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Stochastic continuous timegrowth models that allowfor closed form solutions
Francesco MenoncinStefano Nembrini
Department of Economics and Management
University of Brescia
Italy
WORKING PAPER
Via S. Faustino 74/b, 25122 Brescia – Italy
Tel 00390302988742 Fax 00390302988703
email: [email protected]
WPDEM 4/2016
Stochastic continuous time growth models that allow for closed
form solutions
Francesco Menoncin (1) Stefano Nembrini (2)
(1) Department of Economics and Management, University of Brescia, Italy, [email protected]
(2) Department of Economics and Management, University of Brescia, Italy, [email protected]
Abstract
We find a closed form solution that maximises the expected utility of an agent’s inter-temporal consumption sub-
ject to a stochastic technology, which is a linear combination of AK and Cobb–Douglas technologies. Additionally,
we consider two cases of agent preferences: (i) Constant Relative Risk Aversion (CRRA) preferences, which treat
optimal consumption as a linear function of capital, and (ii) Hyperbolic Absolute Risk Aversion (HARA) preferences,
which treat optimal consumption as an affine function of capital. By establishing a minimum (subsistence) level of
consumption in the HARA model, we are able to create a framework that more accurately represents real-world cir-
cumstances than previous studies have done. Furthermore, for both the CRRA and HARA cases we show the suitable,
consistent stochastic differential equation which describes the capital dynamics. Finally, we perform a numerical sim-
ulation based on the CRRA case and calibrate US data for the HARA case.
Keywords: Dynamic stochastic general equilibrium models; Closed-form solution; HARA preferences; CRRA pref-
erences; Autonomous consumption
JEL classification: E2; O4
1
1 Introduction
In this study we examine the problem that a representative agent faces in maximising the expected present utility of
his/her inter-temporal consumption, subject to a technological constraint on the accumulation of capital. We assume
that technology is a linear combination of AK and Cobb–Douglas models, whose parameters (which measure total
factor productivity, TFP) are allowed to be stochastic. In fact, all of the risk in the economy is captured by a set of
stochastic variables that affect other relevant economic factors, such as TFP, the depreciation rate and volatility of
capital, the size and frequency of the jumps in capital, and the representative agent’s subjective discount rate. Here, we
are dealing with models that allow for a positive long-run growth of capital, and we show this feature by computing
the capital accumulation implied by the optimal consumption strategy (that we are able to find in closed form). Since
some parameters of the production function (measuring the productivity) are allowed to be stochastic, then the optimal
capital accumulation in the long-run could be either enhanced or reduced.
The agent’s behaviour may be described by one of two families of preferences: Constant Relative Risk Aversion
(CRRA) or Hyperbolic Absolute Risk Aversion (HARA). Whereas the subsistence level of consumption is impli-
citly set at zero under CRRA preferences, it is allowed to take on different values under HARA preferences. Thus,
HARA preferences allow for a much wider variety of behavioural descriptions. Additionally, in the case of CRRA,
optimal consumption is a linear transformation of capital, while in the case of HARA it is an affine transformation.
Consequently, HARA preferences are able to account for so-called autonomous (or exogenous) consumption, which
empirical research has found to be significantly different than zero (either positive, as in Weinbaum [2005], Strulik
[2010], Achury et al. [2012], or negative, as in Holt and Laury [2002], Eisenhauer and Ventura [2003], Levaggi and
Menoncin [2013]).
In both cases, we are able to find quasi-explicit solutions for optimal consumption; in the CRRA case, it can be
found in a fully stochastic framework, while in the HARA case it can only be achieved if a given combination of some
of the production function’s parameters is deterministic. Furthermore, both the production function and the stochastic
differential equation for capital accumulation must behave in consistently unique manners in both settings. Moreover,
when autonomous consumption is allowed to be non-zero (under HARA preferences), capital must also have a non-
zero, autonomous level.
This study is inspired by Wälde [2011], who presents stochastic models that yield closed-form solutions for op-
timal consumption. Here, we expand his model in three new directions: (i) under CRRA preferences and AK production
function, we find a closed form solution for optimal consumption allowing for the presence of any number of stochastic
state variables (this allows us to take a wide variety of economic frameworks into account), (ii) under CRRA prefer-
ences we can generalize the production function to a Cobb-Douglas type by obtaining a closed form solution at the cost
of a restriction on parameters: the relative risk aversion parameter must coincide with the output elasticity of capital
(the same assumption made in Feicht and Stummer [2010]), (iii) under HARA preferences, we find a closed form
solution for optimal consumption which is an affine transformation of capital; this time we have to impose a restriction
2
both on the subsistence consumption level and on the shape of the production function.
In order to obtain the solutions to the stochastic differential equations for both forms of preferences, we rely on the
“representation theorem”, which was first proposed by Feynman and Kac and thereafter commonly used in financial
economic literature (Cont and Tankov [2004], Øksendal and Sulem [2007]). In fact, our optimal consumption solution
is written as an expected value that can easily be computed with numerical (Monte Carlo) simulations.
The rest of this study is organised as follows: In Section 2, we present the CRRA preferences case. The subsections
show how we model the stochastic economic environment and the agent’s preferences, as well as how we solve the
dynamic optimal consumption problem. Furthermore, we conduct a numerical simulation to compute the optimal level
of consumption under a simplified AK framework, where factor productivity (A) follows a mean reverting stochastic
process. In Section 3, we solve the optimal consumption problem subject to HARA preferences and, in contrast to the
previous case, we employ a different production function and stochastic differential equation to describe the dynamic
behaviour of capital. We further create two different specifications of this model in order to conduct an empirical
estimation with US consumption and GDP data. Our results indicate that consumption may be effectively modelled as
an affine function of capital. The last section of this study presents our conclusions.
2 CRRA preferences
2.1 The macroeconomic environment
Let us assume that the macroeconomic environment is described by a set of s stochastic state variables, zt , which solve
the following matrix stochastic differential equation (SDE):
dzts×1
= µt (zt)s×1
dt +Ωt (zt)′
s×kdWtk×1
, (1)
where Wt is a set of k independent Wiener processes with means equal to zero and t variances (the prime denotes
transposition).1
Per-capita GDP (Yt (Kt ,zt)) is a function of per-capita capital (Kt ) and, potentially, the state variables (zt ). Further-
more, we assume that the following technology function determines GDP ([Barro and Sala-i Martin, 1999, Chapter
4]):
Yt (Kt ,zt) = At (zt)Kt +Bt (zt)Kγ
t , (2)
where At (zt) and Bt (zt) are (stochastic) variables which measure total factor productivity (TFP). The (positive) para-
meter γ measures non-linearities in the production function. Equation (2) combines two of the most commonly
used production functions, the so-called AK production function (Bt = 0) and the Cobb–Douglas production func-
1The assumption of independence is made without any loss of generality because any set of independent Wiener processes can be transformedinto dependent Wiener processes (and vice versa) by applying the Cholesky decomposition to their variance-covariance matrix.
3
tion (At = 0).
The notation At (zt), Bt (zt) indicates that both functions depend on the state variables zt , but they could also be state
variables themselves. If, for instance, we assume to have two state variables z1,t and z2,t , then we can write At = z1,t
and Bt = z2.t , which is fully compatible with the previous notation.
Remark 1. By combining Equations (1) and (2), we are also able to account for a stochastic labour variable; specifically,
the number of employees (Lt ∈ zt ) can be included in the variable Bt . In fact, a model where the production function is
given as
Yt (Kt ,zt) = At (zt)Kt +bt (zt)Lt (zt)1−γ Kγ
t ,
is merely a particular case of (2) where Bt (zt) = bt (zt)Lt (zt)1−γ (Jones and Manuelli [1990]). This model can be
expressed in terms of per-worker production and capital as follows
Yt (Kt ,zt)
Lt (zt)= At (zt)
Kt
Lt (zt)+bt (zt)
(Kt
Lt (zt)
)γ
.
The amount of GDP which is neither consumed (ct ) nor lost through depreciation (the rate of which is δt (zt)) is
invested (i.e. it coincides with the change in capital). Thus, we may describe capital accumulation with the following
stochastic differential equation:
dKt = (Yt (Kt ,zt)−δt (zt)Kt − ct)︸ ︷︷ ︸investment
dt (3)
+Ktσt (zt)′
1×kdWtk×1
+βt (zt)KtdΠt ,
where σt (zt) is the instantaneous volatility of capital (proportional to the capital level), Πt is a Poisson process whose
(stochastic) intensity (λt (zt)) measures the frequency of jumps, and βt (zt) represents the (stochastic) width of the
jumps (which are also proportional to the capital level). Sennewald and Wälde [2006], Sennewald [2007] provide an
analysis of the mathematical properties and role of jumps in stochastic economic environment models. In regards to
Equation (3), we make the following assumptions:
• the initial amount of capital (Kt0 = K0) is known (deterministic)
• the Poisson process is independent of the Wiener processes
• βt (zt)>−1, this precludes the occurrence of any catastrophic events which would reduce capital to zero. As it
is more likely that the jump component (dΠt ) would negatively affect capital accumulation (in the case of heavy
capital losses), it follows that the size of the jump (βt (zt)) would most likely be negative.
Here, we have used the same risk sources (i.e. Wiener processes) for describing the stochastic components of both
capital (Kt ) and state variables (zt ). This framework does not loose any generality since positive, negative and nil
4
covariance between Kt and zt can all be taken into account through the signs of the elements of matrices Ωt and σt . In
particular, we have
Ct [dzt ,dKt ] = Et [dztdKt ]−Et [dzt ]Et [dKt ] ,
and since dWt and dΠt are assumed to be independent, then Et [dzt ]Et [dKt ] = 0. Thus, we have
Ct [dzt ,dKt ] = Et[(
Ω′tdWt
)(Ktσ
′t dWt
)]= Et
[(Ω′tdWt
)(dW ′t σtKt
)]= Ω
′tEt[dWtdW ′t
]σtKt
= Ω′t (Idt)σtKt = Ω
′tσtKtdt, (4)
where I is the identity matrix of a suitable dimension (and we have used the property that the expected value of the
square of a Wiener differential is dt). If the product Ω′tσt contains only zeros, then the capital and the state variables
are not correlated.
A wide variety of economic environments may be described as special cases of this model. Let us assume we want
to take into account two state variables (zt =[
z1,t z2,t
]′):
1. z1,t which solves
dz1,t = α1 (β1− z1,t)dt +σ1√
z1,tdW1,t ,
thus following a mean-reverting square root-process (converging towards an equilibrium value of β1 at a speed
of α1)
2. z2,t which solves
dz2,t
z2,t= µ2dt +σ2dW2,t +σ2
√√√√ ρ21,2
1−ρ21,2
dW2,t ,
thus following a geometric Brownian motion, where ρ1,2 is the correlation index between z1,t and z2,t and dW1,t
and dW2,t are independent.
Given these variables, we can assume δt (zt) = z1,t and At (zt) = z2,t . Thus, the depreciation rate δt is reverting towards
a constant mean, while the production parameter At , on average, is increasing over time. Furthermore, the shocks on
the two functions At and δt are correlated (through the correlation parameter ρ1,2).
Many other specifications are, of course, possible.
5
2.2 Agent’s preferences
The representative agent is assumed to have CRRA preferences, and as such, they may be described by the following
utility function:
U (ct) =c1−γ
t −11− γ
, (5)
where γ > 0 is the Arrow–Pratt relative risk aversion parameter. The limit of (5) for γ→ 1 coincides with the log-utility
function.
Remark 2. As in Feicht and Stummer [2010], in (5) we use the same γ in the non-linear component of the production
function; this allows us to reach a closed form solution. When γ = 1, the production function (2) collapses to a simple
AK model. Since the output elasticity of capital γ in (2) is estimated to be lower than 1, we are also assuming that the
risk aversion is lower than 1. In empirical evidence the risk aversion is usually found to take values between 0 and 4.
For instance, Arrondel et al. [2010], in stock markets, find that about one third of the consumers/investors participating
to their experiment showed a risk aversion lower than 1.
We take into account the problem of an agent who wants to maximise the utility of his/her intertemporal con-
sumption until his/her stochastic death time τ (defined on the set [t0,+∞[). Thus, the optimisation problem with a
deterministic subjective discount rate (ρ t ) can be written as:
maxcs
Et0
[ˆτ
t0e−´ s
t0ρudu c1−γ
s −11− γ
ds
],
where Et0 [•] is the expected value based on the information set available at time t0. If λt (zt) is the (stochastic)
instantaneous force of mortality, then the survival probability from t through T can be written as:
Prob(τ ≥ T |τ ≥ t) = Et
[e−´ T
t λs(zs)ds]= Et [IT≤τ ] ,
where IT≤τ is the indicator function whose value is 1 if T ≤ τ and 0 otherwise. Accordingly, the previous problem can
be rewritten as:
maxcs
Et0
[ˆ∞
t0Is<τ e−
´ st0
ρudu c1−γs −11− γ
ds
]
= maxcs
Et0
[ˆ∞
t0e−´ s
t0(ρu+λu(zu))du c1−γ
s −11− γ
ds
]
= maxcs
Et0
[ˆ∞
t0e−´ s
t0ρu(zu)du c1−γ
s −11− γ
ds
], (6)
where ρu (zu)≡ ρu +λu (zu).
Our framework is akin to that studied in Hiraguchi [2013] where also a human capital is taken into account (and the
6
same restriction as in Remark 2 is assumed). Nevertheless, the main differences with Hiraguchi [2013] are the following
ones: (i) we allow for a set of any (finite) number of state variables affecting both consumer’s survival probability and
capital accumulation (in Hiraguchi [2013] only At is stochastic and follows a simple geometric Brownian motion), (ii)
we allow for capital to have a stochastic component of its own (σt in Equation (3)) which could be stochastic itself,
(iii) we allow for sudden jumps in the capital (due to crises or depreciations) through a Poisson process in (3).
2.3 Optimal consumption
Problem (6) can be solved through dynamic programming. The resulting optimal consumption is shown in the follow-
ing proposition.
Proposition 1. The consumption level which optimises Problem (6) in the macroeconomic environment described in
Equations (1)–(3) can be written as:
c∗tKt
= Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
, (7)
where: (i) the expected value Eγ
t is computed with respect to a new probability measure under which the Wiener
processes dWt are written as follows:
dWt = (1− γ)σt (zt)dt +dW γ
t , (8)
(ii) the dynamics of the state variables zt is accordingly rewritten as
dzt =(µt (zt)+(1− γ)Ωt (zt)
′σt (zt)
)dt +Ωt (zt)
′ dW γ
t ,
and (iii)
φt (zt)≡ρt +λt +(γ−1)(At −δt)− 1
2 (γ−1)γσ ′t σt −λt (1+βt)1−γ
γ. (9)
Proof. See Appendix A.
The expected value in (7) may be easily computed with Monte Carlo simulations. In fact, it is far easier to calculate
a solution with this method than it is to arrive at one by numerically solving the partial differential equation.
At least to our knowledge, this is the first time a change in probability, as in (8), is used in a macroeconomic
framework.2 It implies that the optimal consumption is computed by solving (or simulating) an expected value after
correcting the dynamics of the state variables by the consumer’s risk aversion parameter γ .
Furthermore, since the conditional expected value in (7) is a stochastic variable itself (it depends on the values of
zt that are stochastic variables), then Proposition 1 allows us to conclude that the ratio between optimal consumption2The literature about the application of change in probability (through Girsanov’s theorem) in finance is well developed. Some examples can be
found, for instance, in Øksendal [2000], Björk [2009].
7
and capital is stochastic over time.
We stress that, with respect to Wälde [2011], here we just need one parameter restriction to find the closed form
solution (7): the risk aversion must coincide with the output elasticity of capital. No other restrictive assumptions on
the macroeconomic framework is required. In particular there can be any number of state variables, that follow any
stochastic process admitting a strong solution. The cost of a restriction on the parameter γ must be paid in order to be
able to enrich the production function and depart from the usual AK approach, which has been proved to be poor in
describing some empirical facts.
Under CRRA preferences, optimal consumption is a linear function of capital, which implies that the autonomous
consumption component would be equal to zero; however, many empirical studies have found evidence contradicting
this assertion. In particular, in Weinbaum [2005], Strulik [2010], Achury et al. [2012] the autonomous consumption
component is found to be positive and in Holt and Laury [2002], Eisenhauer and Ventura [2003], Levaggi and Menoncin
[2013] it is found to be negative.
In the case where there are no state variables (i.e. µt = 0 and Ωt = 0), φt (zt) in (9) is constant and the following
result holds:
Corollary 1. The consumption level which optimises Problem (6) in the macroeconomic environment described by
Equations (1)–(3) with µt = 0 and Ωt = 0 is:c∗tKt
= φ , (10)
where φ is given by (9) when all of the variables are constant.
The results shown in Corollary 1 allow us to draw the following conclusions (for 0 < γ < 1):
• Optimal consumption is positively related to the subjective discount rate (ρ): as the agent becomes ever more
impatient (i.e. ρ increases), his/her optimal consumption (as a fraction of capital) increases, in fact, the accumu-
lation of future capital becomes less important.
• On the other hand, optimal consumption is negatively related to TFP, which is reduced by the depreciation rate
(A−δ ): if capital is accumulated at a greater rate, then a constant level of consumption yields a reduction in the
ratio c∗t /Kt . Thus, a kind of revenue effect prevails (note, though, that if γ > 1, a substitution effect prevails).
• Capital volatility (σ ′σ > 0) and optimal consumption are positively related: if the accumulation of capital be-
comes more volatile, then the agent will increase his/her consumption in order to reduce the accumulated capital
and thus compensate for the greater uncertainty (in fact, total capital volatility is proportional to the level of Kt ).
• Optimal consumption is negatively related to jump size (β ): if the level of consumption is assumed to be constant
and the accumulation of capital has negative (positive) jumps, then the ratio c∗t /Kt will increase (decrease).
• The effect of the frequency of jumps (λ ) on optimal consumption (10), depends on the sign of β . In fact, the
8
derivative of φ with respect to λ is:∂φ
∂λ=
1γ
(1− (1+β )1−γ
),
and thus∂φ
∂λR 0⇐⇒ β Q 0.
Consequently, we can conclude that when the jumps are negative, optimal consumption is positively influenced
by their frequency: if sudden decreases in capital accumulation occur frequently, then consumption increases
(due to the aforementioned revenue effect).
• We are unable to draw concrete conclusions regarding the effects of risk aversion on consumption; in fact, γ
affects the components of the ratio φ differently. If there are no jumps (i.e. λt = 0), we may easily show that:
∂φ
∂γR 0⇐⇒ γ
2 S 2A−δ −ρ
σ ′σ,
indicating that there is a risk aversion threshold below which optimal consumption is a positive function of risk
aversion and above which it is negative function. We know that A−δ is the net accumulation rate of capital, ρ is
the discount rate and σ ′σ is the volatility of capital. If the capital is highly volatile, then consumption will more
likely be decreasing in risk aversion. The intuition is that a very risk averse consumer will try to keep a high level
of capital (by reducing consumption) in order to face the loss that capital could suffer because of the increased
volatility. Furthermore, if the capital accumulated at a very speed rate (i.e. A− δ −ρ is high) it is more likely
that the consumption is increasing in risk aversion. In this case, in fact, the consumer can rely on the growth rate
of capital and finance an increasing consumption even if he is more risk averse.
We can obtain a corollary for the log-utility case by setting γ = 1 (making the production function an AK function).
Corollary 2. The consumption level which optimises Problem (6) in the log-utility case (i.e. where γ = 1) and the
macroeconomic environment described in Equations (1)–(3) is:
c∗tKt
=1
Et
[´∞
t e−´ s
t ρu(zu)duds] , (11)
and by factoring in a constant subjective discount rate (ρt (zt) = ρ) we are able to obtain:
c∗tKt
= ρ.
If we include a stochastic death time in the framework (with ρt (zt) = ρ t + µt (zt)), then the denominator in (11)
9
would equate to the subjective value of an annuity. In fact, we can write:
Et
[ˆ∞
te−´ s
t ρu(zu)duds]
= Et
[ˆ∞
te−´ s
t ρudue−´ s
t µu(zu)duds]
= Et
[ˆ∞
tIs<τ e−
´ st ρududs
]= Et
[ˆτ
te−´ s
t ρududs],
which is the value of an annuity that pays 1 monetary unit every instant until the time of death, τ .
2.4 A numerical simulation
In this subsection we establish a simple framework for examining the behaviour of optimal consumption (7). We
assume that:
• There are no jumps (i.e. λt = 0 or, alternatively, βt = 0)
• The subjective discount rate is constant (ρt = 0.05)
• The production function is given by AK (i.e. Bt = 0), and the parameter γ can take on values greater than 1
because it only measures risk aversion without affecting the production function
• TFP is the only state variable and it follows a mean-reverting square-root process:
dAt = α(A−At
)dt +σA
√AtdWA,t , (12)
wherein the speed of adjustment is α = 0.5, the volatility parameter is σA = 0.01, and the long-term equilibrium
is A = 0.5; we further assume that the process starts at its equilibrium value (i.e. A0 = A)
• The depreciation rate is zero (δt = 0)
• Capital accumulation can be written as
dKt = (AtKt − ct)dt +KtσKdWK,t , (13)
where σK = 0.2 is constant and WK,t is independent of WA,t (i.e. the covariance between TFP and capital shocks
is zero).
While the parameter values have been arbitrarily chosen here, any values may be selected without affecting the quality
of the results.
10
Remark 3. To our knowledge, no other studies have developed a model that allows a state variable (like TFP) to
behave in the manner we establish in Equation (12). Consequently, we assert that our approach constitutes a significant
contribution to the literature as it yields quasi-explicit consumption optimisation solutions for a very wide range of
economic frameworks, which may include any (finite) number of state variables (that may be correlated with the
capital-accumulation-driving risk source).
Under the assumption that the two risk sources, WK,t and WA,t , are independent (which we make merely for the
sake of simplicity as the results in the previous section do not rely on this condition), the new stochastic differential
equation (which is followed by zt in (32)) is the same as the original process (1) (in fact, Ωt (zt)′σt (zt) = 0); in other
words, there is no need to change the probability. Accordingly, the optimal consumption can be given by:
c∗tKt
= Et
[(ˆ∞
te−(
ρ
γ+ 1
2 (1−γ)σ2K
)(s−t)+ 1−γ
γ
´ st Aududs
)γ]− 1γ
.
Remark 4. Since the process for At is mean reverting towards the constant level A, a good approximation of the optimal
consumption could be obtained by letting At ' A for any t. In this case the previous result becomes
c∗tKt
=ρ
γ+
12(1− γ)σ
2K−
1− γ
γA,
which makes sense (and the integral converges) only if c∗tKt
> 0. This conditions holds for any value of γ such that
12+
A−√( 1
2 σ2K +A
)2−2σ2K
(A−ρ
)σ2
K< γ <
12+
A+
√( 12 σ2
K +A)2−2σ2
K
(A−ρ
)σ2
K, (14)
and in both extremes of the interval the optimal consumption tends towards zero.
This expected value cannot be computed in a fully closed form, but we can easily perform a Monte Carlo simulation.
As shown in Figure 1, we are able to compute optimal consumption (c∗t /Kt ) for different levels of risk aversion (γ)
with simulations. We are able to confirm that optimal consumption is a positive (negative) function of risk aversion
when the level of risk aversion is low (high) (as described in the previous section). As risk aversion approaches the
boundaries of interval (14), optimal consumption approaches zero. This highlights one of the central problems posed
by these models; that is, there is no minimum threshold on consumption. Though the model is able to guarantee that
optimal consumption will always be positive (for γ in (14)), it may still take on a value so low that the agent would be
unable to survive. In the next section, we introduce a minimum consumption threshold and explore its effects on our
results.
11
0 5 10 15 20 25
0.00
0.05
0.10
0.15
0.20
0.25
0.30
gamma
Figure 1: Behaviour of optimal consumption ( c∗tKt
) (on the ordinates) with respect to risk aversion γ (on the abscissas),with TFP and capital following (12) and (13), respectively, and λt = δt = Bt = 0, ρt = 0.05. With these values, theinterval in (14) is (0.8962816,25.1037184).
12
2.5 Optimal capital accumulation
Once the optimal consumption is plugged into (3), the differential equation for capital accumulation becomes:
dK∗tK∗t
=
(At (zt)−δt (zt))−Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
dt
+σt (zt)′ dWt +βt (zt)dΠt ,
which implies that capital at any future date (T > t0) can be given by:
K∗T = K0e
´ Tt0
(At (zt )−δt (zt )−Eγ
t
[(´∞
t e−´ st φu(zu)duds
)γ]− 1γ − 1
2 σt (zt )′σt (zt )
)dt
(15)
×e´ T
t0σt (zt )
′dWt+´ T
t0λt ln(1+βt (zt ))dΠt .
Thus, we are able to draw the following conclusions:
Proposition 2. If the initial level of capital is strictly positive (K0 > 0), then the optimal amount (15) will be positive
as well.
The expected rate of capital increase is:
Et
[dK∗tK∗t
]=
At (zt)−δt (zt)−Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
+βt (zt)λt (zt)
dt,
which indicates that capital will increase on average if and only if
At (zt)−δt (zt)+βt (zt)λt (zt)> Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
.
In other words, capital will grow on average if TFP is sufficiently high, despite being reduced by both depreciation and
negative jumps (−βt (zt)) that are weighted by their frequency (λt (zt)).
We may rewrite this last condition as follows:
Proposition 3. The total amount of capital will increase on average over time if and only if
Et
[dK∗tK∗t− c∗t
K∗tdt]> Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
dt.
13
3 HARA preferences
3.1 Agent’s preferences
In this section, we compute a closed form solution for optimal consumption for an agent with HARA preferences. In
this case, the utility function can be written as:
U (ct) =(ct − c)1−γ
1− γ, (16)
where c is a constant parameter, and the Arrow–Pratt absolute risk aversion index is given by:
−∂ 2U(ct )
∂c2t
∂U(ct )∂ct
=γ
ct − c,
which is a hyperbolic function of consumption. The sign of c plays a crucial role in determining the agent’s preferences,
such that:
• If c > 0, the utility function (16) belongs to the decreasing relative risk aversion (DRRA) family and c can be
interpreted as the subsistence level of consumption. In this case, marginal utility approaches infinity as ct → c,
and as such, the optimal level will never be ct = c and will always be greater than c (see Sethi et al. [1992],
Weinbaum [2005], and Achury et al. [2012] for descriptions of the role of subsistence consumption in portfolio
choice as well as Strulik [2010] for its role in economic growth models).
• If c < 0, the utility function (16) belongs to the Increasing Relative Risk Aversion (IRRA) family.3 In this case,
c can no longer be interpreted as the subsistence level of consumption, though it can now capture the behaviour
of an economic agent who becomes ever more conservative as his/her consumption increases. Holt and Laury
[2002], Eisenhauer and Ventura [2003], and Levaggi and Menoncin [2013] find evidence in support of increasing
relative risk aversion.
• If c = 0, the utility function (16) belongs to the CRRA family, which we examined in the previous section.
Given the utility function in (16), the Arrow–Pratt risk aversion index increases (i.e. a consumer becomes more risk
averse) when either γ or c increase. In other words, a consumer with a low γ and consumption level close to c, behaves
exactly like one with a high γ and a consumption level farther from c.
3The same attitude towards risk can be captured by an exponential utility function of the form U (ct) =−e−γct , which can be viewed as the limit
of a HARA utility function: limω→+∞
(ωγ
( ctω+1)
1−ωγ
1−ωγ
)=−e−γct , where γ is the (constant) absolute risk aversion index (which yields an increasing
aversion to relative risk). The quadratic utility function, which is commonly used in financial applications, also belongs to the IRRA family.
14
3.2 Production function
By establishing a minimum level of consumption, we preclude the possibility of capital falling to zero. In fact, a
certain level of production is necessary to support subsistence consumption. In particular, a minimum amount of
capital is needed to finance this consumption as well as depreciation.
We include a modified version of the production function (2) in this framework. In particular, we assume that the
non-linearity of capital starts affecting the production function at any level greater than K (the minimum amount of
capital for subsistence level consumption) according to the following functional form:
Yt = At (zt)Kt +Bt (zt)(Kt −K
)γ+Dt (zt) , (17)
where the term Dt (zt) corrects the linear component and can compensate for part of the distortion caused by introducing
the minimum capital, K.
When capital reaches its minimum level, the amount of production (Yt ) must be able to finance both the minimum
level of consumption as well as capital depreciation so that capital does not fall below K; algebraically:
Yt(K)= c+δt (zt)K,
and, by using (17) we obtain
K =c−Dt (zt)
At (zt)−δt (zt), (18)
which indicates that there is a one-to-one relationship between the minimum amount of capital and the minimum
amount of consumption.
Remark 5. The functional form (17) is the only one compatible with the existence of an optimal consumption in
closed form. The reason being that (17) prevents the capital from falling below the level K which allows to reach the
subsistence consumption level c.
Since K is assumed to be constant,
Assumption 1. The ratio c−Dt (zt )At (zt )−δt (zt )
is assumed to be constant.
This assumption holds, for instance, if Dt , At , and δt are all constant (we make this assumption in the numerical
application).
If capital grows at a rate of ft (Wt ,Πt), and never falls below K, then its value can be represented as:
Kt = K + e ft (Wt ,Πt ),
15
the stochastic differential of which is
dKt = (...)dt +(Kt −K
) ∂ ft (Wt ,Πt)
∂WtdWt
+(Kt −K
)(e ft (Wt ,Πt+1)− ft (Wt ,Πt )−1
)dΠt ,
(the drift component is omitted as it is immaterial to our analysis) which means that both the diffusion and the jump
components of the capital dynamics must be proportional to the difference Kt −K. Thus, the capital dynamics can be
written as:
dKt = (Yt (zt)−δ (zt)Kt − ct)dt (19)
+(Kt −K
)σt (zt)dWt +
(Kt −K
)βt (zt)dΠt ,
where the volatility is proportional to the amount of capital that exceeds the threshold (18). If capital should ever reach
K, it would become deterministic and constant at that level and all of the stochastic components would fall to zero (this
is because of the relationship between c and K that is shown in (18)).
Remark 6. The basis of this framework may easily be traced back to the CRRA preferences case where c = Dt = 0.
In fact, just as in the HARA case, the minimum level of capital (K) goes to zero and the differential equation (19)
coincides with (3) in the CRRA case.
3.3 Optimal consumption
Proposition 4. The consumption optimising Problem (35) (given (19) and under Assumption 1) is:
c∗t = c+(
Kt −c−DA−δ
)Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
. (20)
Proof. See Appendix B.
Corollary 3. When σt , βt , and λt are held constant, optimal consumption (20) can be written as
c∗t = c+(
Kt −c−DA−δ
)φ .
In this new setting, the optimal level of consumption is an affine transformation of capital, which is a significant
accomplishment as most of the previous models that allowed for a closed form solution relied on linear transformations
of capital. If we further assume that B = 0 in (17), we can write optimal consumption as an affine transformation of
16
Utility CRRA HARA
U (ct) =c
1−γ1t −11−γ1
U (ct) =(ct−c)1−γ1
1−γ1
Production function Yt = AtKt +BtKγ2t Yt = AtKt +Bt
(Kt −K
)γ2 +Dt
Restrictions γ1 = γ2 = γ
K = c−DtAt−δt
= constant
Optimal consumption c∗t = Ktκ c∗t = c+(Kt −K
)κ
κ ≡ Eγ
t
[(´∞
t e−´ s
t φu(zu)duds)γ]− 1
γ
Table I: Comparison between the framework (and hypotheses) leading to a closed form solution for optimal consump-tion
GDP:
c∗t = c(
1− φ
A−δ
)+
Dφ
A
(A
A−δ−1)+
Yt
Aφ . (21)
In Table I we present a comparison between the two models presented so far: the one with CRRA preferences and
the one with HARA preferences. In the table we highlight the restrictions needed on the production function parameters
in order to obtain a closed form solution. The HARA preferences are richer because they are able to capture a wider
range of consumer’s behaviour and, nevertheless, they allow for a closed form solution of optimal consumption under
two additional restrictions: the production function must be modified in order to be consistent with the presence of a
non-zero subsistence consumption level (i.e. it must allow for a non-zero parameter K), and the parameter used for this
purpose must be constant and have a precise form (i.e. K = (c−Dt)/(At −δt)).
With respect to other frameworks where a closed form optimal consumption is found, our model is more general
since we allow for the other model parameters (in particular At and Bt ) to follow any stochastic process.
3.4 Optimal capital accumulation
If optimal consumption (20) and the production function (17) are plugged into the capital dynamics equation (19),
optimal capital accumulation can be calculated as follows:
d(
K∗t − c−DA−δ
)(
K∗t − c−DA−δ
)=
A−δ −Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
+Bt (zt)
(K∗t −
c−DA−δ
)γ−1dt
+σt (zt)′ dWt +βt (zt)dΠt ,
17
where c−DA−δ
(as mentioned previously) is constant. In order to solve this differential equation, we must first obtain the
value of Kt ≡(
K∗t − c−DA−δ
)1−γ
through Itô’s lemma:
dKt =
Kt (1− γ)
A−δ − 12
γσt (zt)′σt (zt)−Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
︸ ︷︷ ︸
gt (zt )
dt
+(1− γ)Bt (zt)dt + Kt
((1− γ)σt (zt)
′ dWt +((1+βt (zt))
1−γ −1)
dΠt
).
This is a linear, stochastic differential equation in Kt which has a closed form solution based on the integrating
factor process:
Γt (zt)≡ e12 (1−γ)2 ´ t
t0σs(zs)
′σs(zs)ds−(1−γ)
´ tt0
σs(zs)′dWs−(1−γ)
´ tt0
ln(1+βs(zs))dΠs ,
and as such
Kt =K0
Γt (zt)e´ t
t0(1−γ)gs(zs)ds
+(1− γ)
ˆ t
t0Bs (zs)
Γs (zs)
Γt (zt)e(1−γ)
´ ts gu(zu)duds.
When the production function is linear (i.e. Bt = 0), the above solution can be simplified as follows:
K∗t =c−DA−δ
+
(K0−
c−DA−δ
)e
´ tt0
(A−δ−Eγ
s
[(´∞
s e−´ us φk(zk)dkdu
)γ]− 1γ
)ds. (22)
×e´ t
t0σs(zs)
′dWs+´ t
t0λs(zs) ln(1+βs(zs))dΠs .
Proposition 5. As long as initial capital K0 is greater than c−DA−δ
, optimal capital will never fall below that level.
This proposition confirms the results we presented at the beginning of this section.
Finally, the expected rate of capital increase is
Et
d(
K∗t − c−DA−δ
)(
K∗t − c−DA−δ
)=
A−δ −Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
+λt (zt)βt (zt)
dt,
and from this we are able to obtain the following result:
Proposition 6. Optimal capital (22) increases on average over time if and only if
Et
d(
K∗t − c−DA−δ
)(
K∗t − c−DA−δ
) +c∗t − c
K∗t − c−DA−δ
dt
> Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]− 1
γ
dt.
18
4 Numerical applications
4.1 Numerical application 1
Here, we consider a special case of Equation (17) and use quarterly US data from 1947:1 to 2013:4 (268 observations)
on per-capita GDP and personal consumption expenditures to estimate the model. We deflate the nominal values by
dividing them by the consumer price index (for which the price level in 1947:1 is set to 100). Data are taken from the
Federal Reserve Bank of St. Louis’ FRED database (research.stlouisfed.org). Figure 2 shows the (log) growth rate of
real US GDP.
We assume that:
• Production follows a corrected Cobb–Douglas function (i.e. At = 0 and Bt is constant):
Yt = B(
Kt +c−D
δ
)γ
+D
• There are no jumps (i.e. βt = λt = 0)
• Capital volatility is constant and depends on only one risk source (i.e. σt (zt) = σ ).
Given these hypotheses:
φ =ρ− (γ−1)δ − 1
2 (γ−1)γσ2
γ,
the optimal consumption level is
c∗t = c+φ
(Kt +
c−Dδ
)= c+φ
(Yt −D
B
) 1γ
, (23)
and the capital dynamics is given by
d(
K∗t +c−D
δ
)=
(−(δ +φ)
(K∗t +
c−Dδ
)+B
(K∗t +
c−Dδ
)γ)dt
+σ
(K∗t +
c−Dδ
)dWt .
From this we can easily obtain the dynamics of GDP:
dYt =((
(γ−1)γσ2−ρ−δ
)(Yt −D)+ γB
1γ (Yt −D)2− 1
γ
)dt +σγ (Yt −D)dWt . (24)
The estimation yields the parameter values shown in Table II.
The negative value of σ in Equation (24) indicates that volatility decreases as GDP increases; this is in line with the
information presented in Figure 2, which shows that there was greater volatility before the eighties and lower volatility
19
1947
Q1
1950
Q2
1953
Q4
1957
Q1
1960
Q3
1963
Q4
1967
Q1
1970
Q3
1973
Q4
1977
Q2
1980
Q3
1983
Q4
1987
Q2
1990
Q3
1994
Q1
1997
Q2
2000
Q3
2004
Q1
2007
Q2
2010
Q4
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
Figure 2: Log growth rate of real US GDP between 1947:1 and 2013:4 (268 observations) (research.stlouisfed.org).
20
1947
Q1
1950
Q2
1953
Q4
1957
Q1
1960
Q3
1963
Q4
1967
Q1
1970
Q3
1973
Q4
1977
Q2
1980
Q3
1983
Q4
1987
Q2
1990
Q3
1994
Q1
1997
Q2
2000
Q3
2004
Q1
2007
Q2
2010
Q4
15
20
25
30
35
40
45
50
55
60Real GDPRandom PathsMean Path95 % Confidence Bands
Figure 3: Simulation of 1,000 trajectories (in grey) of Equation (26), based on the parameter values listed in Table III.The actual real GDP values are depicted in black. The dashed black lines represent the 95% confidence bands.
21
Parameter Calibrated value
γ 0.4306363
σ −0.0478583
B 2.1864032
D −0.9235647
δ 0.0202918
ρ −0.0084302
c 16.091919
K =− c−Dδ
−838.5394763
Table II: Parameter values
afterwards.
In Proposition 5, we argue that the optimal capital will never fall below K. Nevertheless, since K is negative in
this situation, the optimal level of capital may be less than zero. This event is highly unlikely, but it cannot be avoided
within the theoretical framework.
In Figure 3 we see that the first real GDP observations are outside of the confidence bands. Despite this, the model
fits the rest of the sample quite effectively.
4.2 Numerical application 2
With the same data used in the previous subsection, we now examine another version of the general model (17). We
assume that:
• Production can be described with a pure Cobb–Douglas function (i.e. At = Dt = 0 and Bt is constant):
Yt = B(
Kt +cδ
)γ
• There are no jumps (i.e. βt = λt = 0)
• Capital volatility is constant and depends on only one risk source (i.e. σt (zt) = σ ).
Given these hypotheses:
φ =ρ− (γ−1)δ − 1
2 (γ−1)γσ2
γ,
the optimal consumption level is
c∗t = c+φ
(Kt +
cδ
)= c+φ
(Yt
B
) 1γ
, (25)
22
Parameter Calibrated value
γ 0.3871407
σ 0.0541073
B 1.8967764
δ −0.021027
ρ 0.0134349
c 16.9262316
K =− cδ
804.9744986
Table III: Parameter values
and the capital dynamics can be given by
d(
K∗t +cδ
)=
(−(δ +φ)
(K∗t +
cδ
)+B
(K∗t +
cδ
)γ)dt +σ
(K∗t +
cδ
)dWt .
From this we can easily obtain the dynamics of GDP:
dYt =
(((γ−1)γσ
2−ρ−δ)
Yt + γB1γ Y
2− 1γ
t
)dt +σγYtdWt . (26)
Upon estimating the model, we find the parameter values shown in Table III.
In this case the volatility σ is, of course, positive because there is no autonomous component in the drift of Equation
(26). The minimum value of capital (K) is now positive, which prevents optimal capital from becoming negative (as
explained in Proposition 5).
The less intuitive result is about the negative sign of the depreciation rate. Nevertheless, this finding is consistent
with the literature (Bu [2006], Hall [2009]) and could stem from a variety of misspecification errors. For instance, the
inflation rate on capital goods may be different than that on consumption goods; as such, different deflators should be
applied to Kt and Yt . If the prices for Kt increase at a lower rate than the average price level, then the value of real
capital will actually increase over time (by means of negative depreciation). In the very simple model that we have used
for this numerical application, we recall that we assumed Bt = 0, while a stochastic Bt is allowed in the general model
we presented in the previous sections. So, we leave to further research the task to find a suitable stochastic process for
describing the productivity parameter Bt in order to suitably compensate the negativity of δ .
Figure 4 shows that, under this specification, almost all of the real GDP observations fall within the confidence
bands. We conclude that this model, which is used to calibrate Equation (26), is preferable to that which was used
to calibrate Equation (24) because it features a lower Akaike Information Criterion (AIC) value (Akaike [1998]). We
highlight that the HARA case is subject to one more constraint on parameters than the CRRA case (as shown in Table
I) and, nevertheless, the shape of the production function is richer and, thus, is also able to capture the actual GDP
23
1947
Q1
1950
Q2
1953
Q4
1957
Q1
1960
Q3
1963
Q4
1967
Q1
1970
Q3
1973
Q4
1977
Q2
1980
Q3
1983
Q4
1987
Q2
1990
Q3
1994
Q1
1997
Q2
2000
Q3
2004
Q1
2007
Q2
2010
Q4
15
20
25
30
35
40
45
50
55
60
65
70
75 Real GDPRandom PathsMean Path95 % Confidence Bands
Figure 4: Simulation of 1,000 trajectories (in grey) of Equation (26), based on the parameter values listed in Table III.The actual real GDP values are depicted in black. The dashed black lines represent the 95% confidence bands.
24
series in a better way.
5 Conclusions
In this study, we used a fully stochastic framework to find a closed form solution that maximised the expected utility
of an agent’s inter-temporal consumption. We found that if the agent’s relative risk aversion was constant, and all of
the parameters were allowed to be stochastic, we could derive a quasi-closed form solution for optimal consumption.
This optimal consumption (presented as a percentage of capital) is calculated as an expected value subject to a new
probability measure that depends on the covariance between the stochastic state variables and capital accumulation.
Finally, we performed numerical calibrations on two specifications of the HARA model in order to show that it could
accurately fit US data.
Our model’s great advantage is that, in the CRRA preferences specification, all of the variables which describe the
economic environment are allowed to follow stochastic Itô processes, without any constraint on either the correlation
between them or the drift and diffusion terms. This provides fellow researchers with a closed form solution for the
optimal consumption level which only requires the calculation of an expected value (with Monte Carlo simulations, for
instance).
Additionally, the HARA specification of our model is able to account for subsistence level of consumption. Con-
sequently, this framework marks a great improvement upon previous models, though we were only able to find a closed
form solution by holding some of the parameters constant. We found that when a subsistence level of consumption
is established, capital accumulation must also remain above a certain level in order to finance both consumption and
capital depreciation.
Though our model accounts for labour, it does not do so by including it as a decision variable. As such, it may be
advantageous for future studies to develop utility functions which depends, not only on inter-temporal consumption,
but also on labour. Furthermore, because of the hypotheses on the parameters, the results obtained in the HARA case
are less general than those found in the CRRA case. Consequently, future research is needed in order to verify whether
these hypotheses can be omitted or relaxed. In particular, a more thorough investigation into the production function is
required to clarify the relationship between minimum capital and minimum consumption.
A Proof of Proposition 1
We may address Problem (6) through dynamic programming. With the value function defined as:
Jt (Kt ,zt) = maxcs
Et
[ˆ∞
te−´ s
t0ρu(zu)du c1−γ
s
1− γds
],
25
the function Jt (Kt ,zt) must satisfy the following Hamilton–Jacobi–Bellman (HJB) differential equation (for the sake
of simplicity, the functional dependencies on zt are omitted in all but the value function):
0 =∂Jt (Kt ,zt)
∂ t−ρtJt (Kt ,zt)+
12
∂ 2Jt (Kt ,zt)
∂K2t
σ′t σtK2
t +µ′t∂Jt (Kt ,zt)
∂ zt(27)
+maxct
[c1−γ
t
1− γ+
∂Jt (Kt ,zt)
∂Kt
((At −δt)Kt +BtK
γ
t − ct)]
+12
tr(
Ω′tΩt
∂ 2Jt (Kt ,zt)
∂ z′t∂ zt
)+Ktσ
′t Ωt
∂ 2Jt (Kt ,zt)
∂ zt∂Kt
+λt (Jt (Kt +Ktβt ,zt)− Jt (Kt ,zt)) ,
where tr is the trace operator. The boundary condition on the HJB equation is:
limt→∞
Jt (Kt ,zt) = 0.
For a reference on how to build the HJB the reader is referred to [Øksendal, 2000, Chapter 11] or [Björk, 2009,
Chapter 19]. The first order condition on (27) for optimising consumption yields:
c∗t =(
∂Jt (Kt ,zt)
∂Kt
)− 1γ
. (28)
Furthermore, the guess function can be written as:
Jt (Kt ,zt) = Ft (zt)K1−γ
t
1− γ+Gt (zt) ,
where Ft (zt) and Gt (zt) must solve (27). Once the guess function is plugged into (27), we obtain two differential
equations, one for Ft (zt) and one for Gt (zt):
0 =∂Ft (zt)
∂ t+(µ′t +(1− γ)σ
′t Ωt) ∂Ft (zt)
∂ zt+
12
tr(
Ω′tΩt
∂ 2Ft (zt)
∂ z′t∂ zt
)(29)
−γFt (zt)φt (zt)+ γFt (zt)1− 1
γ ,
0 =∂Gt (zt)
∂ t+µ
′t∂Gt (zt)
∂ zt+
12
tr(
Ω′tΩt
∂ 2Gt (zt)
∂ z′t∂ zt
)−ρtGt (zt)+Ft (zt)Bt (30)
where
φt (zt)≡ρt +λt +(γ−1)(At −δt)− 1
2 (γ−1)γσ ′t σt −λt (1+βt)1−γ
γ.
These equations’ boundary conditions are:
limt→∞
Ft (zt) = 0, limt→∞
Gt (zt) = 0.
26
Based on the Feynman-Kac theorem (see Appendix C for a short summary of the theorem which can also be found
in Yong and Zhou [1999], Øksendal [2000], Björk [2009]), the solution to (29) can be represented as an expected value
such that (Yong and Zhou [1999], Section 4.2):
Ft (zt) = Eγ
t [Xt ] ,
where the variable Xt solves the Bernoulli differential equation
dXt = γ
(Xtφt (zt)−X
1− 1γ
t
)dt, (31)
with the boundary condition
limt→∞
Xt = 0,
and Eγ
t [•] is the expected value conditional on the information set available at time t and under a new probability. As
dW γ
t is the Wiener process under this new probability, the stochastic differential equation which describes zt can be
written as:
dzt =(µt (zt)+(1− γ)Ωt (zt)
′σt (zt)
)dt +Ωt (zt)
′ dW γ
t , (32)
and since (32) must be equivalent to (1), we obtain the following relationship between the two Wiener processes (as
stated in the Girsanov theorem (Øksendal [2000])):
dWt = (1− γ)σt (zt)dt +dW γ
t . (33)
The unique solution to the Bernoulli differential equation (31) which satisfies the boundary condition is:
Xt =
(ˆ∞
te−´ s
t φu(zu)duds)γ
.
Finally, the solution to differential equation (29) (as simplified to an expectation above) can be represented as:
Ft (zt) = Eγ
t
[(ˆ∞
te−´ s
t φu(zu)duds)γ]
. (34)
Once the value of Ft (zt) is found, the solution to (30) can be obtained by once again applying the Feynman-Kac
theorem as follows:
Gt (zt) = Et
[ˆ∞
tFs (zs)Bs (zs)e−
´ st ρu(zu)duds
],
where a probability change is unnecessary. Nevertheless, it should be noted that Gt (zt) does not play a role in determ-
ining optimal consumption, but is only relevant in calculating the value function.
27
B Proof of Proposition 4
The optimisation problem we address is:
maxcs
Et0
[ˆ∞
t0e−´ s
t0ρu(zu)du (cs− c)1−γ
1− γds
], (35)
where capital evolves according to (19) and the state variables solve (1).
The HJB equation for the value function Jt (Kt ,zt) is:
0 =∂Jt (Kt ,zt)
∂ t−ρtJt (Kt ,zt)+
(Kt −
c−DA−δ
)σ′t Ωt
∂ 2Jt (Kt ,zt)
∂ zt∂Kt
+maxct
[(ct − c)1−γ
1− γ+
∂Jt (Kt ,zt)
∂Kt
((A−δ )Kt +Bt
(Kt −
c−DA−δ
)γ
+Dt − ct
)]
+12
∂ 2Jt (Kt ,zt)
∂K2t
σ′t σt
(Kt −
c−DA−δ
)2
+µ′t∂Jt (Kt ,zt)
∂ zt+
12
tr(
Ω′tΩt
∂ 2Jt (Kt ,zt)
∂ z′t∂ zt
)+λt
(Jt
(Kt +
(Kt −
c−DA−δ
)βt ,zt
)− Jt (Kt ,zt)
),
from which we are able to obtain the first order condition on consumption:
c∗t = c+(
∂Jt (Kt ,zt)
∂Kt
)− 1γ
.
Once c∗t is plugged into the HJB equation, we obtain:
0 =∂Jt (Kt ,zt)
∂ t−ρtJt (Kt ,zt)+
γ
1− γ
(∂Jt (Kt ,zt)
∂Kt
)1− 1γ
+∂Jt (Kt ,zt)
∂Kt
((A−δ )
(Kt −
c−DA−δ
)+Bt
(Kt −
c−DA−δ
)γ)+µ′t∂Jt (Kt ,zt)
∂ zt+
(Kt −
c−DA−δ
)σ′t Ωt
∂ 2Jt (Kt ,zt)
∂Kt∂ zt
+12
tr(
Ω′tΩt
∂ 2Jt (Kt ,zt)
∂ z′t∂ zt
)+
12
∂ 2Jt (Kt ,zt)
∂K2t
σ2t
(Kt −
c−DA−δ
)2
+λt
(Jt
(Kt +
(Kt −
c−DA−δ
)βt ,zt
)− Jt (Kt ,zt)
).
The guess function is
Jt (Kt ,zt) = Ft (zt)
(Kt − c−D
A−δ
)1−γ
1− γ+Gt (zt) ,
which yields differential equations for Ft (zt) and Gt (zt) which are identical to (29) and (30), respectively; the only
28
difference is that, this time, Assumption 1 must hold. The function Ft (zt) is thus given by (34), and consequently,
Proposition 4 is demonstrated.
C Feynman-Kac theorem
Let us assume a vector of stochastic variables Xs solves the (matrix) stochastic differential equation
dXs = µ (s,Xs)dt +Σ(s,Xs)′ dWs,
with Xt = x and s ∈ [t,∞[. Furthermore, we have to solve the following Cauchy problem:
∂v(t,x)∂ t
+
(∂v(t,x)
∂x
)′µ (t,x)+
12
tr(
Σ(t,x)′Σ(t,x)∂ 2v(t,x)
∂x′∂x
)+h(t,x,v) = 0,
whose boundary condition for t = T is v(T,x) = g(x) and h is uniformly Lipschitz continuous. Even if the solution
to this partial differential equation cannot be found in closed form, there exists a representation of the solution (due to
Feynman and Kac) under the form of an expected value as follows
v(t,x) = Et [Y (t,x)] ,
where Y (t,x) is the unique solution of the differential equation
dYs =−h(s,Xs,Ys)ds,
whose boundary condition is YT = g(XT ). A formal demonstration of this result can be found in [Yong and Zhou,
1999, Ch. 7]. When the function h(t,x,v) is linear in v, the solution to the differential dYsds is an exponential function.
It often happens that the term µ (t,x) in the differential equation is different from the original drift in the stochastic
differential equation dXs. In this case, the expected value which solves the Cauchy problem is computed under a new
probability measure.
The demonstration of this result (based on Itô’s lemma) is left to technical papers. Here, we just outline its use-
fulness: instead of numerically solving a partial differential equation, it is possible to find its solution by simulating a
conditional expected value (which is an easier task).
29
D Calibration of numerical applications
D.1 Model 1
The model (24) is calibrated in three steps:
1. We calibrate Equation (24) with a local linearisation method (as shown in Appendix E). Furthermore, once the
θi’s (i ∈ 1,2,3,4,5) are estimated with
dYt =(
θ1 (Yt +θ2)+θ3 (Yt +θ2)θ4)
dt +θ5 (Yt +θ2)dWt ,
we are able to obtain the following values for the parameters:
D =−θ2, γ =1
2−θ3, σ = θ5 (2−θ4) , B =
(θ3
γ
)γ
,
2. Equation (23) is estimated (with OLS), using D, B, and γ (as obtained above):
c∗t = ω1 +ω2
(Yt − D
B
) 1γ
,
as such, we find that c = ω1. The results of this regression are summarised in Table IV.
3. Since both θ1 and ω2 contain δ and ρ , the values of these two parameters may be obtained by solving the
following system: θ1 = γ (γ−1) σ2−δ −ρ,
ω2 =ρ−(γ−1)δ− 1
2 (γ−1)γ σ2
γ,
the unique solution of which is
δ =12(γ−1) σ
2−ω2−θ1
γ,
ρ =
(γ− 1
2
)(γ−1) σ
2 +ω2 +1− γ
γθ1.
D.2 Model 2
The model (26) is calibrated in three steps:
1. Equation (26) is calibrated with the local linearisation method shown in Appendix E. Once we estimate the θi’s
30
Coefficient Estimate Std. Error t-value p-value
ω1 16.9262316
ω2 0.0023134 1.5988035×10−4 14.4698445 2.2868662×10−35
Residual standard error: 0.1260083 with 265 d.f.
Multiple R2: 0.4413719, Adjusted R2: 0.4392638
F-statistic: 209.3764013 with 1 and 265 d.f., p-value: 2.2868662×10−35
Table IV: OLS estimation of the model c∗t = ω1 +ω2
(Yt−D
B
) 1γ
(i ∈ 1,2,3,4) with the following model
dYt =(
θ1Yt +θ2Y θ3t
)dt +θ4YtdWt ,
we are able to obtain the below parameter values:
γ =1
2−θ3, σ = θ4 (2−θ3) , B =
(θ2
γ
)γ
.
2. We estimate Equation (25) with OLS, using B and γ as they are defined above:
c∗t = ω1 +ω2
(Yt
B
) 1γ
,
thus, we find that c = ω1. The results of this regression are presented in Table V.
3. The values of δ and ρ may be obtained by solving the following system:
θ1 = γ (γ−1) σ2−δ −ρ,
ω2 =ρ−(γ−1)δ− 1
2 (γ−1)γ σ2
γ,
the unique solution of which is
δ =12(γ−1) σ
2−ω2−θ1
γ, ρ = γ (γ−1) σ
2−θ1− δ .
31
Coefficient Estimate Std. Error t-value p-value
ω1 16.9262316
ω2 0.0023134 1.5988035×10−4 14.4698445 2.2868662×10−35
Residual standard error: 0.1260083 with 265 d.f.
Multiple R2: 0.4413719, Adjusted R2: 0.4392638
F-statistic: 209.3764013 with 1 and 265 d.f., p-value: 2.2868662×10−35
Table V: OLS estimation of the model c∗t = ω1 +ω2
(YtB
) 1γ
E Local linearisation methodThe local linearisation method consists of locally approximating the drift in the SDE with a linear function. In theShoji–Ozaki method, the drift is allowed to depend on the time variable, t, and the diffusion coefficient is allowed tovary Shoji and Ozaki [1997, 1998]. This method has proven to be more efficient than constant approximation, and isgenerally more stable when dt is large (the observations for which are recorded on a less-than-daily basis).
Consider the SDE:
dXt = f (t,Xt)dt +g(Xt)dWt , t ≥ 0, X0 = x0, (36)
where f is twice continuously differentiable in its second argument and continuously differentiable in its first argument,and where g is continuously differentiable. The drift in (36) can be locally approximated on the interval [s,s+∆s) with
dXt = (LsXt + tMs +Ns)dt +g(Xt)dWt , t ≥ s,
where, if Xs = x, then
Ls =∂ f (x,s)
∂x,
Ms =12
g2 (x)∂ 2 f (x,s)
∂x2 +∂ f (x,s)
∂ s,
Ns = f (x,s)− ∂ f (x,s)∂x
x− sMs.
Based on the transformation Yt = e−LstXt , the stochastic differential equation with respect to Yt can be solved asfollows:
Yt = Ys +
ˆ t
s(Msu+Ns)e−Lsudu+g(Xt)
ˆ t
se−LsudWu.
From this, the discretisation of Xt can be obtained:
Xs+∆s = A(Xs)Xs +B(Xs)Z,
32
where
A(Xs) = 1+f (Xs,s)
XsLs
(eLs∆s −1
)+
Ms
XsL2s
(eLs∆s −1−Ls∆s
),
B(Xs) = g(Xs)
√e2Ls∆s−1
2Ls,
Z ∼ N (0,1) .
Accordingly,
Xs+∆s |Xs = x∼ N(A(x)x,B2 (x)
),
and as a result, the log-likelihood function can be given as
ln p(xs0 , . . . ,xsN
)= ln p
(xs0
)− 1
2
N
∑i=1
(xsi −Esi−1
)2
B2s−1
+ ln(2πB2
s−1)
,
where
E = Xt +f (Xs,s)
XsLs
(eLs∆s −1
)+
Ms
XsL2s
(eLs∆s −1−Ls∆s
).
Once the likelihood function is obtained, we can use a (quasi-)Newton method to maximise it with respect to thevector of parameters (θ ).
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