Can price volatility enhance market power? The case of renewable technologies in competitive electricity markets

Accepted Manuscript Title: Can Price Volatility Enhance Market Power? The Case of Renewable Technologies in Competitive Electricity Markets Author: Irena Milstein Asher Tishler PII: DOI: Reference:

S0928-7655(15)00020-2 http://dx.doi.org/doi:10.1016/j.reseneeco.2015.04.001 RESEN 937

To appear in:

Resource and Energy Economics

Received date: Revised date: Accepted date:

21-1-2014 18-10-2014 5-4-2015

Please cite this article as: Milstein, I., Tishler, A.,Can Price Volatility Enhance Market Power? The Case of Renewable Technologies in Competitive Electricity Markets, Resource and Energy Economics (2015), http://dx.doi.org/10.1016/j.reseneeco.2015.04.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Can Price Volatility Enhance Market Power? The Case of Renewable Technologies in Competitive Electricity Markets

Holon Institute of Technology, 52 Golomb St., Holon 58102, Israel

cr

a

([email protected])

Faculty of Management, Tel Aviv University, Tel Aviv 69978, Israel

us

b

ip t

Irena Milsteina and Asher Tishlerb

corresponding author ([email protected])

an

17/10/2014 Abstract

M

This paper develops a two-stage model with endogenous capacity and operations to assess the practicality of photovoltaic technology (PV) in

d

competitive electricity markets. Applying our model to stylized data of California's electricity market we demonstrate that electricity price spikes are

te

higher and more frequent the higher the PV capacity. Consequently, the average electricity price rises when construction costs of PV capacity decline

Ac ce p

due, for example, to technology improvements, bestowing market power and excessive profits on producers employing fossil-using technologies. We also show that an increase in the number of PV-using firms and higher CO2 tax

reduce consumer surplus.

Keywords: price volatility, market power, electricity, renewable technologies JEL codes: D43, L11, L94, D24 We are grateful to a referee of this journal, Y. Amihud, C. Candelise, B. Horii, I. Spekterman, C.K. Woo and the participants in the BIEE conference on “European Energy in a Challenging World: The impact of emerging markets”, September 2012, Oxford, for valuable comments and suggestions.

1 Page 1 of 63

1. Introduction Generating electricity from renewable energy sources is believed to be one of the main remedies for the fast-increasing problems of greenhouse gases

ip t

(GHG) and air pollution (Tol, 2006; Weyant, de la Chesnaye and Blanford, 2006; Lior, 2010; Friedman, 2011).1 Research on renewable energy suggests, however, that the road to a “green world” is not yet fully paved and the

cr

potential and limits of renewable energy remain insufficiently explored and

understood (Badcock and Lenzen, 2010; Lior, 2010; Trainer, 2010; Blumsack

us

and Fernandez, 2012; Borenstein, 2012). It also points out that the high costs of producing electricity from renewable energy will likely raise electricity

an

prices substantially, unless new technologies of electricity generation are developed and adopted (Odenberger and Johnsson, 2007; Borenstein, 2012). This paper demonstrates that integrating renewable energy such as

M

photovoltaic technology (PV) into the electricity market may indeed lead to higher electricity prices.2 Unlike previous research, we show that higher price

d

volatility, not higher production costs, is the culprit, as it bestows market power on fossil-using electricity producers, and more so the lower the costs of

te

PV capacity (due to PV technology improvements, say) and the greater the number of PV-using producers in the market. More specifically, we show that

Ac ce p

electricity price spikes are higher and more frequent the higher the PV

1

Other means to reduce air pollution and greenhouse gases are, for example, conservation,

demand-side management, better use of transmission and distribution systems and smart grids. In fact, the smart grid enables better use of transmission and distribution systems as well as demand-side management. A significant increase in nuclear power in electricity generation is unlikely in the near future (Lior, 2010; Renewables, 2011; Dittmar, 2012; The Economist, 2012) and new construction of hydroelectric power is limited to specific countries (Renewables, 2011). 2

To simplify the exposition, our model only employs combined cycle gas turbines (CCGT)

and PV plants. Our methodology and results apply equally to other fossil-dependent technologies as well as weather-dependent renewables such as wind, solar-thermal technologies, and sea waves.

2 Page 2 of 63

capacity, and that this phenomenon will be exacerbated by the introduction of CO2 taxes.3 Consequently, the average price paid by electricity customers will likely increase when construction costs of PV capacity decline due, for

ip t

example, to PV technology improvements, yielding market power and excessive profits to electricity producers employing the fossil-using

technology (CCGT).4 We also demonstrate that consumer surplus decreases

cr

whereas overall welfare may increase or decrease when the CO2 tax rate

and/or the number of PV-using producers increase. Finally, we show that the

us

choice of market structure (the number of generation technologies that can be constructed by each producer) may significantly affect price volatility, the

an

average electricity price, industry profits, and welfare.

Our results are derived in the context of a two-stage decision-making process aimed at disentangling the intricate relationships among the costs of

M

capacity construction and electricity production by fossil-using and renewable technologies, the optimal generation capacity mix, and electricity price volatility and price level.5,6 We consider two types of generating technologies:

d

(1) “regular”, fossil-using, technologies such as combined cycle gas turbines

te

(CCGT); and (2) weather-dependent renewable technologies in the form of

Ac ce p

photovoltaic cells (PV). In the first stage of the model, when only the

3

Effective use of renewable energy depends on the tradeoff between the higher cost of

electricity from these sources versus the benefits they deliver in abating local pollution and mitigating greenhouse gases (Borenstein, 2012). Thus, research and public policy debates in the coming decade will likely focus on strategies and technologies aimed at increased conservation and on the development of renewable energy to displace the use of fossil fuels (Lior, 2010; Trainer, 2010; Traber and Kemfert, 2011).

4

The phenomenon of higher price volatility implying higher profits was observed in Tishler

(2008), who demonstrated, for linear demand and constant marginal cost, that the R&D project with the highest variance yields maximal expected profits for risk-neutral firms. 5

Substantial price volatility due to sudden and unexpected change in wind generation is

reported by ERCOT in Texas (Hardy and Nelson, 2010). 6

Analysis of demand volatility in electricity markets with renewable energy is not new. See,

among others, Holland and Mansur (2008) and Chao (2011).

3 Page 3 of 63

probability distribution functions of future daily electricity demands and weather conditions are known, profit-seeking producers maximize their expected profits by determining the capacity to be constructed from each

ip t

technology. In the second stage, once daily demands and weather conditions become known, each producer selects the daily production levels of each

technology subject to its capacity availability (the available capacity of the

cr

renewable technology depends on capacity construction in the first stage of the model, on time of day and on the weather).7, 8

us

The economic process underlying our model is as follows. Electricity production will, in the short term, shift away from fossil-using technologies to

an

PV technology (which features zero marginal cost). In the mid and long term, when the capacity cost of PV declines due, for example, to PV technology improvements, the optimal capacity mix will shift new capacity construction

M

away from CO2-intensive technologies to PV technology and, possibly, cause early retirement of CO2-intensive technologies. Finally, an increase in the share of PV-based electricity production leads to larger and more frequent

d

electricity price spikes during periods in which weather conditions and/or time

te

of day limit the use of PV technology (on days without sunshine, during endof-the-day hours in which electricity production by PV is less effective than

Ac ce p

during the middle of the day). Since electricity demand is inelastic in the short term, electricity price spikes during periods with no sunshine or less-effective PV production will raise the average electricity price, thus bestowing market power on electricity producers employing fossil-using technologies.

7

Like many other studies on the electricity sector, we employ the Cournot model to determine

equilibrium quantities and prices in the second stage of the model, when electricity is sold by all producers to meet market demand (Newbery, 1998; Borenstein and Bushnell, 1999; Green, 2004; Tishler and Woo, 2006; Puller, 2007; Bushnell, Mansur and Saravia, 2008; Murphy and Smeers, 2010). 8

See Chao (2011) and Joskow (2011) on the difficulty of comparing the cost of production

and setting prices of intermittent energy resources.

4 Page 4 of 63

This paper contributes to the literature in several ways. First, it demonstrates that price volatility may be harmful as it can bestow market power on producers, raise the average price to consumers and reduce

ip t

consumer surplus and, possibly, overall welfare. Second, we extend the existing models of endogenous capacity and operations in electricity markets

(Borenstein and Holland, 2005; Borenstein, 2005, 2007; Murphy and Smeers,

cr

2005, 2010; Milstein and Tishler, 2012) to include demand and supply

uncertainties.9 In particular, we extend the analyses in Chao (2011), Joskow

us

(2011) and Milstein and Tishler (2011) by using models which demonstrate the difficulties in pricing energy resources that are available only

an

intermittently, depending on whether or not the sun is shining (or on time of day in which the PV technology is less effective). Third, since the marginal costs of the PV technology are zero, PV becomes the “base” technology and

M

will always be used when the sun is shining, whereas the CCGT technology reverts to the role of the “peaking” technology.10 Thus, the optimal solution is sensitive to the availability of sunshine and the effectiveness of the PV

d

technology. These properties of the model are clarified with stylized data of

te

California's electricity market.

This paper is organized as follows. Section 2 develops the model for a

Ac ce p

market in which each firm can employ only one generation technology (PV or CCGT), and Section 3 extends the model to include firms that can produce electricity by both PV and CCGT technologies.11 We characterize the models

9

Although the analysis becomes more complicated, it is straightforward to show that the

nature of the solution is unchanged when more generating technologies are added (Fan, Norman and Patt, 2012).

10

On this issue see, for example, Fan et al. (2012).

11

There are two reasons for analyzing these two market structures. First, we find in various

countries many small (often very small) PV producers (some of which are households) who cannot afford or have no business capability to build (or own) natural gas generator (which are, usually, "large", most often with capacity in excess of 100 MWH). In this study we account for this phenomenon. Second, the "one technology" model is not nested in the "both

5 Page 5 of 63

of Sections 2 and 3 by employing stylized data in Section 4 and present welfare analyses in Section 5. Section 6 summarizes and concludes. Some of

2. A market with firms employing only one technology 2.1. Set-up

ip t

the analytical results are relegated to Appendices A-D.

cr

Consider two types of generating technologies: (1) PV, denoted S, with high capacity cost and zero variable (marginal) cost; and (2) CCGT, denoted

us

G, which exhibits low capacity cost and high variable (marginal) cost. The market for electricity consists of N identical firms employing technology S

an

and M identical firms employing technology G. Each firm builds generating capacity and then uses it to generate and sell electricity on each day of an operation horizon of T days (e.g. T = 365 for a 1-year horizon).12 Let Pt and

M

Qt denote the electricity price and output on day t. Following Wolfram (1999) Pt  a  bQt   t , N

M

d

and Tishler, Milstein and Woo (2008), daily electricity inverse demand is: (1)

i 1

te

where Qt   QitS   Q Gjt and QitS and Q Gjt denote the production on day t of j 1

Ac ce p

the i-th firm that uses technology S and the j-th firm that employs technology

technologies" model. In fact, under the "one technology" model, PV producers build more capacity, price spikes are higher and more frequent, and the average electricity price (over the year) is higher. This happens because the PV producers cannot hedge and build natural gas capacity. As a result, in the "one technology" model, natural gas producers profit much more than in the "both technologies" model, suggesting that having many small PV (only) producers may not be a good solution for a country (state) that regulates its renewable energy (gives subsidies, say) by providing permits to build PV generation only to small entities (producers). 12

Electricity demand can be defined for any length of time. It is straightforward, for example,

to solve the model for 730 half-days (day- and night-time), 8760 hours or 17520 half-hours of the year. The model assumes that consumers are informed about electricity prices and can respond, at least to some extent, to electricity price changes.

6 Page 6 of 63

G, respectively. The parameters a  0 and b  0 are assumed to be known constants and  t is a random variable accounting for the effect of a random demand factor such as temperature. The variable  t is revealed to the

ip t

electricity producers on day t and the price is determined on each day according to the Cournot model with capacity constraints.13 The (probability)

cr

density function of  t , f (  t ) , is known to the firms when they choose their capacity.

us

Following Murphy and Smeers (2005, 2010) and Milstein and Tishler (2012), the annual production cost of the i-th firm (the j-th firm) employing

C i ( K iS ,QiS )   S K iS  c S QiS C j ( K Gj ,Q Gj )   G K Gj  c G Q Gj

t 1

T

(2a) (2b)

M

T

where QiS   QitS

an

technology S (technology G) and a generator of K iS ( K Gj ) MW of capacity is:

and Q Gj   Q Gjt

denote the annual production of

t 1

d

electricity by firm i and firm j, respectively (note that electricity production is constrained by the available capacity). Capacity cost is US$  S (  G ) per

te

MW-year and variable (marginal) cost is US$ c S ( c G ) per MWH for

Ac ce p

technology S (technology G). Technology G is more expensive in operations, cG  c S  0 , and technology S is more expensive in capacity,  S   G . The

parameters c S , c G ,  S and  G are assumed to be known constants. The availability of PV capacity on day t, t =1,…,T, is conditional on

whether the sun is shining and PV production is effective.14 We suppose that

13

Puller (2007) shows that the conduct of the firms in the restructured electricity market in

California from April 1998 until late 2000 is consistent with a Cournot quantity-setting model. Bushnell et al. (2008) find that a Cournot competition predicted equilibrium prices that are good approximations for actual electricity prices during the summer of 1999 in three US markets. 14

For expositional simplicity we shall use the term "the sun is not shining" to denote the

situation in which PV capacity is unavailable or less effective due to cloudiness or fog or peak

7 Page 7 of 63

the sun is shining on day t with probability . If the sun is shining on day t, all of the PV capacity is available on that day; otherwise the available PV capacity is zero. For expositional simplicity, we also assume that the presence

ip t

of sunshine and  t are independent.15 This assumption reflects our contention that demand variability is mainly driven by time of day and temperature.

cr

Finally, we assume that E(  t )  0 , Var (  t )   2 and c G  a   t .16 The decision process of this two-stage model is as follows:

us

Stage 1: Each of the N + M (risk-neutral) firms decides on its capacity investment, K iS or K Gj , to maximize its expected profits over T days, taking

an

the capacities of the other N+M  1 firms and the probability functions of  t and daily sunshine as given.

Stage 2: Once the firms know  t and the sunshine condition on day t, each

M

firm decides how much electricity to produce (and sell) to maximize its daily operating profit. The firm’s decision is based on the Cournot model, treating the quantity produced by the other N+M  1 firms and the capacity of all N +

d

M firms as given. This stage is repeated T (independent) times.

te

The two-stage model is solved recursively using backward induction. The daily electricity production of each firm is found by solving the second

Ac ce p

stage of the model. The optimal second-stage solutions are then used to determine the expected-profit-maximizing generation capacities in the first stage.

demand at the end of the day when the use of PV is less effective than during noon to 16:00. See Section 4 for detailed discussion on how we apply our model to stylized data of the electricity market of California. 15

In Section 4 we present an example of the model solutions when these two variables are

correlated as well as uncorrelated. 16

If E(  t )    0 , we add µ to the constant a in expression (1), thus setting E(  t )  0 .

8 Page 8 of 63

2.2. Second-stage equilibrium The objective of the i-th firm (which uses technology S) at the second stage of the model is to maximize its operating profits,  it , conditional on  t ,

ip t

the shining of the sun, K iS ( i  1,..., N ), K Gj ( j  1,...,M ), QktS

( k  1,..., N ; k  i ), and Q Pjt ( j  1,...,M ). When the sun is shining, the

cr

decision problem of firm i on day t is:

 it  ( Pt  c S )QitS max S Qit

(3)

us

QitS  K iS , QitS  0 , i  1,..., N

s .t .

If there is no sun on day t (or PV use is ineffective), firm i does not produce on

an

that day.

The objective of the j-th firm (which uses technology G) is to

M

maximize its operating profits,  jt , conditional on  t , K iS ( i  1,..., N ), K Gj ( j  1,...,M ), QitS ( i  1,..., N ), and QltG ( l  1,...,M ; l  j ). If the sun is shining on day t, firm j treats the quantity produced by the N firms employing

d

technology S, and their capacity as given. If there is no sun on day t, firm j

te

relates only to the M  1 firms employing technology G. Formally, the decision problem of firm j on day t is:

Ac ce p

max  jt  ( Pt  c G )Q Gjt G Q jt

Q Gjt  K Gj , Q Gjt  0 ,

s.t .

j  1,..., M

(4)

The Karush-Kuhn-Tucker (KKT) conditions for each firm of each type

and for days on which the sun is shining (capacity G and capacity S can be used) or not shining (only capacity G can be used) are detailed in Appendix A. 2.3. First-stage equilibrium conditions To determine optimal capacities, the i-th firm employing technology S

and the j-th firm employing technology G use the solution of the second-stage to solve their expected-profit maximization problems at stage 1: T

max E [ ∑(  it | K iS ) ]   S K iS , i = 1, ..., N, S Ki

(5a)

t 1

9 Page 9 of 63

T

max E [ ∑(  jt | K Gj ) ]   G K Gj , j  1,..., M , G Kj

(5b)

t 1

where expectations are taken over  t (t = 1, ..., T) and ρ. The expected



E [  it | K iS ]    ( Pt  c S )QitS f (  t )d t , i = 1, ..., N, c S a

c S a



us



E [  jt | K Gj ]    ( Pt  c G )Q Gjt f (  t )d t 

( 1   )  ( Pt  c )Q f (  t )d t , j  1,...,M , G

G jt

(6b)

an

cG a

(6a)

cr

employing technology G on day t are:

ip t

operating profits of the i-th firm employing technology S and the j-th firm

respectively.

The requirement that electricity production, by technology S and

M

technology G, in the second stage of the model cannot exceed the available capacity of each technology (determined in the first stage of the model) implies that the first-stage equilibrium solution should be solved for three

d

possible scenarios at the second stage of the model: (i) firms employing

te

technology G (the peaking technology) enter production when firms using technology S (the base technology) are at full capacity; (ii) firms employing

Ac ce p

technology G enter production when firms using technology S are at less than

full capacity and firms using technology S reach full capacity before firms employing technology G do so; (iii) firms employing technology G enter

production when firms using technology S are at less than full capacity and

they are the first to achieve full capacity. Closed-form solutions of (5) and (6) cannot be obtained in an explicit form for an arbitrary distribution function of

 t , f (  t ) . Following Wang et al. (2007) and Milstein and Tishler (2012), we

assume that  t is uniformly distributed ( f (  t )  1 /(    ) ,    t   ). 17

17

Closed-form solutions of (5a) and (5b) cannot be obtained even when we assume that the

distribution of t is uniform. This assumption, however, facilitates the analytical identification of the scenario for which the reaction functions of (3) and (4) can be derived. It is possible,

10 Page 10 of 63

The

symmetric

equilibrium

solution

(i.e.

K 1S*  ...  K NS*  K S*

and

K 1G*  ...  K MG*  K G* ) for each type of firm is detailed in Appendix A. The derivation of the KKT conditions for each firm of each type, using numerical

ip t

analysis, is detailed in Appendix C.

cr

3. A market with firms employing both technologies

In this section we allow each of the N  M firms to build capacity

us

using either PV (S) technology or CCGT (G) technology or both. Thus, in stage 1 of the model each firm decides on its capacities, taking the capacities of the other N  M  1 firms as given. In stage 2 each firm selects its daily

an

output level (using the Cournot model) subject to its capacity availability, thereby determining the equilibrium market prices. We solve the model

M

recursively using backward induction.

Formally, the objective of the i-th firm in stage 2 is to maximize its operating profits,  it , conditional on  t , whether or not the sun is shining,

d

K iS and K iG ( i  1,..., N  M ), QktS and QktG ( k  1,..., N  M ; k  i ). When the

te

sun is shining, the maximization problem of firm i on day t is: max  it  ( Pt  c S )QitS  ( Pt  c G )QitG

Ac ce p

QitS ,QitG

s .t .

QitS  K iS , QitG  K iG , QitS ,QitG  0 , i  1,..., N  M

(7)

When there is no sun (or PV use is ineffective), the maximization problem of firm i on day t is:

 it  ( Pt  c G )QitG max G Qit

s.t .

QitG  K iG ,

QitG  0 , i  1,..., N  M

(8)

The KKT conditions for each firm and for days on which the sun is

shining (capacity G and capacity S can be used) or not shining (only capacity

though very complicated, to identify these scenarios numerically when t is normally distributed.

11 Page 11 of 63

G can be used) are detailed in Appendix B. The Nash equilibrium in outputs is obtained when these conditions hold simultaneously for all N  M firms. To determine optimal capacities, the i-th firm uses the solution of the

ip t

second-stage reaction functions to solve the following (stage 1) expectedprofit maximization problem: Ki ,Ki

T

E [ ∑(  it | K iS , K iG ) ]   S K iS   G K iG , i  1,..., N  M t 1

(9)

cr

max G S

operating profits of the i-th firm on day t are:

E [  it | K iS , K iG ]  

 ( Pt  c



S



)QitS  ( Pt  c G )QitG f (  t )d t 

S

c a

us

where expectations are taken over  t (t = 1, ..., T) and ρ. The expected

an



( 1   )  ( Pt  c G )QitG f (  t )d t c G a

M

i=1,...,N +M .

(10)

A closed-form solution of (9) cannot be obtained. Assuming that  t is uniformly

distributed,

the

symmetric

equilibrium

solution

(i.e.

d

K 1S*  ...  K NS* M  K S* and K 1G*  ...  K NG* M  K G* ) is derived in an implicit

te

form, using the KKT conditions, in Appendix B. The precise use of the KKT

Ac ce p

conditions in the numerical analysis is described in Appendix D.

4. Model characteristics: Applying the model to stylized data An explicit analytical solution of the optimal capacities in problems

(5), (6) and (9) cannot be obtained even for very simple distribution functions of  t . 18 Thus, to illustrate its real-world relevance, we apply our model to

18

Using the envelope theorem it can be shown analytically that total (CCGT and PV) capacity

is, generally, a decreasing function of the marginal cost of the CCTG technology and the capacity cost of PV. In addition, if the overall optimal capacities in the two market structures that we analyze here are identical, the probability of a price spike on any given day is higher in the market in which firms can construct and use only one technology. We do not report

12 Page 12 of 63

stylized data of the California electricity market. The application is based on two generation technologies (energy types) – natural gas (CCGT) and PV. The total 2012 in-state capacity and generation in California were 74,103 MW and

ip t

198,313 GWh, respectively (CEC, 2013). Natural gas was the main fuel type in the electricity sector in California. It accounted for 61% of the installed capacity and of electricity generation in this state in 2012 (CEC, 2013). 19

cr

Twenty five percent of the installed capacity and 23% of the electricity generation in 2012 were due to nuclear power and hydro, which exhibited very

us

high fixed costs and relatively small marginal production costs. It is unlikely that additional nuclear power will be allowed in California in the near future.

an

Hydro capacity is limited and future additions to this capacity may not be justified due to expansion costs and the warming climate (CEC, 2012, pp. 4145). The 2012 installed capacity of solar energy in California was small

M

(though steadily growing), about 1.2% of total capacity, generating about 0.8% of the electricity in this state during 2012. The 2012-2016 strategic plan of the California ISO aims, however, to increase the share of renewable in

the

total

electricity

d

sources

production

in

2020

to

33%

Ac ce p

fitting exercise.20,21

te

(http://www.caiso.com). Thus, applying our model to California seems to be a

these results in this paper since their policy implications are minor. The formal (analytical) demonstration of these results is available from the authors upon request. 19

Natural gas, hydro and nuclear power are the main generation technologies in California.

Respectively, they account for 61%, 19% and 6% of the 2012 installed in-state electricity generation capacity in California. The balance of the capacity in 2012 is coal (less than 1%), biomass (less than 2%), geothermal (about 4%), wind (7%) and solar (1%).

20

Descriptions and analyses of the California electricity market are available, among other, in

Borenstein and Bushnell (1999), Borenstein, Bushnell and Wolak (2002), Borenstein and Holland (2005), Borenstein (2005, 2007), Bushnell et al. (2008), Borenstein (2012), CEC (2012, 2013). Employing California data in this study is made for convenience and ease of presentation. This study should not be considered as an application with policy recommendations for the electricity market of California.

13 Page 13 of 63

Sun availability (and/or PV production effectiveness) is a central factor in our model, as can be seen in the first stage expected-profit maximization problem (5a)-(5b) (when each firm can employ only one generating

ip t

technology) or (9) (when each firm can employ both generating technologies). The sun is never available during the night and may also be unavailable (or PV production is ineffective) at times during the day. Hence, the PV technology is

cr

never used during the night when the demand for electricity is usually lower than that during the day. This phenomenon is accounted for in the application

us

of our model to the stylized data of California's electricity market by dividing the 24 hours of each day into two sub-periods: 12 hours of day-time in which

an

the sun availability is a random variable (07:00-19:00), and 12 hours of nighttime in which the sun is unavailable (only technology G can be used to generate electricity). Consider, for example, the market structure in which

M

each firm can employ both generating technologies. Denote the inverse demand on day t during the night-time by Pt n and during the day-time by Pt d .

d

Using this setup, we adjust problem (9), which determines the optimal

21

te

capacities that the i-th firm will choose to construct, to account for the

Borenstein and Holland (2005) and Borenstein (2005, 2007) employ a different model of

Ac ce p

endogenous capacity and operations. They assume many 'small' electricity producers and zero delay in construction time. Focusing on the efficiency of real time electricity pricing (RTP) they show that competition in the wholesale and retail markets yields a short run equilibrium in which retailers drive the retail prices for RTP customers to be equal to wholesale prices (which are equal to marginal generation costs) in each period, while the flat retail rate equals to a weighted average of the real time wholesale prices (see Borenstein and Holland, 2005, page 473). Generation capacity is determined by the wholesale competition (zero long run profits). Similarly to our model, larger absolute value of price elasticity implies higher production and lower overall customer bill. Contrary to our model, in which higher absolute value of the price elasticity yields higher capacity (since quantity demanded is higher and peak demand 'determines' capacity which is subject to uncertain demand and cannot adjust instantaneously once construction is finished), the model of Borenstein and Holland (2005) and Borenstein (2005, 2007) yields lower total capacity when the absolute value of price

elasticity is higher (since high peak prices shave peak capacity).

14 Page 14 of 63

different demand functions for electricity during day-time and night-time and the fact that the sun is unavailable during the night-time. That is, we determine the optimal capacities of firm i by solving the following (stage 1) expected  d  S S d G G   S ( Pt  c )Qit  ( Pt  c )Qit f (  t )d t      S K iS   G K iG ∑ c a   d G G n G G t 1  ( 1   )  ( Pt  c )Qit f (  t )d t   ( Pt  c )Qit f (  t )d t    cG  a c G a T





cr

max

KiG ,KiS

ip t

profit maximization problem:

i=1,...,N +M .

us

Table 1 lists descriptive statistics of the hourly electricity use in California during 2011 (http://oasis.caiso.com). The data in Table 1 show that

an

the distribution of the hourly electricity use in 2011 is close to symmetrical, with most of the mass around the mean. Day-time electricity use is higher than night-time electricity use. The distribution of hourly electricity loads in

M

California is very similar to that in many other electricity markets in the world. In fact, Milstein and Tishler (2012; Table 2) show that the distribution of the

d

hourly loads in California is very similar to that of New England, PJM (Pennsylvania, New Jersey and Maryland), ERCOT (Texas) and Israel. Hence,

te

we conjecture that the main characteristics of our application can be

Ac ce p

generalized to those, and other, markets. -------------------------------Insert Table 1 About Here --------------------------------

Computation of the optimal capacities is based on estimates of the

demand parameters, a and b, the cost parameters,  S ,  G , c S , and c G , the parameters of the probability function f (  t ) (α, β), and ρ. The computation of these parameters is based on data from Klein (2010), DOE/EIA (2012), CAISO

(http://oasis.caiso.com),

and

the

tariffs

detailed

by

CPUC

(http://www.cpuc.ca.gov/puc) and PGE (http://www.pge.com/tariffs).22 Setting

22

The average 2011 price for California is 13.79 cents per kWh (DOE/EIA, 2012, Table

5.6.B). The average price of natural gas for the electricity sector is $4.09 per million Btu

15 Page 15 of 63

the average retail price in 2011 at 138 $/MWH and using a (short-run) price elasticity of -0.25 for the daily demand functions for electricity, 23 these estimates are as follows (the superscripts d and n denote day-time and night-

ip t

time, respectively): 24 a d  862 .5 , a n  496 .5 , b d  24.83 , b n  16.01 ,

 G T  205.5 , 25 c S  0 , c G  50 , 26  d  147 .3 ,  n  59 .1 ,  d  147.3

cr

and  n  59.1 . Our base case assumes that the PV to CCGT capacity cost ratio

us

(DOE/EIA, 2012, Table 4.13.A); hence, the marginal energy cost of using natural gas in electricity production is about $50 per MWh. The marginal production cost for firms that employ natural gas includes, in addition to the cost of natural gas, a variable O&M cost of

an

$3.19 per MWh (Table 14 in Klein, 2010). Cost estimates of constructing new generators are available, among other references, in Klein (2010, Table 14). Using the average cost for a new natural gas generator with a capacity of 500 MW we obtain that the annual fixed cost is about

M

$75,000 per MW/year (we assume that the generator will operate for about 20 years and that the real interest rate is 3%). Total distribution and transmission costs in 2010 were about $9 billion (see http://www.cpuc.ca.gov/puc; PGE, 2010; SCE, 2011). These costs are computed from the total allocated revenue requirements of the utilities. These costs may include various

d

components that are already accounted for in the capacity and marginal production costs that

te

we are using in this study. Hence, in this study we estimate that the direct distribution and transmission costs are about 70% of the reported transmission and distribution costs (about $6.3 billion). Sensitivity analysis shows that the main results and conclusions of this study are

Ac ce p

not sensitive to this choice. 23

Table 1 in Lijesen (2007) presents a summary of short term price estimates of electricity

price elasticities, with values ranging from -0.04 to -0.89. Borenstein and Holland (2005) employ a value of -0.1 and Borenstein (2005) uses elasticity values between -0.025 and -0.5 for their assessments of the California electricity market. Note, however, that only a share of the customers in these studies are responding to real time electricity pricing, while we assume here that all customers are subject to prices that vary over the 24 hours of the day (similar argument is made in Borenstein, 2005). Note that the pattern and characteristics of our results do not change for elasticity values between -0.1 and -0.5 (see Figures 2 and 6).

24

See Tishler et al. (2008), Khatib (2010), Lior (2010) and references therein for the costs of

electricity generation by various technologies. 25

For simplicity we employ fixed cost in MW/half-day; that is, it equals 75,000/730.

26

In the computations we added $3.19 per MWh as variable O&M cost (Table 14 in Klein,

2010) to the marginal cost of the CCGT technology.

16 Page 16 of 63

is about 4:1, i.e.  S  G  4 , reflecting the current ratio in the electricity market (data on PV prices are available, among others, in Lior, 2010; Trainer, 2010; Candelise and Gross, 2012; Liebreich, 2012).27 We also assume that ρ = 29

(except where otherwise stated, for simplicity, we

ip t

0.728 and N  M  20

set N = M for the market in which firms can construct and operate only one technology).

cr

Figure 1 presents the industry’s optimal capacity as a function of the capacity cost ratio  S  G . The overall generation capacity increases (or

us

remains unchanged) the lower is the capacity cost of PV (the lower is  S  G due to improvements in the PV technology).30 The distribution of the industry

M

27

an

capacity between the two technologies is very different across the two market

Throughout the paper we vary only the capacity cost of PV since this study is focused on

the expected reduction of PV capacity cost in the future due to technology improvement over time. Hence, we concentrate on the ratio of capacity cost, holding the CCGT capacity cost as

d

fixed. Clearly, our model can be used with any values of PV and CCGT capacity costs. This value seems to be realistic for California, where in the daytime the sun may appear

te

28

partially (rendering PV capacity less effective) or not at all during only 30% of the year, mostly in the winter (mostly at the end of the day). Our model (see the functional form of the cost functions) is not intended to find the optimal

Ac ce p

29

number of producers. It is designed to find the optimal solution for any given number of producers. In principle, one can compute the number of potential producers in the market by invoking zero industry profits (see, for example, Borenstein and Holland, 2005 and Borenstein, 2005). Finding the optimal number of producers in our model requires additional assumptions and additional data which are not readily available. Hence, assessing the optimal number of producers is outside the scope of this paper. Comparison of the model solution for various numbers of producers is detailed later on in this section.

30

Total capacity is determined, mostly, in the peak period during the day-time in which both

technologies are employed (see Figure 4). Since the change in the capacity cost ratio does not affect the marginal electricity cost, it has only a minor effect on overall capacity. That is, the substitution between the capacities of the two technologies is very large, particularly when each producer can internalize the change in the capacity cost ratio (when each producer can employ both technologies).

17 Page 17 of 63

structures. For example, for  S  G  4 , the share of PV capacity (the striped areas of the bars in Figure 1) in the industry’s total capacity is 38% when each firm can employ only one technology and only 26% when each firm can

ip t

employ both technologies. Later on we show that most of the industry profits derive from the CCGT generators, since they are always in operation when prices spike (when the industry is at full capacity), and these spikes are higher

cr

when there is no sun and when the share of PV capacity is higher. Consequently, generators employing CCGT technology have lower market

us

power when all the firms in the market can employ CCGTs (each firm can use both technologies) and, thus, prefer to build more of the more profitable

an

CCGT generators.31

M

-------------------------------Insert Figure 1 About Here --------------------------------

Figure 2 shows the effect of the value of price elasticity on the optimal capacity of each technology, under both market structures, when  S  G  4 .

d

A higher absolute value of price elasticity results in a lower electricity price,

te

which implies a higher quantity demanded and, hence, higher generation capacity of both technologies. That is, overall capacity increases and the share

Ac ce p

of PV capacity (depicted by the striped areas of the bars in Figure 2) in the industry’s total capacity increases from 34% to 41% in the market with firms employing only one technology and from 19% to 31% in the market with

31

The optimal capacity that we obtain here is about 36,000 MW, while total generation

capacity in California in 2012 is 74,103 MW. There are several explanations for this discrepancy. First, the actual generation capacity in California is not optimal and includes a large share of older, as well as intermittent, technologies. Second, most of the households in California are not subject to electricity prices that fluctuate over the day, which prevents the use of marginal prices to reduce capacity.

18 Page 18 of 63

firms using both technologies when the absolute value of the price elasticity rises from -0.1 to -0.4. 32

ip t

-------------------------------Insert Figure 2 About Here -------------------------------Figures 1 and 2 show that the share of PV capacity in the market in

which firms can employ only one technology is larger than that in the market

cr

in which firms are allowed to employ both technologies. This result holds even when the number of firms employing the PV technology is very large.

us

Figure 3 presents the industry’s optimal capacity as a function of the number of producers that use the PV technology (the number of firms employing the

an

CCGT technology is unchanged, i.e. M = 10). The overall capacity increases very slightly as the number of PV-using firms in the market increases, and so does the share of PV capacity (depicted by the striped areas of the bars in

M

Figure 3) in the industry’s total capacity (from 36% (43%) when N = 10 to 37% (44%) when N = 100 and  S  G  5 (  S  G  2 )). That is, the nature

d

of the results of this paper does not change when the number of PV producers

te

is very large. 33 In other words, as we shall elaborate later on, the CCGT producers “set the prices” and reap most of the profits during periods of high

Ac ce p

demand and, particularly, during periods of no sunshine, when the electricity system is at full capacity.

-------------------------------Insert Figure 3 About Here --------------------------------

Figure 4 presents the distribution of daily equilibrium electricity prices,

for both market structures, during the 365 days of the year. Equilibrium prices are stable when production is below full capacity and rise at an increasing rate

32

Extensive sensitivity analyses confirm the robustness of our results for price elasticities

ranging from -0.05 to -0.8. 33

This phenomenon mimics a reality in which many small PV producers generate electricity

locally.

19 Page 19 of 63

once full capacity is reached. The lower is the share of CCGT capacity in total capacity the sooner is full capacity reached.34 Thus, price spikes are larger and more frequent the higher the share of PV capacity due to the declining PV

ip t

capacity cost. Consider the case where  S  G  4 . When each firm employs only one technology, full capacity is reached during the day-time (night-time) on the 246th (31st) day of the year by order of the demand; when each firm

cr

employs both technologies, firms tend to construct more CCGT capacity, and

full capacity is reached on the 268th (234th) day of the year during the day-time

us

(night-time).35

an

-------------------------------Insert Figure 4 About Here --------------------------------

Figure 5 presents the average and maximal electricity prices as

M

functions of  S  G , for the two market structures. If each firm can employ only one technology, the average (maximal) price during the day-time increases from 142 (476) $/MWH when  S  G  5 to 147 (538) $/MWH

Ac ce p

te

d

when  S  G  2 . If each firm can employ both technologies, the average

34

Price spikes occur when both technologies reach full capacity on a sunny day (the value of

the random variable, εt , is sufficiently high in this case) or when the CCGT technology

reaches full capacity on a day without sun ( εt may be “low” in this case) or during the night. 35

The demand for electricity during the night-time, when only CCGTs can produce electricity,

is lower than that during the day-time. Hence, as is shown in Figure 4, price spikes are, generally, much lower during the night-time. Day-time full capacity happens during about 30% of the time. Full capacity occurs more often during the night-time, but as the excess demand during this period is, generally, smaller than that during the day-time, the price spikes are smaller. Currently, most electricity markets, certainly the one in California, have large excess capacity, and use other technologies in addition to CCGT and PV, which reduces the duration and size of price spikes relative to a market in which capacity is at its optimal level (as is the case in our study).

20 Page 20 of 63

(maximal) price during the day-time increases from 113 (355) $/MWH when

-------------------------------Insert Figure 5 About Here --------------------------------

ip t

 S  G  5 to 127 (427) $/MWH when  S  G  2 .36

Figure 6 shows the pattern of electricity prices for several price

cr

elasticities. A higher absolute value of price elasticity implies lower electricity

prices. The phenomenon of a higher average (maximal) electricity price when

us

the capacity cost of PV declines, and the phenomenon of electricity prices being higher when each firm can employ only one technology remain for all values of price elasticities. Figures 6 and 7 confirm that a larger share of PV

an

capacity yields higher electricity price spikes, since all the demand must be met by the CCGT capacity on days without sunshine. That is, the price spikes

M

on days without sunshine endow CCGT producers with monopoly-like power (the ability to raise prices far above marginal cost), particularly when each firm can employ only one generating technology. These phenomena are more

d

pronounced the greater the number of PV-using firms; when  S  G  5

te

(  S  G  2 ) the maximal electricity price during the day-time increases from 476 (538) $/MWH for N = 10 to 485 (546) $/MWH for N = 100 (see Figure

Ac ce p

7). A larger number of PV-using firms in competitive electricity markets leads to a larger share of PV capacity, which, in turn, brings about larger and more frequent price spikes and, consequently, a higher average electricity price for

36

The nature of the results is unchanged if the availability of the PV technology (the case in

which the sun is shining) and  t are correlated. Price volatility tends to be smaller if they are

positively correlated. Consider, for example, 

S

  4 . In the case of market with firms G

employing both technologies, the maximal price during the day is $383 when the correlation between the presence of sunshine and  t is zero (see Figure 5) and $216 when this correlation equals 0.3. The average day-time price equals $118 when this correlation is zero (see Figure 5) and $115 when it equals 0.3.

21 Page 21 of 63

customers. This somewhat unexpected result, which is due to the price spikes

-------------------------------Insert Figure 6 About Here --------------------------------------------------------------Insert Figure 7 About Here --------------------------------

ip t

on days without sunshine, should give the regulator food for thought.

cr

Figure 8 shows that the industry’s production is lower when each firm

can employ only one technology, and it tends to decline as PV capacity cost

us

declines (due to technology improvements in the construction of PV capacity). This result reflects the increase in the average electricity price when PV

an

adoption rises (the share of PV in total production is depicted by the striped areas of the bars in Figure 8) due to the declining PV capacity cost. The model in this paper employs an approximation to the current demand functions for

M

electricity in California. As a result, the total electricity production and the average electricity price (over the 24 hours of the day) obtained by our model

d

are similar to those that we currently observe in California.

9

shows

that

the

industry’s

profit

increases

when

Ac ce p

Figure

te

-------------------------------Insert Figure 8 About Here --------------------------------

 S  G decreases. The increase in profits happens because the cost of

electricity production by PV is lower (due to the decline in PV capacity cost) and, mainly, because the revenues from selling electricity are higher due to the higher and more frequent price spikes (and the relatively inelastic price elasticity, which ensures that producers gain more from higher market prices than they lose from declining electricity production, see Figure 8). The industry’s optimal profits in Figure 9, for  S  G  4 , account for about 9%

and 31% of the industry’s revenues when firms employ both technologies and only one technology, respectively. The profit margin in the first case is reasonable while it is extremely high in the second. That is, our results point out that a market structure in which each firm is restricted to only one

22 Page 22 of 63

technology bestows very high monopoly power on the electricity producers, while letting each firm employ both technologies yields lower average electricity price, higher electricity production and lower industry profits.

ip t

Figure 10 presents industry profits as a function of the number of PV producers. PV producers earn negative profits when  S  G is 3 or higher

(when PV capacity cost is high). Subsidies to PV producers are common in

areas of the bars in Figures 9 and 10).

M

an

us

-------------------------------Insert Figure 9 About Here --------------------------------------------------------------Insert Figure 10 About Here --------------------------------

cr

this situation37 (the positive profits of PV producers are depicted by the striped

5. Welfare implications: Applying the model to stylized data Welfare analysis is required to determine which of the two market

d

structures that we compare in Section 4 is preferable, and to assess the

te

effectiveness of imposing CO2 taxes to reduce greenhouse gas (GHG)

Ac ce p

emissions (see Tol, 2006; Cansino et al., 2010; Friedman, 2011; Borenstein,

37

See, for example, http://energystoragejournal.com/germany-introduces-subsidy-for-pv-and-

storage-systems/;

http://usa.chinadaily.com.cn/china/2013-08/31/content_16933829.htm

(accessed on October 14, 2013). Subsidies to electricity producers which use the PV technology are usually given in the form of fixed payments per unit of (PV) capacity (for a given, pre-approved, amount of capacity) or by purchasing the electricity produced by PV producers at a price (or price schedule over time) which is sufficiently high to cover the total cost (including some positive return) of electricity production by PV over a given period (20 years, say). Since this study determines the solution of a hypothetical, future, electricity market in California, it is not clear which type of subsidies will be used. Hence, we assume that the subsidies, if required, will be set independently from the electricity rate. In fact, we project that in the future subsidies to the PV technology may not be required since the capacity cost ratio of PV to natural gas will decline to less than 3, at which point the PV producers show positive profits.

23 Page 23 of 63

2012). These two issues are the subject of this section. Figure 11 presents the social welfare from electricity generation, for the two market structures that we employ here, as a function of the cost of the PV capacity (consumer

ip t

surplus is depicted by the white area and profits by the gray area). 38 Social welfare is higher when each firm is free to produce electricity by both technologies. This result is due to the lower average electricity price

cr

and consequent higher consumer surplus in this market structure. Though

consumer surplus decreases slightly (since the average electricity price

us

increases) when the capacity cost of PV declines, social welfare increases when each firm constructs and operates both technologies.

an

-------------------------------Insert Figure 11 About Here --------------------------------

Figure 12 presents social welfare as a function of the number of PV-

M

using firms. Counter-intuitively, consumer surplus and social welfare decline, albeit slightly, when the number of PV-using firms in the market increases.

d

This phenomenon is explained as follows. A larger number of PV-using firms leads to higher PV capacity which, in turn, implies higher and more frequent

te

price spikes and, thus, higher average electricity prices (see Figure 7) that

Ac ce p

reduce consumer surplus more than they increase industry profits. -------------------------------Insert Figure 12 About Here --------------------------------

Next, we assess the effectiveness of taxes on CO2 emissions. Levying

CO2 taxes is justified by the social cost that is imposed on consumers by CO2 emissions (Cansino et al., 2010; Borenstein, 2012). CO2 taxes are

controversial and politically difficult to implement even though there is broad agreement on the need to reduce GHG. Assessing such taxes within our model in which both capacity construction and electricity production are endogenous,

38

As the first-best solution in electricity markets may not be attainable (Rogerson, 1990, p.

92), we estimate it as the sum of the industry profits and consumer surplus.

24 Page 24 of 63

may help to determine the level of their effectiveness. In other words, it is important to answer the following question: how will CO2 taxes affect capacity mix, industry profits, consumer surplus and welfare in different

ip t

market structures? Figures 13-15 show the industry capacity, average and maximal electricity prices and social welfare for three different tax rates: $10, $30 and $50 per ton of CO2. Industry capacity decreases, although not by

cr

much, the higher the tax on CO2. As expected, an increase in the tax rate on

CO2 leads to an increase in the share of PV capacity (depicted by the striped

us

areas of the bars in Figure 13) in both market structures since this tax raises the effective cost of natural gas (which emits CO2): setting the tax rate at $50

an

per ton of CO2 increases the share of PV capacity from 38% to 42% when each firm employs only one technology, and from 26% to 31% when each firms employs both technologies. Thus, a CO2 tax is more effective, though

M

not by much, when each firm can construct and employ both technologies. The higher share of PV capacity in the industry’s total capacity leads to higher and more frequent price spikes which, in turn, lead to a higher average (and

d

maximal) electricity prices (see Figure 14).

Ac ce p

te

-------------------------------Insert Figure 13 About Here --------------------------------------------------------------Insert Figure 14 About Here --------------------------------

Figure 15 shows that increasing the tax rate on CO2 increases the tax

payments (depicted by the striped areas in Figure 15), while consumer surplus (depicted by the white areas in the figure) declines (because price spikes are higher and more frequent the higher is the tax rate) and overall welfare increases in response to the increase in the tax on CO2.39,40 Finally, there is

39

Tax payments are about 1% (1%) of total welfare for 10$ CO2 tax and 4% (4.5%) of total

welfare for 50$ CO2 tax when each firm employs only one technology (both technologies). 40

Overall welfare can increase or decrease in this case, depending on the effect of the tax on

consumers and producers. Higher tax reduces consumer surplus since prices are higher. At the

25 Page 25 of 63

very little additional benefit (reduction in health problems, for example) from the reduction in CO2 emissions caused by imposing CO2 taxes (we assume that the damage, per ton of CO2, equals to the tax rate). The direct benefit from

ip t

imposing the CO2 tax is much less than half a percent of social welfare for any of the tax rates that we used in this study, since the overall electricity the CO2 tax.41

us

-------------------------------Insert Figure 15 About Here --------------------------------

cr

production by the fossil-using technology hardly changes in the presence of

an

6. Summary and Conclusions

This paper develops a two-stage decision model with endogenous

M

capacity and operations to assess the outlook and practicality of photovoltaic technology (PV) in competitive electricity markets. The solution of the twostage decision model shows that the integration of PV capacity into the

d

existing electricity system is not simple and requires a better understanding of

te

the nature of intermittently renewable energy. In fact, we demonstrate, at least for the stylized data for California, that the share of PV capacity in total

Ac ce p

capacity will be fairly limited in the near future and a CO2 tax will likely have only a minor effect on CCGT production. The paper shows that electricity price spikes are higher and more

frequent the higher the PV capacity. Consequently, the average electricity price rises when construction costs of PV capacity decline due, for example, to technology improvements, yielding excessive profits to producers employing fossil fuels. That is, higher price volatility bestows market power on fossil-

same time it increases profits due to higher and, possibly, more frequent price spikes. The overall effect depends on the sum of these two effects. It is straightforward to construct scenarios in which overall welfare declines in response to an increase in the CO2 tax. 41

The damages from CO2 are not visible in Figure 15 since they are very small in comparison

to social welfare.

26 Page 26 of 63

using electricity producers, and more so the lower the costs of PV capacity and the larger the number of PV-using producers in the market. We also show that market structure may have a sizeable effect on

ip t

capacity mix, price volatility, average price and welfare. In fact, our results show that letting each firm employ both technologies yields lower average

electricity price, higher electricity production, lower industry profits, higher

cr

consumer surplus, and higher welfare. In addition, an increase in the number

of PV-using firms and a higher CO2 tax rate are likely to reduce consumer

us

surplus and, sometimes, overall welfare. These results are confirmed by an application to stylized data for California’s electricity sector.

an

The paper highlights that tight generating capacity and frequent electricity price spikes in competitive electricity markets are due not only to demand variability over time (day, season and year), the high cost of

M

constructing capacity and the long lead time required to add new capacity, but also to supply uncertainty, an inevitable outcome in markets with substantial intermittent renewable generation capacity (Hardy and Nelson, 2010; Lior,

d

2010; Trainer, 2010; Milstein and Tishler, 2011).

te

To be sure, an independent system operator may mitigate price spikes by maintaining capacity reserves that will not be part of the daily market

Ac ce p

operations. The analysis of this paper underscores the fact that efficient use of renewable capacity requires an integrated approach to the management of electricity markets, one that accounts for modern and properly distributed T&D infrastructure, balancing generation, smart grids, and implementation of appropriate financial and other incentive systems (Hardy and Nelson, 2010; Lior, 2010; Blumsack and Fernandez, 2012, and references therein). Finally, this paper accentuates the importance of regulators

understanding the behavior of the electricity market when considering the promotion of renewable energy or levying a CO2 tax for the purpose of reducing CO2 emissions, particularly with respect to the characteristics of renewable technologies, demand and supply uncertainties, and market structure. The lack of such an understanding may lead to unintended

27 Page 27 of 63

consequences, including rising average market price levels and lower

Ac ce p

te

d

M

an

us

cr

ip t

consumer surplus.

28 Page 28 of 63

References Badcock, J., Lenzen, M., 2010. Subsidies for electricity-generating technologies: a review. Energy Policy, 38(9), 5038-5047.

ip t

Blumsack, S. Fernandez, A., 2012. Ready or not, here comes the smart grid! Energy, 37, 61-58.

cr

Borenstein, S., 2005. The long-run efficiency of real-time electricity pricing. Energy Journal, 26(3), 93-116.

us

Borenstein, S., 2007. Customer risk from real-time retail electricity pricing: bill volatility and hedgability. Energy Journal, 28(2), 111-130. Borenstein, S., 2012. The private and public economics of renewable

an

electricity generation. Journal of Economic Perspectives, 26(1), 67-92. Borenstein, S., Bushnell, J.B., 1999. An empirical analysis of the potential for

M

market power in California’s electricity industry. Journal of Industrial Economics, 47(3), 285-323.

Borenstein, S., Bushnell, J.B, Wolak, F.A., 2002. Measuring market

d

inefficiencies in California’s restructured wholesale electricity market.

te

American Economic Review, 92, 1376-1405. Borenstein, S., Bushnell, Holland, S., 2005. On the efficiency of competitive

Ac ce p

electricity markets with time-invariant retail prices. RAND Journal of Economics, 36(3), 469-493.

Bushnell, J., Mansur, E., Saravia, C., 2008. Vertical arrangements, market structure and competition: an analysis of restructured U.S. electricity markets. American Economic Review, 98(1), 237-266.

California Energy Commission (CEC), 2012. Climate change effects on the high-elevation hydropower system with consideration of warming impacts on electricity demand and pricing, Prepared by University of California, Riverside, Lund University and University of Central Florida. CEC-500-2012-020. California

Energy

Commission

(CEC),

2013.

Energy

Almanac,

http://energyalmanac.ca.gov.

29 Page 29 of 63

Candelise, C., Gross R., 2012. The role of photovoltaics in the UK energy mix: a discussion of recent market and policy developments. A paper presented at the BIEE conference, September 2012, Oxford.

ip t

Cansino, M. J., Pablo-Romero, M.P., Roman, R., Yniguez, R., 2010. Tax incentives to promote green electricity: an overview of EU-27 countries. Energy Policy, 38(10), 6000-6008.

cr

Chao, H-P., 2011. Efficient pricing and investment in electricity markets with intermittent resources, Energy Policy, 39(7), 3945-3953.

us

Dittmar, M., 2012. Nuclear energy: status and future limitations. Energy, 37, 35-40.

an

DOE/EIA, 2012. Electric power monthly, DOE/EIA-0226 (2012/02). Fan, L., Norman, C.S., Patt, A.G., 2012. Electricity capacity investment under risk aversion: a case study of coal, gas, and concentrated solar power.

M

Energy Economics, 34(1), 54-61.

Friedman, L.S, 2011. The importance of marginal cost electricity pricing to

39(11), 7347-7360.

d

the success of greenhouse gas reduction programs. Energy Policy,

te

Green, R., 2004. Did English generators play Cournot? Capacity withholding in the electricity pool. CMI Working Paper 41, Cambridge, MA.

Ac ce p

Hardy, R., Nelson. K., 2010. Green energy outlook. Public Utilities Fortnightly, January 2010, 54-66.

Holland, S.P., Mansur, E.T., 2008. Is real-time pricing green? The environmental impacts of electricity demand variance. Review of Economics and Statistics, 90(3), 550-561.

Joskow, P.L., 2011. Comparing the costs of intermittent and dispatchable electricity generating technologies. American Economic Review: Papers and Proceedings, 100(3), 238-241. Khatib, H., 2010. Review of OECD study into “Projected costs of generating electricity – 2010 edition”. Energy Policy, 38(10), 5403-5408.

30 Page 30 of 63

Klein, J., 2010. Comparative costs of California central station electricity generation technologies, California Energy Commission, CEC-2002009-017-SD.

ip t

Liebreich, M., 2012. Clean energy investment trends. BIEE Conference, September 2012, Oxford.

Lijesen, M.G., 2007. The real-time price elasticity of electricity. Energy

cr

Economics, 29, 249-258.

Lior, N., 2010. Sustainable energy development: the present (2009) situation

us

and possible paths to the future. Energy, 35, 3976-3994.

Milstein, I., Tishler, A., 2011. Intermittently renewable energy, optimal

an

capacity mix and prices in a deregulated electricity market. Energy Policy, 39(7), 3922-3927.

Milstein, I., Tishler, A., 2012. The inevitability of capacity underinvestment in

M

competitive electricity markets. Energy Economics, 34, 62-77. Murphy, F.H., Smeers, Y., 2005. Generation capacity expansion in imperfectly competitive restructured electricity markets. Operations

d

Research, 53(4), 646-661.

te

Murphy, F.H., Smeers, Y., 2010. On the impact of forward markets on investments in oligopolistic markets with reference to electricity.

Ac ce p

Operations Research, 58(3), 515-528.

Newbery, D.M., 1998. Competition, contracts, and entry in the electricity spot market. RAND Journal of Economics, 29(4), 726-749.

Odenberger, M., Johnsson, F., 2007. Achieving 60% reductions within the UK energy system – implications for the electricity generation sector. Energy Policy, 35(4), 2433-2452.

Pacific Gas and Electric Company (PGE), 2010. 2011 General Rate Case, Phase 2, Revenue Allocation and Rate Design, March 22. Puller, S.L., 2007. Pricing and firm conduct in California’s deregulated electricity market. Review of Economics and Statistics, 89(1), 75-87. Renewables, 2011. Global Status Report, Paris: REN21 Secretariat.

31 Page 31 of 63

Rogerson, W.P., 1990. Quality vs. quantity in military procurement. American Economic Review, 80(1), 83-92. Southern California Edison (SCE), 2011. SCE-03 Revenue Allocation

ip t

Proposals. Rosemead, California, October 7. Tishler, A., 2008. How risky should an R&D program be? Economic Letters, 99, 268-271.

cr

Tishler A., Milstein, I., Woo, C.K., 2008. Capacity commitment and price

volatility in a competitive electricity market. Energy Economics, 30,

us

1625-1647.

Tishler, A., Woo, C.K., 2006. Likely failure of electricity deregulation:

an

explanation with application to Israel. Energy, 31, 845-856. The Economist, 2012. Special report: Nuclear energy. March 10.

Tol, R.S.J., 2006. Multi-gas emission reduction for climate change policy: an

M

application of FUND. Energy Journal, Special Issue on MultiGreenhouse Gas Mitigation and Climate Policy, 27, 235-250. Traber, T., Kemfert, C., 2011. Gone with the wind? Electricity market prices

d

and incentives to invest in thermal power plants under increasing wind

te

energy supply. Energy Economics, 33, 249-256. Trainer, T., 2010. Can renewables etc. solve the greenhouse problem? The

Ac ce p

negative case. Energy Policy, 38(8), 4107-4114.

U.S. Energy Information Administration (EIA), 2013. Electric Power Monthly: with data for February 2013; www.eia.gov.

Wang, L., Mazumdar, M., Bailey, M.D., Valenzuela, J., 2007. Oligopoly models for market price of electricity under demand uncertainty and unit reliability. European Journal of Operational Research, 181, 13091321.

Weyant, J.P., de la Chesnaye, F.C., Blanford, G.J., 2006. Overview of EMF21: multi-gas mitigation and climate policy. Energy Journal, Special Issue on Multi-Greenhouse Gas Mitigation and Climate Policy, 27, 132.

32 Page 32 of 63

Wolfram, C., 1999. Measuring duopoly power in the British electricity spot

Ac ce p

te

d

M

an

us

cr

ip t

market. American Economic Review, 89(4), 805-826.

33 Page 33 of 63

Appendix A

ip t

A market with firms employing only one technology Second-stage equilibrium

The first-order conditions in the second stage depend on whether or not the

cr

sun is shining. If the sun is shining on day t, the equilibrium solution is

us

obtained when the following Karush-Kuhn-Tucker (KKT) conditions for each firm of each type are satisfied simultaneously. Condition (A.1) for each firm

technology G: N

M

k i

j 1

an

employing technology S and condition (A.2) for each firm employing

a  2bQitS  b QktS  b  Q Gjt   t  c S  iS   iS  0 ,  iS QitS  0 ,

(A.1)

( K  Q )  0 , K  Q  0 , Q  0 ,   0 ,   0 , i  1,..., N , S it

S i

S i

N

S it

M

S it

S i

M

S i

S i

a  b QitS  2bQGjt  b QltG   t  c G  Gj   Gj  0 ,  Gj QGjt  0 , i 1

l j

( K  Q )  0 , K  Q  0 , Q  0 ,   0 ,   0 , G jt

G j

G j

G jt

G jt

d

G j

G j

G j

(A.2) j  1,...,M ,

te

where iS and Gj are the dual variables for the capacity constraint of

Ac ce p

technology S and technology G, respectively, and iS and  Gj are the dual variables for the non-negativity of QitS and Q Gjt , respectively. If the sun is not shining on day t, the equilibrium solution is obtained when the

following KKT conditions for each firm employing technology G are

satisfied:

M

a  2bQ Gjt  b  QltG   t  c G  Gj   Gj  0 ,  Gj Q Gjt  0 , ( K Gj  Q Gjt )Gj  0 , l j

K  Q  0, Q  0,   0,   0, G j

G jt

G jt

G j

G j

(A.3)

j  1,..., M ,

where Gj is the dual variable for the capacity constraint of technology G, and

 Gj is the dual variable for the non-negativity of Q Gjt .

34 Page 34 of 63

Invoking symmetry, the Nash equilibrium in outputs is obtained when expressions (A.1) and (A.2) or (A.3) hold simultaneously for all N  M firms. The solution of (A.1) and (A.2) is given in Milstein and Tishler (2012)42 and

ip t

the solution of (A.3) is given in Tishler et al. (2008).43

First-stage equilibrium conditions

cr

We assume that  t is uniformly distributed. That is, f (  t )  1 /(    ) ,

us

   t   . The symmetric equilibrium solution (i.e. K 1S*  ...  K NS*  K S* and K 1G*  ...  K MG*  K G* ) is obtained when the following KKT conditions for each type of firm are satisfied simultaneously.44

M

an

Case (i): When K S*  ( c G  c S ) b ,45

See Proposition 1 in Milstein and Tishler (2012), where a base technology, denoted by B, is

d

42

te

PV, and a peaking technology, denoted by P, is CCGT. 43

See Eq. (4) in Tishler et al. (2008).

44

The KKT conditions are built using the proof of Proposition 2 in the Appendix of Milstein

Ac ce p

and Tishler (2012): the KKT conditions in (A.4), (A.5), (A.6), (A.7), (A.8), and (A.9) follow from Eq. (A16a), (A.16b), (A.29a), (A.29b), (A.44a) and (A.44b), respectively, in Milstein and Tishler (2012). The constraint in the maximization problem stems from using the uniform distribution function. That is,   c  a  bK i  bK  i  2bK j  bK  J in (A.4)-(A.7) and, G

S*

S*

G*

G*

  c  a  2bK i  bK i  bK j  bK  J in (A.8)-(A.9). G

45

S*

S*

G*

G*

These cases correspond to the three possible scenarios at the second stage of the model.

Only firms employing technology S produce electricity when daily demand for electricity is

low. When daily demand for electricity is larger, firms employing technology G enter production, provided that firms using technology S are already at full capacity in case (i), or at less than full capacity in case (ii) or case (iii). Case (ii) applies when firms employing technology S reach full capacity before firms employing technology G, whereas in case (iii) firms using technology G reach full capacity before firms employing technology S (see Proposition 1 in Milstein and Tishler, 2012).

35 Page 35 of 63

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  M [( c G  c S  bK G* ) 2  ( c G  c S ) 2 ]   2(    ) S T  2(    )b S  2(    ) S  0 ,  S K S*  0 ,

  [   a  c  bNK S

G

S*

 b( M  1 )K

G*

(A.4)

]  0,

  a  c G  bNK S*  b( M  1 )K G*  0 , K S*  0 , S  0 ,  S  0 ,

   a  c G  bNK S*  b( M  1 )K G*   ( 1   )  a  c G  b( M  1 )K G*   2

ip t

2

2(    ) G T  4(    )bG  2(    ) G  0 ,  G K G*  0 ,

(A.5)

G  [   a  c G  bNK S*  b( M  1 )K G* ]  0 ,

Case (ii): When ( c G  c S ) b  K S*  ( c G  c S ) b  K G* ,

cr

  a  c G  bNK S*  b( M  1 )K G*  0 , K G*  0 , G  0 ,  G  0 ,

us

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  M [( c G  c S  bK G* ) 2  ( bK S* ) 2 ]   2(    ) S T  2(    )bS  2(    ) S  0 ,  S K S*  0 , G

  a  c  bNK G

S*

S*

 b( M  1 )K

 b( M  1 )K

G*

G*

]  0,

(A.6)

 0 , K S*  0 ,  S  0 ,  S  0 ,

an

  [   a  c  bNK S

   a  c G  bNK S*  b( M  1 )K G*   ( 1   )  a  c G  b( M  1 )K G*   2

2

2(    ) G T  4(    )bG  2(    ) G  0 ,  G K G*  0 , G

  a  c  bNK G

S*

S*

 b( M  1 )K

 b( M  1 )K

G*

(A.7)

M

  [   a  c  bNK G

G*

]  0,

 0 , K G*  0 , G  0 ,  G  0 ,

d

Case (iii): When K S*  ( c G  c S ) b  K G* ,

te

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  2(    ) S T  4(    )bS  2(    ) S  0 ,  S K S*  0 , S  [   a  c G  b( N  1 )K S*  bMK G* ]  0 ,

  a  c  b( N  1 )K G

S*

 bMK

G*

 0, K

S*

(A.8)

 0,   0,   0, S

S

Ac ce p

  [   a  c G  bNK S*  b( M  1 )K G* ] 2  N [( c S  c G  bK S* )2  ( bK G* )2 ]  

( 1   )   a  c G  b( M  1 )K G*





2(    )  0 ,  K

G

G

G

  a  c  b( N  1 )K G

G*

S*

2

 2(    ) G T  2(    )bG 

 0 ,   [   a  c  b( N  1 )K

 bMK

G*

G

 0, K

G*

S*

 bMK

(A.9) G*

]  0,

 0,   0,   0, G

G

where S and G are the dual variables for the constraints (stemming from the uniform distribution assumption) for firms employing technology S and

technology G, respectively, and  S and  G are the dual variables for the non-

negativity of K S* and K G* , respectively.

36 Page 36 of 63

Appendix B A market with firms employing both technologies Second-stage equilibrium N M

N M

k i

k i

N M

N M

cr

a  2bQitS  b  QktS  2bQitG  b  QktG   t  c S  iS   iS  0

ip t

If the sun is shining on day t, the KKT conditions for firm i are given by:

a  2bQitS  b  QktS  2bQitG  b  QktG   t  c G  Gi   iG  0 , k i

k i

S it

G i

G it

S i

K  Q  0, K  Q  0, Q  0, Q  0, S i

S it

G i

G it

S it

G it

us

 Q  0 ,  Q  0 , ( K  QitS )iS  0 , ( K iG  QitG )Gi  0 , S i

(B.1)

an

iS  0 , Gi  0 ,  iS  0 ,  iG  0 , i  1,..., N  M

If there is no sun on day t, the KKT conditions for firm i are: N M

 QktG   t  c G  Gi   iG  0 , k i

 iG QitG  0 , ( K iG  QitG )Gi  0 ,

(B.2)

M

a  2bQitG  b

K  Q  0 , Q  0 ,   0 ,   0 , i  1,..., N  M , G i

G it

G it

G i

G i

where iS and Gi are the dual variables for the capacity constraint of

d

technology S and technology G, respectively, and iS and  iG are the dual

te

variables for the non-negativity of Q itS and QitG , respectively. Invoking

Ac ce p

symmetry, the Nash equilibrium in outputs is obtained when expressions (B.1) or (B.2) hold simultaneously for all N  M firms. The solution of (B.1) is given in Milstein and Tishler (2012)46 and the solution of (B.2) is given in Tishler et al. (2008).47

46

See Proposition 3 in Milstein and Tishler (2012).

47

See Eq. (4) in Tishler et al. (2008).

37 Page 37 of 63

First-stage equilibrium conditions The

equilibrium

solution

(i.e.

K 1S*  ...  K NS* M  K S*

and

satisfied:48   [   a  c S  b( N  M  1 )( K S*  K G* )] 2  2( c G  c S )b( N  M  1 )K G*   2(    ) S T  4(    )b  2(    ) S  0 ,

ip t

K 1G*  ...  K NG* M  K G* ) is obtained when the following KKT conditions are

   a  c G  b( N  M  1 )( K S*  K G* )  ( 1   )  a  c G  b( N  M  1 )K G*   (B.3) 2(    ) G T  4(    )b  2(    ) G  0 ,  S K S*  0 ,  G K G*  0 , 2

  [   a  c G  b( N  M  1 )( K S*  K G* )]  0 ,

us

  a  c G  b( N  M  1 )( K S*  K G* )  0 , K S*  0 , K G*  0 ,

cr

2

  0,  S  0,  G  0,

an

where  is the dual variable for the constraint (stemming from the uniform distribution assumption), and  S and  G are the dual variables for the non-

Ac ce p

te

d

M

negativity of K S* and K G* , respectively.

48

The KKT conditions are built using the proof of Proposition 3 in the Appendix of Milstein

and Tishler (2012): the first condition in (B.3) follows from Eq. (A80a) and the second condition in (B.3) follows from Eq. (A.80b). The constraint in the maximization problem is derived by using the uniform distribution. That is:

  c  a  2bK i  bK i  2bK i  bK  i . G

S*

S*

G*

G*

38 Page 38 of 63

Appendix C Numerical first-stage solution: each firm can employ only one technology The optimal capacity of the i-th (j-th) firm employing technology S (G), K S*

ip t

( K G* ), is either an interior or a corner solution. That is, we have to examine whether the KKT conditions, given by (A.4) and (A.5) or by (A.6) and (A.7)

cr

or by (A.8) and (A.9), depending on the case, are satisfied simultaneously at

each one of the six possible solutions49 for each case separately. Since the

us

constraints are identical in the firms’ maximization problems, they are either binding or non-binding concurrently. Therefore,  S  G   .

an

Case (i):

Solution I. K S* and K G* are interior solutions and the constraints in the firms’ maximization problems are unbinding. Thus,   0 ,  S  0 , and

M

 G  0 . K S* and K G* are derived from the following equation system:   [   a  c S  b( N  1 )K S*  bMK G* ] 2  M [( c G  c S  bK G* )2  ( c G  c S )2 ] 

2(    )

G

S*

T 0

 b( M  1 )K



G* 2



 ( 1   )   a  c  b( M  1 )K G



G* 2

(C.1) 

te

   a  c  bNK G

d

 2(    ) S T  0

The numerical solution of (C.1) yields four roots for each decision variable.

Ac ce p

The point ( K S* , K G* ) is optimal if and only if (1) K S* and K G* are real and

non-negative, (2) the constraints are positive at this point, and (3) the condition for case (i) holds, that is, K S*  ( c G  c S ) b . It is straightforward to

show that the second-order conditions hold if and only if the constraints in the firms’ maximization problems hold. Solution II. K S* and K G* are interior solutions and the constraints in the firms’ maximization problems are both binding. Thus,  S  0 and  G  0 .

K S* , K G* , and  are solved from the following system of equations:

49

We do not analyze the trivial solution K

S*

 K

G*

0.

39 Page 39 of 63

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  M [( c G  c S  bK G* ) 2  ( c G  c S ) 2 ]   2(    ) S T  2(    )b  0

   a  c G  bNK S*  b( M  1 )K G*   ( 1   )  a  c G  b( M  1 )K G*   2

2

(C.2)

2(    ) G T  4(    )b  0

ip t

  a  c G  bNK S*  b( M  1 )K G*  0

The numerical solution of (C.2) yields two roots for each decision variable.

cr

The point ( K S* , K G* ) is optimal if and only if (1) K S* , K G* , and λ are real and non-negative, and (2) the condition for case (i) holds, that is,

us

K S*  ( c G  c S ) b . It is straightforward to show that the second-order

conditions hold if and only if the constraints in the firms’ maximization problems hold.

an

Solution III. K S*  0 , K G*  0 , and the maximization problems’ constraints are unbinding. Thus,   0 and  G  0 , and the condition for case (i) always

M

holds. K G* and  S are derived from the following equation system:

  [   a  c S  bMK G* ] 2  M [( c G  c S  bK G* )2  ( c G  c S )2 ]  2(    ) S T  2(    ) S  0



d

 b( M  1 )K

G* 2

 2(    )

G

(C.3)

T 0

te

  a  c

G

The numerical solution of (C.3) yields two roots for K G* . The point (0, K G* )

Ac ce p

is optimal if and only if (1) K G* and  S are non-negative, and (2) the constraints are positive at this point. Solution IV. K S*  0 , K G*  0 , and the maximization problems’ constraints are binding. Thus,  G  0 , and the condition for case (i) always holds. K G* , λ, and  S are derived from the following equation system:

  [   a  c S  bMK G* ] 2  M [( c G  c S  bK G* )2  ( c G  c S )2 ] 

2(    ) S T  2(    )b  2(    ) S  0

  a  c

G

 b( M  1 )K G*



2

 2(    ) G T  4(    )b  0

(C.4)

  a  c G  b( M  1 )K G*  0 The numerical solution of (C.4) yields one root for K G* . The point (0, K G* ) is optimal if and only if K G* , λ, and  S are non-negative.

40 Page 40 of 63

Solution V. K S*  0 , K G*  0 , and the maximization problem constraint is unbinding. Thus,   0 and  S  0 . K S* and  G are derived from the following equation system:

    a  c S  b( N  1 )K S*   2(    ) S T  0

   a  c G  bNK S*   ( 1   )  a  c G   2(    ) G T  2

2

(C.5)

cr

2(    ) G  0

ip t

2

The numerical solution of (C.5) yields two roots for K S* . The point ( K S* , 0)

us

is optimal if and only if (1) K S* and  G are non-negative, (2) the constraints are positive at this point, and (3) the condition for case (i) holds, that is,

an

K S*  ( c G  c S ) b .

Solution VI. K S*  0 , K G*  0 , and the maximization problems’ constraints

equation system:

M

are binding. Thus,  S  0 . K S* , λ, and  G are derived from the following

    a  c S  b( N  1 )K S*   2(    ) S T  2(    )b  0 2

   a  c G  bNK S*   ( 1   )  a  c G   2(    ) G T  2

d

2

(C.6)

te

4(    )b  2(    ) G  0

  a  c G  bNK S*  0

Ac ce p

The numerical solution of (C.6) yields one root for K S* . The point ( K S* , 0) is

optimal if and only if (1) K S* , λ, and  G are non-negative, and (2) the condition for case (i) holds, that is, K S*  ( c G  c S ) b .

Case (ii):

Solution I. K S* and K G* are interior solutions and the constraints in the

firms’ maximization problems are unbinding. Thus,   0 ,  S  0 , and

 G  0 . K S* and K G* are derived from:

41 Page 41 of 63

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  M [( c G  c S  bK G* )2  ( bK S* )2 ]  2(    ) S T  0

   a  c  bNK G

2(    )

G

S*

 b( M  1 )K



G* 2



 ( 1   )   a  c  b( M  1 )K G



G* 2

(C.7) 

T 0

ip t

The numerical solution of (C.7) yields four roots for each decision variable.

The point ( K S* , K G* ) is optimal if and only if (1) K S* and K G* are real and

cr

non-negative, (2) the constraints are positive at this point, and (3) the

condition for case (ii) holds, that is, ( c G  c S ) b  K S*  ( c G  c S ) b  K G* . It

us

is straightforward to show that the second-order conditions hold if and only if the constraints in the firms’ maximization problems hold.

an

Solution II. K S* and K G* are interior solutions and the constraints in the firms’ maximization problems are both binding. Thus,  S  0 and  G  0 . K S* , K G* , and  are solved from the following system of equations: 2(    ) S T  2(    )b  0

M

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  M [( c G  c S  bK G* )2  ( bK S* )2 ]      a  c G  bNK S*  b( M  1 )K G*   ( 1   )  a  c G  b( M  1 )K G*   2

2

(C.8)

d

2(    ) G T  4(    )b  0

te

  a  c G  bNK S*  b( M  1 )K G*  0

The numerical solution of (C.8) yields two roots for each decision variable.

Ac ce p

The point ( K S* , K G* ) is optimal if and only if (1) K S* , K G* , and λ are real

and non-negative, and (2) the condition for case (ii) holds, that is,

( c G  c S ) b  K S*  ( c G  c S ) b  K G* . It is straightforward to show that the

second-order conditions hold if and only if the constraints in the firms’ maximization problems hold. Solution III. K S*  0 , K G*  0 , and the maximization problems’ constraints are unbinding. The condition for case (ii) never holds at this point. Solution IV. K S*  0 , K G*  0 , and the maximization problems’ constraints are binding. The condition for case (ii) never holds at this point. Solution V. K S*  0 , K G*  0 , and the maximization problem constraint is unbinding. The condition for case (ii) never holds at this point.

42 Page 42 of 63

Solution VI. K S*  0 , K G*  0 , and the maximization problems’ constraints are binding. The condition for case (ii) never holds at this point.

ip t

Case (iii): Solution I. K S* and K G* are interior solutions and the constraints in the

 G  0 . K S* and K G* are derived from:   [   a  c S  b( N  1 )K S*  bMK G* ] 2  2(    ) S T  0

cr

firms’ maximization problems are unbinding. Thus,   0 ,  S  0 , and



( 1   )   a  c G  b( M  1 )K G*

us

  [   a  c G  bNK S*  b( M  1 )K G* ] 2  N [( c S  c G  bK S* )2  ( bK G* )2 ]   (C.9)



2

 2(    ) G T  0

an

The numerical solution of (C.9) yields two roots for each decision variable. The point ( K S* , K G* ) is optimal if and only if (1) K S* and K G* are real and

M

non-negative, (2) the constraints are positive at this point, and (3) the condition for case (iii) holds, that is, K S*  ( c G  c S ) b  K G* . It is straightforward to show that the second-order conditions hold if and only if the

d

constraints in the firms’ maximization problems hold.

te

Solution II. K S* and K G* are interior solutions and the constraints in the firms’ maximization problems are binding. Thus,  S  0 and  G  0 . K S* ,

Ac ce p

K G* , and  are solved from:

  [   a  c S  b( N  1 )K S*  bMK G* ] 2  2(    ) S T  4(    )b  0

  [   a  c G  bNK S*  b( M  1 )K G* ] 2  N [( c S  c G  bK S* )2  ( bK G* )2 ]   (C.10)



( 1   )   a  c G  b( M  1 )K G*



2

 2(    ) G T  2(    )b  0

  a  c G  b( N  1 )K S*  bMK G*  0

The numerical solution of (C.10) yields two roots for each decision variable. The point ( K S* , K G* ) is optimal if and only if (1) K S* , K G* , and λ are real

and non-negative, and (2) the condition for case (iii) holds, that is,

K S*  ( c G  c S ) b  K G* . It is straightforward to show that the second-order conditions hold if and only if the constraints in the firms’ maximization problems hold.

43 Page 43 of 63

Solution III. K S*  0 , K G*  0 , and the maximization problems’ constraints are unbinding. The condition for case (iii) never holds at this point. Solution IV. K S*  0 , K G*  0 , and the maximization problems’ constraints

ip t

are binding. The condition for case (iii) never holds at this point. Solution V. K S*  0 , K G*  0 , and the maximization problem constraint is

cr

unbinding. Thus,   0 and  S  0 . K S* and  G are derived from the following equation system:   [   a  c S  b( N  1 )K S* ] 2  2(    ) S T  0

  [   a  c G  bNK S* ] 2  N ( c S  c G  bK S* )2   ( 1   )  a  c G   2(    )

G

us

2

T  2(    )  0 G

(C.11)

an

The numerical solution of (C.11) yields two roots for K S* . The point

( K S* , 0 ) is optimal if and only if (1) K S * and  G are non-negative, (2) the

that is, K S*  ( c G  c S ) b  K G* .

M

constraints are positive at this point, and (3) the condition for case (iii) holds,

Solution VI. K S*  0 , K G*  0 , and the maximization problems’ constraints

te

equation system:

d

are binding. Thus,  S  0 . K S* , λ, and  G are derived from the following

  [   a  c S  b( N  1 )K S* ] 2  2(    ) S T  4(    )b  0

Ac ce p

  [   a  c G  bNK S* ] 2  N ( c S  c G  bK S* )2  ( 1   )  a  c G   (C.12) 2(    ) G T  2(    )b  2(    ) G  0 2

  a  c G  b( N  1 )K S*  0

The numerical solution of (C.12) yields one root for K S* . The point ( K S* , 0)

is optimal if and only if (1) K S* , λ, and  G are non-negative, and (2) the condition for case (iii) holds, that is, K S*  ( c G  c S ) b  K G* .

44 Page 44 of 63

Appendix D Numerical first-stage solution: each firm can employ both technologies

ip t

K S* and K G* , the optimal capacities of the i-th firm employing both technologies S and G, may be interior or corner solutions. That is, we have to simultaneously at each one of the six possible solutions.50

cr

examine whether the KKT conditions, given by (B.3), are satisfied Solution I. K S* and K G* are both interior solutions and the maximization

us

problem constraint is unbinding. Thus,   0 ,  S  0 , and  G  0 . K S* and K G* are derived from: 2(    ) S T  0

   a  c  b( N  M  1 )( K G

K

G*





2

)  ( 1   )   a  c  b( N  M  1 )K

T 0

G



G* 2

(D.1) 

M

2(    )

G

S*

an

  [   a  c S  b( N  M  1 )( K S*  K G* )] 2  2( c G  c S )b( N  M  1 )K G*  

The numerical solution of (D.1) yields four roots for each decision variable. The point ( K S* , K G* ) is optimal if and only if (1) K S* and K G* are real and

d

non-negative, and (2) the constraint is positive at this point. It is

te

straightforward to show that the second-order conditions hold if and only if the constraints in the firms’ maximization problems hold.

Ac ce p

Solution II. K S* and K G* are both interior solutions and the maximization problem constraint is binding. Thus,  S  0 and  G  0 . K S* , K G* , and λ are solved from:

  [   a  c S  b( N  M  1 )( K S*  K G* )] 2  2( c G  c S )b( N  M  1 )K G*   2(    ) S T  4(    )b  0

   a  c G  b( N  M  1 )K S*  b( N  M  1 )K G*  



( 1   )   a  c G  b( N  M  1 )K G*

2



2

(D.2)

 2(    ) G T  4(    )b  0

  a  c G  b( N  M  1 )( K S*  K G* )  0

50

We do not analyze the trivial solution K

S*

 K

G*

0.

45 Page 45 of 63

The numerical solution of (D.2) yields two roots for each decision variable. The point ( K S* , K G* ) is optimal if and only if K S* , K G* , and λ are real, and non-negative. The second-order conditions hold if and only if the constraints

ip t

in the firms’ maximization problems hold. Solution III. K S*  0 , K G*  0 , and the maximization problem constraint is

cr

unbinding. Thus,   0 and  G  0 . K G* and  S are derived from the following equation system:

  [   a  c S  b( N  M  1 )K G* ] 2  2( c G  c S )b( N  M  1 )K G*  

  a  c

G

 b( N  M  1 )K



G* 2

us

2(    ) S T  2(    ) S  0

(D.3)

 2(    ) G T  0

an

The numerical solution of (D.3) yields two roots for K G* . The point (0, K G* ) is optimal if and only if (1) K G* and  S are non-negative, and (2) the

M

constraint is positive at this point.

Solution IV. K S*  0 , K G*  0 , and the maximization problem constraint is

equation system:

d

binding. Thus,  G  0 . K G* , λ, and  S are derived from the following

te

  [   a  c S  b( N  M  1 )K G* ] 2  2( c G  c S )b( N  M  1 )K G*   2(    ) S T  4(    )b  2(    ) S  0 G

 b( N  M  1 )K G*

Ac ce p

  a  c



2

 2(    ) G T  4(    )b  0

(D.4)

  a  c G  b( N  M  1 )K G*  0

The numerical solution of (D.4) yields one root for K G* . The point (0, K G* ) is optimal if and only if K G* , λ, and  S are non-negative. Solution V. K S*  0 , K G*  0 , and the maximization problem constraint is

unbinding. Thus,   0 and  S  0 . K S* and  G are derived from the following equation system:     a  c S  b( N  M  1 )K S*   2(    ) S T  0 2

   a  c G  b( N  M  1 )K S*   ( 1   )  a  c G   2(    ) G T  2

2

(D.5)

2(    ) G  0

46 Page 46 of 63

The numerical solution of (D.5) yields two roots for K S* . The point ( K S* , 0) is optimal if and only if (1) K S* and  G are non-negative, and (2) the constraint is positive at this point.

ip t

Solution VI. K S*  0 , K G*  0 , and the maximization problem constraint is binding. Thus,  S  0 . K S* , λ, and  G are derived from the following equation system: 2

   a  c G  b( N  M  1 )K S*   ( 1   )  a  c G   2

2(    ) G T  4(    )b  2(    ) G  0

  a  c G  b( N  M  1 )K S*  0

(D.6)

us

2

cr

    a  c S  b( N  M  1 )K S*   2(    ) S T  4(    )b  0

an

The numerical solution of (D.6) yields one root for K S* . The point ( K S* , 0) is optimal if and only if K S* , λ, and  G are non-negative.Figure 1. Industry

40 36.69

36.28

36.69

36.36

36.69

36.44

36.69

te

Industry capacity (1000 MW)

d

36.19

M

capacity

Ac ce p

20

0

Legend:

q S/q G=5

q S/q G=4

Each firm employs one technology, S or G Each firm employs both technologies, S and G

q S/q G=3

q S/q G=2

Capacity cost ratio

PV capacity is depicted by the striped areas of the bars. CCGT capacity is depicted by the white area of the bars.

47 Page 47 of 63

Figure 2. Industry capacity as a function of price elasticity ( θ S θ G  4 ) 40

39.44

39.10

ip t

36.69

36.28 33.94

Industry capacity (1000 MW)

33.45

cr

Each firm employs one technology, S or G

Each firm employs both technologies, S and G

an

us

20

0 -0.10

-0.25

-0.40

PV capacity is depicted by the striped areas of the bars. CCGT capacity is depicted by the white area of the bars.

Ac ce p

te

d

Legend:

M

Demand elasticities

48 Page 48 of 63

Figure 3. Industry capacity as a function of N (M = 10) 40 36.28 36.30 36.31

36.36 36.38 36.39

36.44 36.46 36.47

cr

Industry capacity (1000 MW)

ip t

36.19 36.21 36.22

0

N = 10 N = 20 N = 100 N = 10 N = 20 N = 100

q S/q G=4

q S/q G=5

an

us

20

N = 10 N = 20 N = 100 N = 10 N = 20 N = 100

q S/q G=3

q S/q G=2

M

Capacity cost ratio

PV capacity is depicted by the striped areas of the bars. CCGT capacity is depicted by the white area of the bars.

Ac ce p

te

d

Legend:

49 Page 49 of 63

Figure 4. Price distribution Each firm employs one technology, S or G 500

ip t

Each firm employs both technologies, S and G

cr

300

us

US$ per MWH

400

200

day-time 0

1

night-time

365 1

day-time

365 1

night-time

365 1

q S /q G=5

an

100

day-time

365 1

M

q S/q G=4

night-time

365 1

q S/q G=3

365 1

day-time

night-time

365 1

365

q S/q G=2

Ac ce p

te

d

Capacity cost ratio

50 Page 50 of 63

Figure 5. Average and maximal electricity price day-time

600

average: Each firm employs one technology, S or G

538

519

498

night-time

600

average: Each firm employs both technologies, S and G

407

355

200

383 average: Each firm employs one technology, S or G average: Each firm employs both technologies, S and G max: Each firm employs one technology, S or G max: Each firm employs both technologies, S and G

142

144

146

147

113

118

123

127

max: Each firm employs both technologies, S and G

400

231

200

152 126 78

0

q S /q G=4

q S/q G=2

q S /q G=3 Capacity cost ratio

q S/q G=5

170

138 83

258

186

151 90

q S /q G=4 q S/q G=3 Capacity cost ratio

270

199

163 98

q S/q G=2

Ac ce p

te

d

M

an

us

q S/q G=5

0

245

cr

427

US$ per MWH

US$ per MWH

400

ip t

max: Each firm employs one technology, S or G

476

51 Page 51 of 63

Figure 6. Electricity price as a function of price elasticity ( θ S θ G  4 ) day-time 944

night-time

1000

average: Each firm employs one technology, S or G

average: Each firm employs one technology, S or G

average: Each firm employs both technologies, S and G

average: Each firm employs both technologies, S and G

max: Each firm employs one technology, S or G

max: Each firm employs one technology, S or G

max: Each firm employs both technologies, S and G

max: Each firm employs both technologies, S and G

ip t

1000

500

383

364 298

117 104

118 0 -0.1

245 170

243 144

178

-0.25

-0.4

224

138

107 0 -0.1

-0.25

174 131 117 79 -0.4

Demand elasticities

Ac ce p

te

d

M

an

Demand elasticities

83

us

233

459

cr

498

500

US$ per MWH

US$ per MWH

687

52 Page 52 of 63

Figure 7. Electricity price as a function of N (M = 10) day-time

night-time

500

q S/q G=2

q S /q G=3 q S/q G=4 Capacity cost ratio

100

q S /q G=5

q S /q G=3 q S/q G=4 Capacity cost ratio

q S/q G=2

Ac ce p

te

d

M

an

us

q S/q G=5

300

cr

300

100

average: N = 10 average: N = 20 average: N = 100 max: N = 10 max: N = 20 max: N = 100

US$ per MWH

US$ per MWH

average: N = 10 average: N = 20 average: N = 100 max: N = 10 max: N = 20 max: N = 100

ip t

500

53 Page 53 of 63

Figure 8. Industry production 244.08 222.63

241.11 218.39

cr

150

237.89 214.27

200

Each firm employs one technology, S or G

Each firm employs both technologies, S and G

us

100

50

0

q S/q G=4

q S/q G=5

an

Industry production (1000 MWH)

226.87

ip t

246.56

250

q S/q G=3

q S/q G=2

M

Capacity cost ratio

The share of PV in total production is depicted by the striped areas of the bars. The share of CCGT is depicted by the white area of the bars.

Ac ce p

te

d

Legend:

54 Page 54 of 63

Figure 9. Industry profits 15

ip t

13.90

9.79

Each firm employs one technology, S or G

cr

10 7.76

Each firm employs both technologies, S and G 6.36

us

Industry Profits ($ Billion)

11.85

5

4.24

0.52 0

q S/q G=4

q S/q G=5

an

2.25

q S/q G=3

q S/q G=2

M

Capacity cost ratio

Positive profits of PV producers are depicted by the striped areas of the bars.

Ac ce p

te

d

Legend:

55 Page 55 of 63

Figure 10. Industry profits as a function of N (M = 10)

11.85

7.96

10.18

9.99

8.14

us

7.76

5

0

12.22

cr

9.79

10

12.04

ip t

13.90 14.07 14.24

N = 10 N = 20 N = 100

N = 10 N = 20 N = 100

q S/q G=4

q S/q G=5

an

Industry Profits ($ Billion)

15

N = 10 N = 20 N = 100 N = 10 N = 20 N = 100

q S/q G=3

q S/q G=2

M

Capacity cost ratio

Positive profits of PV producers are depicted by the striped areas of the bars.

Ac ce p

te

d

Legend:

56 Page 56 of 63

Figure 11. Social welfare 80

72.10

74.33

72.57

74.84

75.44

73.12

ip t

73.96

60

cr

Each firm employs one technology, S or G 40

us

Each firm employs both technologies, S and G

20

0

q S/q G=4

q S/q G=5

an

Social Welfare ($ Billion)

71.73

q S/q G=3

q S/q G=2

Consumer surplus is depicted by the white areas of the bars. Profits are depicted by the gray areas of the bars.

Ac ce p

te

d

Legend:

M

Capacity cost ratio

57 Page 57 of 63

Figure 12. Social welfare as a function of N (M = 10) 72.10 71.98 71.86

72.57 72.44 72.32

cr

Social Welfare ($ Billion)

60

us

40

N = 10 N = 20 N = 100 N = 10 N = 20 N = 100

q S/q G=4

q S/q G=5

an

20

0

73.12 73.00 72.89

ip t

71.73 71.61 71.49

N = 10 N = 20 N = 100 N = 10 N = 20 N = 100

q S/q G=3

q S/q G=2

Consumer surplus is depicted by the white areas of the bars. Profits are depicted by the gray areas of the bars.

Ac ce p

te

d

Legend:

M

Capacity cost ratio

58 Page 58 of 63

Figure 13. Industry capacity when the tax on CO2 is $10, $30 and $50 per ton 36.17

36.56

35.96

36.29

35.75

36.02

cr

36.69

Each firm employs one technology, S or G 20

us

Each firm employs both technologies, S and G

0

$10 tax

$50 tax

PV capacity is depicted by the striped areas of the bars. CCGT capacity is depicted by the white area of the bars.

Ac ce p

te

d

Legend:

$30 tax

M

no tax on CO2

an

Industry capacity (1000 MW)

36.28

ip t

40

59 Page 59 of 63

Figure 14. Electricity price when the tax on CO2 is $10, $30 and $50 per ton day-time

400

average: Each firm employs one technology, S or G average: Each firm employs both technologies, S and G max: Each firm employs one technology, S or G max: Each firm employs both technologies, S and G

200

ip t

US$ per MWH

396

144

148

155

118

124

134

162

245

200 170

143

138 83

0

$10 tax

0

$50 tax no tax on CO2

$30 tax

179 144 86

$10 tax

262

273

193

207

155 94

$30 tax

165 104

$50 tax

Ac ce p

te

d

M

an

no tax on CO2

251

cr

419 400 383

average: Each firm employs one technology, S or G average: Each firm employs both technologies, S and G max: Each firm employs one technology, S or G max: Each firm employs both technologies, S and G

440

us

498

night-time

600 542

525

507

US$ per MWH

600

60 Page 60 of 63

Figure 15. Industry profits, consumer surplus and tax payments when the tax on CO2 is $10, $30 and $50 per ton 74.33

74.93

72.45

73.04

76.02

73.52

76.97

cr

60

Each firm employs one technology, S or G

us

40

Each firm employs both technologies, S and G

20

0

no tax on CO2

M

$10 tax

an

Social Welfare ($ Billion)

72.10

ip t

80

$30 tax

$50 tax

Ac ce p

te

d

Legend: Industry profits are depicted by the gray areas of the bars. Consumer surplus is depicted by the white areas of the bars. Tax payments are depicted by the striped areas of the bars (above consumer surplus).

61 Page 61 of 63

Table 1: Daily averages and maximal values of electricity use, per hour, in California during 2011 (1000 MWH) Average

hourly use: hourly use:

Average

Maximal

hourly use

hourly use

ip t

Average

day-time

night-time

Mean

27.79

24.81

26.30

Median

27.20

24.04

25.58

Standard deviation

3.42

2.13

2.74

4.10

Minimum

21.42

21.27

21.35

24.27

Maximum

38.72

31.53

35.13

45.57

31.20

Ac ce p

te

d

M

an

us

cr

30.00

62 Page 62 of 63

Highlights:  We employ a two-stage model with endogenous capacity and operations to

electricity markets.

ip t

assess the practicality of photovoltaic technology (PV) in competitive

electricity price increases when PV capacity is higher.

cr

 Price spikes are higher and more frequent and the average (over the year)  Price spikes are higher and more frequent when the regulator introduces

us

CO2 taxes.

 Profits of electricity producers that employ natural gas are higher when PV

an

capacity rises.

 Consumer surplus decreases when the CO2 tax rate and/or the number of

Ac ce p

te

d

M

PV-using producers increase.

63 Page 63 of 63