Firm Valuation in an Environmental Overlapping
Generations Model
Lammertjan Dam∗
April 24, 2006
Abstract
Inter-generational externalities associated with the conservation of the environment
are usually tackled through fiscal policy. The recent increase in socially responsible
investment funds creates a potential role for the stock market to deal with these
environmental externalities. We study this alternative approach in an overlapping
generations model in which agents choose between investing in clean or polluting
technologies. Since agents are short-lived, they do not account for the long-term
effects of pollution. We show that when firm property rights are traded separately
on a forward looking stock market, proper firm valuation can resolve the conflict
between current and future generations.
Keywords: Overlapping Generations, Environmental Quality, Corporate Social Responsibility, Sustainability, Stock Market
JEL classification Codes: D62, E22, G11, G12, Q01, Q20
Paper prepared for the “Young economists sessions”, SURED conference Ascona,
Switzerland, June 5-8, 2006.
∗
University of Groningen, Department of Economics, P.O. Box 800, 9700 AV Groningen, The Netherlands or [email protected], tel: +31(0)50-3637190.
1
Introduction
A problem of growing concern is the threat to the environment resulting from polluting economic activity. From economic theory we know that there is an externality
associated with the conservation of the environment. This externality exhibits two dimensions. First, there is an intra-generational dimension. The environment is a public
good and as such its conservation suffers from the standard free rider problem. Second, there is an inter-generational dimension. Since pollution typically accumulates
in the environment, future generations bear the social costs of the actions of the current generation. Various studies have proposed fiscal policies to manage the long-term
threat of pollution to the environment in order to achieve sustainable development.
This paper proposes an alternative mechanism to deal with the inter-generational
aspect of the pollution externality.
In recent years, not only policy makers, but also large corporations have put sustainable development on their agenda. Corporations publicly report that they engage
in corporate social responsibility (CSR) or sustainability programs. This behaviour
has created the possibility of socially responsible investment (SRI). The idea is that
shareholders do not only care about the cash flows of a project, but also about how
these cash flows are generated. For instance, an investor might oppose to using child
labour or heavily polluting technologies in the production process. Socially responsible investment funds, or “green funds”, create perspectives for the stock market to
function as a tool in dealing with environmental externalities. Typically agents are
short-lived, so they do not internalise the long-term effects of pollution. However, in
the presence of a forward looking stock market, we show that proper valuation can
resolve the coordination failure between current and future generations.
To capture the conflict between generations, we study the environment in a Diamond type overlapping generations (OLG) model, in line with John and Pechinino
(1994, JP). Agents live for two periods. They work when they are young, retire and
derive utility from consumption and environmental quality when they are old. We
adapt the model of JP such that, instead of choosing between consumption and environmental maintenance, agents choose between investing in polluting and clean
2
technologies, where the polluting technology is more productive. This change from a
consumption into an investment decision allows us to introduce and analyse the role
of a stock market. Magill and Quinzii (2003) point out that when firm ownership
rights are traded separately on a stock market, externalities or frictions can make
the value of financial equity different from the value of the real capital goods market.
The introduction of this “missing market” incorporates the negative externality of
pollution in a natural way, as the stock market is forward-looking.
We are not the first to study the threat to the environment in a Diamond-type
OLG model (John and Pechinino, 1994; John et al, 1995; Guruswamy Babu et al.,
1997; Zhang, 1999; Seegmuller and Verchre, 2004; Wendner 2006). However, most of
these studies rely on fiscal policy to overcome the intergenerational conflicts instead
of the mechanism we propose. Moreover, the proposed tax programs are not straightforward, because they require the use of various instruments. The reason is that even
without externalities, the decentralised solution needs not to be Pareto-optimal1 . Finally, the value of financial equity is not necessarily equal to the replacement value
of physical capital when there are externalities, which has not been explored in this
literature.
The introduction of stock markets in an OLG model brings some technical complications that we address in this paper. Various studies discuss the indeterminacy of
asset prices in OLG models (See, for example, Woodford, 1984, Huffman, 1986, Magill and Quinzii, 2003). The indeterminacy is not one resulting from missing markets
or imperfect foresight, but originates from a difference equation. Such a difference
equation has infinitely many solutions, and so it can cause rational price bubbles. In
general the equilibrium paths depend on initial price values, which are usually not
endogenous in the model. When there are multiple steady state equilibria, it may not
be clear to which steady state the economy converges. In our stock market model,
without further restrictions, we face infinitely many possible steady state equilibria.
However, our setting allows us to come up with a solution to this standard problem:
We regard an asset price as the sum of discounted cash flows - as is standard in
1
This is the well known result of Diamond (1965) that agents can over- or under invest in physical
capital compared to the Golden Rule solution.
3
asset pricing theory - instead of technically evaluating prices at two points in time.
Therefore the value of the asset is influenced by all future actions and valuations
of coming generations. Since the current generation takes this into account, we have
implicitly modelled a sequential game. The belief structure of and the solution to this
game is an additional restriction on the economic outcomes. We consequently regard
the investment decisions as Markov state decisions. Instead of letting the equilibrium
path depend on exogenous initial values it depends on endogenous steady state/end
values. This way we determine a unique steady state equilibrium condition, that
leads to the dynamically efficient solution. The fact that a shareholder considers all
future cash and pollution flows when valuing the firm is a subtle adjustment but an
important artefact of the model.
We find that in such a stock market equilibrium, the stock market value of polluting capital is higher than its replacement value, whereas in the absence of a stock
market the value of the firm and its replacement value are the same. This makes investments in the polluting sector relatively less attractive, as the resulting return on
investments will be lower. This mechanism then allocates more resources into clean
investments, resulting in less pollution.
In section 1 we present the core of the model. We show that the resulting equilibrium is indeed dynamically inefficient. Next we turn to the discussion on socially
responsible investment and its consequences for proper valuation and reporting on
firm value. We show that when introducing a stock market, proper valuation resolves
the coordination problem. In section 2 we present the basics of the model and the
valuation of the firm and briefly discuss dynamics. We conclude in section 3.
1
A two sector environmental OLG model
We introduce environmental quality in a standard Diamond-Samuelson overlapping
generations (OLG) model. Pollution, generated by investing in a dirty sector, decreases the stock of environmental quality. We assume that dirty investment is
cheaper than investing in clean technologies that implicitly require more knowledge
and skills. We assume irreversibility: once a good is invested in the dirty technology,
4
it cannot be shifted to the clean sector. In this section we discuss firm behaviour,
consequences for environmental quality, and household choices.
1.1 Firms and environmental quality
We assume two types of firms: firms in the clean sector (C) use capital that saves on
environmental resources and firms operating in the dirty sector (D) not taking into
account environmental considerations. We consider the extreme case where the clean
sector does not pollute at all. Output in both sectors is represented by a linear homogeneous production function F j (K, L) = F (K j , Lj ) where K j denotes the capital
stock and Lj labour used in sector j, j = C, D. Capital invested at t, denoted Itj ,
becomes productive at t + 1. Firms depreciate capital at a uniform rate δ:
j
Kt+1
= (1 − δ)Ktj + Itj , j = C, D
(1)
Total investments are: It = ItD + ItC . The price of output is the numeraire. The firm
hires labour at a rate wt and borrows capital at a rate rt to finance investment at
time t. The owners of the firm maximise the value of the firm V (K j ) subject to (1)
at time t:
max V (Ktj ) =
∞
j
j
X
F (Kt+τ
, Ljt+τ ) − wt+τ Ljt+τ − (1 + rt+τ )It+τ
−1
Qτ
(1
+
r
)
t+i
i=1
τ =1
(2)
j = C, D. We get the familiar necessary first-order conditions:
FKj (Ktj , Ljt ) = rt + δ
(3)
FLj (Ktj , Ljt ) =
(4)
wt
Labour is mobile between the clean and dirty sector, so in equilibrium we have
D
D
FLC (KtC , LC
t ) = FLD (Kt , LT ) = wt . The mobility of labour ensures a single wage
rate, and if we assume a homogeneous production function of degree one, this implies
a single capital-labour ratio and the rental rates are equal in both sectors2 . Because
of constant returns to scale, we can rewrite output: F (K j , L) = f (kt )Lt where f (kt )
is production per capita. We use lower case letters, it , kt , ct , to denote per capita
2
Note that we will make a distinction between the rental rate of capital and the rate of return on savings
in the next section.
5
investment, capital and consumption. Note that we define ktj = Ktj /L, j = C, D and
that these per capita variables are not equal to the capital labour ratios KtC /LC
t and
D
C
KtD /LD
t , but defined such that kt = kt + kt .
Capital in the dirty sector ktD creates contemporaneous pollution. We assume
there is a linear relation between dirty capital and pollution and as a consequence
we are free to choose our unit of account of pollution. We therefore assume that one
unit of dirty capital creates one unit of pollution. Environmental quality Et , a public
good, is modelled as a renewable resource (see JP, 1994):
D
Et+1 = (1 − β)Et − kt+1
+ βE0 ,
(5)
with 0 < β < 1 the rate of natural recovery, and E0 some ‘virgin’ value. Without
pollution, the environmental quality will return to the virgin value E0 .
1.2 Consumers
At each date t a generation of finitely-lived consumers of fixed size L is born. Consumers live for two periods. Consumers supply one unit of labour when young at a
wage rate wt , save all their wages, invest in clean and dirty capital, and consume
when old. This simplification is quite common in OLG models which include environmental quality (see Guruswamy Babu et al. (1997) and JP, 1994). Note also that
we do not focus on how much is saved, but on how savings are invested given the
savings decision. A consumer has preferences defined over consumption ct+1 and the
public good environmental quality Et+1 at old age characterised by a utility function
u(ct+1 , Et+1 ). For ease of computation, we assume a linear additive homothetic utility
function: u(ct+1 , Et+1 ) = u1 (ct+1 )+u2 (Et+1 ). We are aware of the free-rider problems
associated with the provision of public goods, but we wish to abstract from those and
focus on the intergenerational external effects. Therefore we assume there is a form
of coordination to establish optimal provision of the public good for agents alive at
time t. This could be established by their elected representatives through lump-sum
taxes or through investor collaborating in investor groups or “green” funds offered by
financial institutions. In practice projects require large investments, so it is not unlikely that investors collectively decide on these investment decisions. In the absence
6
of this intra-generational externality we can normalise the size of each generation to
one. The representative young agent at time t takes as given the rental rate on capital
and environmental quality at time t. Investing in clean capital is associated with a
sunk cost3 s > 0 per unit of capital kC . This sunk cost is borne by the investor and
is not taken into account by the firm.
Firms are owned by the old generation. The young agent buys the depreciated
C
capital stock (1 − δ)kt from the old and chooses an investment portfolio (iD
t , it ).
C , k D ) to maximise
Using (1) we can see that they are in fact deciding on (kt+1
t+1
lifetime utility:
max u(ct+1 , Et+1 )
subject to:
D
C
+ (1 + s)kt+1
wt = kt+1
(6)
D
C
ct+1 = (kt+1
+ kt+1
)(1 + rt+1 )
(7)
and equation (5). Here, (6) is the budget constraint. The consumer spends its wage on
j
buying the depreciated capital stock and investments (1 − δ)ktj + ijt = kt+1
, j = C, D
C . He retires the next period and
and pays the sunk cost on clean investment skt+1
consumes out of his savings according to (7).
Although both sectors pay the same rental rate for capital, the consumer faces
different net returns, since he has to pay a sunk cost s on clean capital. The net return
D , in the clean sector (1 + r
C
in the dirty sector is 1 + rt+1 =: Rt+1
t+1 )/(1 + s) =: Rt+1 .
It follows that the first order necessary condition of this problem is:
u′Et+1
u′ct+1
=
s
D
C
(1 + rt+1 ) = Rt+1
− Rt+1
1+s
(8)
D − RC
The excess net return on dirty capital Rt+1
t+1 is equated to marginal environ-
mental damage
u′E
t+1
u′ct+1
. If there are no externalities, only investments in the sector
3
We assume a sunk cost for ease of computation. What we require is a higher net rate of return on dirty
capital, so that there is a trade-off between pollution and higher returns. We could have achieved this by
using the assumption of higher productivity in the dirty sector, but qualitatively this makes no difference
to our assumption.
7
with the highest returns will be made. However, since we have external effects, different net returns can co-exist in equilibrium. We define the equilibrium by a sequence
of prices (wt , rt ), consumption, investment and production decisions (ct , ktC , ktD ) for
the sequence of consumers and firms such that each firm maximises its profits, each
consumer maximises utility and markets clear.
1.3 Equilibrium and stability
We can describe the equilibrium by the necessary first-order conditions for firms (3)
and (4), households, and market clearing conditions. Note that labour is not sector
specific, so the wage rate determines a single capital-labour ratio in both sectors: kt .
Using Euler’s theorem we get:
rt = f ′ (kt ) − δ = r(kt )
(9)
wt = f (kt ) − f ′ (kt )kt = w(kt )
s
(1 + r(kt ))
u′ = u′Et
1 + s ct
(10)
(11)
Equation (11) is the first-order condition of the consumer’s maximization problem.
It equates the marginal rate of substitution between consumption and environmental
quality to the marginal rate of transformation. Since consumption is ct = kt (1 +
r(kt )) = c(kt ), we have that (11) defines Et as an implicit function of kt , so Et = φ(kt )
Using this result and equation (5) The whole system can be expressed in terms of
the state variable kt and we obtain a first order nonlinear difference equation in kt :
φ(kt+1 ) +
1+s
w(kt )
kt+1 = (1 − β)φ(kt ) +
+ βE0
s
s
(12)
This equation determines the equilibrium path of the capital stock and consequently all other endogenous variables.
1.4 Dynamics, stability, and welfare analysis
We depict the steady state equilibrium and model dynamics in E − k space in Figure
1. We only consider economies for which we have a stable steady state equilibrium,
for the necessary conditions see Galor and Ryder (1989). Point C is the steady state
8
k=w
E
S
E0
kD=0
F.O.C.
C
V
kC=0
ESS
0
k
k=w/(1+s)
Feasible k
k=w
Figure 1: Steady State equilibrium with mixed capital
solution to the model with state variable k, capital intensity of production. We depict
the virgin value E0 , which can be maintained by not using dirty capital at all (the
line kD = 0). Without clean investment we get the border case kC = 0. All equilibria
need to be in the area between these two lines. Furthermore, we have that if all wages
are spent on clean capital, we end up with an amount of capital k = w/(1 + s). At
the largest the capital stock can become k = w; in this case all wages are spent
on dirty capital. These two lines restrict the feasible set of steady state equilibria
further. The first order condition (11) can be depicted by the dotted lines F OC
and the set of feasible steady states of (12) by the curve S − V . The intersection
of these equations give the Diamond steady state equilibrium. A unique path leads
to C. Point C is by definition in the feasible set of k. In principle any point in the
feasible region can be a Diamond saddle path stable steady state solution to the
decentralised planning problem. But these equilibria can be dynamically inefficient,
9
because the environmental effects of pollution outlive the finitely lived agents. We
derive the Pareto-optimal steady state solution (dynamically efficient). We treat all
generations equally, impose steady state conditions and look for the allocation that
maximises utility. Given such an allocation, it is impossible for some generation to
be better off without compromising the needs for future allocations and hence it is
the only allocation which implies sustainability.
For ease of demonstration, we drop the time subscript: each generation is treated
similarly. Given that the final goods market satisfies: y = f (k) = c + i + skC , we
have consumption c = f (k) − i − skC . We impose the steady state conditions for
D
capital accumulation and environmental quality: iC = δkC , iD = δkD , E = E0 − kβ .
Maximizing utility:
max u(c, E) = max u(f (k) − iC − iD − skC , E0 −
kD
)
β
(13)
we get for the dynamic efficient steady-state solution:
f ′ (k̄) = δ + s
1
su′c (c̄) = u′E (Ē)
β
(14)
(15)
where we denote the steady state dynamically efficient values of k, c and E by k̄, c̄ and
Ē. The first condition resembles the familiar condition for the Golden Rule level of
capital. From the second equation we can see that the marginal cost of environmental
improvement is equated to the benefits. The environment has a natural recuperation
rate of β and so one unit of damage to the environment implies damage of (1 − β) for
the next generation, and (1 − β)2 ,(1 − β)3 , and so on, for the following generations,
which, if infinitely summed, is equal to 1/β. The price of environmental benefits is
equal to giving up s units of consumption, hence the term s.
Equation (15) defines the optimal environmental quality given levels of consumption. If we compare the dynamic efficient solution with condition (11) we can derive
that environmental quality in the steady state of the decentralised economy, E ⋆ , is
related to the dynamic efficient solution Ē:
u′c⋆
(1 + s)β u′c̄
=
u′E ⋆
(1 + r) u′Ē
10
(16)
E
Dynamic efficient
E0
D
C Decentralized
k=w/(1+s)
k=w
k
Figure 2: Dynamic efficient versus decentralised steady state equilibrium
If the sunk cost is lower than the rental rate (which makes sense, otherwise there
would be a negative net return on clean capital) there is over accumulation of pollution. Since the rate of natural recuperation β < 1, it then follows that the coefficient
on the right-hand side is smaller than one, which implies that the social planner
establishes a higher ratio of environmental quality to consumption. It follows that
consumption will be higher and environmental quality will be lower in a decentralised
economy. We depict the difference between the dynamic efficient and decentralised
equilibrium in Figure 2. From the above equation we can see that a tax policy that
achieves the Golden Rule, e.g. f ′ (k̄) = δ + s (and hence r = s) will not drive the
economy to the dynamic efficient steady state. In this case, over accumulation of
capital is resolved, but consumers still do not incorporate the long term effects of
accumulating pollution. The marginal rate of environmental benefits over marginal
rate of costs differs from the dynamic ratio by a factor β. Although short-lived con-
11
sumers equate the marginal costs of pollution to the marginal benefits, they do not
account for the long-term effect of pollution.
Policy implications are not straightforward, since it depends on whether the economy is under or over accumulating capital. We have made a simplifying assumption
on savings behaviour, which almost certainly causes over accumulation of capital. In
such a case, any policy that shifts dirty investments to clean investments is a Pareto
improvement. However, for a more realistic savings behaviour it is also possible that
society is over maintaining environmental quality.
2
Introducing a stock market
The source of the dynamic inefficiency problem in Section 1 is that an individual
does not internalise the future social costs of investments in polluting capital. Since
agents are short-lived, the implicit environmental value of the firm will be evaluated
in the short run only. Furthermore, since there is only a market for real capital
goods, the price of ownership rights of the firm are equal to its replacement costs.
If the ownership rights of dirty capital are traded separately on a stock-market,
market valuation of the firm can solve the inefficiency problem. Pollution due to
dirty production makes the stock market into financial equity that is distinct of the
real capital goods market. This approach is similar to Magill and Quinzii (2003).
However, they consider a different type of friction, namely that capital once installed
is fixed in the firm and cannot be used in a different sector or for consumption
later. Since ownership rights are infinitely-lived, they outlive the current generation
and hence the price of the ownership rights incorporates the discounted stream of
pollution flows.
2.1 Saving and investment in equity
Preferences, technology and ecology are the same as before, but shares of firms are
now traded on a stock-market. Shareholders are residual claimants to profits. Consumers provide financial capital (via an intermediary) and invest in equity, which only
yields a return if profits are made. Savings go to an invisible financial intermediary,
12
that supplies loans to the firms. Besides consumers have direct financial interest in
(dirty) firms via equity investment. Only the dirty sector issues equity. Since there is
no externality associated with the clean sector, there is no reason to make a distinction between issuing equity or borrowing (there is no friction); the value of the firm
is equal to the replacement value4 . St is the amount consumers save via a financial
intermediary, qt the amount they invest in equity of the dirty sector at a price Pt .
Consumers therefore buy claims to profits in the dirty sector.
We assume that all firms finance all capital with one-period loans. This implies
D + (1 + s)k C . Here sk C is the sunk cost of investing in clean capital.
that St = kt+1
t+1
t+1
We assume a single rate of return on savings, namely Rt . Firms maximise the value
of the firm which is similar to maximizing (2), as they take the value of the claim
on profits Pt as given. Since the dirty sector faces the same factor costs but does not
have the sunk costs, it can make a profit, which is paid out as dividend Dt+1 . The
rate of return on savings is equal to Rt =
1+rt
1+s
= RtC and dividends (or profits) are
Dt = sRt ktD = (RtD − RtC )ktD , the difference between the return on dirty and the
return on clean capital. Total return on equity equals next period’s dividends Dt+1
plus the price of the claim Pt+1 .
Consumption ct+1 is now equal to:
ct+1 = St Rt+1 + qt (Dt+1 + Pt+1 )
(17)
The analogue of equation (5) is:
D
Et+1 = Et (1 − β) − qt kt+1
+ βE0
(18)
Consumers know that savings via intermediaries do not generate pollution. However,
owning a firm in the dirty sector will not merely pay out dividends but also generate
pollution. Here we forget about “public good” issues and simply view the negative
effect of pollution as some sort of negative environmental dividend. The consumer
maximises utility at time t + 1 by deciding on savings St and buying stocks qt at time
4
Note that we can have any kind of financing structure for the firm, for instance, all-equity financed,
or a structure where half of the firm’s investments are financed through half of the profits, etc... However,
one can show that the financing structure does not matter for firm valuation, e.g. the Modigliani-Miller
theorem holds and therefore we choose the mathematically most convenient financing structure.
13
t:
D
max u(St Rt+1 + qt (Dt+1 + Pt+1 ), Et (1 − β) − qt kt+1
+ βE0 )
(19)
wt = St + qt Pt
(20)
subject to:
So via savings and shareholdings the consumer optimises consumption and environmental externalities. The first-order necessary condition is:
Rt+1 Pt = Dt+1 + Pt+1 −
u′Et+1
u′ct+1
D
kt+1
(21)
which we can write in a more familiar form:
Pt+1 + Dt+1 −
Pt =
u′E
t+1
u′ct+1
D
kt+1
Rt+1
(22)
The intuition of this pricing equation is that the price of a stock should be equal to
the discounted future price plus discounted future dividends minus the discounted
monetary value of pollution. Setting Pt = 0 gives the first order condition as found
in absence of the stock market. This makes sense, since Pt = 0 implies that the total
value of the firm is equal to its replacement value.
The price sequence Pt should be such that the consumer is indifferent between
investing in stock or saving, or: no consumer wishes to change her share in the firm.
Because all consumers are identical, they all either supply or demand stock and only
that equilibrium exists where supply (or demand) is equal to zero.
2.2 Investments
The decision how much dirty and clean capital the firm adopts is not yet clear. There
are infinitely many combinations of P , kC and kD such that the system is in a steady
state equilibrium. This is related to the well-known problem of indeterminacy of
equilibria in OLG-models. An early survey of these problems is given by Woodford
(1984). Equation (21) relates different prices at two points in time, but it does not tell
us what these prices should be. If we know Pt and Pt+1 in equation (21), we implicitly
know how much the young agent is going to invest in clean and dirty capital. The
14
fact that asset prices are not determined gives us infinite combinations of possible
prices and actions.
Woodford (1984) mentions :“There may be an independent role for the beliefs of
agents in the determination of economic outcomes, even if one restricts one’s attention
to equilibria in which the expectations of all agents are correct.” This is exactly what
we are going to model. First of all, if we iteratively substitute equation (22) led in
itself so that the value of the asset Pt becomes equal to the discounted cash flows
minus the discounted pollution flows in monetary terms. We do not simply look at
prices at two points in time but link all future actions to the current value of the asset
and this is also how the agent perceives it; as a sum of discounted cash flows and
weighted pollution flows. We cannot obtain an explicit functional form for Pt . First
of all, in (21) u′ct+1 is also a function of Pt+1 . Furthermore, as mentioned the price
Pt depends on the sequence Et , Et+1 , Et+2 , ... of the state variable environmental
quality, and on all the actions of future generations, since:
D
D
D
Pt = P (Pt+1 , Et , kt+1
) = P (P (Pt+2 , Et+1 , kt+2
), Et , kt+1
) = ...
∞ Dt+τ −
X
Qτ
=
τ =1
u′E
t+τ
u′ct+τ
D
kt+τ
(23)
i=1 Rt+i
As in standard asset pricing theory, equation (23) can be seen as the fundamental
value of the claim on profits, whereas (22) simply describes a condition for indifference, which potentially generates rational price bubbles. We exclude possible price
bubbles by assuming that the price of the claim is equal to the fundamental value.
This equation shows that the value of the asset depends on future valued pollution
flows. The marginal rate of substitution between environmental quality and consumption of all future generations is affected by current investments in dirty capital.
Therefore, a different way of interpreting our approach is saying that the discount
rate is not constant but affected by the state variable environmental quality.
We have to determine how the supply side of the capital market is going to allocate
investments. Each consumer has one share, q = 1, wt = St + Pt , and as the owners
of the firm they will make sure that investments in clean and dirty capital are done
in their own best interest. For instance, they vote on a shareholder meeting on how
15
much to invest in dirty capital or they instruct the financial intermediary on how
to allocate the loans. Given the structure for Pt , they believe that their actions will
influence the future asset price. Furthermore, we assume that the young have all the
bargaining power in the trade of the asset from the old to the young. The young
therefore set the price Pt . However, the young will also consider that when they are
old, they have to buy the asset for a given future price Pt+1 from the newborn young
generation. They therefore take into consideration how their actions influence the
future price Pt+1 . So we have to solve a sequential (Markov) game5 .
We solve this game in the usual way and we assume that each generation takes
all the future actions and thus the future price Pt+1 as given, optimises utility given
D optimally. We will assume that the optithe state variable Et and decides on kt+1
D depends on the state variable environmental quality only so that
mal decision kt+1
D =k D (E ).
kt+1
t
Pt = P (Pt+1 , Et , kD (Et )) = P (Pt+1 , Et )
(24)
Since we assume the future price is taken as given - when the old sell their assets
they have no bargaining power- we can simply write Pt = P (Et ).
The consumer/shareholder wants to maximise utility given prices. We then can
find the optimal amount of clean and dirty capital by taking another look at the
utility maximization problem, now in terms of clean and dirty capital and the value
of the asset P as a function of environmental quality:
max U (ct+1 , Et+1 )
(25)
subject to
D
ct+1 = Rt+1 St + sRt+1 kt+1
+ P (Et+1 )
(26)
D
Et+1 = (1 − β)Et − kt+1
+ βE0
(27)
wt = St + P (Et )
(28)
D
C
St = kt+1
+ (1 + s)kt+1
(29)
5
Technically, instead of letting the dynamic system depend on exogenous initial values, we determine
an endogenous end-value, namely the steady state value, through the sequential game approach. This is
different from other studies where exogenous initial values determine the outcome of the equilibrium path.
16
C and k D yields the first-order condition:
Optimization with respect to kt+1
t+1
sRt+1 −
′
∂P (Et+1 ) uEt+1
− ′
=0
∂Et+1
uct+1
(30)
We need to know how the price develops in the future, that is we need to know
∂P (Et+1 )
∂Et+1
to get an explicit solution to the problem. The previous generation faced
the same problem, so for the current generation we can find
∂P (Et )
∂Et .
We optimise
utility with respect to Et , since the price P (Et ) must adapt such that the equity
market is in equilibrium:
(1 − β)
u′E
∂P (Et+1 )
∂P (Et )
+ (1 − β) ′ t+1 = Rt+1
∂Et+1
uct+1
∂Et
(31)
where we have used the envelope theorem. Combining (30) and (31), we find that:
∂P (Et )
= s(1 − β)
∂Et
(32)
Evaluating (32) at t + 1 we can substitute this expression in (30) to find
u′Et+1
u′ct+1
= s(β + Rt+1 − 1)
(33)
The stock market takes into account that the environment recuperates with rate β,
instead of treating environmental quality as a flow. Also, if we impose the dynamic
efficient rental rate r = s, this implies R = 1 and from the stock market steady state
it then follows that
u′E
u′c
= sβ; the second condition for dynamic optimality. This shows
that the stock market is able to solve the problem of over- or under accumulation of
environmental quality. However, if R approaches the value of one, it will not result in
the dynamic efficient solution. In this case, investing is merely a storing technology,
and the only steady state equilibrium is one in which there is huge inflation for the
asset prices, since the real interest rate becomes very low. If one wants to rule out
inflation, the only other equilibrium is the so-called monetary equilibrium in which
there is no dirty capital and the asset functions as fiat money. We therefore rule out
R = 1; the fact that this is dynamically efficient is because consumers have no time
preference in our model. For realistic economies we have R > 1.
17
2.3 Steady state and dynamics
We will now derive the steady state equilibrium and dynamic properties of the stock
market economy. First we express equations (21), (26) and (27) in terms of kt , Et
and Pt and rewrite:
Pt+1 = Pt (β + Rt+1 − 1) − (1 − β)[(1 + s)kt+1 − wt ]
(34)
ct+1 = Rt+1 (1 + s)kt+1 + Pt+1
(1 + s)[kt+1 + Pt − wt ]
= Et (1 − β) −
+ βE0
s
(35)
Et+1
(36)
where we have used the budget constraint and the fact that in an equilibrium path
u′E
t+1
u′ct+1
= s(β + Rt+1 − 1) should hold. These three equations can be reduced to two
difference equations since the utility function combines (35) and Et+1 through the
equation
u′E
t+1
u′ct+1
= s(β +Rt+1 −1); this implicitly defines Pt+1 as a function of kt+1 and
Et+1 . Combined with the difference equations (34) and (36) this describes the whole
system. Without making additional assumptions on the utility function we cannot
say much about the dynamics of this system. For ease of demonstration we assume
homogeneous utility of the degree one, such that
u′E
t
u′ct
=
ct
γEt .
Then (35) combined
with the consumer’s first order condition becomes:
Pt+1 = γs(β + Rt+1 − 1)Et+1 − kt+1 (1 + s)Rt+1
(37)
We can then substitute (37) and (37) lagged in (34) and in (36) and rewrite:
1+s
1+s
1−β
kt+1 = γ(β + Rt − 1)Et − kt Rt
+ wt
s
s
s(β + Rt+1 − 1)
1+s
1 + s wt
Et+1 +
kt+1 = [(1 − β) − γ(β + Rt − 1)]Et + kt Rt
+
+ βE0
s
s
s
γEt+1 −
(38)
(39)
This is an implicitly defined, non-linear system of difference equations, with possible
multiple steady states and complex dynamic behaviour. It is possible to derive the
steady states and dynamic properties around the steady states of this system, however
without making additional assumptions on the production technology this is not
straightforward (See for a more general discussion, Zhang, 1999). Assuming a CobbDouglas production technology, for example, can already cause multiple steady state
equilibria, periodically stable solutions and even chaos. One must then rule out the
18
economic implausible steady states and for instance assume that periodic solutions
are not very realistic.
All of these issues can be overcome and it is technically possible to compute and
characterise the dynamics for such a production function, however we take a different
approach and propose a linear production function:
f (kt ) = [(1 + ρ)(1 + s) − (1 − δ)]kt + ω
(40)
This facilitates the demonstration, since the implicit non-linear system then becomes
a linear system. The drawback is that it is not very realistic that labour and capital
are perfectly substitutable. However, we argue that for this exercise it is not of great
significance. First of all, we have already shown that for a general production function
the stock-market can offset the negative externalities that are associated with shortlived consumers. Second, a linear production function implies constant interest rates
and constant wages. In our model, labour is inelastically supplied with respect to
the wage rate as are savings with respect to the interest rate. Therefore, we already
implicitly assumed that the supply side of the labour market and the supply side of
the capital market view the wage rate and the interest rate as a constant. Assuming
this explicitly in a production function is therefore not such a huge price to pay as
it initially seems. Furthermore, the trade-off between dirty and clean capital is made
with respect to the difference in returns and not the level of returns. So, with a linear
production function we maintain the important ingredients of the model, namely the
stock effect of pollution, the trade-off between environmental quality and returns,
and the forward looking equity market.
Assuming the production function above, we have that Rt − 1 = ρ and wt = ω
for all t. We substitute (38) in (39) twice to find the new system of linear differential
equations:
ω(1 + ρ)
+ βE0 (41)
s(ρ + β)
sγ[(1 − β) − (1 + γ)(ρ + β)]
γ(ρ + β)(βsE0 + ω) − (1 − β)ω
=
Et + (1 + ρ)kt +
(42)
(1 + s)(1 + γ)
(ρ + β)(1 + s)(1 + γ)
(1 + γ)Et+1 = Et (1 − β) +
kt+1
19
E0
A
Et=Et+1
SP
kt=kt+1
k
Figure 3: Stock market steady state equilibrium
The steady state solution is given by
ω(1 + ρ)
βE0
+
s(ρ + β)(γ + β) γ + β
(ρ + β − 1)βγsE0
γρ(ρ + β) + β(1 − β)
k̄ =
+
ω
(γ + β)(1 + s)ρ
ρ(1 + s)(ρ + β)(γ + β)
Ē =
and in the steady state the equity price is equal to P̄ =
eigenvalues of the associated matrix are equal to
1−β
1+γ
β(1−β)[ω(1+ρ)−E0 γ(ρ+β)]
.
ρ(ρ+β)(γ+β)
(43)
(44)
The
< 1 and 1 + ρ > 1, so we have a
stable and an unstable root. It means that the steady state is saddle-point stable and
an unique saddle-path exists that leads to the steady state. The phase diagram of
the system is depicted in Figure 3. Again, along the horizontal axes, we have capital
k and along the vertical axes environmental quality E. We have depicted the steady
state solutions of equation (41) and (42) by the Et = Et+1 -line and kt = kt+1 -line.
The steady state solution is in point A. Note that we can in principle also have that
the kt = kt+1 -line is upward sloping or even have a single value for k for every E, e.g.
20
a vertical line. The dynamic forces are depicted by arrows and indicate that there is
a unique saddle point path (dotted line labelled SP) leading to the steady state A.
Point A gives higher utility in the steady state compared to the case where there is
no stock-market, however it does not achieve the dynamic efficient steady state.
In the stock market steady state equilibrium, the replacement value of dirty capital is lower than the market value, as P > 0. Since investing in dirty capital is less
attractive in the stock market equilibrium, more resources will be allocated to the
clean sector, which increases environmental quality. Note that the fact that market
value exceeds replacement value does not imply that we have a price bubble. As
Tirole (1985) pointed out, a price bubble cannot be negative. If we assume steady
state and simply discount the cash-flows without taking the pollution flows into account, we get a value s(ρ + 1)kD /ρ which is higher compared to the market value
s(1 − β)kD /ρ in our stock market equilibrium, implying that we are not dealing with
a price bubble.
3
Conclusions
One of the key issues in achieving sustainable development is managing the impact
of economic activity on the environment. In the last decade, corporations have increasingly put sustainable development on their agendas, creating the possibility of
socially responsible investment. This paper shows that the stock market can play a
role in achieving sustainable development.
We analyse this in an Diamond-type overlapping generations model with shortlived consumers that care about environmental quality, comparable to John and
Pecchenino (1994). A lack of coordination between old and young agents likely leads
to over accumulation of pollution. We show that introducing an equity market that
allows for trade of property rights can generate Pareto improvements, although the
social optimum will not be reached. In a stock market equilibrium, the market value
of a firm using polluting production is higher than the replacement costs of real
capital. Consequently, investments are relatively more attractive in the clean sector
compared to the case without the stock market, which reduces pollution. The intu-
21
ition is straightforward: since the stock market is forward looking, equity allows for
trade in future valued capital, incorporating the welfare loss of pollution of future
generations.
We also address the standard problem of indeterminacy of asset prices in overlapping generations models. If we consider an asset price to be the sum of discounted cash
flows and pollution flows, we can define a sequential game and regard the investment
decisions as Markov state decisions. This way we make the asset price endogenous
and determine a unique steady state equilibrium condition.
Finally, in this paper we have focused on a specific externality, namely the intergenerational problems associated with short-lived agents and a long-lived public
good. However, the idea that proper firm valuation can incorporate negative externalities can be generalised.
22
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