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:
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.
