Unformatted text preview: Final Exam
EC720.01 - Math for Economists
Boston College, Department of Economics Peter Ireland
Fall 2010 Due Tuesday, December 14, 2010 at 11:00am
This exam has three questions on ﬁve pages; before you begin, please check to make sure
your copy has all three questions and all ﬁve pages. Each of the three questions is worth 20
points; as indicated at the beginning of the semester, this 60-point ﬁnal will account for 60
percent of your course grade and the problem sets will account for the remaining 40 percent
of your course grade.
This is an open-book exam, meaning that it is ﬁne for you to consult your notes, textbooks,
and other references when working on your answers to the questions. I expect you to work
independently on the exam, however, so that the answers you submit are yours and yours
1. Growth and Pollution, Part I
This question asks you to use the Kuhn-Tucker theorem to solve a static constrained optimization problem that can be used to think about patterns of pollution and environmental
quality seen when looking across a sample of countries at diﬀerent stages of economic development.
Suppose, in particular, that a country has k > 0 units of physical capital that can be used to
produce output of a single consumption good using any one from a continuum of technologies
indexed by z ∈ [0, 1]. Lower values of z represent “cleaner” technologies that generate less
pollution and higher values of z represent “dirtier” technologies that generate more pollution.
People dislike pollution, but they also like consumption; and because dirtier technologies can
be used to produce output at a lower cost, a trade-oﬀ arises between economic growth and
To make these ideas more speciﬁc, suppose that the country’s k units of capital, when used
with technology z , produce c = Akz units of output, where A > 0 is a parameter governing
the overall level of productivity across all available technologies. Hence, according to this
speciﬁcation, more capital used with dirtier technologies yields more output. Suppose also,
however, that the same k units of capital, when used with the same technology z , generate
p = Akz β units of pollution, where β > 1 is a parameter governing how the use of dirtier
technologies causes environmental quality to deteriorate.
Suppose, ﬁnally, that a representative consumer in this country gets utility
u( c) = c 1− σ − 1
1−σ 1 from consuming c units of output but suﬀers disutility
v ( p) =
when the level of pollution is given by p, where the preference parameters satisfy σ > 0,
B > 0, and γ > 1.
A social planner for this economy takes the stock of physical capital k as given, and chooses
the levels of consumption c and pollution p to maximize the representative consumer’s total
utility subject to the constraints imposed by the description of the technologies from above.
Although there are a number of ways to formalize the social planner’s problem, perhaps
the easiest is to start by observing that because preferences over consumption are such that
limc→0 u (c) = ∞ and because the cleanest technology z = 0 yields no output, the constraints
c ≥ 0 and z ≥ 0 will never bind at the optimum and can therefore be dropped from further
consideration. On the other hand, for certain parameter conﬁgurations at least, it may be
optimal to chose the dirtiest technology z = 1; hence the constraint 1 ≥ z must still be
Substituting the technological relations c = Akz and p = Akz β into the utility and disutility
functions u(c) and v (p) then allows the social planner’s problem to be written as one with a
single choice variable and a single constraint:
(Akz )1−σ − 1
(Akz β )γ subject to 1 ≥ z.
a. Deﬁne (write down) the Langragian for this problem.
b. Next, write down the full set of ﬁrst-order conditions, constraints, nonnegativity conditions, and complementary slackness conditions that, according to the Kuhn-Tucker
theorem, characterize the value z ∗ that solves the social planner’s problem together
with the associated value of the Lagrange multiplier.
c. Now use your results from part (b), above, to solve for the optimal choice of z ∗ as a
function of the model’s parameters: k , A, β , σ , B , and γ .
d. Next, use your result from part (c), above, to obtain an expression that shows how total
pollution depends on the model’s parameters k , A, β , σ , B , and γ when z ∗ is chosen
e. Finally, in order to obtain a sharper economic characterization of your result from part
(d), above, suppose that the preference parameter σ is larger than one: σ > 1. Under
this extra assumption, what does your result imply for the relationship between the
economy’s level of economic development, as measured by its stock of physical capital
k , and its optimal choice of pollution? Looking at a cross-section of countries, under
what circumstances will more developed countries have better environmental quality?
Under what circumstances will more developed countries have worse environmental
2 2. Growth and Pollution, Part II
This question asks you to use the maximum principle to characterize the solution to a
dynamic version of the problem that you just considered in a static context. The dynamics
are added to the model by allowing the social planner to increase the capital stock k over
time by allocating output to investment instead of consumption.
Suppose that time is continuous and the horizon is inﬁnite, so that periods can be indexed by
t ∈ [0, ∞). Let k (t) denote the economy’s stock of physical capital at time t and, extending
the speciﬁcation from above, suppose that by using a technology indexed by z (t) during time
t, the economy produces Ak (t)z (t) units of output but also generates p(t) = Ak (t)z (t)β units
of pollution where, as before, A > 0 and β > 1 are parameters.
Now, however, suppose that in addition to choosing an optimal technology z (t) at each date
t ∈ [0, ∞), the social planner must also choose how to optimally divide up output Ak (t)z (t)
into an amount c(t) to be consumed and an amount Ak (t)z (t) − c(t) to be invested. If, in
addition, the parameter δ , with 0 < δ < 1, measures the depreciation rate of physical capital,
then the stock of capital evolves according to
Ak (t)z (t) − δ k (t) − c(t) ≥ k (t)
for all t ∈ [0, ∞).
With the additional dynamics introduced through physical capital accumulation complicating the problem, assume now that the constraints 1 > z (t) and z (t) > 0 can be ignored,
either because there are ever “dirtier” technologies than the dirtiest z = 1 considered before
or because the conﬁguration of parameters insures that these constraints will not bind at
the optimum. Then, with the speciﬁcation for the utility over consumption
u[c(t)] = c ( t ) 1− σ − 1
1−σ with σ > 0 and the disutility from pollution
v [p(t)] =
p( t ) γ
with B > 0 and γ > 1 the same as before, the social planner’s problem can be stated as
follows: given the initial capital stock k (0), choose continuously diﬀerentiable functions c(t),
z (t), and k (t) for all t ∈ [0, ∞) to maximize
c ( t ) 1− σ − 1
[Ak (t)z (t) ]
subject to ˙
Ak (t)z (t) − δ k (t) − c(t) ≥ k (t) for all t ∈ [0, ∞), where ρ > 0 denotes the discount rate.
3 a. Write down the Hamiltonian for the social planner’s problem.
b. Next, write down the ﬁrst-order conditions and the pair of diﬀerential equations that,
according to the maximum principle, characterize the solution to the social planner’s
c. The conditions you derived for part (b), above, should form a system of equations
involving the not only optimal choices of c(t), z (t), and k (t) but also the associated
values for a new variable, corresponding to the Lagrange multiplier on the capital
accumulation constraint that you would have introduced into the problem if you had
decide to solve it using the Kuhn-Tucker theorem instead. Use one of your optimality
conditions to eliminate this new variable from the system, so as to obtain a smaller set
of three equations involving only the model’s parameters and the three variables c(t),
z (t), and k (t) that enter into the original statement of the planner’s problem.
d. The smaller system you derived for part (c), above, implies that starting from any
initial value of k (0), there are unique values of c(0) and z (0) that place the economy
on a saddle path that converges to a steady state, in which c(t) = c∗ , z (t) = z ∗ , and
k (t) = k ∗ for all t. Use your answers from part (c), above, to derive a system of three
equations that can be used to solve for the steady-state values c∗ , z ∗ , and k ∗ in terms
of the model’s parameters: A β , δ , σ , B , and γ . Note: you don’t have to actually solve
for the steady-state values; just write down the three-equation system that would, in
principle, allow you to ﬁnd these values.
3. Stochastic Growth
This problem asks you to use the guess-and-verify method to characterize the solution to a
version of the stochastic growth model.
Suppose that time is discrete and the horizon is inﬁnite, so that periods can be indexed by
t = 0, 1, 2, .... Let output produced during each period t = 0, 1, 2, ... be given by yt = At kt ,
where kt is the beginning-of-period capital stock and At is a random shock to productivity.
By consuming ct units of output at each date t = 0, 1, 2, ..., the representative consumer gets
β t ln(ct ),
t=0 where the discount factor satisﬁes 0 < β < 1. Letting δ , with 0 < δ < 1, denote the
depreciation rate for physical capital, the evolution of the capital stock is described by
At kt + (1 − δ )kt − ct ≥ kt+1 ,
a constraint that must hold for all t = 0, 1, 2, ... and all possible realizations of the productivity shock At at each date t = 0, 1, 2, .... 4 Assume, ﬁnally, that the productivity shock At gets realized at the very beginning of period t, before ct and kt+1 are chosen, and that the total return on capital, accounting for
Rt = At + 1 − δ
follows the ﬁrst-order autoregressive process
ln(Rt+1 ) = (1 − ρ) ln(R) + ρ ln(Rt ) + εt+1 , where the parameter ρ, satisfying 0 < ρ < 1, measures the degree of serial correlation in the
productivity shock, the parameter R, satisfying R > 0, determines the average value of Rt ,
and where εt+1 is a serially uncorrelated shock, satisfying Et εt+1 = 0 for all t = 0, 1, 2, ...,
that gets realized at the beginning of period t + 1. Hence, when ct and kt+1 are chosen, Rt
is known but Rt+1 is still viewed by the representative consumer as random.
The representative consumer’s problem can now be stated as: given the initial capital stock
k0 , choose contingency plans for ct , t = 0, 1, 2, ..., and kt , t = 1, 2, 3, ..., to maximize
β t ln(ct ), t=0 K
subject to the constraints (which are more conveniently rewritten in terms of Rt )
Rt kt − ct ≥ kt+1
for all t = 0, 1, 2, ... and for all possible realizations of Rt at each date t = 0, 1, 2, .... a. Write down the Bellman equation for the representative consumer’s problem.
b. Next, guess that the value function takes the time-invariant form
v (kt , Rt ) = E + F ln(kt ) + G ln(Rt ) where E , F , and G are unknown constants to be determined, and, using this guess,
write down the ﬁrst-order and envelope conditions that help characterize the consumer’s optimal decisions.
c. Use your results from part (b), above, to solve for the constant F in terms of the model’s
parameters β , R, and ρ.
d. Use your results from parts (b) and (c) to derive a pair of equations that can be used
to construct the optimal choices for ct , t = 0, 1, 2, ..., and kt , t = 1, 2, 3, ..., for a given
value of the initial capital stock k0 , a given set of realizations for Rt at each date
t = 0, 1, 2, ..., and for given values of the model’s parameters β , R, and ρ. 5 ...
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This note was uploaded on 02/29/2012 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.
- Fall '09