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Asset prices and climate policy∗
Larry Karp†
Armon Rezai‡
December 30, 2015
Abstract
Climate-related damages can reduce future productivity of capital
assets. Climate policy, by mitigating these damages and protecting
future productivity, can increase today’s price of capital, increasing
current wealth and current welfare. Asset markets transfer future
policy-related benefits to currently living generations, regardless of the
concern these people have for future generations. A political economy
equilibrium involving self-interested agents can support a significant
level of abatement. Previous models of climate change have ignored
this asset price effect by implicitly assuming a fixed price of capital.
Keywords: Climate externality, overlapping generations, climate policy, generational conflict, dynamic bargaining, Markov perfection, adjustment costs.
JEL, classification numbers: E24, H23, Q20, Q52, Q54
∗
Larry Karp thanks the Ragnar Frisch Centre for Economic Research for financial support. Armon Rezai benefited from the WU Visiting Fellowship and the Austrian Science
Fund (grant no: J 3633-G11) and thanks the Department of Agricultural and Resource
Economics at UC Berkeley for its hospitality. This research was made possible by a generous collaborative research grant under the Berkeley-Austria program of the Austrian
Marshall Plan Foundation.
†
Department of Agricultural and Resource Economics, University of California, Berkeley, email: [email protected]
‡
Department of Socio-Economics, Vienna University of Economics and Business, email:
[email protected]. Also affiliated with the International Institute for Applied Systems
Analysis, IIASA.
1
Introduction
Climate policy requires costly abatement. Due to inertia in the climate
system, those currently alive obtain few of the direct environmental benefits of their abatement. Integrated Assessment Models (IAMs) support the
view that current generations must care about their successors in order to
undertake meaningful climate policy (Stern, 2006; Nordhaus, 2008). However, asset markets link future productivity changes to current asset prices.
Climate policy that protects the environment can increase future capital productivity. Operating through asset markets, climate policy can raise current
wealth, benefitting currently living agents (Oates, 1972). Asset markets can
induce self-interested asset owners to undertake substantial abatement.
Most IAMs use an Infinitely Lived Agent (ILA) model in which factors
of production, the climate, and policy jointly determine production of a perfectly fungible composite commodity that can be either consumed or invested.
In these models, the production possibility frontier (PPF) is a straight line
in consumption-investment space; choice of units determines the slope of this
line, making the price of investment (and of capital) exogenous. In order to
study the effects of climate policy on asset prices we need (i) a model with
buyers and sellers of capital, and (ii) an endogenous price of capital. We
obtain the first of these features by replacing the ILA model with a Diamond
(1965) Overlapping Generations (OLG) model, and we obtain the second by
replacing the linear PPF with a strictly concave PPF.
After production occurs in a period, the old agent, who owns capital, sells
remaining capital to the young agent. These agents care about the price of
capital. A deterioration in the environmental stock, e.g. a higher mean temperature, lowers the productivity of capital. Climate policy that protects
the environment increases the future productivity of capital, potentially increasing its current price and benefiting the old agent.1 A young agent with
a sufficiently high elasticity of intertemporal substitution also benefits from
a higher asset price. In this situation, both agents benefit from a climate
policy that increases the price of capital, even if they have no concern for
their successors’ welfare.2 If young agents live long enough to suffer from
1
Recent studies show that asset markets capitalize environmental characteristics (Chay
and Greenstone, 2005; Bushnell, Chong, and Mansur, 2011). Bansal and Ochoa (2011)
and Blavers et al. (2012) show that asset returns respond to temperature increases.
2
Climate policy can benefit currently living agents via other mechanisms. For example,
policies that reduce greenhouse gasses can also reduce local pollutants, creating near-
1
climate change, they obtain additional benefit from a policy that reduces
future damages.3
Temperature and atmospheric and oceanic carbon stocks change slowly,
relative to capital’s depreciation rate. This difference in speed suggests that
the asset price effect might be empirically unimportant, even if it exists. The
asset price effect vanishes if intra-period capital depreciates at 100%, as in
some IAMs (e.g. Golosov et al 2014). In order to assess the importance
of asset prices, we numerically study a model whose climate component is
calibrated to DICE, using a 6% annual depreciation rate for capital. In
this framework, and in the absence of bequest motives, the political economy
equilibrium emissions reduction is about half of the socially optimal reduction. The numerical results suggest that asset markets create important
incentives to internalize future benefits, even in the absence of altruism.
The standard climate policy narrative emphasizes the costs of climate
policy to current generations.4 That narrative usually ignores asset price
effects, or discusses these only in the context of the “stranded assets”, whose
value is reduced by climate policy. Fossil fuel companies, the prime example
of stranded assets, constitute one of the world’ largest asset classes, currently
worth about $5 T (trillion) (Bullard 2014). However, the world stock market
capitalization exceeds $69 T, and the value of total financial assets is above
$284 T (Witkowski (2015). Fossil fuel companies comprise a small part of
world financial assets. In considering the wealth effect of climate policy, it
is important not to place undue weight on the narrow asset class harmed by
policy. Fossil fuel owners are politically powerful, but climate policy has the
potential to increase the value of assets writ large, benefiting people with a
diversified portfolio. The failure to recognize this possibility may exaggerate
the political difficulty of implementing meaningful climate policy.5
term health benefits. In addition, rebalancing society’s investment portfolio, by reducing
saving of man-made capital and increasing saving of environmental capital, can benefit all
generations (Foley, 2009; Rezai et al., 2013).
3
Generations differ only modestly in their opinion about the reality of climate change
and the value of policies to address it. Almost 99% of Americans state that protecting
future generations is an important reason for protecting the environment (Feldman et al.
2010, Jones et al. 2014). The absence of altruism in our model is not descriptive, but it
shows that limited environmental protection may not require altruism.
4
In focusing on current generations’ incentive to shift costs to our successors, we ignore
people’s incentives to shift costs on to other nations. Both intergenerational and crosscountry free-riding incentives arise in a multi-region OLG setting (Karp, 2015).
5
Trade liberalization provides a useful analogy. While recognizing that trade reform
2
2
Related literature
Our model builds on three strands of research. Adjustment costs explain the
endogeneity of the price of capital. An overlapping generations model makes
it possible to include an asset market and also to examine intergenerational
conflict. A political economy model imbedded in a dynamic game provides
a basis for determining policy in the absence of a social planner.
Adjustment costs: Adjustment costs can explain the sluggish adjustment in investment and other macroeconomic data (Eisner and Strotz,
1963; Lucas and Prescott, 1971; Sargent, 1978). These costs lead to statedependent asset prices in an OLG economy with productive assets (Lucas,
1978; Huberman 1984; Huffman 1985, 1986; Labadie 1986a). These papers
ignore the intergenerational link arising from the asset price, and the associated policy incentives. We use estimated adjustment costs from Shapiro
(1989) and Hall (2004) for our calibration. Chirinko, (1993), Hamermesh and
Pfann (1996), and Caballero (1999) provide additional empirical evidence of
adjustment costs.
Environmental OLG models: Overlapping generations models distinguish between intra- and intergenerational transfers across time. Bovenberg
and Heijdra (1998) show that the issuance of public debt can support Pareto
improving environmental policy. We exclude public debt, social security,
and other means of intergenerational transfers. Karp and Rezai (2014) study
asset prices in an OLG model without a climate component or investment
(i.e., for a fixed capital stock). There are many other resource-based OLG
models, but these do not consider the effect of environmental policy on the
price of privately owned, endogenously evolving capital, the focus of our research (Howarth and Norgaard, 1992; John and Pecchenino, 1994; Gerlagh
and Keyzer, 2001; Williams et al., 2014).
Determination of equilibrium policy: Currently living agents, not an
infinitely-lived social planner, choose current abatement in our model. We
use a probabilistic voting model to determine policy in each period (Lindbeck
and Weibull, 1987; Persson and Tabellini, 2000). The currently living old and
young generations are part of a sequence of pairs of generations in a dynamic
game. We study the Markov perfect equilibrium (MPE) to this game. Hassler
et al. (2003), Conde-Ruiz and Galaso (2005) and Klein et al. (2008) use this
harms some sector-specific factors, economists are careful to emphasize that the aggregate
effect of trade reform increases national income.
3
type of political economy model to determine intergenerational redistributive
policies and/or the provision of a public good.
3
The Model
Composite-commodity Ramsey IAMs use a linear production possibility frontier (PPF) between a consumption and an investment good, making the asset
price exogenous. Agents can buy old capital at price  or convert the composite commodity to new capital. With positive investment, agents are indifferent between the two ways of buying capital (saving), so the equilibrium
asset price equals the price of the numeraire good,  = 1.
Convex adjustment costs in converting the composite good to the investment good lead to a strictly concave PPF and an endogenous asset price.
Investment  uses  ()  units of the composite commodity, so adjustment
costs equal ( () − 1) . For  () ≡ 1, adjustment costs are 0 and the PPF
between the consumption and the investment good is a line with slope -1.
For 0 ()  0, adjustment costs are positive and the PPF is strictly concave.
Firms that produce  (equivalently, transform the composite good into
) create congestion in their sector. Each firm behaves as if its marginal
production (transformation) cost is constant at  ≡  ( ); but this cost
increases with aggregate , unless adjustment costs are 0. The price of new
capital equals average instead of marginal aggregate production costs, creating a static market failure. Our analtytic results do not depend on the
micro-foundations of the concavity of the PPF. Appendix B.1 discusses alternative micro-foundatons and also explains the rationale for the congestion
assumption.
In each period, a cohort of constant size,  ≡ 1 is born. Agents live
two periods and maximize their lifetime utility. The young agent has labor
income, the wage  , spends  on consumption, and purchases share 
of the remaining end-of-period capital at a cost of   (1 − ) ;  is the
depreciation factor. She also purchases new capital,  , at the total cost   .
In period  + 1, the agent earns the factor payment on her capital stock
+1 ( (1 − ) +  ), and obtains revenue from selling her depreciated capital
stock, +1 (1 − )( (1 − ) +  ). The old agent spends all her income on
consumption, +1 .
Agents take prices  ,  , and  as given and have rational point expec-
4
tations of +1 and +1 . The young agent’s maximization problem is
max
 ≥0  +1
 ( ) +   (+1 ) subject to
 ≤  −   (1 − ) −  
+1 ≤ (+1 (1 − ) + +1 ) ( (1 − ) +  ) 
(1)
The optimal saving decision,  , satisfies the condition
 ≡
 0 ( )
(+1 (1 − ) + +1 )

=

0
  (+1 )

(2)
which states that the marginal rate of intertemporal substitution equals the
marginal rate of transformation. The right side gives the number of consumption units a young agent obtains in the next period by reducing consumption by 1 units today and buying an additional unit of capital. This ratio
equals the marginal rate of intertemporal substitution,  , 1 plus the interest
rate. In equilibrium, demand for old capital equals the inelastic supply, so
 ≡ 1 ∀.
The first-order condition for investment in new capital,  , is the complementary slackness condition:
¾
½
(+1 + +1 (1 − ))
 0 ( )
≥
⊥ { ≥ 0} 
(3)
  0 (+1 )
( )
The return on new capital earns the interest rate if investment is positive.
If that rate of return is below the interest rate, agents do not create new
capital:  = 0. If   0, the first inequality in equation (3) holds with
equality and conditions (2) and (3) imply no-arbitrage:
  0 ⇒  = ( )
(4)
We restate the optimality conditions for the agent’s decisions, (2) and (3),
as an asset price equation:
 =
+1 + +1 (1 − )
for   

(5)
where  is the last period, after which the asset has no value.
Privately owned capital and a public good, the environment, are the two
endogenously changing state variables. Production-related emissions harm
5
the environment, lowering future productivity of both capital and labor. The
climate state variable,  , equals the increase of the atmospheric stock of 2
relative to its pre-industrial level. This increase causes climate change, which
lowers economic output by the factor  ( ), with (0) = 1, 0  0 and
00  0 for   0. Absent abatement, output equals  ( )  (  ), where
 (  ) is an increasing, concave, constant returns to scale function. Output
creates carbon emissions, raising future levels of  .
We define the Business as Usual (BAU) carbon intensity (units of emissions per unit of output) as the level that maximizes the current value of
output and minimizes abatement costs. We use a stationary setting in which
the BAU carbon intensity is a constant, . Environmental policy obliges
firms in period  to abate the fraction  ∈ [0 1] of emissions  ()  ( ).
Abatement is costly, reducing output by the factor Λ (), with Λ0  0,
Λ00  0 for   0. Under BAU, abatement costs are 0 and minimized,
so Λ (0) = Λ0 (0) = 0. Net output of the composite commodity, , and
emissions, , equal
 = (1 − Λ()) ()  ( ) and  =  (1 − )  ()  ( ) 
Because the composite good (the numeraire) is also the consumption
good, real factor returns equal nominal factor returns. All markets are competitive and the representative firm hires labor and capital to equate a factor’s
wage or rental rate and its marginal product:
 = (1 − Λ()) ()  ( ) and  = (1 − Λ()) ()  ( ) 
(6)
An increase in the stock of 2 (higher ) or in the stringency of environmental policy (higher ) reduces the factor returns, leaving their ratio unchanged.
With constant decay rates  for capital and  for the atmospheric carbon,
the transition equations for the stock of atmospheric carbon and capital are
+1 = (1 − ) +  (1 −  )  ( )  (  )
(7)
+1 = (1 − ) + 
with the initial stocks, 0 and 0 , given. By altering future climate stocks,
current abatement can alter the asset price, potentially creating incentives
for non-altruistic agents to abate. Table 1 collects the names of functions.
 ()
Λ ()
 ()
Ω

damage
abatement cost
single period utility
welfare
interest factor
Table 1: Names of functions
6
The political economy equilibrium determines climate policy, resulting
in a sequence of emissions standards. Given climate policy, we have the
definition
Definition 1 A competitive equilibrium at , with initial condition  and  ,
is a sequence of the carbon and capital stocks and asset price, {+  +  + }−
=0 ,
satisfying the optimality conditions of the young agents’ and the firms’ decision problems (2)-(3) and (6) respectively, the asset market equilibrium
(5),and the transition equations in (7).
Assumption 1 (“Iso-elasticity”) Technology and preferences are isoelastic:
1−
 ( ) =   1− , Λ () =   , () =  1−−1 , with 0    1,   0,
  1, and  ≥ 0 (but  () = log() for  = 1).
Labor obtains the share  of output;  is the fractional reduction in output
caused by eliminating emissions ( = 1);  is the elasticity of abatement
costs; and  the inverse of the elasticity of intertemporal substitution (IES).
Lemma 1 provides the formulae for equilibrium welfare for a given climate
policy.
Lemma 1 Under Assumption 1 and for a given climate policy and  6= 1,
equilibrium lifetime welfare in period  is Ω for the young and Ω for the old
agent, with
Ω
≡
( )
+
 (+1 )
and
Ω ≡ ( ) =
( )−
1
 −
(1 − )
=
1−
1−
[( + (1 − ) ) ]1− − 1

1−
(8)
(9)
The Appendix contains proofs; we discuss the case  = 1 in section 4.1.
4
Climate Policy, Asset Price and Welfare
Climate policy diverts resources from consumption or savings to mitigation,
imposing a cost for generations that implement the policy. In an ILA economy, agents benefit from their climate investment in the future. In an OLG
economy with linear PPF and no bequest motive, the current old generation
7
has no reason to reduce emissions below BAU (corresponding to  ≡ 0).
The current young generation might benefit from a current action that reduces climate-related damages when they are old, but because of the inertia
in the climate system, that direct benefit is likely small. The asset market
transfers some future benefits to the current period; this increase in wealth
might offset the cost of climate policy.
This section examines the welfare effects of a small level of current abatement,  =  with   0 and  ≈ 0. A higher value of  represents a stricter
climate policy in the current period.6 Here we take as given future levels of
abatement; the next section endogenizes policy, taking into account future
responses to current policy.
Abatement in the current period lowers current output, lowering the current returns to labor and capital. Because abatement costs are minimized
under BAU, the first unit of abatement has a zero first order effect on current output and factor returns: Λ0 (0) = 0. However, the perturbation has a
first-order effect on current emissions and possibly on investment, causing a
change in future carbon and capital stocks. We have
Proposition 1 For   1 and a predetermined level of the environmental
stock and capital, a small level of abatement ( =  with   0 and  ≈ 0)
increases the old generation’s welfare if and only if this policy raises the asset
price:
¯
¯
Ω ¯¯
 ¯¯
0⇔
 0
 ¯=0
 ¯=0
For  = 1, climate policy has zero first order effect on the old generation’s
welfare.
The next proposition shows that a policy-induced increase in the asset
price benefits the young agent if and only if  ∈ (0 1), i.e. when the elasticity
of intertemporal substitution is large. We provide intuition for this result,
using the income and substitution effect to explain the necessity of  ∈ (0 1).
To this end, suppose that the policy-induced increase in  does increase the
young agent’s welfare.
6
As is standard, we approximate the welfare effect using the first order term of the
Taylor expansion around  = 0. This experiment holds future abatement levels fixed.
Our results do not change if, instead of considering the perturbation of only current policy,
we consider the perturbation of a sequence of abatement levels μ̄ ∈ R− , ̄ ≥ 0. In that
case, climate policy is μ̄.
8
The young’s current consumption is  =  −  +1 and next period
consumption is +1 =   +1 , implying the budget constraint +1 =
 ( −  ). The hypothesis that the policy-induced increase in  benefits
the young implies that the higher price increases  .7 The increase in 
creates an income effect that encourages higher consumption when young,
and a substitution effect that encourages lower consumption when young.
However, the small abatement policy at  creates only a second order reduction in aggregate output at , while it leads to a first order increase in the
old agent’s consumption at  (by Proposition 1). Therefore, the policy must
lead to a decrease in the young agent’s consumption at . Consequently,
the substitution effect must dominate the income effect. For the iso-elastic
utility function, the substitution effect dominates the income effect if and
only if   1. Thus,  ∈ (0 1) is necessary for the young agent to benefit
from a policy-induced increase in the asset price.
Proposition 2 Assume that investment is positive on the BAU trajectory
and that the consumption-investment good PPF is strictly concave (0 ()  0
for   0). (i) If  ∈ (0 1), environmental policy increases welfare of the
current young if and only if the asset price rises:
¯
¯
Ω ¯¯
 ¯¯
0⇔
 0
 ¯=0
 ¯=0
(ii) If   1, environmental policy increases welfare of the current young if
and only if policy causes the asset price to fall:
¯
¯
Ω ¯¯
 ¯¯
0⇔
 0
 ¯=0
 ¯=0
(iii) Climate policy has the same qualitative effect on welfare of the two generations alive in the first period if  ∈ (0 1); climate policy has opposite
effects on their welfare if   1.
Today’s asset price depends on all future prices, making it difficult to
analytically sign the effect of policy on the asset price. For logarithmic utility
and for a linear model, however, we show that a small increase in abatement
7
The policy has only a second order effect on  . If the policy causes   to fall,
the budget constraint pivots counter-clockwise around the  intercept (equal to  ),
necessarily lowering the young agent’s welfare, contrary to our hypothesis.
9
in period  strictly increases the welfare of one of the two types of agents
alive in that period, has no effect on the other agent’s welfare, and strictly
increases welfare for the agent born in the subsequent period. The noteworthy
result is that the asset market gives currently living agents an incentive to
impose climate policy, despite their lack of altruism and the lack of any other
mechanism for transferring welfare from the future to the current period.
4.1
Logarithmic utility
For logarithmic utility, income and substitution effects cancel: Equilibrium
saving is a constant fraction of income and the asset price is positive function
of saving.
Lemma 2 Under Assumption 1 with  = 1,

 and ()  =  ( ) with  0  0
1+
Lemma 2.i is standard: with logarithmic utility, an agent saves a constant
fraction of her income. An increase in the wage shifts out demand for savings,
increasing the price of capital (part ii).
()  +1 =
Proposition 3 Under Assumption 1 with  = 1, a small period-0 climate
policy: (i) has only a second order effect on the welfare of the old agent in
period  = 0, (ii) increases welfare of the agent born in  = 0, and (iii)
increases welfare of the agent born in the next period,  = 1.
Because the small abatement policy has a zero first order effect on the
current wage, it also has zero first order effect on the current asset price,
and therefore a zero first order effect on the old agent’s welfare. The policy
has a zero first order effect on the young agent’s saving, but it creates a
first order reduction in next-period pollution stock, raising the next-period
wage and asset price.8 These changes increase welfare of the current and the
next-period young. The agent born at  = 2 inherits a larger capital stock,
which tends to increase her welfare. However, with a fixed abatement policy
at  = 1, the higher  = 2 capital stock implies higher  = 1 emissions. We
cannot determine the welfare effect, for the agent born at  = 2, of the  = 0
perturbation.
8
The statement that there is a first order change in 1 and only a second order change
in 0 is consistent with equation 5. The change in  0 offsets the change in 1 .
10
4.2
A linear model
To obtain another perspective, we consider a linear model, replacing Assumption 1 with
Assumption 2 (“Linearity”) Preferences, the production technology, damages, and average adjustment costs are linear, and emissions do not depend
on the climate stock:  () = ,  ( ) =   +  ,  =  (1 − )  ( ).
Output is (1 − Λ ()) (( − )  + ( − )), so factor returns are  =
(1 − Λ ())( − ) and  = (1 − Λ ())( − ) with      0 and
  1.
With linear utility ( = 0), the interest rate equals the pure rate of time
preference ( = 1 ). Using equation (8), the lifetime welfare of the young
generation is
Ω |=0 =  − 1 + 
(10)
The policy has only a second order effect on the period-0 wage. Therefore, climate policy has zero first order effect on welfare for the current young agent.
The policy alters the asset price, altering the young agent’s consumption profile. With infinite elasticity of intertemporal substitution ( = 0), however,
the agent has no consumption smoothing incentive; the reallocation of consumption across the life-cycle has no effect on welfare. Equation (10) holds
in every period. Therefore, policy benefits future young agents if and only if
it increases their wage.
The conditions for a competitive equilibrium reduce to a system of linear
difference equations. The economy in period 0 moves from BAU (0 = 0) to
a small climate policy (0  0 and small). To eliminate time dependence
after period 0, we assume a constant policy,  = ̄, for  ≥ 1. For the
limiting case as the time horizon,  becomes infinite, we have
Proposition 4 Under Assumption 2, with  = ̄ constant for  ≥ 1, given
any equilibrium in which capital and pollution remain bounded, a small reduction in emissions at  = 0: (i) has a zero first order effect on the current
( = 0) young agent’s welfare, (ii) increases the current asset price and the
current old agent’s welfare, and (iii) increases the next-period ( = 1) young
agent’s welfare.
The fall in period-0 emissions lowers period-1 pollution stock, raising the
period-1 wage. But because the policy raises the asset price, it also increases
11
current investment and the next-period capital stock. The current policy
therefore increases emissions in the next period. We cannot determine the
policy’s welfare effect on agents born at  ≥ 2, because the period-0 policy
might lead to a higher pollution stock for some   1.
4.3
Discussion
The analysis above considers the first order effect of a small climate policy
at  = 0, holding fixed all subsequent policies. For both the logarithmic case
and the linear model, the policy weakly increases the welfare of the agents
alive at  = 0, and strictly increases the welfare of the agent born in the next
period. These results are independent of the magnitude of the pure rate
of time preference and the capital depreciation rate, provided that agents
attach some value to their own future consumption, and capital does not
depreciate fully within a period.
Under logarithmic utility, agents invest a constant fraction of their income. Because the policy has no effect on the period-0 wage or the (predetermined) stock of capital, the policy has no effect on the period-0 asset
price. Climate policy therefore has zero first order effect on welfare for  = 0
old agents. The policy increases the  = 1 asset price, strictly increasing
utility at  = 1 of the agents born at  = 0. Therefore, the policy strictly
increases the lifetime welfare of agents born at  = 0.
For the linear model, the asset price equals the sum of future rental rates,
discounted by the rate of pure time preference. The  = 0 old generation
captures the policy-induced capital gains, and the policy has zero first order
effect on welfare for the  = 0 young generation.
In both of these cases, the  = 0 policy strictly increases welfare of agents
born at  = 1. However, for neither of these cases can we guarantee that
the policy benefits agents born in subsequent periods. The change in the
trajectory of capital between  = 1 and the time of their birth alters the
stock of pollution that they inherit. Without additional structure, we cannot
determine the policy’s effect on these agents’ welfare.
5
Political Economy Equilibria
We compare outcomes under BAU, a social planner who cares about all future
generations, and in a political economy with self-interested agents. Our goal
12
is to assess the effect of asset markets on incentives to internalize future
damages. In all settings, non-strategic price-taking agents choose savings,
consistent with the endogenously determined asset price equation (5). Under
BAU, abatement is fixed at zero, minimizing abatement costs. The social
planner chooses the abatement policy to maximize the present discounted
infinite stream of welfare, using agents’ discount factor (the standard bequest
motive). In the political economy equilibrium, currently living agents choose
the abatement level to maximize their joint welfare, ignoring their successors’
welfare. To maintain transparency and a tight connection with our analytics,
we use a stationary model (no growth in technology or population) with a
single climate state variable.
5.1
The political economy setting
In the political economy setting, we assume that the current abatement decision is the equilibrium to a probabilistic voting model. In each period, young
and old agents cast their vote for one of two political parties. Each party
presents a program consisting of the current level of abatement, , and an
exogenous ideological component. Agents in the two generations currently
living care about their consumption-related welfare induced by this value of
, and about the ideological component. Their preference for ideology is
orthogonal to their utility of consumption. Agents within a group have the
same random preference for consumption-related utility relative to ideology,
but the two groups may have different preferences. A group with stronger
relative preference for ideology has fewer swing voters — those who can be
swayed by the choice of . The political parties have less incentive to choose
a  that appeals to a group with fewer swing voters. In equilibrium, the
political parties choose the same endogenous platform, , and each has equal
chance of winning the election (Lindbeck and Weibull, 1987; Perrson and
Tabellini, 2000; Hassler et al., 2005).
With a probabilistic voting model, the equilibrium policy, , maximizes
a political preference function equal to the weighted sum of the agents’
consumption-related welfare, Ω and Ω , where the weights depend on the
number of swing voters in the two groups. We assume that the two generations have the same relative preference for consumption and ideology, so they
have equal weights: the political preference function at  equals Ω + Ω .
The payoff-relevant state variable is the pair (   ). The equilibrium
savings decision and abatement rate jointly determine current consumption,
13
utility levels, and the next period state variable, (+1  +1 ). We consider
a stationary Markov Perfect Equilibrium (MPE) to the game involving the
sequence of policymakers. A MPE is a mapping from the state variable in period , (   ), to politicians’ abatement policy and young agents’ saving policy in that period. Denote the abatement policy mapping as  =  (   ).
The Nash condition to the game across periods requires that given agents’
belief that + =  (+  + ) for   0, the political economy equilibrium that determines the current abatement is  =  (   ). Investment
decisions are privately made; they satisfy the investment rule  =  ( ).
Current abatement decisions can affect the forward-looking asset price
(equation (5)). The asset price also determines the distribution of welfare
between the old and the young. We do not want the current climate policy
to be used as an instrument for income redistribution across currently living
agents, and therefore assume that the policymaker takes the current investment decision as given in choosing current abatement. The political economy
maximizes current generations’ joint welfare, conditional on subsequent generations following the equilibrium decision rule. The current decision rule
does not maximize a weighted sum of all generations’ welfare, and therefore
is not Pareto efficient.
Given a policy function  =  (   ), the current wage and rental rate
are function of the state variables:  (   ) and  (   ). Asset owners
receive revenue from rent and asset sales. Via the dependence of factor prices
on , the equilibrium asset price is a functional in ,  = Ψ (   ), that
satisfies the recursion in equation (5). The political economy equilibrium in
period  maximizes the sum of the lifetime welfare of agents at :
max Ω + Ω

(11)
subject to equation (1) and (7). For   0, using  = ( ) (equation (4))
and  = Ψ (   ), we obtain the equilibrium investment rule
 = −1 (Ψ (   )) 
We use equations (8) and (9) to write the agents’ lifetime welfare as
functions of the primitives of the model,  ( ),  (), and  (), and the
definitions  and  . Agents at time  take the functions  (+1  +1 ) and
Ψ (+1  +1 ) as given, but they are endogenous to the problem. Equation (5)
recursively determines the function  = Ψ (   ). We obtain a numerical
14
solution to this equilibrium problem using the collocation method and Chebyshev polynomials (Judd, 1998; Miranda and Fackler, 2002); see Appendix B.3
for details. Given a candidate policy function,   (   ), we numerically approximate Ψ (   ) and −1 (Ψ (   )) and solve the maximization problem
(11) to obtain the mapping
̂  (   ) = arg max (Ω + Ω ) 

This equality is a mapping from   to ̂  . A MPE is a fixed point of this
mapping. Appendix B.3 discusses uniqueness.
5.2
The social planner’s problem
The social planner has the same pure rate of time preference, applied to
all future generations, as agents have for their own future utility, and the
planner maximizes the discounted sum of utility of all generations. The social
planner’s problem is
∞
X
¡
¢
max
 Ω+ + Ω+
{  }
=0
subject to equations (5), (7),  = ( ), with initial conditions 0 , 0 .
This social planner faces the same constraints as in the political economy setting, but has a different objective. The social planner respects the
endogeneity of the asset price and recognizes the distributional implications
of the asset price that her program {   } induces. Except in the limiting
case of linear utility ( = 0), the planner would like to use lump sum transfers between agents currently living, in order to equate their marginal utility.
Because her policy menu does not include these transfers, she operates in a
second best setting. The planner today would like to choose future actions
partly with a view to their effect on the distribution of current utility. In a
subsequent period, she does not care about distribution in previous periods.
Therefore, her preferences are in general not time consistent. We consider a
MPE for this planner which is, of course, time consistent.
The planner’s dynamic programming equation is
¡
¢
 (   ) = max Ω+ + Ω+ +  (+1  +1 )
(12)
 
subject to equations (5), (7),  = ( ) . The equilibrium decision rules are
the functions  =  (   ) and  =  (   ) that maximize the right side
15
of equation (12). The algorithm is essentially the same as for the political
economy equilibrium, except that here we need the additional function, ,
and computationally we find the investment rule instead of the asset price
function. Although our planner can choose two actions in period , (   ),
the private agents’ forward-looking behavior imposes a constraint, eliminating one degree of freedom, and implying that the planner effectively chooses
a single instrument, just like political parties in the political equilibrium.9
5.3
Calibration
Our stationary one-climate-state model, based on Assumption 1, cannot exactly reproduce a more complicated nonstationary model with multiple climate state variables. Our social planner scenario is not intended to provide
policy advice on the socially optimal abatement trajectory. We care about
the difference in abatement and investment trajectories across scenarios, not
their absolute levels. For the model to be informative about the significance
of asset markets, we remain as close as possible to standard IAMs, within
the confines of the stationary, single climate state setting. Where possible,
we therefore calibrate the model using DICE-07.
 = 07
= 2
 = 06
 = 088
discount
factor
inverse
IES
labor
share
capital
depreciation
 = 0062
 = 0126
BAU carbon
intensity
carbon
decay rate
 = 0056
 = 28
 = 4 × 10−7
abatement
cost parameters
damage
parameter
 = 264
TFP
 = 00003
adjustment
cost parameter
Table 2: Parameter names and baseline values
Our calibration assumes moderate rates of capital depreciation, adjustment costs, and damages, and somewhat expensive (relative to DICE-07)
9
Given that the planner takes future policy rules as given and understands the linkage
of prices across time (equation (5)), she understands that the current asset price depends
on next period state variables, i.e.  = ̃ (+1  +1 ). The function ̃ (+1  +1 ) equals
the right side of equation (5). Because current actions determine next period states, we
can write this function as  =  (   ) ≡ ̃ (+1  +1 ), supressing the dependence of
 on the current state variables. This relation and the constraint  = ( ) implies
( ) =  (   ), and therefore eliminates one degree of freedom for the planner.
16
mitigation technology. Table 2 collects parameter names and baseline numerical values.
Agents live for 70 years, and one time period lasts 35 years. Our baseline
uses an elasticity of intertemporal substitution of 0.5, so  = 2. Agents
discount the future at 1% , implying  = 07. We scale nominal units by
109 2010 USD ($T). Year 2010 capital stock, 0 , is roughly 200 $ (Rezai
et al., 2012). Yearly world output is roughly 63 $ , so output in one 35-year
period (in our stationary setting) is 0 = 35 ∗ 63 $ ∼
= 2200 $ (CIA, 2010).
Given the initial endowments of capital and labor (the latter normalized to
1), 0 , and  = 06, total factor productivity is calibrated to  = 264. We
set capital depreciation to 6%, above the mean of 4% for 2010 of the
Penn World Table and below the 10% used by Nordhaus (2008); our
depreciation rate implies  = 088. The sensitivity analysis also considers
depreciation rates of 4% and 8%.
We measure the carbon stock, , in parts per million volume (ppmv).
Currently, 836   are burnt per year (BP Statistical Review of World
Energy, 2011), corresponding to an annual increase in atmospheric 2 of
392 . With a yearly world output of 63 $ , this implies a carbon
dioxide emission intensity  = 392
≈ 0062 
. The actual increase in
63
$
atmospheric CO2 concentration in 2010 was 242  (NOAA, 2010), implying dissipation was 15 . The corresponding depreciation factor
15
equals 388
= 00038 %, implying  = 1 − (1 − 00038)35 = 0126. This
number is close to the mean of the (00025%/yr, 00055%/yr) range of the
implied dissipation rates of carbon in DICE-07 (Rezai, 2010).
The calibration of the abatement cost and damages follows Nordhaus
(2008), where the abatement cost elasticity is  = 28. The parameter 
equals the share of GDP necessary to abate all emissions (Λ(1) =  1 = ).
In DICE-07, it costs 54% to abate all emissions today, 09% in 30 decades
and 04% in 60 decades. We set  = 0054, a constant, in order to work
with a stationary model. Consequently, future abatement is more expensive
in our model than in DICE, tending to raise current abatement and reduce
future abatement (“flattening the policy ramp”).
In DICE, damages are a function of temperature, and carbon stocks
change temperature with a lag. In our model, damages depend on the
single climate state, atmospheric carbon stock, ; emissions in one period
17
cause damages in the next period.10 As in DICE, we assume that doubling
the carbon stock relative to preindustrial levels reduces national income by
3%. This assumption and the damage function,  ( ) = (1 +  2 )−1 , imply
 = 4 × 10−7 .11
A standard, quadratic formulation of the total cost of adjustment is
( ) = 2 (  )2 . Hall (2004) finds average adjustment costs of  = 091.
Shapiro (1989) reports  between 8 and 9 for a quarterly estimation and proportional to net output, and that the error of misspecification under convex
adjustment costs is small. (See also Cooper and Haltiwanger, 2006).12 Due
to convergence problems in the numerical simulations, we adopt adjustment
cost per unit of invest () = 1 + 2  where  ≥ 0 measures the concavity
of the production possibility frontier between the consumption and the investment goods. Marginal adjustment costs equal . Given the similarity
to Shapiro’s formulation, we base our calibration on his estimates. Shapiro’s
marginal adjustment costs are    ≡   . The units of the adjustment
cost parameters are [] = ($  )2 and [] = $  35  as marginal
adjustment costs are unitless. Accounting for differences in time scales, we
have  =    (4 35) = 000033 with Shapiro’s data  = 33.13 We take our
benchmark as  = 00003, implying that economy-wide marginal adjustment
10
Ricke and Caldeira (2014) estimate that most of the warming effect of current emissions, and thus most of the temperature-related damage, occurs within a decade. Allen et
al. (2009) estimate that the maximum warming effect occurs after many decades. IAMs
also disagree about the lag between emissions and damage response. In Nordhaus’ (2008)
DICE, the maximum temperature response occurs after about 60 years. In Golosov et
al (2014), the maximum damage response occurs within a decade. With a 35 year time
step, and current emissions affecting next period damages, our model represents a middle
ground.
11
We can compare the damage functions in our model and in DICE-07, using the equilibrium relation between temperature,  , and stocks, :  = 3 log[280] log[2]. Our
damage function is lower than in DICE for temperatures below   2◦  but rises more
steeply beyond this point. The two damage functions differ by less than one percent for
temperatures below 33◦ .
12
Note that Shapiro uses gross and Hall net investment. Shapiro’s original estimates
are proportional to GDP, with marginal adjustment costs equal to   , and have to be
adjusted to the time frequency used.
13
The values for  and  are taken from Hall (2001, appendix C). Shapiro also tests an
alternative specification of marginal adjustment costs:     ((1 − )) with 1 −  = 1 −
00175 = 09825 the depreciation factor per quarter,   = 025 the estimated coefficient
and  = 200. The transformation of the alternative specification is  =   ((1−)) =
025 33  (4 35 200) = 000029. Shapiro’s annual depreciation rate of 68% is close to our
baseline of 6%.
18
costs are 0.03% of investment. Hall’s average cost parameter, which is about
half of Shapiro’s, serves as lower bound in our sensitivity analysis.14
5.4
Numerical results
The numerical results support our analytic results: asset prices create incentives for current generations to reduce emissions, despite their indifference to
future agents’ welfare. In the political economy setting, current generations
vote for substantial abatement, reducing emissions by 20% — 30% of BAU
levels. Reduced emissions lead to lower trajectories of atmospheric carbon,
lower damages, and higher levels of capital stock and investment.
Figure 1 shows the trajectories of capital stock, atmospheric carbon, investment and mitigation under BAU, the political economy, and the social
planner. Capital stock initially increases rapidly in all scenarios. Under BAU
(zero mitigation, red, short-dashed), the higher emissions resulting from the
higher capital stock lead to higher carbon stocks and damage. These damages reduce both the funds available for investment (output) and the return
to capital, lowering equilibrium investment. Eventually the stock of capital,
and thus emissions, fall, slowing the growth in the carbon stock. Carbon
levels stabilize at 1625 ppmv which translates to an equilibrium temperature
increase of 76◦ .
The social planner (blue, long-dashed) abates 40-80% of emissions. The
trajectories of capital and investment are similar under BAU and the social planner for the first century, but later diverge, lagging the divergence
in the two carbon trajectories. Lower levels of emissions and atmospheric
carbon support a higher equilibrium capital stock and output. Equilibrium
temperature rises to 35◦ , significantly higher than in DICE. Future abatement costs are higher in our stationary model than in DICE, due to the lack
of exogenous decreases in abatement costs and carbon intensity; the higher
emissions and abatement costs imply higher steady state carbon stocks and
damages.
The trajectories under BAU and under the social planner sandwich the
political economy (green, solid) trajectories. Equilibrium abatement in the
political economy is slightly less than half of the level under the social planner.
14
Limited substitutability between labor and capital increase the estimated adjustment
cost parameters by a factor of 2.5 (Hall, 2004). Figure 4 in Appendix B.1 shows that
our baseline adjustment cost parameter (and even much larger values) lead to production
possibility frontiers that are nearly indistinguishable from straight lines.
19
k $T
800
e ppmv CO2 1500
1300
1100
900
700
500
600
400
200
0
100 200 300 400 500
t yrs
0
$Tyr
20
100 200 300 400 500
t yrs
Μ of emissions
1.0
0.8
15
0.6
10
0.4
5
0.2
0
100 200 300 400 500
t yrs
0
100 200 300 400 500
t yrs
Figure 1: Equilibrium trajectories for selected variables: BAU (red dotted),
Markov-Perfect (green solid), Social Planner (blue dashed). In the MarkovPerfect equilibrium 20-40% of emissions are mitigated, under social planner
40-80%.
Steady state damages equal about 4% of output under the social planner, 25%
of output in the MPE, and 42% of output under BAU.
Figure 2 presents the welfare gains under the social planner and political economy equilibria as percent increases over BAU, for current (Ω0 and
Ω0 ) and future generations (Ω12 ). In the political economy equilibrium,
current generations choose policy to maximize their joint welfare. To focus
on welfare effects across periods, we adopt the convention that the young in
period 0 compensate the old generation at the BAU welfare level, and keep
the remaining surplus, so ∆Ω0 ≡ 0. The accompanying table shows numerical values for the first five generations. In the political economy equilibrium,
where generations care only for their own welfare, policy increases aggregate welfare for those currently living in every period. The social planner
puts positive weight on the welfare of future generations and imposes tighter
emission standards, leading to losses for generations alive today. Even the
generation born in the next period (Ω1 ) is worse off than under the political
20
Figure 2: Equilibrium life-time welfare of current (Ω0 , Ω0 ) and future (Ω1 ,
Ω2 , ...) generations relative to BAU: MPE (green solid), SP (blue dashed).
For the comparison initial surplus is distributed to current young (Ω0 ), the
old generations is compensated (Ω0 = 0). Under MPE all generations gain,
under social optimum current young (Ω0 ) lose.
economy equilibrium, but they are already better off than under BAU. All
subsequent generations (those born in 2080 or later) are best off under the
social planner policy, because of its higher capital stock and lower carbon
stock.
Robustness If  = 1 or  = 0, asset prices are irrelevant; these two
parameters therefore distinguish our model from standard IAMs. To test the
sensitivity of our results, we vary these parameters. The “weak” scenario
reduces  so that marginal adjustment cost falls from 0.03% to 0.015% of
investment cost, and increases annual depreciation from 6% to 8%. These
changes move the model closer to a standard IAM. The “strong” scenario
increases  so that adjustment costs equal 0.045% of investment costs, and
annual depreciation is 4%. Figure 3 shows that these parameter changes
lead to large changes in equilibrium investment but smaller changes in abatement. Lower (higher) depreciation and higher (lower) adjustment costs lead
to higher (lower) investment, asset prices, and mitigation. The robustness
of abatement relative to investment and the asset price is surprising and is
potentially due to the time-consistency property of the game.
21
Figure 3: Political economy equilibrium trajectories for baseline (solid) and
"weak" (dashed below) and "strong" (dashed above) sensitivity scenarios.
Left panel: investment. Right panel: abatement.
6
Conclusion
General equilibrium models of climate change assume that a composite commodity can be transformed costlessly into either a consumption or an investment good. This assumption fixes the price of capital, giving currently
living old agents no (selfish) incentive to support policies that increase future
returns to capital. If the production possibility frontier between the consumption and the investment goods is strictly concave, the price of capital
is endogenous. Here, asset owners potentially have an incentive to support
climate policy in order to increase the value of their assets.
Young people are more likely than the old to be alive to enjoy the direct
benefits of climate policy, possibly giving them a selfish reason to support
that policy. In addition, the policy-induced increase in the asset price can
benefit the young as well as the old. Climate policy lowers current wages (and
returns to capital), but climate policy can increase young people’s lifetime
welfare by raising their future income and consumption. If people have
a sufficiently high elasticity of intertemporal substitution, the interests of
old and young agents are aligned: they both benefit from a policy-induced
increase in asset price.
Our model uses many of the standard assumptions from climate economics, apart from the introduction of a concave production possibility frontier and overlapping generations. Our numerical results suggest that the asset
price effect identified analytically is large enough to be important. In our
22
baseline, a political economy equilibrium achieves almost half of the abatement as a social planner, despite the fact that agents in the political economy
setting place no value on their successors’ welfare, whereas the social planner
discounts future generations at the same rate as people discount their own
future.
The climate policy narrative embraces the view that climate policy imposes costs on current generations. We agree that an ethical treatment of
unborn generations requires sacrifices by those currently alive. Nevertheless,
it is important to recognize that climate policy potentially increases current
wealth. Models that make asset prices exogenous overlook this possibility, and thereby overstate the cost of climate policy to current generations.
When the relation between asset values and climate policy is discussed, it is
usually in the context of stranded assets, those likely to be harmed by climate policy. Fossil fuels companies are worth about $5 T, a large number,
but a small fraction of the world’s financial wealth. Climate policy has the
potential to increase the value of assets writ large, benefiting people with a
diversified portfolio. The failure to recognize this possibility exaggerates the
political difficulty of implementing meaningful climate policy.
23
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27
A
A.1
Appendix
Proofs
Proof. (Lemma 1) The old generation consumes all of its income, so  =
( + (1 − )  )  , which implies equation (9). To obtain equation (8) we
begin with the definition of the young agent’s lifetime welfare
Ω ≡ ( ) +  (+1 )
³  ´

With isoelastic utility,  = 1 +1
. Using this result in equation (5) gives


µ

 =  +1

¶−
¡  ¢1−

+1
(+1 + (1 − )+1 )
=  +1

+1
( )− +1
where the second equality follows from the second constraint in equation (1).
Rearranging this equation gives
¡  ¢1−
( )− +1
+1

= 

Using this expression for the utility of the agent who is old in period  + 1,
we write the lifetime welfare for the agent who is young in period  as
µµ
µ
¶¶¶
( )− +1
1

 1−
Ω =
−1+

−1
( )
1−

( )− 
1
=
( +  +1 ) −
(1 − )
1−
1−
1
( )−
 −
(1 − ) 
=
1−
1−
where the last equality follows from the first constraint in equation (1).
Proof. (Proposition 1) Both claims follow from inspection of equation (9),
together with the facts that  is predetermined and the policy has only a
second order effect on the current return to capital.
Proof. (Proposition 2) Using equation (8) (and ignoring the constant) we
write the young agent’s lifetime welfare as
Ω () =
( )−
 
1−
28
Differentiating with respect to , using 0 = ̄0 , we have
h
i

dΩ ()
̄
 −
 −−1   

=
(
)
−

(
)



d
1− 
  


Evaluating this expression on the BAU trajectory, where  = 0, and using

= 0, this equation simplifies to
Proposition 1 to set 


|=0
¯
µ
¶
dΩ () ¯¯
̄
−−1   
  

=−
d ¯=0
1−
  
The wage and consumption are both positive (  0 and   0). The
assumption that investment is positive and that there is friction, together

with the second part of equation (4) implies 
= 10  0. The first constraint

in equation (1) and the fact that agents take average adjustment cost as

exogenous implies  = −  0. By assumption, ̄  0. Consequently,
µ
¶
µ  ¯ ¶

dΩ () ¯¯
= sign
for 0    1, ̄  0
sign
d ¯=0

µ  ¯ ¶
µ
¶
dΩ () ¯¯

sign
= −sign
for   1, ̄  0
d ¯=0

The third statement follows from the first two statements and from Proposition 1.
Proof. (Lemma 2) The budget constraint for the agent who is old in period

 + 1 requires +1
= +1 + (1 − )+1 . Using this relation and the fact that
+1
 =
+1

under logarithmic utility, equation (5) becomes



=⇒
 =   (+1 + (1 − )+1 ) =  +1
+1
 =
 +1

Using the last equation and the young agent’s budget constraint at time ,
we have
 =  −  +1 =
 +1

1
⇒  +1 =
 ⇒  =
 

1+
1+
1
of her
With logarithmic utility, the agent spends a constant fraction 1+
wage on first period consumption and saves the rest. Using the fact that
29
 = ( ), or  = −1 ( ), we obtain an implicit function for  .
function,  =  ( ), solves
¤
£

 
 +1 =  (1 − ) + −1 ( ) =
1+
This
(13)
Replacing  with  ( ), using the fact that −1 is an increasing function,
and totally differentiating the last equation in (13) implies that  0 ()  0: an
exogenous increase in the current wage causes an increase in the equilibrium
price of capital.
Proof. (Proposition 3) (i) A small policy at  = 0 has zero first order effect
on the wage at  = 0. By Lemma 2.ii, this policy therefore has zero first
order effect on the asset price at  = 0. By Proposition 1, the policy has
zero first order effect on the old agent’s welfare at  = 0.
(ii) By Lemma 2.i and the fact that the policy has a zero first order effect
on wage at  = 0, the policy has zero first order effect on the  = 0 young
agent’s consumption, and thus on her savings, 1 . However, the policy
creates a first order reduction in the  = 1 pollution stock, leading to a first
order increase in both 1 and 1 , and (by Lemma 2.ii) on the equilibrium
value of 1 . The policy therefore leads to a first order increase in her period
 = 1 consumption, 1 = (1 + (1 − )1 ) 1 , and thus in her lifetime welfare.
(iii) The period-0 policy increases 1 by reducing the period-1 pollution
stock. By Lemma 2.i, the period-0 policy therefore increases the consumption, 1 , of the agent born at  = 1. This agent’s consumption in the subsequent period equals 2 = (2 + 2 (1 − )) 2 . Because the policy increases
1 , it also increases investment, 1 , and therefore increases 2 . The period-0
policy lowers 1 without altering 1 , so the policy lowers the subsequent pollution stock, 2 . The reduction in 2 and the increase in 2 both increase the
equilibrium wage, 2 . By Lemma 2.ii, the policy therefore increases 2 , and
thus increases 2 2 . The equilibrium condition for the rental rate implies
2 2 = (1 −  (2 )) Λ (2 ) (1 − ) 21− 
Because the policy has increased 2 and reduced 2 , it increases 2 2 . Therefore, the policy also increases consumption, at  = 2, of the agent who was
born at  = 1. Consequently, the policy increases this agent’s welfare.
Proof. (Proposition 4). We invoke the turnpike theorem in letting  → ∞.
(i) This result is an immediate consequence
of equation (10),  = (1 −
¯
Λ(̄) ¯
= 0.
Λ (̄))( − ), and the fact that ̄ ¯
̄=0
30
(ii) (sketch; see Appendix
B.2 for details) Using Proposition 1 we need
¯
0 ¯
 0. We confirm this inequality by examining
only establish that  ¯

0 =0
the equilibrium conditions, a system of three linear difference equations in
  . Price is a forward looking variable, and the equilibrium price is
linear in  . Using a linear trial solution and the assumption that the
system determining   is stable, yields the inequality.
(iii) This result follows from equation (10),  = (1 − Λ ())( − ),
1
Λ0 (0) = 0, and 
= − (0 + )  0.
0
B
Reviewers’ Appendix (not for publication)
This appendix discusses alternative micro-foundations for the concave PPF,
it provides the detailed proof of Proposition 4.ii, and it contains details of
our numerical approximation.
B.1
Alternative microfoundations for the PPF
We use adjustment costs to obtain a concave PPF in order to have a tractable
model, and because of the extensive theoretical and empirical literature on
these costs. We introduce congestion in the investment sector, so that the
price of the investment good equals average instead of marginal costs, +0 .
If price equals marginal (greater than average) costs, there must be a sectorspecific factor receiving rent in the investment sector. That change would
add little additional descriptive power, but it would immensely complicate
the model. It would require an additional state variable (the sector-specific
asset), along with its price and endogenous evolution. Therefore, a tractable
model using adjustment costs appears to require the additional assumption
that price equals average costs. This type of “congestion assumption” is
widespread in economics; for example, it is the basis for much of the literature
on endogenous growth.
Of course, there are many other ways to generate a strictly concave PPF,
without relying on adjustment costs. Perhaps the most obvious of these is a
general equilibrium model such as Heckscher-Ohlin-Samuelson (HOS). With
this model, and Cobb Douglas production in the consumption and investment
sectors, we obtain a concave PPF whenever the factor shares in the two
sectors are not equal. If we additionally assume that the two sectors have
the same damage and abatement cost functions, we can calibrate the model
31
using the two factor shares instead of the single adjustment cost parameter
(as in Section 5.3). The HOS calibration is perhaps more transparent than
the adjustment cost calibration, because the factor shares can be observed,
whereas the adjustment costs must be estimated.
The HOS model (like the adjustment cost model) leads to an explicit
expression for the price of investment as a function of the level of investment,
the stocks of capital and labor, the climate state variable, and the abatement
decision. That function, however, is considerably more complicated than
equation (4), the price in our adjustment cost model; the resulting analysis
would also be much more complicated.
We also prefer the simple adjustment cost model over the HOS alternative,
because (except for the linear model in Section 4.2) the wage/rental rate in
the former, but not in the latter, is independent of the level of abatement and
the climate state variable. The effect of abatement and the climate state
on relative factor returns seems like an interesting research question, and
has, to the best of our knowledge, not been studied in the IAM framework.
However, this issue is tangential to our focus on asset prices. In order to be
able to compare our results to those of standard IAMs, we want to keep as
close as possible to the standard framework, apart from the introduction of
asset prices. This objective militates against the HOS or a similar general
equilibrium model.
It is worth emphasizing that our numerical calibration implies a nearly
linear PPF: we do not require “much concavity”. Figure 4 shows the production possibility frontier with  = $63 T. under our baseline adjustment
cost parameter,  = 00003, (dotted) and for two larger values,  = 0003
and  = 003 (solid and dashed). Our baseline adjustment cost parameter
(and even much larger values) lead to production possibility frontiers that
are nearly indistinguishable from straight lines. Even this low level of curvature makes the asset price important enough to lead to substantial levels of
abatement.
B.2
Proof of Proposition 4.ii
Using Proposition 1 we need only establish that
¯
0 ¯
 ¯
0 =0
 0.
Under
Assumption 2, the dynamics of the competitive equilibrium, equations (5)
and (7), from periods  = 1 onward, reduce to the following system of linear
difference equations (with ∆ = 1 − ,  = 1 − , and  = 2 to simplify
32
C 60
50
40
30
20
10
0
0
10
20
30
40
50
60
Investment
Figure 4: Our baseline calibration ( = 00003): green dotted. Increase
adjustment cost parameter by a factor of 10: solid red. Increase adjustment
cost parameter by a factor of 100: red dash.
notation):
 =  [(1 − Λ (̄))( − +1 ) + ∆ +1 ]
+1 = ∆  +  ( − 1)
+1 =   +  (1 − ̄) (  + ) 
(14)
The initial conditions for this system are 1 and 1 . These values depend
on the state variables and the policy at time 0, 0 , 0 , and 0 . In this
linear model, where at  the states  and  are predetermined, and  is
“forward looking”, the equilibrium  is a linear function of  and  :  =
 +   +   . The coefficients    depend on model parameters,
including the constant ̄. Substituting the trial solution,  = +  +  ,
into system 14 we obtain
⎞
⎛
⎞
⎛

¡
¢
¢ 
¡
 = 1   · ⎝  ⎠ =  1   ⎝  ⎠ with


⎞
⎞
⎛
1 + 2

⎝  ⎠ = ⎝  ∆ (∆ +  ) +   (1 − ̄) ( ∆ −  (1 − Λ (̄))) ⎠

 ∆ ( +  ) −   (1 − Λ (̄))
⎛
with
1 =  (1 − Λ (̄)) +  ∆ +  ∆  ( − 1)
2 =   (1 − ̄) ( ∆ −  (1 − Λ (̄))) 
33
Equating coefficients yields
⎛
⎞
⎞
⎛


⎝  ⎠ =  ⎝  ⎠


(15)
We show that there is a unique negative value of  that satisfies this system:
the asset price declines with capital and the pollution stock. (Other roots
of the system are either complex or positive.)
The last two equations in system 15 are independent of . Solving the
last equation for  gives
[ ] = −
(1 − Λ (̄))

1 − ∆( +  )
(16)
with the implication that   0 ⇔ 1 − ∆( +  )  0. Substituting the
expression for  into the second equation of system (15), we obtain
0 = Υ [ ] ≡
h
³
 −   ∆ (∆ +  ) −  (1 − ̄)  (1 − Λ (̄)) 1 +
Define
∆
1−∆(+ )
´i

Θ = Υ × [1 − ∆( +  )] 
For
 6=  ≡
1−∆  
 0,
∆
Υ( ) = 0 ⇔ Θ ( ) = 0
Θ, is a cubic and thus has one or three real roots (as does Υ). The cubic Θ
can be written
3  3 + 2  2 + 1  + 0 with
¡ 2 2 2¢
∆   0
∆ (∆(∆ + ) − 2)  0
(1 − ∆)(1 − ∆2 ) − ∆2 (1 − ̄)(1 − Λ)
 (1 − Λ) (1 − ̄)  0
¡
¢
¡
¢ ¡
¢
The sign pattern of 3 1 1 0 is either + − + + or + − − + .
In either case, there are two sign differences between consecutive coefficients.
Θ
3
2
1
0
=
≡
≡
≡
≡
34
By Descartes’ rule of signs, there exist either two or zero positive roots. (In
a knife-edge case, there is a single positive root.) To apply the corollary
of Descartes’ rule of signs, multiply ¡the coefficients ¢of odd
¡ powered terms
¢
by −1. This gives the sign pattern − − − + or − − + + .
In either case, there is one change in signs. The corollary states that the
number of negative roots is either the number of sign changes (one), or fewer
than that number by a multiple of 2. Because there cannot be a negative
number of negative roots, we conclude that there is a unique negative root.
We cannot rule out the existence of up to two roots with positive values for
.
The system is unstable for  ≥  ; therefore, if  is positive and the
system is stable,  ∈ (0  ). Substituting  =  +   +   into the
last two equations in system (14) gives
+1 = ∆  +  ( +   +   − 1)
+1 =   +  (1 − ) (  + ) 
The Jacobian of this system (with respect to the state variables,  and  )
is
¶
µ
∆ +  

=
(1 − ) 
Stability requires that all eigenvalues of  lie within the unit circle. A weaker
necessary condition requires the sum of the eigenvalues or the trace of ,
 () = ∆ +  +  , to be less than 2. For the parameter restrictions
0 ≤ ∆ ≤ 1 (capital decays) and   0 (positive discount factor) we have
 ≥  =⇒  ()  2:
 ()| = = ∆ +  + 
1
1−∆  
=∆+
 2
∆
∆
Because  () is increasing in  , we rule out all positive roots  +   .
Therefore, any equilibrium root (which we have shown to exist) satisfies
   ; consequently,   0.
The next step obtains the current ( = 0) asset price as a function of the
next period pollution stock. Using equation (5) and  =  +   +   ,
we have
0 =  [(1 − Λ (0 ))( − 1 ) + ∆ 1 ]
=  [(1 − Λ (0 ))( − 1 ) + ∆ ( +  1 +  1 )]
=  [(1 − Λ (0 ))  + ∆] +  ∆1 −  [(1 − Λ (0 ))  − ∆] 1 
35
We now use the relation +1 = ∆  +  ( − 1) to eliminate 1 and write
0 =  [(1 − Λ (0 ))  + ∆] +  ∆ (∆ 0 +  (0 − 1))
− [(1 − Λ (0 ))  − ∆] 1 
Solving for 0 gives
0 =
1
×
(1− ∆)
[ [(1 − Λ (0 ))  + ∆] +  ∆ (∆0 − ) −  [(1 − Λ (0 ))  − ∆] 1 ]
With convex abatement cost Λ () =      1, we have Λ0 (0) = 0 and
only need to consider the effect of 0 on 0 via 1 . Using the transition
equation of  in (14), we establish
¯
1
0 ¯¯
= 4
 0 with
(17)
¯
 0 =0
0
4 ≡ −

 ((1 − Λ (̄))  − ∆)
=
 0 and
(1 −  ∆)

1
= − (0 + )  0
0
B.3
Numerical Approximation
The possibility of multiple equilibria arises for two kinds of reasons; our
numerical results suggest that neither of these reasons are important for
our model. First, for a given candidate   (+1  +1 ), there might be
multiple solutions to the political economy problem at time . Because
the function   (+1  +1 ) is a polynomial approximation, we cannot rely
on curvature properties to guarantee uniqueness; instead, we depend on the
numerical algorithm. Second, the infinite horizon (required in our stationary
setting) generically raises the possibility of non-uniqueness, a standard result
in dynamic games where there is an “incomplete transversality condition”.
However, our algorithm works backwards, beginning with a scrap function to
represent the last period; we iterate until convergence, so that the algorithm
selects a solution that is close to the “limit equilibrium”, as the horizon
goes to infinity. We confirm that the converged equilibrium is insensitive to
changes in the scrap function.
36
We approximate  (+1  +1 ) and Ψ (+1  +1 ) as polynomials in +1
and +1 , and find coefficients of those polynomials so that the solution to
the maximization of the joint welfare
[( + (1 − ) ) ]1− − 1
+
max

1−
(
)
[+1 + (1 − )+1 +1 ]1− − 1
[ −  +1 ]1− − 1
+
1−
1−
subject to
+1 = (1 − ) +  (1 −  )  ( )  (  ) 
+1 = (1 − ) + −1 ( )
(18)
with  ,  given and −1 () the inverse function of (), approximately equals
 (   ). For the BAU and MPE solutions, we use 81-degree Chebyshev
polynomials evaluated at 9x9 Chebyshev nodes on the [200 700] interval for
 and [200 1000] for MPE and [100 1350] interval for BAU interval for  (the
interval for  increases under BAU due to its higher steady state). At each
node the recursion defining Ψ (   ) 
−
[(+1  +1 ) + Ψ (+1  +1 ) (1 − )]1−
+1
 = Ψ (   ) = 

[ (   ) − Ψ (   ) +1 ]−
is satisfied.
condition,
For BAU,  = 0.
"
−
 ( −  +1 )

1−
For the MPE we have the optimality
1−
 +
(19)
[( + (1 − ) ) ]
1−
−1
#
=0
(20)
and the Nash equilibrium condition  =  (   ). In addition, system 18
must be satisfied. The first order condition 20 takes  as given, instead
of using the right hand side of equation (19). This procedure means that
current generations do not use the mitigation instrument for redistributional
purposes.
Starting with an initial guess for the coefficients of the approximations
of Ψ (·) and  (·), we evaluate the right side of equation (19) for at each
node. Using these function values, we obtain new coefficient values for the
37
Figure 5: BAU solution: upper Asset price, Ψ (·), and policy,  (·), function; lower deviation from true asset value outside of approximation nodes
("Residuals) and %-change of coefficients between iterations ("̂")
approximation of Ψ (·). We then use the optimality condition (20) to find
the values of  at the nodes; we use those values to update the coefficients
for the approximation of Ψ (·). We repeat this iteration until the coefficients’
relative difference between iterations falls below 10−6 . See chapter 6 of Miranda and Fackler (2002) for details. Figures 5 and 6, graph the asset and
mitigation functions, the differences (the “residuals”) between the right and
left sides of equations (19) and (20), and the %-change in the coefficients of
the approximated function between iterations. Residuals equal 0 at the nodes
because we set both the degree of the polynomial and the number of nodes
equal to . We choose  = 81 to ensure that residuals are at least 5 orders
of magnitudes below the solution values on the approximation interval.
For the social planner’s problem, we approximate  (+1  +1 ),  (+1  +1 )
and  (+1  +1 ) as polynomials in +1 and +1 , and find coefficients of
38
Figure 6: MPE solution: upper Asset price, Ψ (·), and policy,  (·), function;
lower approximation from true asset value outside of approximation nodes
("Residuals") and %-change of coefficients between iterations ("̂")
39
those polynomials so that the solution to the maximization of the welfare
( − ( )+1 )−
[( + (1 − )( )) ]1− − 1
 +
+   (+1  +1 )
max
 
1−
1−
subject to
+1 = (1 − ) +  (1 −  )  ( )  (  ) 
+1 = (1 − ) +  
(21)
with  ,  given, approximately equals  (   ) and  (   ). At each node,
the value function recursion is
( − ( )+1 )−
[( + (1 − )( )) ]1− − 1
 (   ) =
 +
+  (+1  +1 ) 
1−
1−
(22)
the social planner’s optimality conditions are
i
h
( −( )+1 )−
[( +(1−)( )) ]1− −1

 +
+   (+1  +1 ) = 0

1−
1−
h
i
(23)
( −( )+1 )−
[( +(1−)( )) ]1− −1


+
+


(


)
=
0

+1
+1

1−
1−

For the social planner, we use 40-degree Chebyshev polynomials evaluated
at 10x4 Chebyshev nodes on the [200 900] interval for  and [100 350] interval
for . Starting with an initial guess for the coefficients of the approximations
of  (·) ,  (·) and  (·), we evaluate the right side of equation (22) for at each
node. Using these function values, we obtain new coefficient values for the
approximation of  (·). We then use the optimality conditions (23) to find the
values of  and  at the nodes; we use those values to update the coefficients
for the approximation of  (·). We repeat this iteration until the coefficients’
relative difference between iterations falls below 10−6 . Figure 7 graphs the
value, investment, and mitigation functions, the differences (the “residuals”)
between the right and left sides of equations (19) and (20), and the %-change
in the coefficients of the approximated function between iterations. Residuals
equal 0 at the nodes because we set both the degree of the polynomial and the
number of nodes equal to . We choose  = 40 to ensure that residuals are at
least 5 orders of magnitudes below the solution values on the approximation
interval.
The graph of the Production Possibility Frontier
40
Figure 7: SP solution: upper Value,  (·), and investment function,  (·),
(upper); middle mitigation policy function,  (·), and deviation from true
value function value outside of approximation nodes ("Residuals"); lower
%-change of coefficients between iterations ("̂")
41