Effects of electricity price volatility and covariance on the firm s investment decisions and long-run demand for electricity C. Brandon (cbrandon@andrew.cmu.edu) Carnegie Mellon University, Pittsburgh, PA December 31, 2004 Abstract We examine how firms investment decisions, and hence their long run electricity demand, are affected by two likely features of a deregulated electricity market price volatility and covariance with other input factor prices. Building on Abel s (1983) investment model, we show that uncertainty from price volatility leads to an increase in investment when the electricity prices are either unrelated or positively correlated with other factor prices. A negative correlation between electricity prices and other input prices, however, causes price volatility to lead to a decrease in investment. Implications for electricity market design are discussed. 1 Introduction Deregulation, by letting prices be determined in a market, may increase price volatility. Such volatility creates uncertainty for customers and may encourage suppliers to turn to dynamic pricing schemes that pass on volatility to their customers. For example, following the deregulation of the electricity markets in California, small commercial customers of San Diego Gas and Electric (SDGE) saw the charges for electricity generation fluctuate between 3.242 cents per kw and 23.045 cents per kw with the space of a year. In addition, the experience of deregulation in California has led to calls for dynamic prcing of electricity (Borenstein et al. 2002, Smith and Kiesling, 2003). Apart from increasing price volatility, deregulation may also cause electricity prices to become correlated with the prices of other factors of production. In Previously titled The Reaction of Small Firms to Fluctuating Electricity Prices The author gratefully acknowledges financial support from the Carnegie Mellon Electricity Industry Center. Additional thanks go to Greg Katsapis and his collegues at San Diego Gas & Electric for their valuable help. 1
California, for example, about a third of SDGE s small firm customers use both natural gas and electricity. Natural gas has also been the primary fuel for instate electricity generation, accounting for, by 2002, 48.7% of total generation (DOE 2004). Thus firms who use both natural gas and electricity may find their prices correlated. To date, few studies have examined the effects of electricity price volatility and covariance on firm investment decisions and long-run electricity demand. We address this gap by developing a general theoretical model. We use data from SDGE to illustrate the potential for significant correlation between wages and electricity prices and for variation in this correlation across different industries. We also discuss the potential implications of our model for market design. 2 Model Our model is a generalized version of Abel s (1983) model of investment under uncertainty, which analyses the behavior of a profit maximizing firm facing a stochastic price for its output. Although this model has been faulted for ignoring the role of irreversibility in investment decisions, it provides a convenient framework to discuss how co-movements and volatility in input prices affects investment. To simplify analysis we restrict ourselves to three inputs; a semi-fixed input, capital and two variable factors of production (x 1, x 2 ). The firm in our model has a Cobb-Douglas production function and faces exogenous, stochastic input prices for the variable factors of production. These prices follow a multi-dimensional geometric Brownian motion. The output consists of a single homogenous good sold in a competitive international market i.e. regional input price shocks do not increase the firm s output price. The firm seeks to maximize expected profit over time subject to the following constraints: dk t = (I t δk t ) dt (1) dp t /p t = σ p dz p (2) dp 1,t /p 1,t = σ p1 dz p1 (3) dp 2,t /p 2,t = σ p2 dz p2 (4) dz i dz j = ρ ij dt, i, j {p t, p 1,t, p 2,t }. (5) Subject to these constraints, firm s optimization problem is described by: V (K t, p t, p 1,t, p 2,t ) = [ ps X1,sK α s X γ 2,s p 1,sX 1,s τis φ ] p 2,s X 2,s e r(s t) ds max X 1,X 2,I t (6) Where K t, X 1,t, X 2,t denote capital and the two variable factors of production. I t is the amount of investment at time t, while the costs of the new investment 2
is captured by τi φ t where φ > 1, τ > 0 are both model parameters and. The unit costs of X 1 and X 2 are p 1, p 2,t respectively, while p t is the price of the firm s output. r is the time discount factor. 1 ρ i,j 1, ρ = 1 if i = j, while ρ p,p1 = ρ p,p2 = 0. φ is a constant whose value is assumed to be greater than unity due to the presence of adjustment costs. Dynamic programming methods show that the optimal solution requires: rv (K t, p t, p 1,t, p 2,t ) dt = [ ] max p t X α X 1,X 2,I t 1,tK t X γ 2,t p 1,tX 1,t τi φ t p 2,t X 2,t dt + E t dv (7) Assuming that the value function is a function of all the state variables, K t, p t, p 1,t, p 2,t we can apply the multi-dimensional Ito s lemma (see the appendix) and substitute for dk t to get the following expression for X 2,t dv. E t dv = (V K (I t δk t ) + p 1,t p 2,t V p1p 2 σ p1 σ p2 ρ p1p 2 ) dt ( 1 + 2 p2 t V pp σp 2 + 1 2 p2 1,tV p1p 1 σp 2 1 + 1 ) 2 p2 2,tV p2p 2 σp 2 2 dt (8) Substituting the above for E t dv in the condition for optimality yields the following first order conditions with respect to X 1,t, X 2,t and I t : αp t X α 1 1,t K t X γ 2,t p 1,t = 0 (9) γp t X α 1,tK t X γ 1 2,t p 2,t = 0 (10) φτi φ 1 t + V K = 0 (11) Algebraic manipulation of the first order conditions yields the optimal values of the control variables solely in terms of the state variables and the parameters of the model: [( ) ( ) γ ] 1/(1 α γ) αpt γp1,t X 1,t = K t (12) p 1,t [( ) ( γpt X 2,t = p 2,t 2,t 2,t γp 1,t ) α ] 1/(1 α γ) K t (13) I t = ( ) 1/(φ 1) VK (14) φτ At the moment we do not know V K. However, by substituting for the optimal levels of the control variables in our condition for optimality, we obtain a second order differential equation that defines the value function, V (K, p, p 1, p 2 ) 1 [ (p2 ) γ (p1 ) α rv = (1 α γ) K γ α ] 1 1 α γ + (φ 1) τ ( ) φ/(φ 1) VK φτ V K δk + p 1 p 2 V p1p 2 σ p1 σ p2 ρ p1p 2 + p2 2 V ppσ 2 p + p2 1 2 V p 1p 1 σ 2 p 1 + p2 2 2 V p 2p 2 σ 2 p 2 (15) 1 We dropped the time subscript to aid clarity, since no meaning is lost by doing so 3
To get a closed form solution for the value function, we now need to assume constant returns to scale technology. In doing so, the above expression simplifies to: rv = ( ) γ p2 γ ( p1 ) α K + (φ 1) τ α ( ) φ/(φ 1) VK V K δk φτ +p 1 p 2 V p1p 2 σ p1 σ p2 ρ p1p 2 + p2 2 V ppσ 2 p + p2 1 2 V p 1p 1 σ 2 p 1 + p2 2 2 V p 2p 2 σ 2 p 2 (16) It can be readily checked that the solution to this equation takes the following form: where (1 ) x = 2 2 σp 2 α (α + ) 2 2 σp 2 1 ( φ z = φ 1 V (K, p, p 1, p 2 ) = mh y + nhk (17) h = p 1 α p y = 1 p γ φ φ 1 2 (18) (19) n = γ γ α α r + δ + x (20) m = 1 (φ 1) r + z (21) σp 2 2 αγ 2 σ p 1 σ p2 ρ p1p 2 (22) γ (γ + ) 2 2 ) 2 x + φ ( ασp 2 1 + γσp 2 2 σp) 2 2 2 (φ 1) 2 (23) By substituting these values into (12-14) we obtain the optimal values for X 1, X 2 demand and the level of investment. X 1,t = p 1 t X 2,t = p 1 t ( α p 1,t ( γ p 2,t ) 1 γ ) 1 α I t = ( γ p 2,t ( α ( nh φτ p 1,t ) γ Kt (24) ) α Kt (25) ) 1/(φ 1) (26) Note that since 1 γ = α +, inspection reveals that the relationship between optimal X 2 demand and the demand for X 1 is what we would expect, namely X 2,t = γp 1,t αp 2 X 1,t The reader is welcome to check that when we set σ p1, σ p2, γ = 0 and = 1 α our answers for optimal X 1 and investment match those in Abel s (1983) model. 4
2. 2.1 Effects of input price volatility on investment and the long run demand for X 2 Having derived the expression for optimal investment above we see that volatility affects investment by affecting the value of n. Since φ > 1 we know that investment is increasing in n. In turn, since input price volatility only enters the expression for n through the value of x, we know that anything that increases the value of x will decrease the optimal level of investment (and vice versa). Thus the sign of the partial derivatives of σ p1 and σ p2 determine whether an increase in input price volatility will increase or decrease investment. x α (α + ) = σ p1 2 σ p1 αγ 2 σ p 2 ρ p1p 2 (27) x γ (γ + ) = σ p2 2 σ p2 αγ 2 σ p 1 ρ p1p 2 (28) These results are consistent with those of the original model. So long as the correlation between the price of X 2 and the price of X 1 is positive, an increase in input price volatility reduces the value of x, increases the risk to expected profits and thus boosts investment. However, a negative relationship between the price of X 1 and the price of X 2 could potentially reverse this effect. For example investment decreases with increased X 2 price volatility if the covariance between p 1 and p 2 is negative so that: 1 γ γ ( σp1 σ p2 ) < ρ p1p 2 (29) In other words, if the X 2 price becomes significantly more volatile than the price of X 1, and the two are negatively correlated, the effect will be a reduction in investment. This is quite different to the results of the original model where increased price volatility always increased investment. Similarly, investment decreases with increased volatility in p 1 if the covariance between p 1 and p 2 is negative enough that: 1 α α ( σp2 σ p1 ) < ρ p1p 2 (30) But what might cause a negative correlation between p 1 and p 2? In our model the firm competes in an international market, thus if the costs of the firm s foreign competitors remain unchanged any increase in X 2 price will force firms to cut costs or go out of business, either could depress p 1. Alternatively, if X 2 became very cheap, firms would exploit the cost advantages to boost output and thus would bid up p 1. 2 Abel uses where we have used φ and γ where we have used τ 5
3 An empirical example: correlation between electricity prices and wages To see if this perverse effect could be anything beyond a theoretical anomaly, and to see if the effect of input price covariance could have a significant effect, we decided to look at the labor costs and electricity prices faced by firms in deregulated electricity markets. We obtained data from SDGE on their small non-residential customers for the period from September 1999 to September 2002. We examined the correlations between wages and the commodity cost of electricity paid by manufacturing firms. We looked at different two digit SIC codes to see if there were differences in the relationship between electricity prices and wages across industries. While most industries have positive correlations between wages and electricity prices in this period, firms in SIC 28 (chemical products) experienced strong negative correlation. Depending on when they were billed each month, the correlation between wages and electricity prices ranged from -0.492 to -0.446, while the standard deviations in wages ranged from 0.242-0.252. The range for the deviation in electricity price was from 0.033-0.039. Thus, even if we use the most conservative values α > 0.265 is enough to ensure the second inequality holds true. If α > 0.265 is a fair characterization for these firms, then our model suggests that any increase in wage volatility would result in a decrease in investment. In our model, electricity price volatility affects electricity demand only in the long run, through its effect on investment. Looking at (25) reveals that electricity consumption is linear in the capital stock. Thus any increase in the capital stock increases electricity consumption while anything to decrease it leads to a drop in electricity demand. Thus the long term effects of a permanent increase in electricity price volatility on electricity consumption will depend on whether it increases investment (in which case demand will rise) or not. The above results may be modified by the inclusion of irreversibility in investment. Irreversibility dampens the positive impact of profit uncertainty on investment through the value of the option to delay investment. The inclusion of input price variance and co-variance, however, affects the magnitude of the uncertainty associated with profits. Therefore the inclusion of irreversible investment does not alter the arguement that input price covariance and volatility must be considered when determining the effects of uncertainty on investment. 4 Implications Real time pricing (RTP), where retail prices reflect changes in the wholesale price in real time, could involve dramatic price volatility for consumers. While the California experience was unusual and partly due to price manipulation, the SDGE tariff wasn t a RTP, but rather a weighted average of the RTP. Yet the commodity costs per kw still fluctuated from a little over 3 cents to 23 cents a kw. Without providing easy and accessible hedging options for small customers (which would effectively undermine the very incentives RTP is trying to create) 6
our model shows that this could cause significant problems for some firms. RTP may be an attractive option only for firms who can easily smooth consumption. Our model also suggests an interesting trade-off. Deregulating electricity is expected to reduce the average price consumers pay for their power. On the other hand, in the absence of accessible hedging options for small businesses, it also exposes these firms to greater electricity price uncertainty. Our model suggests that how firms respond may vary across industries and that if electricity prices fail to fall much, not all electricity consumers will be better off if they are exposed to fluctuating prices. The key implication is that different market designs can affect not just the average market clearing price but also price volatility and how it may co-vary with other factor prices. This in turn can have a significant impact on the purchasing firms investment plans and thus its long run demand for different inputs. We believe that this implies that the context of the market becomes important. Different industries will have different technological options and different market structures for their input and output markets, so that any changes to the market structure of one input will affect different industries in different ways. 5 References Abel, Andrew B. (1983): Optimal Investment Under Uncertainty, American Economic Review, 13: 228-233. Borenstein, Severin, Michael Jaske, and Arthur Rosenfeld, (2002): Dynamic Pricing, Advanced Metering, and Demand Response in Electricity Markets, Hewlett Foundation Energy Series, available at http://www.ef.org/documents/dynamicpricing.pdf DOE (2004) http://www.eia.doe.gov/cneaf/electricity/st profiles/california.pdf Smith, Vernon L. and Lynne Kiesling: Demand, Not Supply Wall Street Journal, New York, N.Y.: pg. A 10. August 20, 2003 Zarnikau, Jay (1990): Customer Responsiveness to Real-Time Pricing of Electricity, Energy Journal, 11(4), 99-116. A Multidimensional Ito s lemma applied to a geometric Brownian motion Let f (X 1, X 2,..., X n ) be a function where the variables X 1, X 2,..., X n follow GBM of the form: dx i (t) = µ i X i (t) dt + σ i X i (t) dw i (t), (31) 7
with the correlation coefficients defined as: dw i dw j = ρ ij dt, i, j = 1, 2,..., n. (32) 1 ρ ij 1 and ρ ij = 1 when i = j. Then applying Ito s lemma we get: n df = f µ i + f X i t + 1 n σ 2 2 f i 2 i=1 i=1 X 2 i + 2 f ρ ij σ i σ j dt + X i X j i j n i=1 σ i f X i dw i (33) B A note on the data The manufacturing wage data came from the Labor Market Information for California website. Wages were calculated as average hours * average hourly earnings. We used information for firms located in the San Diego MSA from archived hours and earnings data files. These files are available to the public at http://www.calmis.ca.gov/htmlfile/subject/indh&e.htm The firms in our sample did not all face the same electricity prices. This is because SDGE bills its customers based on a bill cycle. The bill cycle number determines what day the customer s meter will be read for each month. The result of this arrangement is that different firms have bills that cover different periods. As a result, all variables (including wages) have to be weighted accordingly to ensure a fair comparison. We include a table with the meter reading dates so that readers can see how the data was adjusted. Information on the commodity charge (PX) that customers paid came from SDGE s website and is currently available at http://www2.sdge.com/eic/px/pxweekly.cfm. Finally, we chose to ignore two additional complicating factors that were not essential to our arguements. Firstly, SDGE actually uses 22 bill cycle numbers, the last code represents customized billing arrangements, which were not observable. Secondly, customers located within the city limits of San Diego also had to pay a 1.9% tax so their actual PX costs were higher. 8
Table 1: Dates Meters Read Month Bill 1 2 3 4 5 6 7 Group 8/1999 8/3 8/4 8/5 8/6 8/9 8/10 8/11 9/1999 9/1 9/2 9/7 9/8 9/9 9/10 9/13 10/1999 10/4 10/5 10/6 10/7 10/8 10/11 10/12 11/1999 11/2 11/3 11/4 11/5 11/8 11/9 11/10 12/1999 12/2 12/3 12/6 12/7 12/8 12/9 12/10 1/2000 1/3 1/4 1/5 1/6 1/7 1/10 1/11 2/2000 2/2 2/3 2/4 2/7 2/8 2/9 2/10 3/2000 3/3 3/6 3/7 3/8 3/9 3/10 3/13 4/2000 4/3 4/4 4/5 4/6 4/7 4/10 4/11 5/2000 5/2 5/3 5/4 5/5 5/8 5/9 5/10 6/2000 6/1 6/2 6/5 6/6 6/7 6/8 6/9 7/2000 6/30 7/5 7/6 7/7 7/10 7/11 7/12 8/2000 8/2 8/3 8/4 8/7 8/8 8/9 8/10 9/2000 8/31 9/1 9/5 9/6 9/7 9/8 9/11 10/2000 10/2 10/3 10/4 10/5 10/6 10/9 10/10 11/2000 10/31 11/1 11/2 11/3 11/6 11/7 11/8 12/2000 11/30 12/1 12/4 12/5 12/6 12/7 12/8 1/2001 1/2 1/3 1/4 1/5 1/8 1/9 1/10 2/2001 1/31 2/1 2/2 2/5 2/6 2/7 2/8 3/2001 3/2 3/5 3/6 3/7 3/8 3/9 3/12 4/2001 4/2 4/3 4/4 4/5 4/6 4/9 4/10 5/2001 5/1 5/2 5/3 5/4 5/7 5/8 5/9 6/2001 5/31 6/1 6/4 6/5 6/6 6/7 6/8 7/2001 6/29 7/2 7/3 7/5 7/6 7/9 7/10 8/2001 7/31 8/1 8/2 8/3 8/6 8/7 8/8 9/2001 8/29 8/30 9/4 9/5 9/6 9/7 9/10 10/2001 10/1 10/2 10/3 10/4 10/5 10/8 10/9 11/2001 10/30 10/31 11/1 11/2 11/5 11/6 11/7 12/2001 11/30 12/3 12/4 12/5 12/6 12/7 12/10 1/2002 1/2 1/3 1/4 1/7 1/8 1/9 1/10 2/2002 1/31 2/1 2/4 2/5 2/6 2/7 2/8 3/2002 3/4 3/5 3/6 3/7 3/8 3/11 3/12 4/2002 4/2 4/3 4/4 4/5 4/8 4/9 4/10 5/2002 5/1 5/2 5/3 5/6 5/7 5/8 5/9 6/2002 5/31 6/3 6/4 6/5 6/6 6/7 6/10 7/2002 7/1 7/2 7/3 7/8 7/9 7/10 7/11 8/2002 8/1 8/2 8/5 8/6 8/7 8/8 8/9 9/2002 8/30 9/3 9/4 9/5 9/6 9/9 9/10 9
Table 1 continued Month Bill 8 9 10 11 12 13 14 Group 8/1999 8/12 8/13 8/16 8/17 8/18 8/19 8/20 9/1999 9/14 9/15 9/16 9/17 9/20 9/21 9/22 10/1999 10/13 10/14 10/15 10/18 10/19 10/20 10/21 11/1999 11/11 11/12 11/15 11/16 11/17 11/18 11/19 12/1999 12/13 12/14 12/15 12/16 12/17 12/20 12/21 1/2000 1/12 1/13 1/14 1/18 1/19 1/20 1/21 2/2000 2/11 2/14 2/15 2/16 2/17 2/18 2/22 3/2000 3/14 3/15 3/16 3/17 3/20 3/21 3/22 4/2000 4/12 4/13 4/14 4/17 4/18 4/19 4/20 5/2000 5/11 5/12 5/15 5/16 5/17 5/18 5/19 6/2000 6/12 6/13 6/14 6/15 6/16 6/19 6/20 7/2000 7/13 7/14 7/17 7/18 7/19 7/20 7/21 8/2000 8/11 8/14 8/15 8/16 8/17 8/18 8/21 9/2000 9/12 9/13 9/14 9/15 9/18 9/19 9/20 10/2000 10/11 10/12 10/13 10/16 10/17 10/18 10/19 11/2000 11/9 11/10 11/13 11/14 11/15 11/16 11/17 12/2000 12/11 12/12 12/13 12/14 12/15 12/18 12/19 1/2001 1/11 1/12 1/15 1/16 1/17 1/18 1/19 2/2001 2/9 2/12 2/13 2/14 2/15 2/16 2/20 3/2001 3/13 3/14 3/15 3/16 3/19 3/20 3/21 4/2001 4/11 4/12 4/13 4/16 4/17 4/18 4/19 5/2001 5/10 5/11 5/14 5/15 5/16 5/17 5/18 6/2001 6/11 6/12 6/13 6/14 6/15 6/18 6/19 7/2001 7/11 7/12 7/13 7/16 7/17 7/18 7/19 8/2001 8/9 8/10 8/13 8/14 8/15 8/16 8/17 9/2001 9/11 9/12 9/13 9/14 9/17 9/18 9/19 10/2001 10/10 10/11 10/12 10/15 10/16 10/17 10/18 11/2001 11/8 11/9 11/12 11/13 11/14 11/15 11/16 12/2001 12/11 12/12 12/13 12/14 12/17 12/18 12/19 1/2002 1/11 1/14 1/15 1/16 1/17 1/18 1/21 2/2002 2/11 2/12 2/13 2/14 2/15 2/19 2/20 3/2002 3/13 3/14 3/15 3/18 3/19 3/20 3/21 4/2002 4/11 4/12 4/15 4/16 4/17 4/18 4/19 5/2002 5/10 5/13 5/14 5/15 5/16 5/17 5/20 6/2002 6/11 6/12 6/13 6/14 6/17 6/18 6/19 7/2002 7/12 7/15 7/16 7/17 7/18 7/19 7/22 8/2002 8/12 8/13 8/14 8/15 8/16 8/19 8/20 9/2002 9/11 9/12 9/13 9/16 9/17 9/18 9/19 10
Table 1 continued Month Bill 15 16 17 18 19 20 21 Group 8/1999 8/23 8/24 8/25 8/26 8/27 8/30 8/31 9/1999 9/23 9/24 9/27 9/28 9/29 9/30 10/1 10/1999 10/22 10/25 10/26 10/27 10/28 10/29 11/1 11/1999 11/22 11/23 11/24 11/26 11/29 11/30 12/1 12/1999 12/22 12/23 12/27 12/28 12/29 12/30 12/31 1/2000 1/24 1/25 1/26 1/27 1/28 1/31 2/1 2/2000 2/23 2/24 2/25 2/28 2/29 3/1 3/2 3/2000 3/23 3/24 3/27 3/28 3/29 3/30 3/31 4/2000 4/21 4/24 4/25 4/26 4/27 4/28 5/1 5/2000 5/22 5/23 5/24 5/25 5/26 5/30 5/31 6/2000 6/21 6/22 6/23 6/26 6/27 6/28 6/29 7/2000 7/24 7/25 7/26 7/27 7/28 7/31 8/1 8/2000 8/22 8/23 8/24 8/25 8/28 8/29 8/30 9/2000 9/21 9/22 9/25 9/26 9/27 9/28 9/29 10/2000 10/20 10/23 10/24 10/25 10/26 10/27 10/30 11/2000 11/20 11/21 11/22 11/24 11/27 11/28 11/29 12/2000 12/20 12/21 12/22 12/26 12/27 12/28 12/29 1/2001 1/22 1/23 1/24 1/25 1/26 1/29 1/30 2/2001 2/21 2/22 2/23 2/26 2/27 2/28 3/1 3/2001 3/22 3/23 3/26 3/27 3/28 3/29 3/30 4/2001 4/20 4/23 4/24 4/25 4/26 4/27 4/30 5/2001 5/21 5/22 5/23 5/24 5/25 5/29 5/30 6/2001 6/20 6/21 6/22 6/25 6/26 6/27 6/28 7/2001 7/20 7/23 7/24 7/25 7/26 7/27 7/30 8/2001 8/20 8/21 8/22 8/23 8/24 8/27 8/28 9/2001 9/20 9/21 9/24 9/25 9/26 9/27 9/28 10/2001 10/19 10/22 10/23 10/24 10/25 10/26 10/29 11/2001 11/19 11/20 11/21 11/26 11/27 11/28 11/29 12/2001 12/20 12/21 12/24 12/26 12/27 12/28 12/31 1/2002 1/22 1/23 1/24 1/25 1/28 1/29 1/30 2/2002 2/21 2/22 2/25 2/26 2/27 2/28 3/1 3/2002 3/22 3/25 3/26 3/27 3/28 3/29 4/1 4/2002 4/22 4/23 4/24 4/25 4/26 4/29 4/30 5/2002 5/21 5/22 5/23 5/24 5/28 5/29 5/30 6/2002 6/20 6/21 6/24 6/25 6/26 6/27 6/28 7/2002 7/23 7/24 7/25 7/26 7/29 7/30 7/31 8/2002 8/21 8/22 8/23 8/26 8/27 8/28 8/29 9/2002 9/20 9/23 9/24 9/25 9/26 9/27 9/30 11