# I. Introduction

Restructuring a region’s electricity sector often leads to the creation of an independent system operator (ISO) of an electric grid, a competitive wholesale market for generation, a regulated market for transmission and distribution (T&D), and a competitive retail market for end-use consumption (Sioshansi, 2013). In the US, an electric grid’s hourly wholesale market price is determined by an ISO’s centralized market operation (Woo et al., 2019). Generation companies (Gencos) compete for sales to wholesale buyers, including local distribution companies and retail service providers (Woo et al., 2019).

A wholesale electricity market’s hourly prices ($/MWh) are highly volatile, with occasionally sharp spikes (Eydeland & Wolyniec, 2003). An important strand of the literature on retail pricing under volatile wholesale market prices is the pass-through of retail prices based on wholesale market price changes. The pass-through is partial, increases with wholesale price volatility, increases with rising retail competition, and declines with customer search costs (Brown, Tsai, et al., 2020; Deller et al., 2021; Fabra & Reguant, 2014; Hartley et al., 2019).

Partially driven by its forward-looking per MWh procurement cost (Brown, Tsai, et al., 2020), an electricity retailer’s fixed price offer (FPO) depends on vertical integration that enables economies of coordination, scale, and scope (Arocena, 2008; Fetz & Filippini, 2010). A substantive research question thus arises: does vertical integration * always* reduce an electricity retailer’s FPO? Underlying this question’s real-world relevance and policy importance is the ubiquity of fixed price plans (Brown, Tsai, et al., 2020) and mergers of large Gencos and retailers (Brown, Zarnikau, et al., 2020; Brown & Sappington, 2021).

The question’s commonly received answer is “yes,” because vertical integration tends to reduce a retailer’s FPOs (Brown & Sappington, 2021; de Bragança & Daglish, 2017; Mydland, 2020). However, this answer may not hold when residential customers are segmented by consumption size and price sensitivity. As demonstrated below, our answer is based on the profit-maximizing pricing behaviour of a Gentailer (GT) affiliated with a Genco and a non-Gentailer (NGT) unaffiliated with a Genco.

# II. A non-Gentailer’s Optimal FPO

Consider a risk-neutral NGT’s FPO of *F*_{1}($/MWh) for a 12 -month contract period. To meet its contractual obligation, a NGT procures in hour at dayahead market (DAM) price *P*_{h} ($/MWh). Mirroring the time sequence of retail market transactions, is the *ex-post* MWh amount bought * after contract commencement* by a NGT customer. However, when making its FPO, a NGT does not know and and can only set based on its expectation of and subsequently realised in the FPO’s contract period.

Suppose a NGT’s average customer’s *ex-post* linear demand is:

\[Q_{1 h}=\alpha_h+\beta_1 F_1+\omega_h,\tag{1}\]

where

is the time-dependent intercept, is s price response to a $1/MWh change in and is a random disturbance with zero mean and finite variance. The ex-post profit per customer in the contract period is:\[\pi_1=\sum_h\left(F_1-P_h\right) Q_{1 h}.\tag{2}\]

Let (Mood et al., 1974), a NGT’s expected profit per * contracted* customer is:

\[ \begin{align} \mathrm{E}\left(\pi_1\right)&=\sum_h \mathrm{E}\left[\left(F_1-P_h\right) Q_{1 h}\right]\\ &=\sum_h\left[\left(F_1-\mu_{P h}\right) \mu_{Q 1 h}-\rho \sigma_{P h} \sigma_{Q 1 h}\right] \text {, } \end{align} \tag{3} \]

with (Woo, Moore, et al., 2016), and under the linear demand assumption.^{[1]}

We assume the probability of a customer contracting with a NGT is

), where marketing effort of a NGT; FPO of a GT; and marketing effort of a GT. We further assume decreases with and but increases with and This is because when a NGT lowers and raises it increases the probability of customer signup. However, this probability declines when a GT reduces and raises As the probability for a GT is mirrors the market equilibrium of retailer choices made by residential customers.The unconditional expected profit of a NGT * before contract signing* is:

\[ \kappa_1={J} \mathrm{E}\left(\pi_1\right)-{C}_1, \tag{4} \]

where *C*_{1} is a NGT’s marketing cost that increases with *M*_{1}. The first order conditions for *F*_{1} > 0 and *M*_{1} > 0 that maximize *κ*_{1} are:

\[ \frac{\partial \kappa_1}{\partial M_1}=E\left(\pi_1\right) \frac{\partial J}{\partial M_1}-\frac{\partial C_1}{\partial M_1}=0; \tag{5} \]

\[ \frac{\partial \kappa_1}{\partial F_1}=J \frac{\partial \mathrm{E}\left(\pi_1\right)}{\partial F_1}+\mathrm{E}\left(\pi_1\right) \frac{\partial J}{\partial F_1}=0 . \tag{6} \]

Equation (5) states that at maximum *κ*_{1}, E(*π*_{1}) ∂*J*/∂*M*_{1} = marginal benefit should equal ∂*C*_{1}/∂*M*_{1} = marginal cost of a NGT’s marketing effort. Hence, when a NGT has a high marginal marketing cost, it employs low marketing effort.

Using equation (3), we determine:

\[ \begin{align} \frac{\partial \mathrm{E}\left(\pi_1\right)}{\partial F_1}&=\sum_h\left[\mu_{Q 1 h}+\left(F_1-\mu_{P h}\right)\left(\frac{\partial \mu_{Q 1 h}}{\partial F_1}\right)\right]\\ &=\sum_h\left[\mu_{Q 1 h}+\left(F_1-\mu_{P h}\right) \beta_1\right]. \end{align} \tag{7} \]

Equations (6) and (7) jointly inform a NGT’s optimal FPO

\[ {F}_1=-\left(\frac{\mu_{Q 1}}{\beta_1}\right)+\mu_{P}-\Delta_1, \tag{8} \]

where

average of a NGT’s expected hourly sales per contracted customer; average of a NGT’s expected hourly per MWh procurement costs; andWe expect *∆*_{1}≈ 0 because ∂*J*/∂*F*_{1}≈ 0, as a $1/MWh (= 0.1 cent per kWh) increase in *F*_{1} is unlikely to cause NGT customers to switch and contract with GTs. Hence, equation (8) simplifies to:

\[ {F}_1=-\left(\frac{\mu_{Q 1}}{\beta_1}\right)+\mu_{P}. \tag{9} \]

Equation (9) reveals a NGT’s optimal *F*_{1} price contains a forward premium of (*F*_{1} – *μ _{P}*) = − (

*μ*

_{Q}_{1}/

*β*

_{1}) > 0. This premium increases with average consumption size measured by

*μ*

_{Q}_{1}but declines with

*ex-post*demand’s price sensitivity measured by

*β*

_{1}.

To illustrate *F*_{1}’s calculation, suppose *μ _{Q}*

_{1}= 0.0014 MWh

^{[2]},

*β*

_{1}= -0.0002

^{[3]}, and

*μ*= $50/MWh

_{P}^{[4]}. The resulting optimal

*F*

_{1}is $[(0.0014 / 0.0002) + 50] = $57/MWh, closely matching the FPO data (net of T&D charges) available from Texas’s Power to Choose website operated by the Texas Public Utilities Commission (Brown, Tsai, et al., 2020).

The forward premium embodied in *F*_{1} is $(57 – 50) = $7/MWh, which is 14% of *μ _{P}* = $50/MWh. Should

*β*

_{1}be -0.0003, the optimal

*F*

_{1}would decline to $[(0.0014 / 0.00003) + 50] = $54.7/MWh. Hence, a NGT’s optimal

*F*

_{1}and forward premium decline when NGT customers consume less or become more price sensitive under smart metering (Jessoe & Rapson, 2014).

# III. A Gentailer’s optimal FPO

Consider a risk-neutral GT affiliated with a Genco that uses natural gas as the marginal fuel (Woo, Moore, et al., 2016; Zarnikau et al., 2019, 2020). A GT’s expected hourly per MWh procurement cost in the contract period is E(*X _{h}*) =

*μ*<

_{Xh}*μ*, where

_{Ph}*X*= min(

_{h}*P*,

_{h}*HR × G*), where

_{h}*HR*= marginal heat rate (MMBtu per MWh) of natural-gas-fired generation and

*G*= spot natural gas price ($/MMBtu) in hour

_{h}*h*(Woo, Horowitz, et al., 2016). This is because a GT would procure from the wholesale market if the hourly wholesale market price

*P*is less than own-generation’s hourly marginal fuel cost of

_{h}*HR × G*.

_{h}A GT’s unconditional expected profit is:

\[ \kappa_2=(1-J) \mathrm{E}\left(\pi_2\right)-C_2, \tag{10} \]

where *π*_{2} is analogously defined as *π*_{1} and *C*_{2} is a GT’s cost of marketing effort. Using the same steps for deriving equation (9), a GT’s optimal FPO is:

\[ {F}_2=-\left(\frac{\mu_{Q 2}}{\beta_2}\right)+\mu_{X}, \tag{11} \]

where

average of a GT’s expected hourly sales per contracted customer; price coefficient of a GT customer’s ex-post linear demand, and average of a GT’s expected hourly per MWh procurement costs.To calculate *F*_{2}, we assume *μ _{Q}*

_{2}= 0.0028 MWh,

*β*

_{2}= -0.0001 and

*μ*= $30/MWh. Sharply different from those used to compute a NGT’s

_{X}*F*

_{1}, these assumptions are made to demonstrate the combined price effect of customer size,

*ex-post*demand’s price sensitivity, and per MWh procurement cost. The resulting

*F*

_{2}is $[(0.0028 / 0.0001) + 30] = $58/MWh, embodying a forward premium of $28/MWh, that is 93% of

*μ*= $30/MWh. Despite the $20/MWh difference between

_{X}*μ*and

_{P}*μ*,

_{X}*F*

_{2}is $1/MWh higher than

*F*

_{1}= $57/MWh, which is found in the last section. Further,

*F*

_{2}’s large forward premium encourages vertical integration observed in Europe and the US (Brown & Sappington, 2021).

# IV. FPO Comparison and Market Segmentation

The FPO difference between a NGT and GT is

\[ D=F_1-F_2=A+B, \tag{12} \]

where *A* = (*μ _{Q}*

_{2}/

*β*

_{2}) - (

*μ*

_{Q}_{1}/

*β*

_{1}) and

*B*= (

*μ*-

_{P}*μ*) > 0. Affirming the commonly received answer in Section 1, equation (12) shows

_{Χ}*D*> 0 if

*A*> 0 or

*B*exceeds |

*A*| when

*A*< 0

^{[5]}.

Equation (12) also shows the necessary condition for *D* < 0 is

\[ \mathrm{A}=\left(\frac{\mu_{Q 2}}{\beta_2}\right)-\left(\frac{\mu_{Q 1}}{\beta_1}\right)<0, \tag{13} \]

implying that a GT’s optimal *F*_{2} can exceed a NGT’s optimal *F*_{1} when GT customers consume more electricity and are less price sensitive than NGT customers. When *F*_{1} < *F*_{2}, it indicates retail market segmentation, with relatively large customers preferring a GT and relatively small customers preferring a NGT.

A plausible explanation for retail market segmentation is that, when selecting a retailer, relatively large customers are more concerned with a retailer’s reputation and financial strength than relatively small customers. Moreover, a NGT likely has a higher marginal marketing cost than a GT that enjoys economies of coordination, scale, and scope. Consequently, a NGT tends to rely more on price competitiveness than a GT to attract customer signups.

# V. Conclusion

Our paper suggests that an electricity retailer’s residential FPO contains a large forward premium that encourages vertical integration. However, a GT’s FPO can exceed that of a NGT when GT customers have higher usage and are less price sensitive than NGT customers. Hence, a proposed merger of a big Genco and a big NGT requires close regulatory scrutiny to sustain the retail market’s price competition.

After a small change in

*F*_{1}, the hourly demand’s variance is var(*Q*_{1h}+*β*_{1}∆*F*_{1}). As*β*_{1}∆*F*_{1}is non-stochastic, var(*Q*_{1h}+*β*_{1}∆*F*_{1}) = var(*Q*_{1h}). Hence, the change in*F*_{1}does not affect var(*Q*_{1h}) =*σ*_{Q}_{1h}^{2}.The value of 0.0014 MWh = 1.4 kWh = (1,000 kWh ÷ 720 hours per month) matches monthly consumption of ~1,000 kWh per customer.

Implied by

*β*_{1}= -0.0002,*ε*= -0.0002 × $100 per MWh ÷ 1 MWh = -0.02 is the own-price elasticity of monthly retail consumption at the total retail price of ~$100/MWh which includes T&D charges. This implied*ε*matches the short-run elasticity estimates reported by Li et al. (2021).Changing

*μ*_{P}from $50/MWh to $55/MWh or $45/MWh would alter the numerical results below. However, it would not qualitatively affect our answer to the policy question posted in Section I.For example, if

*β*_{2}=*β*_{1}= -0.0002, then*F*_{2}will become $47, lower than the*F*_{1}obtained in Section 2.