Renewable Grid Power – Appendix B
Case Study on Renewable Grid-Power Electricity
Appendix B: Overview of the Model
From: Fischer, Carolyn and Richard G. Newell. “Environmental and Technology Policies for Climate Change and Renewable Energy.” RFF Discussion Paper 04-05, April 2004.
We develop a unified framework to assess the six different
policy options for reducing greenhouse gas emissions
and promoting the development and diffusion of renewable
energy. The stylized model is deliberately kept simple
to highlight key features. It includes two sectors,
one emitting and one non-emitting, and both are assumed
to be perfectly competitive and supplying an identical
product, electricity.
Fossil fuel production is assumed to be the marginal
technology, setting the overall market price; thus,
to the extent that renewable energy is competitive,
it displaces fossil fuel generation. The model has two
stages. Electricity generation, consumption and emissions
occur in both, while investment in knowledge takes place
in the first stage and through technological change
lowers the cost of renewable generation in the second.
An important assumption is that firms take not only
current prices as given, but they also take prices in
the second stage as given, having rational expectations
about those prices.
To allow for consideration of the length of time it takes
for innovation to occur, and for the lifetime of the
new technologies, let the first and second stages be
made up of 
and 
years, respectively. For simplicity, we assume that
no discounting occurs within the first stage; this assures
that behaviour within that stage remains identical.
However, let 
represent the discount factor between stages. It is
possible to allow for discounting in the second stage
by altering to reflect such discounting; in that case can be thought of as “effective” years.
Emitting Fossil Fuels Sector
The emitting sector of the generation industry relies on fossil fuels and is denoted with superscript 
. Total output from the emitting sector is 
in year 
. Marginal production costs 
are assumed to be constant with respect to output and weakly decreasing in emissions intensity 
, up to some natural rate, 
. This form allows for a trade-off between emissions intensity and higher costs (i.e., a carbon abatement cost function).
Two policies affect the fossil fuel sector directly: an
emissions price and an output tax (which may be explicit
or implicit, as with the portfolio standard discussed
below). Let 
be the price of emissions (i.e., emissions tax or equilibrium
permit price) and 
be the tax on fossil fuel generation at time ,
respectively. Other policies that stipulate quantity
standards, such as renewable portfolio standards and
emission performance standards, will be specified in
the next section, as they require some modifications
to the generalized model.
Profits for the representative emitting firm are:
where Pt is the price of electricity. The firm maximizes
profits with respect to output and emissions intensity,
yielding the following first-order conditions:
be the price of emissions (i.e., emissions tax or equilibrium
permit price) and
be the tax on fossil fuel generation at time ,
respectively. Other policies that stipulate quantity
standards, such as renewable portfolio standards and
emission performance standards, will be specified in
the next section, as they require some modifications
to the generalized model.
Profits for the representative emitting firm are:
Thus,
as shown in equation (2), the price of emissions 
determines the emission rate. The corresponding marginal
costs of output are then constant (including the output
tax (
)
and the price of the emissions embodied in that output
). Thus, the
fossil fuel sector is the “marginal technology”
– as long as fossil fuel generation occurs, the
competitive market price must equal the sum of these
marginal costs, as shown by equation (3).34
Total
emissions, 
,
are the product of the emission rate and fossil fuel
output:
In
the absence of a price on emissions, the first-order
condition for emission intensity implies 
.
Let the solution to this equation be ,
the baseline emission rate, and the corresponding baseline
price of electricity generation be , 
.
The
Non-Emitting Renewable Energy Sector
Another
sector of the industry generates without emissions by
using renewable resources (wind, for example); it is
denoted with superscript 
.
Annual output from the renewables sector is 
.
The costs of production 
are
assumed to be increasing and convex in output, and declining
and convex its own knowledge stock 
,
so that 
where
lettered subscripts denote derivatives with respect
to the subscripted variable.35
Furthermore, since marginal costs are declining in knowledge
and the cross-partials are symmetric, 
.
Note that we have simplified considerably by assuming
there is technological change in the relatively immature
renewable energy technologies, but none in the relatively
mature fossil fuel technologies. While it is of course
not strictly true that fossil fuel technologies will
experience no further technological advance, incorporation
of a positive, but slower relative rate of advance in
fossil fuels would complicate the analysis without adding
substantial additional insights.
The
knowledge stock 
is a function of cumulative R&D, 
,
and of cumulative experience through “learning
by doing” (LBD), 
,
where 
.
Cumulative R&D increases in proportion to annual
investment in each stage, 
,
so 
.
Cumulative experience increases with total output during
the first stage, so 
.
Research expenditures, 
,
are increasing and convex in the amount of new R&D
knowledge generated in any one year, with 
,
and 
.
An important issue is whether research and experience
are substitutes, in which case
,
or complements, in which
case 
.
Two
price-based policies are directly targeted at renewable
energy: a renewable energy production subsidy (
),
and a renewable technology R&D subsidy in which
the government offsets a share ()
of research expenditures.
In
our two-stage model, profits for the representative
non-emitting firm are:
The firm maximizes profits with respect to output in
each stage and R&D investment, yielding the following
first-order conditions:
Rearranging, we get:
As
shown in equation (6), the renewable energy sector produces
until the marginal cost of production equals the value
it receives from additional output, including the market
price, any production subsidy, and the contribution
of such output to future cost reduction through learning
by doing (note that the last term in equation (6) is
positive overall).36 Second-stage output does not generate a learning benefit,
so there is no related term in equation (7). Meanwhile,
as shown in equation (8), the firm also invests in research
until the discounted returns from R&D equal investment
costs on the margin.
Consumer
Demand
Renewable
energy generation and fossil fuel production are assumed
to be perfect substitutes. Let 
be the consumer demand for electricity, a function of
the price, where 
.
In equilibrium, total consumption must equal total supply,
the sum of fossil fuel and renewable energy generation:
.
In
this model, fossil fuels are the marginal technology
(by the assumption of flat marginal costs), and their
generation costs determine the price of electricity.
Fossil fuel output is therefore equal to the residual
after profitable renewable energy is produced: 
.
Thus, any increase in renewable energy production “crowds
out” fossil fuel production.
Consumer
surplus is therefore 
.
Thus, the change in consumer surplus due to the renewable
energy policy in this partial equilibrium model is:
Welfare
Policies
also have implications for government revenues, which
we denote as 
We assume that these revenues are raised or returned
in a lump-sum fashion. The change in these transfers
equals the tax revenues net of the cost of the subsidies:
Environmental
damages are a function of the annual emissions and the
length of each stage. To be able to accommodate both
for flow and stock pollutants, we write this function
in a general form:
The
change in welfare due to a policy is the sum of the
changes in consumer and producer surplus, net of the
change in environmental damages and revenue transfers
from the subsidy or tax:
Note
that constant marginal costs in the fossil fuels sector
implies zero profits, so 
.
However,
welfare is unlikely to be the only metric for evaluating
policy. Other indicators may be total emissions, consumer
surplus, renewable energy market share, and so on. General
equilibrium factors – like interactions with tax
distortions, leakage, or other market failures –
can also be important for determining welfare impacts.
Political economy constraints may also be important
for determining policy goals. To the extent that these
unmodelled issues are present, this partial equilibrium
presentation of welfare within the sector will not reflect
the full social impacts; still, it represents a useful
baseline metric.
Response
of Renewable Energy to Changes in Prices, Output and
R&D
While
policies can affect both the market price of energy
and the renewable energy subsidy, the renewable energy
producer ultimately cares about the total price it receives
for generation in each period, which we define as:
This
Appendix derives the comparative statistics for the
response of the renewable energy sector to changes in
these prices and in the price it pays for research.
The main results are as follows. First, renewable energy
output in each period is increasing with the price received
in that period.
(i.e., 
.)
Output in the second stage is also increasing in knowledge,
since marginal costs are lowered (
).
Next,
R&D is increasing in the second-stage price, since
higher prices imply more renewable output, which implies
more scope for profits from reduced costs. Similarly,
to the extent there is learning by doing, first-stage
output is increasing in the second-stage price, for
the same reasons. Unsurprisingly, R&D is also increasing
in its own subsidy, since effective investment costs
to the firm decrease.
The
harder questions regard how first-stage output responds
to R&D, and vice versa – in other words, how
LBD and R&D interact. While both are increasing
in second-stage output, the incidence across the two
forms of knowledge accumulation depends on the degree
of their substitutability or complementarity. That substitutability
also determines whether they respond in the same or
opposite directions due to changes in the first-stage
price and in the R&D subsidy, since those changes
affect the relative prices of LBD and R&D.
If
R&D and LBD are complements, first-stage production
will tend to increase with investment in R&D. That
means an increase in the R&D subsidy will also increase
first-stage renewable generation. Similarly, an increase
in first-stage renewable energy prices can also increase
R&D, if it is complemented by more LBD.
On
the other hand, if R&D and LBD are substitutes in
knowledge production, then more R&D makes LBD less
productive, given any output level. But the increase
in second-stage output resulting from lower costs due
to more R&D also tends to make LBD more valuable.
First-stage production may then increase or decrease
with investment in R&D. But a strong substitution
effect means that a larger R&D subsidy will decrease
first-stage production, and a larger subsidy to first-stage
production will decrease R&D investment, as R&D
and LBD crowd each other out. These interactions will
be important determinants of policy effects, since different
policies have different implications for the prices
of output in the first and second stages and the cost
of R&D.
Policy
Scenarios
As
developed in the modelling section, renewable energy
production depends on the price received by that sector
and the cost of R&D investment. Fossil fuel energy
production depends on the amount of renewable sector
output and the price of electricity, and emissions intensity
depends on the price of emissions. Different policies
vary in their effects on these different prices, resulting
in different market equilibria. As we will see, the
policies therefore provide varying incentives for emissions
reduction along these different margins – emissions
intensity, energy conservation, and renewable energy
output – leading to a divergence in their relative
efficiency.
No
policy
We
have defined 
as
the baseline emissions rate and 
as
the baseline electricity price, so in the absence of
policy (i.e., 
),
the first-order conditions for production imply that
output prices equal this baseline price in both markets
and over time: 
.
We assume that an interior solution exists; that is,
that some wind energy is viable without any policy.
A sufficient condition would be that 
.
However, wind production could occur even if marginal
production costs are higher than the price in the first
stage, as long as the value of learning by doing for
lowering second-stage costs is sufficient.
Fixed-price
policies
We
look first at three policies that directly set prices:
an emissions price, a renewable production subsidy,
and a tax on fossil-based production.
Emissions
price
With
a direct price for emissions – via either an emissions
tax or a tradable emissions permit system – the
fossil fuel sector has an incentive to lower its emission
rate until the marginal cost of reduction equals the
emissions price 
.
The market price of electricity reflects the total marginal
cost of fossil generation, inclusive of the embodied
emissions cost as well as higher marginal production
costs: 
(see
equation (3)). Without other subsidies, the renewables
sector receives the market price for electricity ( 
),
and the price increase promotes greater renewable energy
generation in both stages. The prospect of more output
in the second stage increases knowledge investment incentives
in the renewable sector, both for R&D and learning.
The higher market price also means consumers have added
incentive to conserve. Thus, the emissions price provides
efficient incentives for achieving a given emissions
reduction goal as it provides equalized incentives for
emission reduction along all three margins – emissions
intensity, output reduction (via price increase) and
renewable energy production.
Renewable
energy production subsidy
Under
a renewable production subsidy, since there is no direct
price on emissions, there is no reduction in fossil
emissions intensity, and 
,
as in the no-policy scenario. While the market price
of electricity remains unchanged, and thus provides
no incentive for energy conservation, the effective
price received by the renewable energy sector rises
by the amount of the subsidy, so that 
.
In this way, the renewables subsidy crowds out fossil
fuels generation in both stages and reduces emissions.
Fossil
fuel production tax
The
analytic structure of a fossil fuel production tax is
similar to the renewables subsidy, except that it is
a rise in the consumer price of electricity, rather
than a direct subsidy, that raises the price received
by renewables. Thus, both the market price and the effective
price received by the renewable energy sector rise by
the amount of the tax: 
.
Although no incentive exists to reduce emissions intensity,
to the extent that demand falls due to higher prices,
fossil output and emissions will be lower than under
an equivalent renewable energy subsidy.
Renewable
energy technology R&D subsidy
Without
a price on emissions or subsidy/tax on output, output
prices in both markets equal the baseline price (
). The primary effect of the R&D subsidy is to increase
research expenditures and lower future renewable costs,
crowding out some fossil fuels generation in the second
stage. The R&D policy provides no incentive for
reduction in fossil emissions intensity or energy conservation
through an electricity price increase.
Regarding
incentives for technological change, in the appendix
we show that an increase in R&D can encourage learning
either by making it more productive if R&D and learning
are complements, or by inducing a sufficient expansion
in second-stage output. On the other hand, if they are
substitutes R&D could discourage learning. In the
latter case, although an R&D subsidy would increase
renewable energy generation in the second stage, renewable
output will be lower in the first stage relative to
the baseline. The time path of emissions would tilt
in the opposite direction, rising in the first stage
and falling in the second. In the absence of a learning
effect (
),
the R&D subsidy would do nothing for first-stage
emissions.
Rate-based
policies
Two
additional, rate-based policies familiar to the electricity
generation sector are portfolio standards and tradable
performance standards. A portfolio standard requires
a certain percentage of generation to come from renewable
energy sources. A tradable performance standard mandates
that average emissions intensity of all generation not
exceed a standard. Both policies create effective taxes
on fossil fuel generation and subsidies for renewable
energy sources. However, those prices are not fixed,
as in the previous policies, but rather adjust endogenously
according to market conditions to achieve the targeted
rate.
Endogenous
prices raise additional issues with respect to innovation
incentives. Essentially, as increased knowledge brings
down the costs of renewable energy, the standards become
less costly to meet, which becomes reflected in the
implicit taxes and subsidies. The question is how firms
in the renewable energy sector perceive these price
changes. Do they recognize the impact of their innovation
decisions on second-stage prices? Do they myopically
expect prices to remain unchanged? Or do they expect
the future prices, but take them as given, as competitive
firms?
Given
our starting assumptions of a representative, perfectly
competitive firm, we will proceed with the latter assumption.
This view is most appropriate for describing firm-specific
innovation in a sector of many small, competitive firms.
These assumptions may be strong, and exploring alternatives
will be an important extension, in particular to incorporate
spillover effects. A long literature recognizes the
differences in incentives depending on the structure
of markets for output and for innovation.37
But one must begin somewhere, so we examine the logical
starting point of price-taking firms with rational expectations.
Renewable
energy portfolio standard
We
model the portfolio standard as a requirement that 
%
of generation be from renewable energy sources in each
stage (i.e., no banking allowed). We assume that responsibility
lies with the emitting industry to satisfy the portfolio
constraint. Thus, the fossil fuel producer must purchase
or otherwise ensure at least units
of renewable energy for every 
units of fossil fuel generation, or 
“green
certificates” for every unit generated.
In
equilibrium, the incentives correspond to a combination
of the fossil fuel production tax and renewable energy
subsidy cases. Assuming this constraint binds, the renewable
energy sector receives a subsidy per unit output equal
to the price of a green certificate, 
,
where “^” denotes equilibrium values under
the portfolio standard. The effective tax per unit of
fossil-fuelled output under this policy, 
,
is then proportional to the effective subsidy to the
renewable energy producer:
(14)
The
implicit tax and subsidy are determined competitively
by the market to meet the portfolio constraint. The
resulting market price of electricity is
,
while the price received by the renewables sector is
.
The
portfolio standard provides no incentive to lower the
emissions intensity of fossil fuels, but crowds out
fossil fuel generation by implicitly taxing it and subsidizing
renewables compared to the market price. The rise in
consumer prices is positive (unlike a pure renewables
subsidy where it is zero), but only a fraction of the
rise in the effective price received by renewables (whereas
a fossil energy tax would fully pass this increase on
to consumers). Thus the portfolio standard results in
only modest energy conservation incentives.
Another
important difference is that, if the portfolio standard
is fixed as we have assumed, the implicit tax and subsidy
decline over time as renewable energy costs fall due
to technological change. This occurs because the implicit
tax/subsidies reflect the shadow cost of meeting the
renewable production constraint, and this shadow cost
declines as the cost of renewable production declines.
Emission
performance standard
While
a portfolio standard requires a certain percentage of
renewable energy, a performance standard requires an
average emissions intensity of all generation. With
a tradable performance standard of 
,
the emitting firm must buy emission permits to the extent
that its emission rate exceeds that standard. The price
of emissions at time t, 
,
will now be determined by a market equilibrium, denoted
by “~”. All firms are in effect allocated permits
per unit of output, which leads to an implicit subsidy
of 
per
unit of output. Thus, if the standard is binding, the
fossil fuel sector will be a buyer of permits costing
per
unit of output, and the renewable sector will be a seller
of permits valued at 
per
unit of output.
Thus,
the emissions performance standard corresponds to a
combination of an emissions price 
and
a generation subsidy for both renewable and fossil energy
producers, where
(15)
The
equilibrium values are determined in conjunction with
the previous market-clearing conditions for energy supply
and demand, along with the additional constraint that
(16)
The
resulting market price of electricity is
, reflecting both the higher cost of achieving lower
emissions intensity, and the cost to fossil fuel producers
of emissions in excess of the standard. The price received
by the renewables sector is 
, which also includes the revenues they gain from permit
sales (i.e., the implicit subsidy).
Note
that the price received by renewables is the same as
with an equivalent pure emissions price (i.e., if 
),
assuring the same amount of renewable energy. The incentive
to lower emissions intensity is also the same for the
fossil fuel sector in that case. However, the consumer
price is lower by the output subsidy, 
,
and the resulting larger total output is filled by additional
fossil fuel generation, meaning that total emissions
are higher.
As with the portfolio standard, a fixed performance
standard implies a subsidy that changes over time. In
this case, as costs fall in the second stage, the expansion
of renewable energy allows fossil fuel sector emissions
to increase. Some of this will arise from greater production,
and some from increased emissions intensity, as the
permit price falls.
© 2013