Left and RightHand Marginal Costs
In the example of Figure 1-6.2, the marginal cost of production goes from $30 on the left of 10 GW to infinity on the right of 10 GW. This is a double complication. Not only does marginal cost change abruptly, it becomes infinite. The present
Right- and left-hand marginal costs.
10,000 MW Q
10,000 MW
10,000 MW Q
10,000 MW
definitions can be illustrated more clearly with a less pathological marginal-cost curve.
Figure 1-6.3 shows the total cost curve and the marginal cost curve of a simple market. The discontinuity is the jump in marginal cost at the 10 GW output level. To the left of 10 GW the marginal cost is $20/MWh, while to the right it is $40/MWh. But what is the marginal cost precisely at 10 GW? It is undefined, but, as every textbook would confirm, the answer is not MC = $20/MWh.
To formalize this definition, it is useful to consider the mathematics of the total cost curve shown at the left of Figure 1-6.3. To the left of 10 GW, its derivative (slope) is $20/MWh, while to the right its slope is $40/MWh. But the mathematical definition of a derivative breaks down at 10 GW, and since marginal cost is just the derivative of total cost, the definition of marginal cost also breaks down at this point. Mathematics does define two very useful quantities at the 10-GW point, the left-hand derivative (slope) and the right-hand derivative (Courant 1937, 199-201). These are, of course, $20 and $40/MWh, respectively. Because marginal cost is just the derivative, it is natural to define left-hand marginal cost (MCLH) as the left-hand derivative, and right-hand marginal cost (MCRH) as the right-hand derivative. Other points along the total cost curve also have left and right-hand derivatives, and these are just equal to the normal derivative. Similarly, MCLH and MCrh are normally equal to each other and equal to standard marginal cost, MC.
The marginal-cost range, MCR, is defined as the range of values between and including MCLH and MCRH. This definition is motivated by the idea that marginal cost cannot be pinned down at a point of discontinuity but can reasonably be said to lie somewhere between the savings from producing one less and the cost of producing one more unit of output.
Definitions Left-hand marginal cost (MCLH)
The savings from producing one less unit of output. Right-hand marginal cost (MCRH)
The cost of producing one more unit of output. When this is impossible, MCRH equals infinity.
The marginal-cost range (MCR)
The set of values between and including MCLH and MCRH.
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