On the Supply Side of the Capital Goods Market:
Roles that Maintenance and Repair Can Play
Junkan Li
The Ohio State University, Department of Economics
li.6118@buckeyemail.osu.edu
A Simple Model with Analytical Solutions
September 8, 2019
1
Introduction
Below is a simple partial equilibrium model, which, for now, only contains two types of agents:
Consumption Goods Producers (called CP hereafter) and Investment (Capital Goods) Producers
(called IP hereafter). The simple model is of finite horizon, 3 periods of time specifically. All the
decisions are about buying and selling capital goods and maintenance goods. The simple model
keeps most of key features in the expanded version of model: a) Substitution between capital goods
and maintenance goods for both IP’s and CP’s; b) A forward-looking capital pricing formula where
the existence of maintenance matters; c) An endogenous amplification mechanism due to the possible default of IP’s. This simple toy model is presented to provide insights for understanding the
significance of maintenance and repair (M&R hereafter) in modern macroeconomics, rather than
to numerically match the data in the real world.
Although simple, the model results indicate that introducing M&R has potentials in explaining
or resolving the following macroeconomic questions: a) The high volatility of investment and
investment-like durables; b) pro-cyclicity of M&R expenditure and maintenance/investment ratio; c) pro-cyclicity of capital good prices in micro level and counter-cyclicity of relative price of
capital in macro level. Overall, this model is offering us some inspiring ideas for answering one
of the big questions in macroeconomics: what are considered as important on the supply side of
capital goods market and how shall we model these?
2
Model
This is a finite-horizon economy, with two types of agents: CP’s and IP’s. The measure of each
type of agents is normalized to be 1. CP’s are homogeneous and IP’s are heterogeneous. There are
three goods in the market 1) consumption goods produced by CP’s by using capital goods only; 2)
capital goods, produced by IP’s and sold in the fore-market of capital goods market; and 3) maintenance goods, produced by the IP’s and sold in the after-market of capital goods market. The
consumption goods are treated as numeraire. This simple paper is focusing on modeling relations
between capital goods and maintenance goods. Following the conventional assumption in the micro
literature, I assume CP’s are price takers in both fore-market and after-market; IP’s are, on the
other hand, price makers.
1
Compare to most of previous literatures, what is new in this paper is the way I model the M&R,
which further determines the transitions of capital goods that CP’s hold. Therefore, I shall start
with M&R.
2.1
Maintenance and Repair
Each period, consumption goods producers utilize capital goods for production, and by the end of
each period some parts get broken. The maintenance & repairing technology here after is modeled
as replacement the broken parts of old capital goods with the new parts sold by the original capital
supplier. An old capital good can be used for production in next period only if all the broken parts
has been replaced by new parts and this paper does not distinguish old but functional capital from
new capital. Therefore, given the marginal cost of buying a piece of new capital p, repairing cost
cm , and the price of new parts pm , the consumption goods producer would like to repair the old
capital goods only if the repairing cost is lower than cost of purchasing new capital; mathematically,
if maintenance & repairing is the better choice, then the number of non-operative parts x should
satisfies:
x ≤ m∗ (pd , pm ) ≡
pd − cm
, with cm > 0, pm > 0, and pd > cm
pm
(1)
Another assumption in this paper is that CP’s are allowed to trade-in capital goods to their producers. For the capital they do not want to repair, they can sell them to IP’s and IP’s are capable
to find the operative parts and utilize them. Both of CP’s and IP’s have incentive to make that
deal. However, since IP’s are the price makers, they will for sure set the price to the minimum level
to which CP’s won’t reject; that is zero. Therefore, the M&R behaviors of CP’s can be summerized
in the table below:
Dealing
Behavior
M&R
Trade-in
with an Old Capital Good
Condition
Benefit
∗
x ≤ m (pd , pm ) pd
x > m∗ (pd , pm ) 0
with x Broken Parts
Cost
Quantity
x · pm − cm x
0
1−x
To derive the aggregation result, the probability distribution of x is required. Assume that each
complete capital good consist of m parts and here m is normalized to be 1. By the end of each
period, there will be at least m parts broken for sure, and the number of parts broken falls within
[m, 1] stochastically with a probability measure, H : B([m, 1]) → [0, 1]. Unlike McGrattan and
Schmitz (1999), Boucekkine and Ruiz-Tamarit (2003), Boucekkine et al. (2010), and Albonico
et al. (2013), who directly assume the functional relation between depreciate rate of capital and
maintenance expenditure, this paper makes assumption on the density (h) of H. To in line with
the former literatures, this paper assume the following functional form:
h(x) =
ξ
m
, with ξ =
and x ∈ [m, 1]
2
x
1−m
The depreciate rate of capital goods are endogenously determined:
Z pd −cm
pm
δd (pd , pm ) = 1 −
dH(x)
m
2
(2)
(3)
Let D be the capital goods stock held by CP’s. Then the quantity of parts demanded, M and the
quantity of parts traded-in, R are as follows:
Z
M (pd , pm , D) = ρm (pd , pm ) · D =
pd −cm
pm
xdH(x) · D
(4)
(1 − x)dH(x) · D
(5)
m
Z
R(pd , pm , D) = ρr (pd , pm ) · D =
2.2
m̄
pd −cm
pm
Investment Goods Producers
There are two types of IP’s in the market; one is good and the other is bad. Bad IP’s suffer s fixed
probability (q) of default at the beginning of each period due to high debt burden; while good IP’s
do not. If one IP defaults on its debt, the firm will be forced to liquidate all capital and leave the
market permanently. The default is assumed to be exogeneous in this simple model. Let j ∈ {g, b}
denotes the type of IP’s, which is also known by CP’s.
I assume there is no cost for IP”s to transfer from capital goods to parts or from parts to capital
goods, which means capital goods and maintenance goods are perfect substituts in terms of production. Hence, sum of the parts an IP produces and gained from trade-in should equal the sum
of new capital goods sold in the fore-market and maintenance goods sold in the after-market.
According to the results in the previous section, once pd and pm are determined, both the quantity
of maintenance goods demanded and the quantity of trade-in parts are determined. I assume the
production technology is K α N 1−α ; thus, K α N 1−α is total number of parts that the IP produces;
ρr (pd , pm )·D is the number of parts that CP’s have traded in. Therefore, the number of new capital
goods sold is K α N 1−α − (ρm (pd , pm ) − ρr (pd , pm )) · D. For convenience, I denote Q(K, N, pd , pm , D)
be the quantity of new capital goods sold in the fore market, hence:
Q(K, N, pd , pm , D) = K α N 1−α − (ρm (pd , pm ) − ρr (pd , pm )) · D
(6)
State Variables, j ∈ {b, g}
Kj
Dj
ξj
Capital stock of IP’s
Capital goods stock in market
Indicator of default: 0 means default; 1 means not
Action Variables, j ∈ {b, g}
State transitions: if ξ j = 1, j ∈ {b, g}
0
K j = (1 − δk ) · K j + I j
(7)
3
Ij
Nj
Pdj
j
Pm
Investment of IP’s
Labor hired by IP’s
Price of new capital goods
Price of maintenance goods
0
Dj = 1 − δd (pjd , pjm ) · Dj + Q(K j , N j , pjd , pjm , Dj )
(8)
Reward function, dividend if ξ b = 1, j ∈ {b, g}
fd (I j , N j , pjd , pjm , K j , Dj ) = γ · pjd · Q(K j , N j , pjd , pjm , Dj ) − w · N j
− κ · K j − φk (I j , K j ) − I j + pjm · ρm (pjd , pjm ) · Dj
(9)
Here, w is exogenous wage rate; κ is proportional cost of production; and γ ∈ (0, 1) reflects the
selling cost for new capital. φk is the adjustment costs for capital with functional form:
φk (I, K) = ck ·
2.3
I2
K
(10)
Consumption Goods Producers
CP’s hold the capital goods produced by good and bad IP’s (Dcg and Dcb ). For CP’s, these are two
types of capital. However, this simple model leaves the only difference between Dcg and Dcb be their
producers ability to provide maintenance in the future.
Since the policies of maintenance are already known, the only action that is undetermined is
investment.
State Variables, j ∈ {b, g}
Dcj
Capital produced by type j IP’s
Action Variables: j ∈ {b, g}
Ycj
Investment bought from type j IP’s
State transitions, j ∈ {b, g}
0
Dcj = (1 − δd (pjd , pjm )) · Dcj + Ycj
(11)
Here, δd (·) presents maintenance technology, defined in previous section.
Reward function, profit
fc ({Ycj , Dcj }j∈{b,g} ) =
X Dcj − pjd · Ycj − pjm · ρm (pjd , pjm ) · Dcj
j∈{b,g}
(12)
− cm · 1 − δd (pjd , pjm ) · Dcj
Here, I assume CP’s adopt AK production technology. Let C be the quantity of consumption
goods produced, then C = Dc .
4
2.4
Timeline
Among the measure 1 IP’s, θ of them are good, 1 − θ of them are bad. At each period, CP’s
utilize capital to produce consumption goods, and at the end of period T0 , some capital breaks
out. For next period’s production, CP’s also have to make decisions on purchasing new capital
and maintaining old capital. IP’s produce maintenance goods and new capital goods using their
capital stock. I assume the capital goods produced by different IP’s are not compatible; thus, each
IP actually form a small sub-market, and they monopolize their own sub-market. Therefore, the
IP’s also decide prices to maximize their profit.
In this simple model, there are only three periods: T0 , T1 and T2 .
Period 0: The economy starts at T0 . The CP’s are endowed with D0b capital produced by bad IP’s
and D0g capital produced by good IP’s. Good IP’s and bad IP’s are all endowed with K0 capital.
And we assume IP’s and CP’s are initially ”n-to-1” paired.
Period 1: At T1 , q of the bad IP’s will have to default on their debt and exit the economy for
ever. The measure of bad IP’s now becomes (1−θ)(1−q). All the good IP’s will stay in the market.
Period 2: At T2 , all IP’s exit the economy at the beginning of T2 , thus there is no market for
capital goods. CP’s will keep producing consumption goods this period and exit the market at the
end of T2 .
2.5
Maximization Problems
Consumption goods producer
Vc (D0b , D0g ) =
max
1
X
j
{Yc,t
} t=0,1 t=0
j∈{b,g}
j
j
g
b
+ Dc,2
)
β t · fc ({Yc,t
, Dc,t
}j∈{b,g} ) + β 2 · (Dc,2
(13)
Subject to
j
j
j
Dc,t+1
= 1 − δd (pjd,t , pjm,t ) · Dc,t
+ Yc,t
,
t = 0, 1; j ∈ {b, g}
(14)
Investment goods producer: good
Vd (K0g , D0g , g)
=
max
{Itg ,Ntg ,pgd,t ,pgm,t }t=0,1
1
X
β t · fd (Itg , Ntg , pgd,t , pgm,t , Ktg , Dtg )
(15)
t=0
Subject to
g
Kt+1
= (1 − δk ) · Ktg + Itg ,
t = 0, 1
g
Dt+1
= 1 − δd (pgd,t , pgm,t ) · Dtg
(16)
+ Q(Itg , Ntg , pgd,t , pgm,t , Ktg , Dtg ),
5
t = 0, 1 (17)
Investment goods producer: bad
Vd (K0b , D0b , b) =
1
X
max
{Itb ,Ntb ,pbd,t ,pbm,t }t=0,1
β t (1 − q)t · fd (Itb , Ntb , pbd,t , pbm,t , Ktb , Dtb )
(18)
t=0
Subject to
b
Kt+1
= (1 − δk ) · Ktb + Itb ,
t = 0, 1
b
Dt+1
= 1 − δd (pbd,t , pbm,t ) · Dtb
(19)
+ Q(Itb , Ntb , pbd,t , pbm,t , Ktb , Dtb )
2.6
t = 0, 1 (20)
Market Equilibrium
The CP’s and IP’s maximize their value functions according to the maximization problems described
above given the initial values. Both fore-market and after-market are clear. Due to the Walras’s
Law, we only require:
α
j
Yc,t
= (Ktj ) (Ntj )
1−α
− ρm (pjd,t , pjm,t ) − ρr (pjd,t , pjm,t ) · Dtj ,
j ∈ {b, g}, t = 0, 1 (21)
3
Analytical Solutions
The central task of solving this model is to solve the prices pm and pd . According to the results in
the maintenance section, it is equivalent to consider pd and m∗ , rather than pd and pm , in solving
this simple model. The reason for using m∗ is that it provides convenience in taking derivatives. I
solve the prices using backwards induction, starting with prices in T1 .
Since, all IP’s will exit the economy at T2 , there is no difference between the good and the bad,
and no IP’s will make any investment at T1 , All IP’s will focus on the dividends at T1 . Hence, the
following FOC’s holds:
I1j : 0 = I1j
(22)
R mj

mj1
: 0 = −γ ·
pjd,1 h(mj1 )
N1j : 0 = (1 − α)γ · pjd,1
+
(pjd,1
(K1j )α
(N1j )α
1
− cm ) h(mj1 ) −
−w
m
xdH(x)
(mj1 )2


(23)
(24)
Given that the CP’s also know that all IP’s will exit at T2 , the price of new capital goods in T1 can
be determined:
pjd,1 = β
(25)
6
Therefore, if interior solution can be obtained, mj1 and pjm,1 should be:
mj1 = e
(1−γ)β−cm
+ln(m)
β−cm
(26)
β − cm
pjm,1 =
e
(27)
(1−γ)β−cm
+ln(m)
β−cm
Next, I calculate the prices of good IP’s at T0 . Since the good IP’s will stay in the economy at
T1 , the capital goods bought from them at T0 can be used for more than one period given proper
maintenance. Similar to what has been modeled in other durable goods literatures, CP’s evaluation
now is also related to expected prices of capital goods and maintenance goods at T1 , which can be
clearly seen in the FOC’s of CP’s problem:
g
Yc,0
: pgd,0 = β · 1 − cm · 1 − δd (pgd,1 , pgm,1 ) − pgm,1 · ρm (pgd,1 , pgm,1 )
+ β 2 · 1 − δd (pgd,1 , pgm,1 )
(28)
Given the results in previous sections, the forward-looking pricing formula can be written explicitly:
!
Z mg
Z mg
1
1
1
g
pd,0 = β + β(β − cm )
dH(x) − g ·
xdH(x)
(29)
m1 m
m
Given the results of T1 , pgd,0 is fully determined by parameters. For other actions, the behaviors of
IP’s are fully captured by the FOC’s:
I0g
g 1−α
g α−1
g
)
−
κ
)
(N
(K
+
β
αγ
·
p
1
1
d,1
K0g


R mg
0
xdH(x)
m

mg0 : 0 = −γ · pgd,0 h(mg1 ) + (pgd,0 − cm ) h(mg0 ) −
(mg0 )2
!
g
α
∂fd,1
(1 − α)γ · (K0g )
g
g
N0 : 0 =
pd,0 + β ·
−w
α
∂D1g
(N0g )
I0g : 0 = −2ck ·
(30)
(31)
(32)
where:
g
∂fd,1
∂D1g
=
∂fd (I1g , N1g , pgd,1 , pgm,1 , K1g , D1g )
∂D1g
=
γpgd,1
Z
m̄
mg1
(1 − x)dH(x) +
β − cm
− γpgd,1
mg1
Z
mg1
xdH(x) (33)
m
Note that if a capital goods can only be used for one period, the price should be β. Therefore,
given the probability of bad IP’s default at T1 , the pricing formula for bad IP’s can be written as
follows:
!
Z mb
Z mb
1
1
1
b
dH(x) − b ·
xdH(x)
(34)
pd,0 = β + β(1 − q)(β − cm )
m1 m
m
7
Therefore, the FOC’s of bad IP’s can be derived as follows:
b α−1
I0b
b (K1 )
: 0 = −2ck · b + (1 − q)β αγ · pd,1 b α−1 − κ
K0
(N1 )


R mg
0
xdH(x)
m

mb0 : 0 = −γ · pgd,0 h(mg1 ) + (pgd,0 − cm ) h(mg0 ) −
(mg0 )2
!
α
b
∂fd,1
(1 − α)γ · (K0b )
b
b
pd,0 + (1 − q)β ·
N0 : 0 =
−w
α
∂D1g
(N0b )
I0b
(35)
(36)
(37)
where:
b
∂fd,1
∂D1b
=
∂fd (I1b , N1b , pbd,1 , pbm,1 , K1g b, D1b )
∂D1g
=
γpbd,1
Z
m̄
(1 − x)dH(x) +
mb1
β − cm
− γpgd,1
b
m1
Z
mb1
xdH(x) (38)
m
Note that
(δk · k)2
k
(x − ζ · b)2
φb (b, x) = cb
b
1
1−α2
w
, with γ0 = 1, γ1 = 1.1
n(k, i) = γi
α
α2 · p · k 1
φk (k, i) = i · ck
(39)
(40)
(41)
Here, φk and φb are the adjustment costs for capital and debt, respectively, and n is labor.
CP’s
D
Capital of CP’s
K
D
ξ
IP’s
Capital of IP’s
Capital goods stock in market
Indicator of default
Action Variables
CP’s
Y
M
Investment of CP’s
Maintenance goods
Yd
I
Pd
Pm
IP’s
New capital goods sold
Investment of IP’s
Price of new capital goods
Price of maintenance goods
State transitions if i = 1
k 0 = (1 − δk )k + δk k
(42)
d0 = k α1 n(k, i)α2 + (1 − δd )d
(43)
8
b0 = x
(44)
Here, n denotes labor.
Reward function, dividend
f (k, d, b, x, i) = p · k α1 n(k, i)α2 − κ1 · k − κ0
− w · n(k, i) − pm · d − φk (k, i) − φb (b, x) − pb · b + qi · x (45)
where
(δk · k)2
k
(x − ζ · b)2
φb (b, x) = cb
b
1
1−α2
w
, with γ0 = 1, γ1 = 1.1
n(k, i) = γi
α
1
α2 · p · k
φk (k, i) = i · ck
(46)
(47)
(48)
Here, φk and φb are the adjustment costs for capital and debt, respectively, and n is labor.
Bellman equation
V (k, d, b) = max
f (k, d, b, x, i) + iβV (k 0 , d0 , b0 )
(49)
x≥0
i∈{0,1}
Here, V (k, d, b) denotes the maximum attainable present value of current and future dividends,
given that, at the beginning of the period, the firm is in good standing with lenders and possesses
capital stock k, durable stock d, and debt b. The value function if the firm is not in good standing
with lenders, because it has defaulted, is, of course 0.
Parameters
9
Symbol
p
w
α1
α2
κ1
κ0
pm
pb
ck
δk
δd
cb
ζ
β
γ0
γ1
q0
q1
Value
1.183
1.5
0.6
0.3
0.55
0.05
0.02
0.9892
7.5
0.1
0.2
0.06
0.6
0.964
1.0
1.1
0.2
β
Definition
price of durable good, calculated in other part of the project
wage rate
share of capital in durable production
share of labor in durable production
fixed cost of production
proportional cost of production
maintenance profit rate
debt payment, due to the tax shield, it’s between β and 1
capital adjustment cost parameter
capital depreciation rate
durable goods depreciation rate
debt adjustment cost paprameter
zero adjustment cost debt policy
discount factor
labor coefficient, if default next period
labor coefficient, if repay next period
price of new debt, default
price of new debt, repay
Given the formulation above, it appears to me that you have incorrectly titled κ0 and κ1 . κ0
is a constant term in your reward function, and thus indicates a fixed cost. κ1 appears to be a
unit cost of capital maintenance or something related, yielding a total cost this is proportional to
capital stock. Please revise the names in the table above as needed. Also, the name you give to pb
is not very clear. It appears to be some kind of rate; if so, revise the name to this variable in the
table above.
4
Numerical Solution
We will compute an approximation to the value function via the method of collocation. Specifically,
we approximate the value function using a linear combination of N three-dimensional Chebychev
polynomials φj :
X
V (k, d, b) =
cj φj (k, d, b).
(50)
j
We will then chose a series of N nodes (kj , dj , bj ) and compute the unknown coefficients cj by
solving the N nonlinear equations simultaneously




X
X
cj 0 φj 0 (kj , dj , bj ) = max f (kj , dj , bj , x, j) + iβ
cj 0 φj 0 (k 0 , d0 , b0 ) .
(51)
x≥0 

0
0
j
j
i∈{0,1}
j = 1, 2, . . . , N
I am going to propose a derivative-free algorithm to solve the collocation equation that should
prove to be fast and stable. However, I am not going to explain the algorithm now. We first need
to get on the same page about the model.
10
5
Questions
1. When I substituted your investment policy equation into the capital transition equation, the
latter reduced to k 0 = k (see equation 1). This makes no sense.
To simplified the exogenous investment policy, I assume fixed investment rate: i = ξ · k.
ξ can be other numbers.
I don’t understand you answer. You have not changed equation (1). It still reads k 0 = k. This
means that k is fixed and fully exogenous to the model, and thus not a true state variable.
The model cannot be solved unless you specify its value, and you have not done so. I would
like to proceed to advise you on how to solve the model numerically. However, I cannot do
so until you resolve how you want to treat k. If it is truly unchanging, then tell me what
its value is, and revise the model formulation above to exclude any mention of k as a state
variable.
2. When I substituted your investment policy equation into the capital adjustment cost equation, it reduced to a linear function of k (see equation 5). Is this what you intended?
No, this is true only when fixed investment rate is assumed. When we fully solve the model,
investment policy should have some curvature.
I also don’t fully understand you answer. You have not changed equation (5) above. If you
want to proceed with equation (5) as written, fine ... just confirm that this is the case. If
not, then edit equation (5) to what you believe it should say.
3. You state then when the firm decides to default the following period, it will announce this
publicly, making it known to lenders. You then reduce the cost of capital q from β ≈ 0.9 to
0.2. Why would the firm’s cost of capital fall if it publicly announces an intent to default the
following period? Indeed, why would any lender lend any amount to the firm, given that it
has announced its intent to default.
The price of debt q falls because the return of the debt falls when firm default. If the
firm declares default, then creditors will liquidate the firm’s capital. If the liquidation value
is 0, then q should be zero; if the liquidation value is positive, then q > 0. Ideally, the liquidation value should be a function of capital stock, but I failed to solve the model with this
assumption. Therefore, I assume q = 0.2 instead.
Also, here I assume that the financial intermediaries has full information, thus they know
the expected return of firm’s debt and can price the debt rationally. It is different from ”announce to default the following period”. Later, I will introduce households who do not have
such information.
I will not question your reasoning on this matter further, and continue to treat q as it is
specified above.
6
Comments
1. I am unwilling to let i denote the transition indicator, since eventually I will have to solve
the investment. and i is commonly used to denote investment in macroeconomics literatures.
11
I understand that the intended audience for your research will be most comfortable with i
used to indicate investment. However, as the subtitle “Implementation Notes” above conveys,
this document is designed to guide us in developing working numerical code. In numerical
work, i is universally used as a counter or discrete variable, and in developing working code
for your model, we are going to use it to represent the default choice. If, as proceed, you
reintroduce investment to the model (which for now has disappeared due to substitution) you
will need to find a different variable name for it. However, this does not commit you to using
that variable name in the narrative of your dissertation or working paper. You can, in your
narratives, use i to indicate investment.
2. We can first solve the simpler version with the assumption qi = q̄. But in the baseline model,
qi is indeed a function of i since firms’ future default decisions can be rationally expected by
the lenders. And this makes the model more challenging and interesting.
I am not sure what to make of this comment. Keep in mind that all we want to do right
now is to specify a very basic version of model that retains only the most essential features
of the model you ultimately wish to solve, so we can proceed to developing working code.
Once we have developed working code for the simple model, we can begin to introduce more
complicated features.
3. For the derivative-free method, I have tried the N elder − M ead method to solve the optimal
decisions at each grid node. I found it is not stable. Is the method in your mind related to
Nelder-Mead method?
No. The derivative-free method I am going to propose for solving your model has nothing to
do with Nelder-Meade. This will all become clear later, once we are ready to develop initial
working code, at which time I will explain the matter further. However, we cannot proceed to
writing numerical code until we have a viable, simplified, fully parameterized model to work
with. And, as the discussion above makes clear, we have not yet reached this point.
Finally, you have sent me some of your code, but I do not know what you hope to have me
do with it. The code is not even minimally documented, and neither have you provided side notes
(like the implementation notes in this document) that explains what the code is supposed to do.
Moreover, the parameters you specify in your code do not conform to those used in this document.
You must develop discipline when you write code. The code must documented sufficiently to give
the reader some sense of what each code segment is supposed to do. Moreover, your code should
be accompanied by side notes, like these implementation notes, that describe what the code is
designed to do more clearly. Keep in mind that writing implementation notes to accompany your
code will not be wasted work. In your dissertation and working papers, you will need to explain
your numerical procedures, and most of the language in your implementation notes will have to
appear in your paper, albeit perhaps in abbreviated form or as an appendix. Your implementation
notes must be sufficiently clear and detailed for me to fully understand what the code is designed
to do. If I cannot understand what are doing, readers of your dissertation and papers certainly
not be able to understand. In any case, I am ignoring the code you sent me, not only because you
have failed to document it properly, but because I cannot see how it can help us solve the model
numerically.
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