily a cost, but an optimal equilibrium value. We also easily understand that to preserve the feasibility of the solution, if we want to produce an additional unit of said product, we must correspondingly reduce the production of other products, and that the notion of marginal cost corresponds, therefore, to a valuation balance which takes into account possible substitutions between flows.

## 2 – Marginal cost: for which applications?

The notion of marginal valuation introduced previously can be generalized to any limiting resource in the system and corresponds to the improvement in the economic function which would result from the relaxation of the resource constraint of a unit. Here are some examples :

- Capacity constraint: the interpretation of the marginal cost is immediate, and corresponds to the improvement in the margin at the optimum, if we had an additional unit of capacity of a particular unit
- Limit of availability of a raw material: if there is a maximum availability of a raw material and this limit is reached at the optimum, the marginal cost of said constraint corresponds to the difference between the optimal valuation that ‘we can obtain from the processing of an additional unit of this material (therefore its marginal value), and the purchase price of this material
- Limit of outlet for a finished product: similarly, the marginal cost corresponding to a maximum limit on the sale of a product is equal to the difference between its selling price and its optimal production cost (its own marginal cost).
- Material balance constraint: we are here in the case introduced in 1. By applying the definition, this is the variation in the economic function which would result from the relaxation of the material balance constraint by one unit. Now this constraint is written: Production – use >= 0

Relaxing this constraint by one unit would amount to using one unit of this flow, without having produced it. This results in a gain which corresponds to the optimal valuation of this flow, as explained in the first paragraph.

## 3 – Dual value in linear programming and marginal cost: what is the link?

The analysis of marginal costs in an optimization solution constitutes a very powerful tool for understanding the economic weight of different constraints and limiting resources. It turns out that this information is immediately available when solving a linear programming model, in the form of “dual variables”. Indeed, to each problem of optimizing the use of resources (called primal) there naturally corresponds a problem of valuing the same resources (dual problem), and vice versa, and the resolution of one immediately gives the solution of the other. Marginal costs therefore correspond to the optimal valuation of limiting resources (their dual value).

## 4 – Emission constraints: a special case to carefully consider

Like any constraint, emission constraints have a marginal cost. If we make it possible to exceed the regulatory limits by resorting to the purchase of emission rights, then the marginal cost is nothing else than the price of the ton of CO2. On the other hand, if the limits are not allowed to be exceeded, then the marginal cost represents the real economic cost that would be incurred in the case of a tightening of the limit by one unit.

Finally, a particular way of formulating the emissions problem makes it possible to calculate the marginal rate of substitution or the marginal CO2 content corresponding to any product.