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A joint initiative of Ludwig-Maximilians University’s Center for Economic Studies and the Ifo Institute
CESifo Conference Centre, Munich
Area Conferences 2012
CESifo Area Conference on
Energy and Climate Economics
9 – 10 November
Pareto-improving climate policy: the role of
asset prices
Larry Karp and Armon Rezai
CESifo GmbH · Poschingerstr. 5 · 81679 Munich, Germany
Tel.: +49 (0) 89 92 24 - 14 10 · Fax: +49 (0) 89 92 24 - 14 09
E-mail: [email protected] · www.CESifo.org
Draft— not for distribution
Pareto-improving climate policy: the role of
asset prices
Larry Karpy
Armon Rezaiz
October 23, 2012
Abstract
We combine a one-commodity model of climate change with an
OLG structure to study the intergenerational e¤ects of climate policy.
Previous models of climate change make assumptions that render the
market for capital trivial and …x the price of the capital good. By
introducing adjustment costs, the price of capital is endogenous in our
model and captures future pro…tability of the asset. Climate policy
a¤ects the future pro…tability of assets and therefore their current
valuation. This neglected asset price creates the possibility of Paretoimproving climate policy.
Keywords: Climate externality, overlapping generations, climate
policy, generational con‡ict, dynamic bargaining, Markov perfection
JEL classi…cation numbers: E24, H23, Q20, Q52, Q54
Larry Karp thanks the Ragnar Frisch Centre for Economic Research for …nancial support. Armon Rezai bene…ted from the WU Visiting Fellowship and thanks the Department
of Agriculatural and Resource Economics at UC Berkeley for its hospitality.
y
Department of Agricultural and Resource Economics, University of California, Berkeley, and the Ragnar Frisch Center for Economic Research, email: [email protected]
z
Department of Socio-Economics, Vienna University of Economics and Business, email:
[email protected]
1
Introduction
Existing Integrated Assessment Models (IAMs) emphasize the welfare tradeo¤ between agents living in di¤erent periods. In these models, agents currently alive sacri…ce curent consumption to improve the environment, bene…tting people living in the future. A strand of this literature considers
multiple regions, taking into account the geographical distribution of abatement costs and climate-related damages. We depart from this tradition, examining instead climate policy’s intra- and inter-period distributional e¤ects
across owners of di¤erent factors of production. These owners correspond to
di¤erent generations in an overlapping generations (OLG) framework. We
consider both a political economy equilibrium and the outcome under a standard intertemporal social planner.
Most IAMs models make two assumptions. The …rst, (“no friction”)
posits a linear transformation between a consumption and an investment
good. The second (“separability”) requires that a reduction in emissions or
a worsening environmental stock reduces the marginal productivity of both
capital and labor, without changing the ratio of marginal productivities.
Taken together, these assumptions exclude the possibility that climate policy
has di¤erent e¤ects on owners of capital and labor, leaving only the possibility
that policy e¤ects di¤er across regions or over time. Our one-region onecommodity Overlapping Generations (OLG) model drops “no friction” but
maintains “separability”. The separabillity assumption implies that a change
in an environmental stock or a change in emissions have qualitatively similar
e¤ects on the returns to capital and to labor. However, by relaxing the
“no friction” assumption, a fundamental di¤erence between the two factors
emerges. Capital, unlike labor, is an asset that can be transferred across
generations. The price of capital, as distinct from its single period rate
of return, re‡ects anticipated changes in future returns. A policy change
creates capital gains or losses. When the old generation owns capital and the
young generation owns labor, climate policy can have di¤erent e¤ects on the
di¤erent generations alive in a period.
The “no friction”assumption states that each unit of output in a period
can be consumed or it can be used to increase the capital stock by one unit
in the next period. An agent can increase her holding of capital by buying
one unit at the price p, or she can create one unit by using one unit of
the composite commodity, which has a price normalized to 1. The price p
clears the market for capital. If investment is positive, then agents have to
1
be indi¤erent between the two ways of obtaining capital, so the equilibrium
price must be p = 1. The market for capital is trivial, and usually not even
discussed.
If it is not possible to eat capital, then there is a non-negativity constraint
on investment. In a period when this constraint is binding, agents are willing
to give up less than one unit of the composite commodity to obtain one unit
of capital, so the equilibrium price of capital is strictly less than 1. If it is
possible to eat capital (i.e. the non-negativity constraint does not exist), or
if the constraint happens never to bind, then the price of capital is identically
1. Such a model has nothing to say about the e¤ect of environmental policy
on asset prices. In a model with a single stock of capital, the possibility
that the non-negativity constraint binds seems remote, particularly because
capital depreciates.
A non-trivial capital market requires friction in the transformation between the consumption and the investment goods. Karp and Rezai, 2012
consider an extreme version of friction: the stock of capital is …xed, or changes
exogenously (e.g. due to exogenous technical change). In this case, it is not
possible to convert the consumption good to capital, and the price of capital is endogenous; it responds to changes in policy and to policy-induced
changes in the environmental stock. That model is a polar opposite to the
standard model: the former takes quantity as exogenous and price as endogenous, and the latter does the reverse. Both of these extreme assumptions
are restrictive.
We consider an alternative that nests the two special cases. The capitallabor composite, the environmental stock and the ‡ow of emission are inputs
that produce a consumption good and an investment good.1 If the production possibility frontier for these two goods is linear, we have the standard
model without friction. If the production possibility frontier is strictly concave, the opportunity (marginal) cost of the investment good, in terms of
foregone consumption, rises with the investment level. Convex adjustment
cost produces an equivalent model; in this setting the cost (in units of the
consumption good) of investment is a convex function of investment. An extensive theoretical and empirical literature explains and measures adjustment
costs. If gross investment in period t is It , the price of a unit of investment
1
We follow the convention of referring to emissions as an input. It is actually one of
three joint outputs, together with the consumption and the investment goods. The joint
output function can be inverted to write joint output of the consumption and investment
goods as a function of the environmental stock, emissions, and the capital-labor composite.
2
is pt = A (It ), where the function A (It ) is related to the adjustment cost.
The no-arbitrage condition described above now implies that the equilibrium
price of capital equals pt : an agent can obtain a unit of capital by buying an
existing unit at price pt or by buying a new unit at price A(It ).
Even without adjustment costs, equilibrium investment depends on expectations of future environmental stocks and future environmental policies.
However, adjustment costs make the asset price depend on expectations of
these future variables. Adjustment costs therefore give current asset owners a stake in the future return on capital. This mechanism operates only
to the extent that current capital does not fully depreciate within a period.
No analagous mechanism exists for labor: workers can sell their labor ‡ow,
but they cannot indenture their progeny. We explore the relation between
primitives such as the depreciation rate, the elasticity of intertemporal substitution and adjustment costs, and the welfare e¤ects of exogenous policy
changes. We also examine how the primitives a¤ect endogenous policy in a
political economy setting.
2
Literature Review
Standard climate policy evaluation uses the in…nitely-lived representative
agent (ILRA) model; Heal (2009) reviews this literature. In the ILRA setting, intergenerational con‡ict manifests as di¤erences in consumption over
time. In a …rst-best world, the cost of mitigation falls on present generations
while the bene…t accrues to those in the future. Karp (2009) discusses this
"sacri…ce" view of conventional analysis.
There already exist two challenges to the conventional view that climate
policy requires sacri…ces by those alive today. First, by correcting multiple
market failures jointly, it is possible to reduce climate change without reducing current consumption (a “win-win” opportunity). Second, there may be
opportunities to rebalance society’s investment portfolio, reducing saving of
man-made capital and increasing saving of environmental capital in a way
that leaves all generations better o¤ than under BAU (Foley 2009, Rezai et
al. 2012). For the later challenge to hold, e¢ ciency gains of climate policy
need to be distributed in a Pareto-improving manner. Bovenberg and Heijdra (1998, 2002) and Heijdra et al. (2006) show that the issuance of public
debt can be used to achieve intergenerational transfers, leading to Pareto improvements; they also examine the distributional impacts of pro…t, wage, and
3
lump-sum taxes. Our contribution emphasizes the role of asset price e¤ects
and shows that Pareto-improving tax policy can be implemented and sustained through an endogenous political process. In particular, climate policy
can improve current generations’welfare even in the absence of a government
that uses bonds to distribute income across generations. In addition, we assume that agents discount the welfare of other (present or future) agents. To
this end, we use the overlapping generation framework of Samuelson (1958)
and Diamond (1965).
Within the broader context of the economics of renewable resources, there
is a longer tradition of evaluating the feasibility of sustainble (non-decreasing)
consumption paths in such OLG economies. Building on Kemp and van Long
(1979), Howarth and Norgaard (1990) and Mourmouras (1993) demonstrate
that a social planner can implement welfare-improving conservation measures
relative to BAU in a model with environmental externalities and capital accumulation. Howarth and Norgaard (1992) and Krautkraemer and Batina
(1999) analyze welfare aspects of sustainable consumption paths in OLG
models. John and Pecchenino (1994) and John et al. (1995) discuss the transitional and steady state ine¢ ciencies due to intergenerational disconnectedness in the presence of private goods with negative externalities. Marini and
Scaramozzino (1995) analyze the intertemporal e¤ects of environmental externalities and optimal, time-consistent …scal policy. In the narrower context
of climate change, OLG models have been used by Howarth (1996, 1998),
Rasmussen (2003), and Leach (2009) to gage the magnitude and distribution of the welfare e¤ects of predetermined climate action. The advantages
of OLG relative to ILRA models are also discussed in Gerlagh and van der
Zwaan (2000, 2001). Laurent-Lucchetti and Leach (2011) introduce induced
innovation as an additional form of market failure in an OLG model of climate change. All papers …nd that welfare-implications crucially depend on
the speci…c policy design.
Previous models of climate change and of environmental policy ignore the
role of asset prices. They do so because their assumptions render the market
for capital trivial. Huberman (1984), Hu¤man (1985), and Labadie (1986)
use adjustment costs to capital in an OLG model to allow for non-constant
asset prices. Their analyses of adjustment-cost based theories of asset pricing,
however, ingore the idea that asset prices establish a link between future and
present generations; a link which can provide an incentive for non-altruistic
current generations to improve the welfare of those in the future.
Our political economy model is based on a Markov Perfect equilibrium
4
(MPE) in a game amongst a succession of generations. This framework
follows earlier papers that study games involving intergenerational redistribution and/or the provision of a public good (Hassler et al. 2003, 2005
and 2007, Conde-Ruiz and Galaso 2005, Klein et al. 2008, Bassetto 2008).
Ours is the …rst paper to use this kind of political economy setting to study
equilibrium climate policy.
3
The Model
Asset markets provide a mechanism that transfers future bene…ts of climate
policy to the generations who implement that policy today. We use an OLG
model with a stock externality and adjustment costs, extending Karp and
Rezai (2012) by including depreciation and endogenous investment, leading to endogenous changes in capital. Capital and the environment are the
two endogenously changing stocks. Capital is privately owned and the environment is a public good. Economic activity produces emissions which
cause the environment to deteriorate and thereby lower future productivity
of both capital and labor. Environmental policy can decrease equilibrium
investment, reducing the asset price and harming the old generation, the
current owner of these assets. The current young generation might bene…t
from future improvements in the environmental stock, and from the lower
asset price. Generations alive at the same period might have opposing interests despite the fact that the cost of climate policy falls symmetrically on
their respective factor payments. The e¤ect on aggregate welfare depends on
the relative strengths of the o¤setting e¤ects.
The focus on the con‡ict between current generations is heightened by
our assumption that agents care only about their own lifetime welfare. This
extreme assumption makes it less likely that current environmental policy
bene…ts those currently alive. It also allows us to bypass the debate on the
ethics of discounting the well-being of future generations. Even when those
currently alive have no concern for their successors, we …nd that climate
policy sometimes creates a Pareto improvement and can be implemented
and sustained in a political economy equilibrium.
In each period t a cohort of constant size, L 1, is born. We suppress
time subscripts when convenient. Agents live two periods; they maximize
their intertemporal additive utility, and have no bequest motive. In the
…rst period the representative agent earns a wage income, wt . The price
5
of the consumption good is normalized to 1. The fraction 1
of capital
depreciates in a period. The young agent spends ct on consumption and
purchases share st of the existing capital stock at the end of the period, at
a cost of st pt (1
)kt . She also purchases it units of new capital, at the
cost A(it )it . In the “no friction”world, A is the identity function, and with
convex costs A is an increasing function. In the second period she derives
income from holding the asset: She obtains the factor payment on her capital
stock rt+1 (st (1
)kt + it ). She also earns the proceeds from selling a share
st+1 of her depreciated capital stock, (1 )(st (1 )kt +it ), valued at current
prices, pt+1 . Given that she is non-altruistic, she spends all her income on
consumption, ct+1 .
The young agent’s maximization problem is
max u(cyt ) +
it 0; st
u(cot+1 )
subject to
cyt
cot+1
wt st pt (1
)kt A(it )it and
rt+1 (st (1
)kt + it ) + st+1 pt+1 (1
)(st (1
)kt + it ).
Agents take wt , pt , and the aggregate level of investment and therefore
A(it ) as given and have rational point expectations of rt+1 , st+1 , pt+1 . The
young agent dedicates all of her time to working and the old agent manages
the manufacturing …rm. The labor and commodity markets are competitive
and clear in each period.
Assuming an interior solution, the …rst-order condition for st is
u0 (cyt )( 1)pt (1
)kt + u0 (cot+1 )(1
)kt (rt+1 + st+1 pt+1 (1
)) = 0
This equation implies
(rt+1 + st+1 pt+1 (1
u0 (cyt )
=
o
0
u (ct+1 )
pt
))
(1)
and states that the cost of giving up a unit of consumption for saving today has to equal the bene…t of increasing consumption by the compounded
value tomorrow. The intertemporal marginal rate of substitution equals the
intertemporal marginal rate of transformation for existing capital. In this
equation the maximizing household only takes into account the marginal
6
rates of substitution of her own consumption today and tomorrow. For linear utility, equation (1) reduces to the Keynes-Ramsey rule.
This relation has to hold for any value of st . Given that the old generation
selling the asset has a price elasticity of 0, in equilibrium st = L1 1 8t.
Similarly, the …rst-order condition for it is:
0
u0 (ct )( 1)A(it ) + u0 (ct+1 ) [rt+1 + st+1 pt+1 (1
or
u0 (ct )
(rt+1 + st+1 pt+1 (1
A(it )
))
u0 (ct+1 ):
)] ;
(2)
The intertemporal marginal rate of substitution is no less than the intertemporal marginal rate of transformation for newly installed capital. If the rate
of return on capital was very low, agents would want to eat capital instead
of investing. In that case, equation (2) would hold with inequality. The
non-negativity of investment then binds. If the rate of return on holding the
asset is large enough, the agent invests in new capital. The price for existing
capital then equals the cost of producing new capital. Conditions (1) and
(2) imply
it > 0 ) pt = A(it ):
(3)
Because the supply of existing capital by the old generation is …xed, all
capital stock is sold in each period. The transition equation for capital stock
is
kt+1 = (1
)kt + it :
(4)
Economic activity creates emissions. The Business as Usual (BAU) emissions intensity (the number of units of emissions per unit of output) is constant at . Environmental legislation takes the form of a possibly time
varying abatement rate, 2 [0; 1], de…ned as the fraction of BAU emissions
that …rms are required to abate. “Potential output”, F~ (k; l; x), a function of
capital,labor and the environmental stock, x, equals the amount of the composite commodity that would be available if no factors were used to abate
emissions. The function ( ) equals the fraction of potential output devoted
to abatement, and (1
) equals the number of units of emissions per unit
of output.
7
Actual output of the composite commodity, y, and actual emissions, z,
are
output: y = (1
( ))F~ (k; l; x)
emissions: z = (1
) F~ (k; l; x):
The emissions per unit of output is
1
1
( )
(5)
and the cost per unit of emis-
( )F~ (k;l;x)
(1 )F~ (k;l;x)
sions is
= (1( ) ) . Competitive markets ensure that factors of
production are paid their marginal post-abatement product:
( ))F~l (k; l)
w = (1
r = (1
( ))F~k (k; l)
(6)
Emissions change the environmental stock, which alters the productivity
of factors. Our second state variable is the level of carbon in the atmosphere
in excess of pre-industrial levels, x. Emissions increase this stock, and the
stock decays at a constant rate ":
xt+1 = (1
")xt + (1
) F~ (kt ; lt ; xt ):
(7)
A higher stock of carbon lowers potential output:
F~ (k; l) = D (x) F (k; l) ;
(8)
with D(0) = 1, and for x > 0, D0 (x) < 0 and F constant returns to scale in
capital and labor. Actual output equals D (x) (1
( ))F (k; l); This function is separable in the capital-labor composite, F , and in the environmental
variables.
Equations (2), (3), (4) and (7) describe the evolution of the economy
for a given emission policy. We are interested in how changes in this policy
a¤ect welfare of current and future generations. In particular, we want to
investigate the role of asset prices. In order to do so, we restate the optimality
conditions for the agent’s decisions, (1) and (2), as an asset price equation,
u0 (cot+1 )
(rt+1 + pt+1 (1
pt =
u0 (cyt )
u0 (cy )
)) :
(9)
With jt = u0 (cto ) , the interest factor faced by generation t, forward
t+1
subsitution yields the price of today’s capital stock as the discounted sum of
8
future marginal pro…ts.
pt = jt 1 [rt+1 + (1
1
X
i
Q
1
=
jt+h
(1
1
)pt+1 ] = jt 1 [rt+1 ] + jt 1 jt+1
(1
) [rt+2 + (1
)i rt+i+1 :
)pt+2 ]
(10)
i=0 h=0
As functional forms we de…ne a Cobb-Douglas production function and
isoelastic utility and abatement functions:
F~l (k; l) =
k1
l ;
;
( )=
u(c) =
c1
1
1
(11)
with 0 > > 1, > 0, > 1, and
0 (but u(c) = log(c) for = 1). is
a convex increasing function with range [0; ] over the domain 2 [0; 1] and
(0) = 0 (0) = 0.
Using the fact that utility has a constant elasticity of substitution, we
can simplify the young agent’s lifetime welfare by substituting the …rst order
condition back into the maximand. (See appendix for derivation of the second
equality below.)
Uty = u(cyt ) +
=
wt (wt
u(cot+1 ) =
p~t (kt (1
1
wt (wt
wt c~t
1
=
1
) + ~{t ))
1
p~t kt+1 )
1
1
(12)
where c~t , ~{t , and p~t represent the equilibrium decisions and values. Lite-time
welfare of the old agent reduces to
Uto = u(cot ) =
4
)~
pt )kt ]1
[(rt + (1
1
1
:
(13)
E¤ects of Climate Policy on the Asset Price
and Welfare
In this section we examine the distributional e¤ect of a small exogenous
tightening of the emission standards relative to BAU. Asset prices play a
crucial role in determining the e¤ects of climate policy on life-time welfare of
agents in equations (12) and (13). Under BAU, the environmental standard
9
is identically 0. Consider an arbitrary non-negative standard trajectory, the
vector , with element j
0. Strict inequality holds for at least one j,
including j = 0. The index j denotes the number of periods in the future, so
j = 0 denotes the current period. A non-negative perturbation of the zero
BAU policy is = " , with " 0 the perturbation parameter. A larger "
therefore is equivalent to a stricter climate policy.
Such a policy directly lowers post-abatement output and, thereby, the
returns to labor and capital. Given the assumption that the …rst unit of
abatement does not create any costs, 0 (0) = 0, the …rst order e¤ect on
factor returns, of a small perturbation, is zero. Such a perturbation, however,
does have a …rst-order e¤ect on the stock of carbon in the next period which
translates into higher factor returns in the future. Due to its importance for
the results that follow, we state this as
Proposition 1 (i) For a predetermined level of the environmental stock, a
small increase in the emission standard has no …rst order e¤ect on the current
wage and rental rate. (ii) Above pre-industrial levels of atmospheric carbon,
a lower carbon stock has a positive …rst order e¤ect on the wage and the
rental rate.
Through asset prices, consumption and investment decisions of today’s
generation depend on consumption and investment decisions made by future
generations. In particular, today’s decision depends on the interest rate, jt =
u0 (ci;t )
, which is a function of next period’s asset price, pt+1 . Movements
u0 (ci;t+1 )
in the asset price determine whether the current old generation bene…ts from
climate policy. The old generation derives an income from the returns to
and the sale of the asset. The return is not a¤ected by a small change in the
standard as stated in proposition (1). With a …xed kt , the asset price and
the old generation’s welfare change in the same direction.
Corollary 1 The current old generation bene…ts from a small increase in
the emission standard if and only if this leads to an increase in the asset
dW o
t
price, d"t
> 0 , dp
> 0.
d" "=0
"=0
Similarly, we can ask whether the lifetime welfare of the young generation increases under climate policy. The relation between the change in the
asset price and the change in the young generation’s welfare depends on the
intertemporal elasticity of substitution, the value of .
10
Proposition 2 (i) If 2 (0; 1), environmental policy increases welfare of the
t
t
> 0 , dp
> 0.
current young if and only if the asset price rises: dLT
d" "=0
d" "=0
(ii) If > 1 environmental policy decreases welfare of the current young if
t
t
and only if the asset price falls: dLT
> 0 , dp
< 0.
d" "=0
d" "=0
Corollary 2 Interests of current generations coincide under
con‡ict under > 1.
2 (0; 1), but
Climate policy a¤ects future stocks of carbon and capital and has ambiguous e¤ects on current welfare. Lower future carbon stocks make future
capital more productive, increasing rt+i+1 , thus tending to increase the right
side of the asset price equation (10). However, the current tax lowers the
current wage, which tends to reduce current investment, it , tending to lower
the current asset price.
It is not possible to determine the sign of the asset price change analytically as in general the price today is a function of all future prices. However,
there exist two special cases in which we can solve the model analytically. By
assuming (i) linear utility or (ii) logarithmic utility, we make equilibrium decisions essentially independent of the in…nite asset price sequence. For these
cases we …nd that a small increase in the emission standard always constitutes a Pareto improvement. In section (5) we employ numerical methods
to determine the signs and the relative magnitudes of welfare changes for a
range of parameter values and emission standards. The Pareto-improving
quality of climate policy generalizes to this larger range.
4.1
The case of linear utility
Under linear utility = 0 the interest rate is …xed at the pure rate of time
preference. The asset price equation (9) simpli…es to
pt = [rt+1 + (1
)pt+1 ] =
1
X
i
(1
)i 1 rt+i :
(14)
i=1
Given that a small tightening of the emission standard has zero …rst-order
e¤ects on factor payments, but a positive …rst order e¤ect on the future stock
of the carbon, we can establish the following proposition.
Proposition 3 Under linear utility, climate policy increases the asset price.
11
In order to …nance the increase in the asset price, …rst period consumption
of the young agent necessarily falls. To establish whether they can increase
their overall welfare, second period consumption has to be taken into account.
Under linearity of the utility function, the higher asset price in the second
period allows for a second period consumption level that fully compensates
the initial asset price increase, assuming that both levels remain positive.
Because the elasticity of substitution is now in…nite agents are willing to
forgo exactly of one unit of consumption to purchase a discounted unit of
the consumption in the next period. This implies that their life-time welfare always equals the wage. The investment and purchase decisions merely
distribute the given level of utility between the two periods of their lifetime.
Using equation (12),
LTt j
=0
= wt :
By increasing the future resource stock, the tax also a¤ects future generations.
Proposition 4 Under linear utility, the …rst order welfare e¤ect of climate
policy (i) is zero for the current young generation and (ii) positive for future
generations.
Corollary 3 Some climate policy constitutes a Pareto improvement under
linear utility.
Given that a perturbation around BAU has a zero …rst-order direct e¤ect,
but a positive …rst-order indirect on factor payments through the improvement of the environment, we are able to extend the results of Karp and
Rezai (2012) to an o¤-the-shelf IAM. Note that these …ndings are independent of the levels of depreciation and the discount rate as long as neither is
prohibitive.
4.2
The case of log utility
Optimal saving behavior is determined by income and substitution e¤ects.
With elasticity of substitution of 1, income and substitution e¤ects cancel
out; the intertemporal decision problem becomes independent of the interest
rate (de la Croix and Michel 2002). Instead of discounting the utility that
the asset provides to all future generations, young agents are only willing to
12
expend a …x share of their income on it. This reduces to the time horizon
in consideration to the next period only and the transfer of policy-induced
future wealth is extremely limited. However, we can show that even in this
case, there is no sacri…ce involved in climate policy.
Under logarithmic utility the asset price in equation (9) simpli…es to
pt kt+1 =
1+
wt :
(15)
The asset price is a function of the wage: pt = f (wt ) with f 0 > 0 (for a
derivation see appendix A.4). Given that the asset price is an increasing
function of today’s wage and today’s wage is unaltered by a (su¢ ciently
small) tightening of the emissions standard, we …nd that
Proposition 5 For logarithmic utility, climate policy has no …rst order e¤ect
on the current asset price.
Corollary 4 For logarithmic utility, climate policy has no …rst order e¤ect
on the current old generation’s welfare.
Since the young generation sets the price by its decision rule to spend
a constant fraction of its wage on consumption and saving, climate policy
has a zero …rst order e¤ect on the …rst period equilibrium. Climate policy,
however, does have a …rst order e¤ect the return on these savings in the next
period. There is a positive …rst-order e¤ect on next period’s return to capital
and asset price, f (wt+1 ), through the improvement of the environment.
Proposition 6 For logarithmic utility, climate policy increases the welfare
of present and future young generations’lifetime welfare.
Corollary 5 Some climate policy constitutes a Pareto improvement under
logarithmic utility.
We are able to show that for the special cases of linear and logarithmic
utility, some climate policy relative to BAU necessarily provides a Pareto
improvement. This result rests on the fact that a small perturbation around
BAU creates no …rst-order static e¢ ciency losses, but yields the bene…t of
shifting the economy to a growth trajectory with lower environmental degradation. Under linear utility, the asset prices equals the sum of future marginal pro…ts of the asset discounted by the rate of pure time preference. The
13
policy-induced revaluation of the asset accrues to the old generation; current
and future young generations bene…t from the improvement in the returns to
their labor. The complete transfer of future pro…tability to the agent holding the asset at the announcement of the policy is broken under logarithmic
utility. Agents are only willing to dedicate a constant fraction of their wage
to asset purchases. The asset price is fully determined by current stock and
policy variables, leading to smaller gains for current generations from climate
policy than under linear utility. These results are independent of the value of
the rates of pure time preference and capital depreciation as long as attach
some value to their own future consumption and capital does not depreciate
fully.
We are able to derive analytical results for speci…c values of the intertemporal rate of substitution. In general, equilibrium decisions of saving and investment are in‡uenced by the agent’s desire to smooth consumption across
periods. We use numerical methods to obtain results for settings in which
this is the case. This also allows us to consider the equilibrium sequence of
emission standards. In particular, we specify a political economy structure
in which current generations choose an emission standard recognizing that
future generations have the same ‡exibility.
5
Political Economy Equilibria
So far we have not speci…ed how decisions on the environmental standard
are made; we took a small tightening of the standard as a given. Here we
assume that in each period the current young and old generations bargain
over the tax in order to maximize the sum of their lifetime welfare. In this
sense, the bargaining is e¢ cient. Agents recognize that future generations
have the same ‡exibility; in particular, the emission standard in future periods are conditioned on the future value of the state variables, capital and
atmospheric carbon. Agents currently alive are able to in‡uence future policies by in‡uencing the state variables that they bequeath to the future, but
current agents cannot choose future policies. That is, we consider a stationary Markov Perfect equilibrium (MPE) in the dynamic game amongst the
succession of generations. The transfer from the future to the present occurs
via the asset price; future policy decisions a¤ect this price. The cost, to
those currently alive, of the e¢ ciency in bargaining, is a possible loss of commitment ability, relative to bargaining environments where friction makes it
14
harder to change policies. Since we are interested the e¤ect of policy decisions on asset prices rather than the actually political economy progress of
reaching such an agreement, we do not specify the particulars of the game
such as the factors that determine the division of surplus.
In our stationary model, agents condition the choice of the current standard on the stocks of capital and atmospheric carbon. The MPE consists
of a policy function mapping the state variables into the reduction of emissions. If agents in the current period believe that future agents will use that
policy function, and if it is optimal for current agents to also set the current standard equal to the value returned by that function, then we have
a MPE. Hassler et al. (2003, 2005, 2007), Conde-Ruiz and Galaso (2005),
Battaglini and Coate (2007), Klein et al. (2008), and Bassetto (2008) also
study MPE in political economy settings. Hassler et al. (2005), page 1339,
note that the probabilistic voting model described in Lindbeck and Weibull
(1987) and Perrson and Tabellini (2000) provides an explanation for an equilibrium decision that maximizes current agents’joint welfare. In our model,
with two types of agents of equal measure, voting models are not particularly
useful; nevertheless, the assumption that these two agents play a bargaining
game remains compelling. Current decisionmakers are constrained by the
equilibrium decision rules of their successors. This constraint means that the
equilibrium standard need not be, and in fact is typically not, Pareto e¢ cient, just as in Battaglini and Coate (2007). We contrast our MPE policies
with the Pareto e¤ecient policy rules of a social planner.
5.1
Markov Perfect Equilibria
Our goal is to …nd the equilibrium stationary policy function, denoted t =
M (kt ; xt ). The Nash condition requires that given agents’belief that t+i =
M (kt+i ; xt+i ) for i > 0, the equilibrium decision for the agents choosing
the current tax is t = M (kt ; xt ). Equation (9) states that ownership of
the asset entitles the owner to pro…ts and revenue from the sale of the
asset after production. By purchasing the asset from the old in period t,
the agent who is young in period t obtains the utility derived from profits and asset sales when she is old. Denoting the wage as W (kt ; xt ) =
f1
[M (kt ; xt )]g Z(xt )Fl (kt ; lt ) and the rental rate as
`R(kt ; xt ) = f1
[M (kt ; xt )]g Z(xt )Fk (kt ; lt ), we de…ne and simplify the
15
asset price function recursively using
k [R(kt+1 ; xt+1 ) +
(kt ; xt ) = t+1
[W (kt ; xt )
(kt+1 ; xt+1 ) (1
(kt ; xt ) kt+1 ]
)]1
:
(16)
The bargaining equilibrium in period t is the solution to the optimization
problem
max t Uto + Uty =
= max
subject to
1
t
1
8
<
:
+(1
[kt (rt +pt (1
1
))]1
1
+
wt (wt pt kt+1 )
1
1
=
[kt ((1
[ t ])Z(xt )Fk (kt ; lt ) + (kt ; xt ) (1
))]1
[ t ])Z(xt )Fl (kt ; lt ) [(1 M (kt ; xt ))Z(xt )Fl (kt ; 1)
(kt ; xt ) kt+1 ]
2
(17)
kt+1 = (1
)kt + it
xt+1 = (1 ")xt + (1
) F~ (kt ; lt )
pt = A(it ):
(18)
Equation (17) states that the objective is to maximize the sum of the the
lifetime utility of the current old and the current young generation.
The primitives of the model lead to explicit expressions for the functions F (x; l), Z (x), and A (i). Equation (16) recursively determines the
function (kt ; xt ). Agents at time t take the functions M (kt+1 ; xt+1 ) and
(kt+1 ; xt+1 ) as given, but they are endogenous to the problem. We obtain a
numerical solution using the collocation method and Chebyshev polynomials
(Judd, 1998; Miranda and Fackler, 2002); see Appendix ??.
5.2
Calibration
We calibrate the model to represent the world economy. Agents live for 70
years and one time period lasts 35 years. We assume that agents have an
elasticity of intertemporal substitution of 12 , = 2, and discount the future
by 1%=yr , = 0:7. We scale nominal units by 109 2010 USD ($T). Current
capital, K0 , stock is roughly 200 $T (Rezai et al., 2012). Yearly world output
is roughly 63 $T , output in one (35-year) period is Y0 = 35 63 $T = 2200 $T
(CIA, 2010). Given initial factor endowments, Y0 ; and = 0:6, total factor
productivity is calibrated to
= 264. Capital depreciation is about at
6%=yr, = 0:9.
16
9
=
;
Currently, 8:36 Gt C are burnt per year (BP Statistical Review of World
Energy, 2011). This corresponds to an increase in atmospheric carbon of
3:92 ppmv. With a yearly world output of 63 $T , this implies a carbon
0:062 ppmv
. The actual increase in
dioxide emission intensity = 3:92
63
$T
atmospheric CO2 concentration in 2010 was only 2:42 ppmv (NOAA, 2010).
Hence, dissipation was 1:5 ppmv. This yields a depreciation factor equal to
1:5
= 0:0038 ppmv=yr, " = 0:126. This number is close to the mean of the
388
(0:0025%/yr, 0:0055%/yr) range of the implied dissipation rates of carbon in
DICE-07 (Rezai, 2010).
We follow Nordhaus (2008) in the calibration of the abatement cost function, = 2:8. equals the share of GDP necessary to abate all emissions.
In most IAMs this a exogenous function of time. In DICE-07, it costs 5:4%
to abate all emissions today, 0:9% in 30 decades. 0:4% in 60 decades. We
set = 0:056. The damage function is calibrated so that a doubling of
pre-industrial carbon causes a 3% output loss: Z[xt ] = (1 + ax2t ) 1 with
a = 4 10 7 .
We assume average adjustment costs to be linear
A(I) = 1 + b
I
2
where b
0 is the adjustment cost parameter. There is little evidence for
this parameter and our speci…cation makes adjustment cost share in total
investment dependent on the stock of investment. For b = 0; a unit of capital
costs one unit of the composite commodity. In a …rst approximation we set
b = 0:0002. This implies with an investment share of 20%, that adjustment
costs amount to 4% of investment. Our calibration assumes rapid depreciation of capital and moderate adjustment costs. In a future step, we plan to
reduce parameter uncertainty through sensitivity analysis around this value.
Equation system (19) summarizes the parameter values:
= 264; = 0:6; = 0:9; " = 0:126; = 0:062; = 2;
= 0:7; = 0:056; = 2:8; a = 4 10 7 ; b = 0:0002
5.3
(19)
Preliminary Results
Our numerical simulations con…rm our analytic results: asset prices create
incentives for current generations to improve the welfare of those in the future
despite the fact that they completely discount the future past their life-time.
17
In a dynamic bargaining setting, present generations agree on substantial
emission standards yielding reductions of up to 30%. This allows for lower
equilibrium values of atmospheric carbon and climate damage and higher
levels of capital stock and investment. Figure 1 presents the time pro…le of
these variables on the 35-year time scale. The …gure also presents life-time
welfare comparisons of current and future generations relative to BAU. While
the social planner imposes a sacri…ce on current generations, the MarkovPerfect equilibrium policy distributes the e¢ ciency gains of climate policy
more equally such that all generations bene…t.
Table 2 present the numbers for the …rst 5 generations. Given that our
bargaining framework does not specify how current generations split the gains
or losses from changes of the asset price, we assumed that the old (t=1) are
compensated by the current young (t=2) to keep their welfare constant.
All asset prices accrue to the current young generation. The social planner
imposes tighter emission standards leading to losses for generations either
alive today. While generations born in the next period (t=3) are already
better o¤ than under BAU, they are still worse o¤ relative to MPE. All
future generations are better o¤ under the social planner policy than under
either no policy (BAU) or the bargaining solution (MPE).
This is clearly still work in progress. In future steps we want to improve
the numerical calibration and conduct a sensitivity analysis around key parameters.
18
Capital
Atm. Carbon
x ppmv
k $T
800
1000
600
800
400
600
200
400
t
5
10
15
t
20
5
Investment
10
15
20
Abatment
$T
700
600
500
400
300
200
100
0.6
0.5
0.4
0.3
0.2
0.1
t
5
10
15
t
20
5
10
15
20
WtLT
1.30
1.25
1.20
1.15
1.10
1.05
t
5
10
15
20
Figure 1: Equilibrium trajectories for selected variables and life-time welfare
of current and future generations: BAU (red), MPE (green), SP (blue)
19
1
2
3
4
5
MPE
1.
1.00034
1.00342
1.01107
1.02424
SP
1.
0.989911
1.00362
1.01668
1.04472
Figure 2: Welfare relative to BAU: t=1 current old, t=2 current young, t>2
future young
References
Bassetto, M. (2008): “Political economy of taxation in an overlappinggenerations economy,”Review of Economic Dynamics, 11, 18–43.
Bovenberg, A. L., and B. J. Heijdra (1998): “Environmental tax policy
and intergenerational distribution,” Journal of Public Economics, 67(1),
1–24.
(2002): “Environmental Abatement and Intergenerational Distribution,”Environmental & Resource Economics, 23(1), 45–84.
Conde-Ruiz, J., and V. Galasso (2005): “Positive arithmetic of the welfare state,”Journal of Public Economics, pp. 933 –955.
Copeland, B. R., and M. S. Taylor (2009): “Trade, Tragedy, and the
Commons,”American Economic Review, 99(3), 725–49.
de la Croix, D., and P. Michel (2002): A Theory of Economic Growth.
Cambridge University Press.
Diamond, P. A. (1965): “National Debt in a Neoclassical Growth Model,”
The American Economic Review, 55(5), 1126–1150.
Farmer, K., and R. Wendner (2003): “A two-sector overlapping generations model with heterogeneous capital,” Economic Theory, 22(4), 773–
792.
Foley, D. K. (2009): “Economic Fundamentals of Global Warming,” in
Twenty-First Century Macroeconomics: Responding to the Climate Challenge, ed. by J. M. Harris, and N. R. Goodwin, pp. 115–126. Edward Elgar.
20
Galor, O. (1992): “A Two-Sector Overlapping-Generations Model: A
Global Characterization of the Dynamical System,” Econometrica, 60(6),
1351–1386.
Gerlagh, R., and B. C. C. van der Zwaan (2000): “Overlapping generations versus in…nitely-lived agent: The case of global warming,”in The
long-term economics of climate change, ed. by D. Hall, and R. Howarth,
pp. 301–327. JAI Press.
(2001): “The e¤ects of ageing and an environmental trust fund in an
overlapping generations model on carbon emission reductions,”Ecological
Economics, 36(2), 311–326.
Guersnerie, R. (2004): “Calcul economique et developpement durable,”
Revue economique, 55(3), 363–382.
Hassler, J., JV Rodriguez Mora, K. Storeletten, and F. Zilibotti (2003): “The survival of the welfare state,” American Economic
Review, 93(1), 87 –112.
Hassler, J., P. Krusell, K. Storeletten, and F. Zilibotti (2005):
“The dynamics of government,” Journal of Monetary Economics, 52,
1331–1358.
Hassler, J., K. Storeletten, and F. Zilibotti (2007): “Democratic
public good provision,”Journal of Economic Theory, 133, 127 –151.
Heal, G. (2009): “Climate Economics: A Post-Stern Perspective,”Climatic
Change, 96, 275–297.
Heijdra, B. J., J. P. Kooiman, and J. E. Ligthart (2006): “Environmental quality, the macroeconomy, and intergenerational distribution,”
Resource and Energy Economics, 28(1), 74–104.
Hoel, M., and T. Sterner (2007): “Discounting and Relative Prices,”
Climatic Change, 84, 265–80.
Howarth, R. B. (1991): “Intertemporal equilibria and exhaustible resources: an overlapping generations approach,” Ecological Economics,
4(3), 237–252.
21
Howarth, R. B. (1995): “Sustainability under Uncertainty: A Deontological Approach,”Land Economics, 71(4), 417–427.
Howarth, R. B. (1996): “Climate Change And Overlapping Generations,”
Contemporary Economic Policy, 14(4), 100–111.
(1998): “An Overlapping Generations Model of Climate-Economy
Interactions,”Scandinavian Journal of Economics, 100(3), 575–91.
Howarth, R. B., and R. B. Norgaard (1990): “Intergenerational Resource Rights, E¢ ciency, and Social Optimality,”Land Economics, 66(1),
1–11.
(1992): “Environmental Valuation under Sustainable Development,”
American Economic Review, 82(2), 473–77.
Huberman, G. (1984): “Capital asset pricing in an overlapping generations
model,”Journal of Economic Theory, 33(2), 232–248.
Huffman, G. W. (1985): “Adjustment Costs and Capital Asset Pricing,”
Journal of Finance, 40(3), 691–705.
(1986): “Asset Pricing with Capital Accumulation,” International
Economic Review, 27(3), 565–82.
John, A., and R. Pecchenino (1994): “An Overlapping Generations
Model of Growth and the Environment,” Economic Journal, 104(427),
1393–1410.
John, A., R. Pecchenino, D. Schimmelpfennig, and S. Schreft
(1995): “Short-lived agents and the long-lived environment,” Journal of
Public Economics, 58(1), 127–141.
Judd, K. (1998): Numerical Methods in Economics. MIT Press, Cambridge,
Massachusetts.
Kemp, M. C., and N. v. Long (1979): “The Under-Exploitation of Natural Resources: A Model with Overlapping Generations,” The Economic
Record, 55(3), 214–221.
Klein, P., P. Krusell, and J. Rios-Rull (2008): “Time-consistent public policy,”Review of Economic Studies, 75(3), 789–808.
22
Koskela, E., M. Ollikainen, and M. Puhakka (2002): “Renewable Resources in an Overlapping Generations Economy Without Capital,”Journal of Environmental Economics and Management, 43(3), 497–517.
Krautkraemer, J. A., and R. G. Batina (1999): “On Sustainability and
Intergenerational Transfers with a Renewable Resource,”Land Economics,
75(2), 167–184.
Labadie, P. (1986): “Comparative Dynamics and Risk Premia in an Overlapping Generations Model,”Review of Economic Studies, 53(1), 139–52.
Laurent-Lucchetti, J., and A. J. Leach (2011): “Generational Welfare
under a Climate-Change Policy with Induced Innovation,” Scandinavian
Journal of Economics, 113(4), 904–936.
Leach, A. J. (2009): “The welfare implications of climate change policy,”
Journal of Environmental Economics and Management, 57(2), 151–165.
Linbeck, A., and J. Weibull (1987): “Balanced budget redistibution as
political equilibrium,”Public Choice, 52, 273–297.
Marini, G., and P. Scaramozzino (1995): “Overlapping Generations
and Environmental Control,” Journal of Environmental Economics and
Management, 29(1), 64–77.
Miranda, M. J., and P. L. Fackler (2002): Applied Computational Economics and Finance. MIT Press, Cambridge, Massachusetts.
Mourmouras, A. (1991): “Competitive Equilibria and Sustainable Growth
in a Life-Cycle Model with Natural Resources,”The Scandinavian Journal
of Economics, 93(4), 585–591.
(1993): “Conservationist government policies and intergenerational
equity in an overlapping generations model with renewable resources,”
Journal of Public Economics, 51(2), 249–268.
Nordhaus, W. D. (2008): A Question of Balance: Economic Modeling of
Global Warming. Yale University Press, New Haven.
Persson, T., and G. Tabellin (2000): Political Economics - Explaining
Economic Policy. MIT Press, Cambridge.
23
Rasmussen, T. (2003): “Modeling the Economics of Greenhouse Gas
Abatement: An Overlapping Generations Perspective,” Review of Economic Dynamics, 6(1), 99–119.
Rezai, A., D. K. Foley, and L. Taylor (2011): “Global Warming and
Economic Externalities,”Economic Theory, forthcoming.
Samuelson, P. A. (1958): “An Exact Consumption-Loan Model of Interest
with or without the Social Contrivance of Money,”The Journal of Political
Economy, 66(6), 467–482.
Schneider, M., C. Traeger, and R. Winkler (2010): “Trading o¤
generations: In…nitely-lived agent versus OLG,”Working Paper.
Traeger, C. (2011): “Sustainabililty, limited substitutability and nonconstant social discount rates,”Journal of Environmental Economics and
Management, forthcoming.
US-NIPA
(2010):
“Bureau
of
Economic
Analysis,
National Income and Product Accounts Table 2.3.5. Personal
Consumption
Expenditures
by
Major
Type
of
Product,”
http://www.bea.gov/national/nipaweb/index.asp.
24
A
A.1
Proofs
Derivation of Lifetime welfare 12
Proof. Under isoelastic utility, equation (9) can be restated as
1
c~t+1
c~t+1
(rt+1 + (1
)pt+1 ) =
c~t
c~t kt+1
=ct+1 =kt+1
pt =
1
1
1
c~1t+1 =
1
pt
c~t kt+1
where the tilde above a decision variable denotes its equilibrium value. Using
this expression for the welfare level of the young agent in her second period,
we can simply life-time welfare to
1
LTt =
(c1t
1
c~t
=
1
+
1
c1t+1 ) =
1
c~
(~
ct + pt kt+1 ) = t
1
(ct1
+ c~t kt+1 pt )
wt
where the last equality follows from the …rst period’s equilibrium budget
constraint of the young agent.
A.2
Proof of Proposition 1
Proof. The old generation’s remaining lifetime welfare consists of the utility
it obtains from current consumption,
Wto (") =
1
1
(cot )1
=
1
))]1
[kt (rt + pt (1
1
We di¤erentiate each term in the sum by i = " i , recognizing that
direct e¤ect on the rental and the price. We use i = " i , so d i =
i
h
dWto (")
@pt
@rt
o
=
(c
+
(1
)
)
k
t
t @
t
d"
@
t
t
Along the BAU trajectory, " = 0. The expression simplies to
dWto (")
d"
= (cot )
kt
"=0
25
t
0 + (1
)
@pt
@ t
(20)
:
has a
i d".
i
with cot , kt and (1
sign
A.3
) positive and
t
non-negative. It follows that
dWto (")
@pt
= sign
d"
@ t
"=0
o
dWt (")
= 0
d"
"=0
for
t
>0
otherwise:
Proof of Proposition 2
Proof. The young generation’s lifetime equals,
LTt (") =
wt c~t
:
1
(21)
We di¤erentiate each term by i = " i , recognizing that i has a direct e¤ect
on the wage and consumption. We use i = " i , so d i = i d".
h
i
dLTt (")
1 @~
@wt
ct @it @pt
t
=1
c~t + ( )wt c~t
d"
@
@it @pt @
t
t
Along the BAU trajectory, " = 0. The expression simplies to
dLTt (")
d"
with wt , c~t , and
that
sign
sign
@it
@pt
=
"=0
t
1
positive,
t
0
wt c~t
ct
1 @~
@it @pt
@it @pt @ t
non-negative, and
dLTt (")
@pt
= sign
d"
@ t
"=0
dLTt (")
@pt
=
sign
d"
@ t
"=0
dLTt (")
= 0
d"
"=0
@~
ct
@it
negative. It follows
for 0 <
for
t
< 1,
> 1,
t
t
>0
>0
= 0:
we treat the cases of linear ( = 0) and logarithmic ( = 1) utility in appendices ... (linear case) and A.4.
26
A.4
Proof of Proposition of equation (15)
Proof. Under logarithmic utility, equation (9) can be restated as
pt =
c~t =
wt
pt kt+1 =
c~t
c~t
(rt+1 + (1
)pt+1 ) =
c~t+1
kt+1
=ct+1 =kt+1
pt kt+1
pt kt+1
wt
1+
1
wt
=
1+
pt kt+1 =
c~t
Under logarithmic utility, the agent expends a constant fraction of his wage
on …rst period consumption and saving. The interest rate has no in‡uence
on these decisions. Using the fact that pt = A(it ), we de…ne pt implicitly
pt kt+1 =
pt (1
with A
1
)kt + A 1 (pt )
an increasing function.
pt = f (wt )
+
27
=
1+
1+
wt
wt