The withholding equilibrium

This section will show a possible N-player Nash equilibrium with each power gener-ator withholding capacity in order to increase individual profits. The analysis is based on a paper by Lave and Perekhodtsev looking at power generation in the California ISO area from 2001.⁶⁴⁷

a) Assumptions of the Lave and Perekhodtsev model

The analyzed N-player Nash equilibrium is found based on a continuous linear sym-metric model. This model works with some simplifying assumptions as compared to real electricity markets:⁶⁴⁸

▪ Completely inelastic industry demand is assumed, as well as

▪ complete information of all market participants.⁶⁴⁹

▪ Furthermore, for the generating firms continuous marginal cost of production is assumed instead of stepwise functions. This assumption permits the use of func-tions in the model instead of multiple units. The simplification is justified in the case of power supply, since the average size of the units offered by a generator is negligible in comparison to its total supply.⁶⁵⁰

▪ Finally, symmetry of the generating firms is assumed which helps to determine the Nash equilibrium and avoids accounts for capacity constraints of generators.⁶⁵¹

In order for the industry marginal cost to resemble the real one shown in the first chapter (merit order mechanism)⁶⁵², increasing concave marginal cost functions for each generator are used, such that the marginal cost of a single generator will be

p = dc dx and

with N being the total number of generators in the market, industry marginal cost will be

647 Lester B. Lave and Dmitri Perekhodtsev, “Capacity withholding equilibrium in wholesale electricity mar-kets”, CEIC working paper CEIC-01-01.

648 Ibid, 5-6.

649 Ibid, 1.

650 Ibid, 6.

651 Ibid, 7.

652 See the first chapter, section D.II.4.b) of this work.

dC dX =

dc dxN .

Accordingly, with D being the industry demand, the market-clearing price will be

p = dDdc

N .

Therefore, each of the generators faces an equal demand share d of

d = D N.

b) Withholding Nash equilibrium

The effect of withholding some inframarginal amount of capacity x by one of the generators, assuming that all the other generators´ behavior remains unchanged, results in a gap in supply:

X – x < D.

The residual demand has to be made up for by a shift of the vertical demand line to the right by the amount x, resulting in a higher market-clearing price.⁶⁵³ The individual de-mand lines for each generator shift to the right by an amount x/N. Therefore, total dede-mand for the generator who withheld capacity will change to

d_new = d_old - x + x N.

The choice of the inframarginal unit of capacity to withhold depends on the marginal cost:

It is most profitable for the generator to withdraw the most expansive inframarginal ca-pacity, Δx. However, since the generator foresees the right shift of the demand line by Δx/N as a consequence of him withdrawing capacity, he would prefer to withdraw the formerly ultra marginal capacity Δx, which is, after the shift of the demand line, the most expansive inframarginal unit. Therefore, the generator withholds both:

Δx and Δx(N – 1)/N.⁶⁵⁴

653 This is the ratio of the characteristic of physical capacity retention treated extensively at the beginning of this chapter. See section B.I. for details and a graphical analysis of the case. With regard to the model applied here, see Lester B. Lave and Dmitri Perekhodtsev, “Capacity withholding equilibrium in wholesale electricity markets”, CEIC working paper CEIC-01-01, 7.

654 Ibid.

Assuming the generator reasoning this way, market price will increase by Δx/N. The extra profit (∏extra) earned from withholding the inframarginal capacity is

π_extra (∆x) = D - ∆x (N – 1) N

∆x N , as opposed to a sacrificed profit (∏sac) of

π_sac (∆x) = ∆x² (N – 1)² 2N² . The total change in profit Δ∏ is therefore

∆π = D ∆x

N² - ∆x²N - 1 N² - 1

2 ∆x² (N - 1)² N² . This quadratic function has its maximum at

∆x = D N²- 1.⁶⁵⁵

This amount is hence being retended by producers. Having derived the optimal behavior for a single producer, we can now turn to the examination of a market equilibrium Δx*

with all the agents withholding capacity Δx* and nobody wanting to deviate from this choice. It is assumed that N – 1 market participants already withheld the capacity Δx* in the way described above. The question becomes therefore, what would be the optimal capacity for the nth agent to withhold.

Lave and Perekhodtsev show, that due to N – 1 agents each already having withhold Δx*, the industry demand line changed to

D´ = D + Δx*(N – 1).

Thus, the optimal amount to withdraw from the market for the nth agent is

∆x = D + Δx*(N – 1) N²- 1 .⁶⁵⁶

655 For the derivation of the profit functions, please refer to ibid, 8. The maximum of the quadratic profit func-tion is found by calculating the first derivative of this funcfunc-tion with regard to Δx, which is, after combining and rearranging terms, ^d∆π_d∆x = D + ∆x - ∆xN²

N² , and equalizing it to zero.

656 For the derivation of the profit functions, please refer to ibid, 9.

In order to find the withholding equilibrium for the market, Δx has to equal Δx*. The solution to the resulting equation in terms of Δx* delivers the amount of capacity being withhold by each generator in equilibrium:

∆x* = D

N (N – 1).⁶⁵⁷ Total withholding in the market is therefore

∆X* = D N - 1.⁶⁵⁸

As a result of the above analysis, Lave and Perekhodtsev find that for any demand D faced by the industry, the total withholding is D/(N – 1). Hence, the industry inverse supply function changes to

p = X N - 1.⁶⁵⁹

As compared to the competitive case where supply equals marginal cost (p = X/N), this means that “whatever the demand is, the resulting price will always be by N/(N – 1) higher in the case of linear marginal cost” if capacity is held back.⁶⁶⁰