Modeling Climate Policy Instruments in a Stackelberg Game with ...

28 3 MODEL DEVELOPMENT AND ANALYSIS
The amount of total energy E is a function of fossil and renewable energy
E fos and E ren , respectively. Although a CES function is common to combine
both energy forms, I use a linear composition to study the consequences of the
absence of renewable energy for climate policy instruments. 31
The Optimization Problem Good producing firms seek to maximize their
profit:
max
{K Y ,L,E} Π Y
subject to:
Π Y = Y − rK Y − ¯wL − p¯
E E (26)
Y = (a 1 Z σ1 + (1 − a 1 )(A E E) σ1 ) (1/σ1) (27)
Z = (a 2 K σ2
Y + (1 − a 2)(A L L) σ2 ) (1/σ2) (28)
E = E fos + E ren (29)
˙K Y = I Y − δK Y (30)
Optimizing Conditions Due to the strict concaveness of the production
function in each input factor, there exists an interior maximum of Π Y in (K Y , L, E)
that can be found by derivating Π Y respect to the input factors and setting the
derivatives equal to zero: 32 r(1 + τ KY ) = Y ′ K Y
(31)
¯w = Y ′ L (32)
¯p E = Y ′ E (33)
Further Considerations Because of the linear homogeneity 33 of the CES
function optimal profits are zero (Arrow et al., 1961).
3.1.3 Fossil Energy Producing Firms
Description of the Sector Fossil energy firms produce final energy E fos
from the two input factors fossil resources R and capital K E . Labor is neglected
since it is no essential production factor and may not bring new insights for the
policy analysis. To give a consistent description of the sectoral disaggregated
economy I chose a CES function in analogy to the production sector. 34
31 Popp (2006a) and Grimaud et al. (2007) use a CES function that treats both energy forms
as imperfect substitutes. Van der Zwaan et al. (2002) justify this by the existence of niche
markets that make at least a small energy production always efficient. In contrast, Edenhofer
et al. (2005) treats in MIND all energy forms as perfect substitutes by linear combination.
32 Applying the Maximum Principle of dynamic optimization to an intertemporal objective
function yields the same conditions as the static optimization problem, because no intertemporal
decisions are made.
33 A function f(q 1 , ...,q n) is called homogene of degree a if f(λq 1 , ...,λq n) = λ a f(q 1 , ..., q n).
Linear homogene means homogene of degree one.
34 Several other approaches are common: Popp (2004, 2006a) and Grimaud et al. (2007) use
a linear production function of fossil resources divided by an (decreasing) carbon efficiency
variable. Nordhaus and Yang (1996) neglect the production of energy completely and assume
carbon emissions as side-effect of production that decreases in the presence of technological
change. In contrast, Edenhofer et al. (2005) use a CES function the same way as I will