Model

Consumers: Consumers purchase a quantity of electricity Z _t in each period t. Furthermore, they demand a greater quantity of electricity in certain periods; for instance, consumers need more electricity during the middle of the day more than at night. At the same time, consumers are willing to shift their consumption from one period to another in response to a shift in prices. Overall, these characteristics can be captured using a standard CES utility function. Hence, we consider a representative consumer with the utility function

(1)$$U = \left({\mathop \sum \limits_t \alpha_tZ_t^\phi } \right)^{1/{\rm \phi }}{\rm \phi } \lt 1$$

where σ = 1/(1 − ϕ) is the intertemporal elasticity of substitution for electricity consumption. The restriction on ϕ implies we have σ > 0; additionally, since electricity consumption increases utility, we must have α _t > 0 for all t. For simplicity, we also define $\sum _t\alpha _t = 1$, so that a 1 percent increase in electricity consumption in all periods increases utility by 1 percent; consequently, we then have α _t ∈ (0, 1). The budget constraint is given by

(2)$$I = \mathop \sum \limits_t p_tZ_t$$

where p _t is the price of electricity in period t and I is the income. Our representative consumer maximizes utility against this budget constraint; the first order conditions of this problem imply:

(3)$$Z_t = \left({\displaystyle{{\alpha_t} \over {\,p_t}}} \right)^\sigma \displaystyle{I \over P}$$

(4)$$P = \mathop \sum \limits_t \alpha _t^\sigma p_t^{1-\sigma } $$

where P is the price index. Note that this model naturally does not allow for blackouts in equilibrium, since the price of electricity in any period gets arbitrarily large as the quantity of energy consumed in that period approaches 0. Furthermore, note that prices must be positive; although this is sometimes violated in reality, we do not believe that this assumption significantly affects our analysis.

Firms: Secondly, we have firms maximizing profit by picking an optimal set of energy inputs. In reality, electricity markets are fairly competitive, so we can model the set of firms by using a single representative firm that sets marginal revenue equal to marginal cost.

We let X _i represent the quantity of energy technology i, and we define its output per unit in period t as ξ _1,t. So, for example, if i is solar power, X _i would be the number of solar panels and ξ _i,t may be kWh generated per solar panel in period t. Consequently, the energy generated in period t, Z _t, is given by $\sum _i\xi _{i\comma t}X_i$. To simplify notation, we have

(5)$${\bi X}\equiv \left({\matrix{ {X_1} \cr \vdots \cr {X_n} \cr } } \right)\comma \; {\bi Z}\equiv \left({\matrix{ {Z_1} \cr \vdots \cr {Z_m} \cr } } \right)\comma \;\ {\bi p}\equiv \left({\matrix{ {\,p_1} \cr \vdots \cr {\,p_m} \cr } } \right)\comma \; {\bi \xi }\equiv \left({\matrix{ {\xi_{1\comma 1}} & \ldots & {\xi_{1\comma m}} \cr \vdots & \ddots & {} \cr {\xi_{n\comma 1}} & {} & {\xi_{n\comma m}} \cr } } \right)$$

where we have n technologies, m periods, and Z ≡ ξ^TX.

A key assumption of our model is that the time scale is short enough such that the input vector X does not vary with time and the output per unit matrix ξ is exogenous. This is important because the time scale of our model must be fairly granular to study the effects of intermittency. So, for instance, a reasonably short two-period setting would be t ∈ {peak, off-peak}. Since the overall time frame is a single day, producers cannot modify the quantity of the technologies they deploy, so X must be fixed. Furthermore, this time frame is short enough to assume that ξ is exogenous; that is, technologies like coal power cannot significantly modify their output within a day, so ξ can be treated as a given set of constants. These two implications, that X cannot change over time and that ξ is exogenous, are important for parsimony, because they prevent more complicated dynamics from entering our model. And, they also make intermittency economically relevant, because having X fixed and ξ exogenous means that producers have no way of compensating for intermittent output; therefore, they must view the intermittency of technologies like solar and wind as a trade-off.

Furthermore, assuming a short time scale makes it simple to differentiate intermittent technologies. In this setting, we define an intermittent technology as a technology where output per unit varies with t. Specifically, this variation is exogenous and captured through ξ. We further narrow our article to deterministic variation as opposed to stochastic variation; hence, elements of the matrix ξ are constants. In total, we only need to consider ξ to determine whether a technology is intermittent. Additionally, in this context, we define intermittency as variation in electricity output over time. Since the set of inputs X are fixed over time, we can know whether the overall supply of electricity is intermittent based on total output Z = ξ^TX.

Now, we consider the firm's problem. Our representative firm chooses X to maximize total profit; it does so by setting marginal profit equal to marginal cost. Helm and Mier (Reference Helm and Mier2019) note that past literature has argued for concave cost functions in the energy market due to effects such as economies of scale and learning-by-doing; on the other hand, standard cost functions are generally convex. So, like Helm and Meir, we take an intermediate approach by using a linear cost function. Specifically, the total cost of the input bundle X is given by $\sum _ic_iX_i\equiv {\bi c}^T{\bi X}$ where c _i is the cost per unit of X _i. Then, total profit is given by

(6)$$\Pi = {\bi p}^T{\bi Z}-{\bi c}^T{\bi X}$$

To simplify the algebra, we set the number of technologies equal to the number of periods (n = m). Additionally, we further require that the output per unit of each technology is unique and nonnegative in each period; in other words, the output per unit of one technology is not a linear combination of those of the other technologies in our set. This then implies that ξ is of full rank and therefore invertible. Now, maximizing profit, we find the first order condition:

(7)$$\displaystyle{{\partial \Pi } \over {\partial {\bi X}}} = 0\ \Rightarrow \ {\bi p} = {\bi \xi }^{{-}1}{\bi c}$$

Combining this FOC with the demand equation (equation 3) allows us to find the equilibrium. Generally, the equilibrium results for any number of technologies (n) are analytic, but they are difficult to interpret algebraically due to the number of parameters involved.

Empirical Methodology

In order to better understand the practical implications of our model, we empirically estimate its parameters and study its implications numerically. We are particularly interested in estimating σ, since it determines the importance of intermittency. As shown earlier, when σ is large, consumers care less about when they get their electricity and are thus more likely to adopt intermittent technologies. On the other hand, when σ is small, consumers prefer to smooth their electricity consumption and therefore prefer less intermittent sources. Hence, ensuring that we have accurate estimates of σ is important for understanding how consumers decide between fossil and renewable energy. The other parameters in our model, c, ξ, and α are of secondary interest, since they are easier to obtain directly.

Recall the demand equation from earlier

(8)$$Z_t = \left({\displaystyle{{\alpha_t} \over {\,p_t}}} \right)^\sigma \displaystyle{I \over P}$$

where P is the price index and I is income. Retail customers pay fixed rates each month for electricity, hence p _t is constant within each month, and we expect that income I does not vary significantly on a daily basis. Consequently, all variation in intramonthly demand is due to the share parameter α and the elasticity σ. But this creates a problem; since retail consumers do not face prices that vary each hour, we cannot estimate σ on an hourly basis.

A number of other articles have approached this problem using data from real-time pricing experiments.Footnote ⁵ In such experiments, consumers of electricity face prices that vary on an hourly basis; this makes it possible to estimateσ. Articles that use these experiments include Schwarz et al. (Reference Schwarz, Taylor, Birmingham and Dardan2002), Herriges et al. (Reference Herriges, Baladi, Caves and Neenan1993), and King and Shatrawka (Reference King and Shatrawka1994). The latter two articles estimate σ to be around 0.15 while the article by Schwarz et al. obtains estimates around 0.03. All three articles study real-time electricity pricing programs for industrial consumers using similar methodologies. Additionally, Aubin et al. (Reference Aubin, Fougere, Husson and Ivaldi1995) also provide estimates of the σ but using a different methodology; their results find an elasticity of substitution below 0. Under a CES structure, this is problematic, because it would imply upward-sloping demand curves. Finally, Mohajeryami et al. (Reference Mohajeryami, Moghaddam, Doostan, Vatani and Schwarz2016) also empirically estimate the share parameters for a CES function of this form, but they do not estimate the elasticity of substitution.

Overall, the past literature has estimated σ by running regressions on the CES demand equation, but we are concerned that this approach suffers from endogeneity. That is, producers may intertemporally substitute electricity output, which essentially means that there is a supply equation affecting prices. For instance, during the oil crisis of 1973, refineries increased gasoline stocks, expecting future prices to be higher (Adelman Reference Adelman1995). The existence of such behavior implies that estimates of σ would be biased downward unless we properly control for endogeneity. This is particularly important because whether σ is closer to 0.1 or 1 significantly changes the practical implications of our model.

So we take a different approach by using a supply instrument to identify the CES demand parameters. Specifically, we use coal prices, which affect the supply of electricity but not the demand. Furthermore, we estimate σ on a monthly basis. This decision is primarily due to data limitations, since we do not have access to the proprietary data on real-time pricing experiments, which the past literature has used. Although we are interested in understanding intertemporal substitution over a shorter time scale (since intermittency plays a larger role in shorter periods), estimates of σ on a monthly basis may still be applicable on a smaller time frame. For instance, Schwarz et al. (Reference Schwarz, Taylor, Birmingham and Dardan2002) estimate σ on a daily and hourly basis and find fairly close results; similarly, Herriges et al. (Reference Herriges, Baladi, Caves and Neenan1993) also find no significant difference in their estimates of σ for these two intervals. That is, while a daily basis is 24 times larger than an hourly basis, the estimates for σ, surprisingly, do not appear to change. Hence, we expect our estimates of σ on a monthly basis to not be far from estimates on shorter time scales. At the same time, our estimates of σ will likely be larger than that of the literature, because we are controlling for endogeneity. We now define our econometric methodology in detail.

Theory: We begin with the demand equation from our general model:

(9)$$Z_t = \left({\displaystyle{{\alpha_t} \over {\,p_t}}} \right)^\sigma \displaystyle{I \over P}$$

(10)$$P = \mathop \sum \limits_t \alpha _t^\sigma p_t^{1-\sigma } $$

For any pair of electricity outputs Z _t and Z _s, we have:

$$\displaystyle{{Z_t} \over {Z_s}} = \left({\displaystyle{{\alpha_t\;p_s} \over {\alpha_s\;p_t}}} \right)^\sigma$$

Taking logs on both sides and letting i represent different observations, we may rewrite this in a form more suitable for estimation.

(11)$$\ln \lpar {Z_{t\comma i}/Z_{s\comma i}} \rpar = {-}\sigma \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + \sigma \ln \lpar {\alpha_{t\comma i}/\alpha_{s\comma i}} \rpar $$

Our data differentiates consumption for each state in the United States, so we let i refer to a particular state. Additionally, most consumers pay monthly fixed rates for electricity, so we can, at most, estimate this equation on a monthly basis; hence, t and s refer to different months. Also, note that each observation only corresponds to a single state i; this is because consumers within each state can substitute consumption across time, but consumers in different states do not substitute consumption with one another.

In order to estimate this σ, we further modify this equation. Firstly, note that we cannot observe the demand shifter α _t,i directly, so we must replace the α terms with a set of controls that may be responsible for shifts in demand. So, still in general terms, our regression equation is now

(12)$$\ln \lpar {Z_{t\comma i}/Z_{s\comma i}} \rpar = {-}\sigma \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + \gamma _{t\comma i}A_{t\comma i} + \gamma _{s\comma i}A_{s\comma i} + u_i$$

where A represents set of controls for changes in demand while u _i is a normal error term. Note that the control A _t,i replaces σln(α _t,i) and likewise for the period s term; this substitution is valid because the ln(α _t,i) ∈ ℝ and the σ term is simply absorbed into the estimated coefficient γ _t,i. For the demand controls themselves, we consider degree days and the difference in months between periods t and s. Firstly, we use degree days rather than temperature due to the aggregation of the data. A degree day is defined as the difference between the average temperature for a day and a base temperature—our data uses 65°F (18°C). Cooling degree days (CDDs) and heating degree days (HDDs) further split this measure into deviations above and below the base temperature. That is, if the average temperature of a day is x°F, its CDD is max{0, x − 65} and its HDD is max{0, 65 − x}. Since these measures are linear, CDDs and HDDs can be aggregated without losing information. This does not hold true for temperature; averaging temperature over a month causes daily variation to be lost. Secondly, the demand for electricity may rise over time. Hence, we include, as a control, the difference in months between time t and s; this is represented by Δ_t,s. Finally, this panel requires us to consider fixed effects for each state, so we use a fixed effects panel regression. In total, the demand equation is:

(13)$$\eqalign{\ln \lpar {Z_{t\comma i}/Z_{s\comma i}} \rpar = & -\sigma \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + \gamma _{t\comma i}A_{t\comma i} + \gamma _{s\comma i}A_{s\comma i} + \eta \Delta _{t\comma s} + u_i \cr = & -\sigma \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + \gamma _{t\comma i}\lpar {{\rm CD}{\rm D}_{t\comma i} + {\rm HD}{\rm D}_{t\comma i}} \rpar \cr & + \gamma _{s\comma i}\lpar {{\rm CD}{\rm D}_{s\comma i} + {\rm HDD}s\comma i} \rpar + \eta \Delta _{t\comma s} + u_i} $$

Still, this equation may suffer from bias, since producers can also substitute production over time. To address endogeneity concerns, we define the following supply equation

(14)$$\ln \lpar {Z_{t\comma i}/Z_{s\comma i}} \rpar = \beta \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + \zeta \ln \lpar {C_{t\comma i}/C_{s\comma i}} \rpar + v_i$$

where C _t,i is the average cost of coal used for electricity generation in state i at time t and v _i is a normal error term. Coal prices are independent of the electricity demand error term u _i, since residential consumers generally do not use coal for electricity generation; on the other hand, shocks in the price of coal are linked with the supply of electricity. Hence, coal price is a theoretically valid instrument. In total, the reduced form equation is given by:

(15)$$\ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar = \lpar {\beta + \sigma } \rpar ^{{-}1}\lpar {\gamma_{t\comma i}A_{t\comma i} + \gamma_{s\comma i}A_{s\comma i} + \eta \Delta_{t\comma s}-\zeta \ln \lpar {C_{t\comma i}/C_{s\comma i}} \rpar + u_i-v_i} \rpar $$

where A _t,i consists of CDDs and HDDs at time t.

Finally, we also consider a semiparametric specification. That is, we allow the error terms u _i and v _i to be non-normal and place the demand controls and instruments in unknown functions. So, overall, we have:

(16)$$\ln \lpar {Z_{t\comma i}/Z_{s\comma i}} \rpar = {-}\sigma \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + f\lpar {A_{t\comma i}\comma \;A_{s\comma i}\comma \;\Delta_{t\comma s}} \rpar + u_i$$

(17)$$\ln \lpar {Z_{t\comma i}/Z_{s\comma i}} \rpar = \beta \ln \lpar {P_{t\comma i}/P_{s\comma i}} \rpar + g\lpar {\ln \lpar {C_{t\comma i}/C_{s\comma i}} \rpar } \rpar + v_i$$

where f and g are unknown, bounded functions. We restrict cov(u _i, v _i) = 0 but allow for the controls and instruments to be correlated. The advantage of this specification is that we can account for the controls or instrument having any nonlinear effects on the regressands. In order to estimate these equations, we use a procedure based on Newey (Reference Newey1990) that we describe in further detail in Appendix C.

Data: We collect monthly data from the EIA (2019a) on retail electricity prices and consumption for the 48 contiguous U.S. states from 2010 to 2018. Also from the EIA, we obtain data on the average cost of coal for electricity generation for each state and month.Footnote ⁶ We deflated both electricity and coal prices over time using the PCEP Index provided by the US BEA (2019). Finally, we collect data on HDDs and CDDs from the NOAA Climate Divisional Database (2019) for the same panel. Then, we merge these three data sets and trim 1 percent of outliers for a total of 818 observations for each month and state. We provide descriptive statistics for this data in Table 1. We use this preliminary data set to construct the data required for our regressions. That is, each observation in our estimation equation belongs to a set (t, s, i) consisting of two time periods and a state. Hence, we construct each row in our regression data set using unique combinations of t, s where t ≠ s for each state i. This gives us a total of 6817 observations. All in all, each observation in our regression data set consists the following variables: state (i), date 1 (t), date 2 (s), the log difference in electricity consumption between month 1 and month 2, the log difference in the price of electricity, the log difference in the price of coal, the number of CDDs for each date, the number of HDDs for each date, and the difference in months between dates 1 and 2.

Table 1. Descriptive Statistics

Revisiting Past Models

To start, we consider how a variable elasticity of substitution affects past models of the energy sector. As an example, consider the results of Acemoglu et al. (Reference Acemoglu, Aghion, Bursztyn and Hemous2012). Firstly, they find that the short-run cost of policy intervention is increasing with the elasticity of substitution between clean and dirty technologies.Footnote ¹⁵ Additionally, their article states that the cost of delaying intervention is increasing with the elasticity of substitution. Secondly, Acemoglu et al. argue that, when the discount rate and elasticity of substitution between clean and dirty energy (e) is sufficiently low, a disaster cannot be avoided under laissez-faire.Footnote ¹⁶ And, finally, Acemoglu et al. find that, “when the elasticity of substitution is high … a relatively small carbon tax is sufficient to redirect R&D towards clean technologies.”

Based on our results, each of these statements has a complementary interpretation in terms of intermittency. For instance, the first statement suggests that the cost of policy intervention is decreasing with the intermittency of clean technologies; the cost of delaying intervention is decreasing with intermittency as well. This is because the elasticity of substitution between renewables and fossil fuels becomes small when renewables are more intermittent. So in other words, delaying policy intervention will be the most costly in regions with access to clean, non-intermittent energy (such as hydro and geothermal energy). Secondly, when the discount rate is sufficiently low and the intermittency of clean energy is sufficiently high, a disaster cannot be avoided under laissez-faire. The intuition here is that, even if clean energy became relatively cheap, its intermittency would prevent it from adequately substituting for fossil energy. Finally, Acemoglu et al. argue that the size of a carbon tax should be inversely proportional to the elasticity of substitution and should decrease over time. Recall that, in our model, the elasticity of substitution e changes with relative prices; when fossil fuels are relatively cheap, e is large. Hence, this implies we need a relatively small carbon tax early on to spur research. As technological change makes renewables cheaper, fossil fuels will become relatively more expensive so e will fall; thus, the carbon tax will need to increase over time. These dynamics (a rising carbon tax) contrast with the results of Acemoglu et al., who numerically show that an optimal carbon tax should decrease over time when e is sufficiently large. A potential resolution to these contradictory solutions may be to take an intermediate route and maintain a constant carbon tax over time.

Carbon Taxes and Their Distributive Consequences

Our results also suggest that carbon taxes can have important distributional consequences. Earlier, we showed that the elasticity of substitution between renewable and fossil fuels (e) is non-constant and falls with the intermittency of present generation. This further implies that the elasticity of demand for generation technologies is non-constant as well. Specifically, the elasticity of demand for non-intermittent energy increases and then decreases with respect to its own price. Using the earlier numerical example of solar and coal power, we show this explicitly in Figure 2. As the price of coal power rises significantly beyond its current price, demand gets more elastic because its price becomes the primary factor disincentivizing its use. But initially, a rise in the cost of coal power causes its demand to become inelastic. This is because a rise in the price of coal shifts generation towards solar; consequently, generation becomes more intermittent and demand for coal becomes less price-sensitive since it is needed to help smooth electricity generation. This relationship should be of concern for policy makers considering a pollution tax. Consumers in regions without access to clean, non-intermittent energy will have the most inelastic demand for fossil fuels; hence, they will bear the greatest welfare losses from a tax on pollution/carbon. On the other hand, consumers in regions that have access to dispatchable, renewable energy such as hydropower will see smaller welfare losses; the demand for fossil fuels in these regions will not be as inelastic, because dirty generation can be replaced with non-intermittent, clean generation. All in all, if a carbon tax were implemented federally and its revenue were distributed equally, it may nevertheless function as an inequitable, between-state welfare transfer due to differences in the availability of renewable energy technology. This same argument would apply to carbon quotas.

Notes: These results were obtained using the following parameters: α _t = 0.6, α _s = 0.4, ξ ₁ = (1, 1), ξ ₂ = (1, 0.1), c ₁ = 104.3, c ₂ = 60, σ = 0.5. We generate these results by finding the optimal quantity of coal, X ₁, over a range of percent changes in its price c ₁. Then, on the y-axis, we plot the log difference in X ₁ divided by the log difference in its price. This is equivalent to the price elasticity of demand for X ₁.

Figure 2. The Price Elasticity of Demand for Coal Power

The Case for Subsidizing Battery Research

However, there are alternative policies that can mitigate this distributional side effect. One such policy is a research subsidy for improving battery technologies. Reducing battery costs, improving their storage capacity, and reducing their energy loss can allow intermittent renewables to far more easily substitute for traditional, fossil energy. To understand the magnitude of this effect, we again provide a numerical example using solar and coal. We aim to understand how batteries effect substitutability, so we consider a parsimonious model where batteries are used to shift a portion of solar energy output from its high-output period to its low-output period.Footnote ¹⁷ Specifically, we initially parametrized solar with ξ ₂ = (1, 0.1), implying that it functions at 100 percent of its potential capacity during the peak and at 10 percent of its potential capacity during the off-peak; this is far from matching consumer demand. So we represent solar output with batteries using ξ ₂ = (0.95, 0.15) and ξ ₂ = (0.90, 0.20); this is equivalent to transferring 5 percent and 10 percent of solar output. We plot our results in Figure 3. It is immediately clear that making solar output more consistent through the day results in a large increase in the elasticity of substitution e _1,2. Moreover, e _1,2 no longer tapers off around 4 as solar becomes the dominant source of electricity. Rather, with batteries, cost plays a much larger role in determining the optimal quantity of each technology even when solar is relatively cheap. Overall, shifting even a small fraction of solar output using batteries can significantly improve solar power's substitutability with coal.

Notes: Technology 1 is coal and technology 2 is solar. The legend in the upper subplot also applies to the lower subplot. The elasticity of substitution between technology 1 and 2 is given by the slope of the upper subplot, and it is graphed in the lower subplot. These results were obtained using the following parameters: α _t = 0.6, α _s = 0.4ξ ₁ = (1, 1), ξ ₂ = (1, 0.1), c ₁ = 104.3, c ₂ = 60. Furthermore, we set the parameter for the intertemporal elasticity of substitution for electricity consumption equal to our estimate σ̂ = 0.8847. We generated these numerical results with the same procedure used for Figure 1 We repeated this procedure with ξ ₂ = (0.95, 0.15) and ξ ₂ = (0.90, 0.20) to simulate the effects of shifting solar power output using batteries.

Figure 3. The Effect of Battery Storage on the Elasticity of Substitution between Solar and Coal

Consequently, batteries can complement reductions in the cost of intermittent renewables and mitigate the distributional side effects of a carbon tax. As shown in Figure 3 the elasticity of substitution between solar and coal rises significantly as solar becomes less intermittent; this implies that a change in the relative price of solar would have a far greater effect on solar adoption if batteries were employed with solar. Hence, if policy makers aim to promote the use of renewables, they should subsidize research that reduces the cost of renewables as well as research that improves battery technology. Using both policy instruments can be more effective than either alone. Additionally, the second benefit of batteries is that they mitigate the distributional problems of a carbon tax. This is because, by reducing the intermittency of renewables, batteries make demand for fossil fuels less inelastic. We specifically show this elasticity would change with another numerical example in Table 3; as in Figure 3, we model batteries that shift solar output towards the off-peak. We then estimate the elasticity of demand for coal power around its initial price; this elasticity is decreasing at an increasing rate with the percent shift in solar output. The practical implications of this are straightforward. Researching and implementing battery technology can reduce the welfare losses from a carbon tax. Moreover, regions without clean, consistent renewables would benefit in particular, because they will be able to combine intermittent technologies with better energy storage to help transition away from fossil fuel energy. In short, subsidizing battery research can reduce some of the unintended distributional consequences of carbon taxes while mitigating environmental damage.

Table 3. The Effect of Battery Storage on the Elasticity of Demand

Learning-by-Doing and Intermittency

Learning-by-doing, originally introduced by Arrow (Reference Arrow1992), refers to an increase in productivity or quality as firms acquire experience producing a particular good. In the context of renewable policy, a common argument is that knowledge from learning-by-doing spills over across firms, and thus governments should subsidize renewable adoption (Borenstein Reference Borenstein2012).

Our results suggest that renewable subsidies aimed at correcting the positive externalities of learning-by-doing should vary geographically. This is because our model shows that the substitutability of renewable energy depends on their intermittency and the preexisting technologies used for generation in a particular area. And both intermittency and existing technologies vary by region; for instance, the intermittency of wind power may vary due to regional differences in climate. Consequently, the level of subsidies needed to promote the deployment of renewables can vary geographically.

But are geographic differences in substitutability economically significant? As an example, consider equivalent solar panels deployed in two different regions. In the first region, the solar panels generate energy at 100 percent of their capacity during peak hours and at 10 percent during the off-peak; on the other hand, in the second region, the same solar panels produce at 90 percent during the peak and 20 percent during the off-peak due to geographic differences. This is similar to the earlier exercise where we studied how substitutability would change if we shifted a portion of solar output toward the off-peak using batteries. Specifically, we can see how the elasticity of substitution differs between the “(100 percent, 10 percent)” region and “(90 percent, 20 percent)” region using Figure 3. That is, if the regions are otherwise equivalent, solar is far more substitutable for fossil energy in the (90 percent, 20 percent) region due to its slightly smoother output profile. Specifically, at the prices used for the numerical simulation earlier (c _coal = 104.3, c _solar = 60), the point elasticity of demand for solar power is −8.9 in the (100 percent, 10 percent) region and 12.8 in the (90 percent, 20 percent) region; this is given in Table 3. Consequently, the level of subsidies needed to promote solar adoption in the former region would need to be about 43 percent larger than those in the latter region due to differences in intermittency.

Along with geographic variance in intermittency, preexisting generation technologies also vary by region and can create large differences in substitutability between alternative power generation feedstocks. For instance, consider two regions where the first primarily uses natural gas and the latter uses coal power. It is easier to promote the adoption of intermittent renewable energy like solar in the first region, since the output of electricity from a natural gas plant can be deliberately changed within an hour. Again, the intuition is similar to that from the battery storage example where smoothing solar output increased its substitutability. In this case, solar and natural gas are strong substitutes, since natural gas can fill gaps in solar's intermittent generation; the same idea would apply to other intermittent renewables such as wind. On the other hand, coal is a base load technology that is much less flexible in its intra-daily output. So, in the region relying on coal power, increasing the share of solar power would necessarily create greater intermittency in the overall electricity supply. This means solar and coal are much weaker substitutes than solar and natural gas. More generally, the elasticity of substitution between intermittent renewables and fossil energy varies significantly based on type of fossil energy in question. This variation in substitutability creates differences in the efficacy of subsidies/taxes aimed at promoting renewable adoption. Therefore, regional differences in the preexisting technologies used for generation can be economically significant and should be accounted for when designing energy policy aimed at correcting externalities from learning-by-doing.

Adjusting Future Models

Given that traditional assumption of a CES relationship between renewable and fossil energy appears to be inaccurate, we offer two alternative approaches for future models. The first approach is to directly integrate our model within a given framework. The second approach is to assume a variable elasticity of substitution (VES) structureFootnote ¹⁸ between clean and dirty energy. These two approaches have their own advantages/disadvantages and serve different purposes.

Direct Integration: The first approach is better suited for numerical models. For example, our model of the energy sector can be embedded within a Computational General Equilibrium (CGE) model. CGEs, specifically ones designed to study energy and environmental policy, tend to assume a CES relationship between generation technologies and treat electricity as a homogeneous good. On the other hand, our representation of the energy sector directly models the source of imperfect substitutability between generation technologies—intermittency. Consequently, embedding our model within a CGE would improve its predictive accuracy by giving it more realistic microfoundations.

Furthermore, implementation can be done with relatively few modifications to standard CGE. To start, CGEs already use CES utility to model consumer preferences. In order to integrate our method of modeling electricity consumption, a CGE practitioner need only to rewrite preferences as:

(20)$$U = \left({\mathop \sum \limits_{i\ne elc} \alpha_iZ_i^\eta + \alpha_{elc}{\left({\mathop \sum \limits_T^{t = 1} \alpha_{elc\comma t}Z_{elc\comma t}^\phi } \right)}^{\eta /\phi }} \right)^{1/\eta }$$

where the intertemporal elasticity of substitution for electricity consumption is given by σ = 1/(1 − ϕ), the elasticity of substitution between all consumer goods is given by 1/(1 − η), the consumption of each good is given by Z _i, and the consumption of electricity in each period t is given by Z _elc,t. This can be simplified to:

(21)$$U = \left({\left({\mathop \sum \limits_{i\ne elc} \alpha_iZ_i^\eta } \right) + \alpha_{elc}Z_{elc}^\eta } \right)^{1/\eta }$$

(22)$$Z_{elc} = \left({\mathop \sum \limits_T^{t = 1} \alpha_{elc\comma t}Z_{elc\comma t}^\phi } \right)^{1/\phi }$$

where Z _elc is an aggregate electricity good. Next, we consider the production functions for electricity production in each period. These are simply linear, since the total electricity produced in a period is equal to the sum of electricity produced by each technology in that period. That is, we have

(23)$$Z_{elc\comma t} = \mathop \sum \limits_i \xi _{i\comma t}X_i$$

where X _i is a generation technology and ξ _i,t is its output in period t. The parameters for ξ _1,t must be estimated using additional data. For fossil fuels, ξ _i can remain constant over time. For intermittent renewables, ξ _i,t may be set to empirical estimates of the average output per unit of X _i in period t.Footnote ¹⁹ Additionally, the supply-side equations for building capacity in each technology can be modeled using the usual CGE approach; total cost is given by the sum of costs for each input factor, and total quantity is given by a CES of the input factors used to build capacity (X _i). Finally, since these functional forms are equivalentFootnote ²⁰ to those used in standard CGEs, they can be calibrated using the usual procedures and a social accounting matrix. So overall, implementing our model within a CGE can be done in a fairly simple way.

VES Approximation: The second approach is to use a variable elasticity of substitution function to model the relationship between clean and dirty energy. A VES function would approximately capture the dynamics of e, the elasticity of substitution between clean and dirty technology while keeping a theoretical model relatively simple. The latter is because assuming a VES structure between clean and dirty energy only requires considering one more parameter than a traditional CES function. To be more precise, the VES function is defined in Revankar (Reference Revankar1971) as

(24)$$Z = \gamma X_1^{\omega \lpar {1-\delta \rho } \rpar } \lpar {X_2 + \lpar {\rho -1} \rpar X_1} \rpar ^{\omega \delta \rho }$$

(25)$$e = 1 + \beta \left({\displaystyle{{X_1} \over {X_2}}} \right)$$

(26)$$\beta = \left({\displaystyle{{\rho -1} \over {1-\delta \rho }}} \right)$$

$$\gamma \gt 0\comma \;\;\omega \gt 0\comma \;\;0 \lt \delta {\vscale 180%\langle} {1\comma \;0 \le \delta \rho \le 1\comma \;\;\left({\displaystyle{{X_2} \over {X_1}}} \right)} {\vscale 180%\rangle} -\beta$$

where ω, δ, ρ, and γ are parameters, Z is the output, X ₁ and X ₂ are the inputs, and e is the elasticity of substitution between X ₁ and X ₂. In the context of an electricity sector, we can think of Z as electricity output, while X ₁ and X ₂ represent dirty and clean inputs. Note that e is a linear function of X ₁/X ₂; this relationship allows a VES function to approximate the effects of intermittency. To show why, we plot e against its VES approximation $\hat{e}$ for solar and coal in Figure 4. This approximation was fit by running OLS against the numerical results for e while holding the intercept fixed at 1.Footnote ²¹ We can see that, as either the capacity of solar or coal dominates the market, the approximation becomes weaker. But overall it seems to mimic the shape of the true elasticity of substitution e implied by our model of intermittency. Hence, a VES function is a reasonable alternative to a CES function—it better models intermittency while also maintaining parsimony.

Notes: Technology 1 is coal and technology 2 is solar. The purple, dash-dots line represents a linear approximation of e _1,2 for σ = 0.8847 with a fixed intercept of 1. These results were obtained using the following parameters: α _t = 0.6, α _s = 0.4, ξ ₁ = (1, 1), ξ ₂ = (1, 0.1), c ₁ = 104.3, c ₂ = 60. Furthermore, we set the parameter for the intertemporal elasticity of substitution for electricity consumption equal to our estimate σ̂ = 0.8847. In order to generate these numerical results, we first found the optimal quantities of X over a range of prices c ₁*ε(0.5c ₁, 2c ₁). Then, we obtained estimates of the elasticity of substitution by numerically differentiating ln(X ₁/X ₂) with respect to −ln(c ₁, c ₂).

Figure 4. The VES Approximation of the Elasticity of Substitution between Solar and Coal

One potential objection to using the VES function is that it assumes the elasticity of substitution e approaches 1 when the majority of electricity output comes from renewables (X ₁/X ₂ → 0). And a VES structure assumes linear e, which does not seem to be the case in Figure 4. However, both these assumptions may still be fairly reasonable in the right context. That is, suppose renewable energy was highly intermittent and that the consumer's elasticity of substitution σ = 1. Firstly, this results in renewable and fossil energy entering a Cobb-Douglas relationship (e = 1) as X ₁/X ₂ → 0, which supports the VES assumption of an intercept of 1. Secondly, no matter the relative prices, whenever the renewable technology is highly intermittent, e becomes perfectly linear with respect to X ₁/X ₂; this supports the VES assumption of linearity. The proofs for both are given in Appendix A.C. Hence, in these cases, a VES approximation can perfectly match e. Additionally, even when the consumer's elasticity of substitution is not 1, e may still be approximately linear; this is shown in Figure B4 where we consider our estimated $\hat{\sigma } = 0.8847$. In short, the assumptions made by a VES structure may nevertheless be accurate in certain cases.

Overall, using a VES function is a reasonable alternative to directly modeling intermittency. It allows theorists to derive policy implications from intermittency without relying too heavily on numerical models. Furthermore, it keeps theory simple, since the elasticity of substitution can be represented as a linear function of a single parameter β. And this parameter can be estimated econometrically, so theoretical models assuming a VES structure can offer empirically based results. Finally, it may even be of use for CGE modelers. Its structure is simple enough to be calibrated with a given set of data, while its implementation can produce simple but reasonably accurate models of energy sectors with intermittency.

Robustness

As a robustness check, we reproduce the estimates of e shown in Figure 1 across a larger range of values for σ. This allows us to evaluate the extent to which our results for e are driven by our estimate of σ. Specifically, we consider parametrizations of σ closer to estimates from the literature. Herriges et al. (Reference Herriges, Baladi, Caves and Neenan1993) and King and Shatrawka (Reference King and Shatrawka1994) estimated σ to be between 0.1 and 0.3, while estimates by Schwarz et al. (Reference Schwarz, Taylor, Birmingham and Dardan2002) were between 0.02 and 0.04. Additionally, we consider σ = 1 and σ = 2, which is closer to our OLS and 2SLS estimates. We estimate e using values of σ around these values and plot our results in Figure 5.

Notes: This figure is identical to Figure 1, but it considers a large range of values for σ. Technology 1 is coal and technology 2 is solar. The purple, dash-dots line represents a linear approximation of e _1,2 for σ = 0.8847 with a fixed intercept of 1. These results were obtained using the following parameters: α _t = 0.6, α _s = 0.4, ξ ₁ = (1, 1), ξ ₂ = (1, 0.1), c ₁ = 104.3, c ₂ = 60. Furthermore, we set the parameter for the intertemporal elasticity of substitution for electricity consumption equal to our estimate σ̂ = 0.8847. In order to generate these numerical results, we first found the optimal quantities of X over a range of prices c ₁*ε(0.5 c ₁, 2 c ₁). Then, we obtained estimates of the elasticity of substitution by numerically differentiating ln(X ₁/X ₂) with respect to −ln(c ₁, c ₂).

Figure 5. The Elasticity of Substitution Between Solar and Coal (Robustness Check)

Using values of σ closer to our OLS and 2SLS estimates causes e to be much higher. This is because the OLS and 2SLS results have σ higher than our partially linear specification. This suggests that intermittency is much less of a problem for substitutability if we parametrize our model using the 2SLS results. However, as argued in the “Results” section, the partially linear model is robust to functional form misspecifications and nonlinearities, while 2SLS is not.

When using the literature's estimates of σ, which are much lower than ours, we find lower values of e. The estimate for e near the parametrized value of c is 10.29 when σ = 0.8847 (our estimate) and 8.93 when σ = 0.1. Furthermore, the difference between the implied elasticity e when σ = 0.1 and when σ = 0.01 is fairly small. Furthermore, the difference between the results implied by our estimate of σ and that of the literature gets bigger when solar is relatively cheaper (left-hand side of the plot); this is because of the intermittency effect, as discussed earlier.

Although these differences from using the literature's estimates appear to be large, they do not negatively affect the implications of our results. Thus far, we have discussed how intermittency complicates environmental policy. The literature's estimates of σ are lower than our own; in other words, the literature finds that consumers care strongly about smoothing their electricity consumption. Consequently, if one puts more weight on the literature's estimates, then intermittency is more important than we have argued with our results. That is, when σ is lower, it is harder to substitute between intermittent renewable and reliable fossil energy; this is shown in Figure 5, where e is lower at all relative prices log (c ₂/c ₁). As a result, using the literature's estimates of σ makes the distributional consequences of energy policy more pronounced and research into batteries more important. Hence, the policy implications from our article are strengthened when using the literature's estimates of σ.