TRENT UNIVERSITY
DEPARTMENT OF ECONOMICS
Dr. M. Arvin
ECONOMICS 4250H -- Winter
2010-2011
FINAL EXAMINATION
April 2011
Instructions; please read carefully:
There are three sections in this examination.
You are to attempt TWO questions from Section A. If you do all three questions in Section A, only
the first two will be marked. Each question is Section A is worth 20 marks.
In addition, you must answer the question in Section B and ONE of the two questions in Section C.
If you answer both questions in Section C, only the first one will be marked.
Each question in Sections B and C is worth 30 marks.
The parts in each question are not necessarily worth the same.
As your course outline states, sharing of calculators is not allowed. Paper and electronic
dictionaries, blackberries, mobile phones, and other devices are not permitted in the examination
room.
In answering each question you must explain all your steps carefully. Unsupported answers will
receive little credit.
You are not allowed to consult any notes or books during the examination period. Calculators are
allowed.
Please make sure your answers are neat and legible and that you have your name and student
number on the front of all your examination booklets. Please answer each question in a separate
exam booklet.
You may keep the question paper.
There is a time limit of THREE (3) HOURS. Allocate your time between the questions wisely.
Good luck!
SECTION A
1. Consider a problem of maximizing utility
n

U   Qi i
i 1
subject to the budget constraint
n
Y   Pi Qi
i 1
where Y, Pi , and αi (i = 1,… , n ) are parameters. Take a natural log transformation
of this utility function. Write your Lagrangian function and the first-order conditions
after this transformation. Use your FOCs to find optimal Qi .
2. Suppose that a firm has its policies determined by a manager whose objective is to
maximize sales, i.e., revenue, without letting profit drop below some fixed level m.
To spice up the problem, let us add an advertising cost A dollars. Let R(y, A) denote
the firm’s revenue as a function of production level y and advertising expenditure A.
Assume that the firm is bound by past contract to advertise by at least n much
(expressed in dollars). Let C(y) denote the cost of manufacturing y units of output.
Obviously the firm’s choice variables are y and A . Assume R / A  0 .
i)
ii)
iii)
Write down the information given above in the form of a non-linear
programming problem.
Specify a sufficiency theorem to solve above and write your Kuhn-Tucker
conditions.
Assume the solution entails a non-zero level of output. Further suppose n is
non-zero. By using this information and using some of your Kuhn-Tucker
conditions developed in ii), prove that the profit realized is the minimal profit
allowed.
3.
Let x and y denote the biomass of two species respectively. (Biomass simply means
the total quantity or weight of an organism in a given area.) Assume that the two
species compete for food, so that the rate of growth of each species is negatively related
to the biomass of the other species. Suppose functional forms showing evolution of x
and y are:
x  x(1  x  2 y )
y  y (1  y  2 x) .
[Note: Obviously x, y, x , and y are each a function of time t.]
i) Find all the possible intertemporal equilibria.
ii) Using a phase diagrammatic technique, discuss the stability/instability of the interior
equilibrium. (Note: For full marks, all trajectories must clearly be marked and the
mathematical derivation shown and explained in some detail. Poorly drawn
diagrams will be penalized.)
SECTION B
1. There are two parts in this question. Answer both parts. (Each part is an independent
question.)
a)
Consider an individual whose only income is the interest generated from his past stock
of savings. Let the market rate of interest be constant at 100r% and let his stock of
savings at time t be denoted by k(t) . Hence his income at time t is equal to rk(t).
At any instant of time, t , the individual spends her income on consumption (c (t) )
and savings (s (t) ). Hence rk (t) = c (t) + s(t). Since k(t) denotes the stock of savings
t
at time t, k (t )   s (t ' )dt ' k 0 , and k(t )  s(t ) . Assume the individual has
0
k (0)  k 0 , k (T )  kT , and her utility function specifically is u (c(t ))  a[c(t )] v ,
where a>0 and 0<v<1. The individual wishes to maximize her life time utility. So
suppressing the time script t , her problem is to maximize
T v
 0 ac dt
Subject to
k  s ,
k (0)  k 0 , k (T )  kT ,
where c = rk – s, a>0, and 0<v<1.
i)
By solving the above optimal control problem, find the sign for the slope of the
optimal consumption profile for this individual and provide an intuition for your
result.
ii) Find an expression for optimal c .
iii) Describe how you would go about finding an expression for optimal k and s . (A
description in words is all that I am looking for here.)
iv) Is the second-order condition for this problem satisfied? Explain how that might
be the case.
b)
Suppose you have to use the algebraic method to determine the type of intertemporal
equilibrium a system has. Suppose there is a case of saddlepath stability. Tell me what
signs your eigenvalues take.
SECTION C
1.
There are two parts to this question. Answer both parts. (Each part is an independent
question.)
a)
Given a utility function
2

U   Qi i
i 1
show that both U and the positive monotonic transformation V = ln (U) have the
same MRS.
b)
Consider the following non-linear system of dynamic equations describing the
evolution in the capital stock (K) and pollution (P) of an economy.
.
K  K (s K  1   )
.
P  K  P
[As an aside, the first equation comes from a model where Y is output, and the
production function is given by Y  K  . Savings (s) in this model are a constant
proportion of output; and δ is the rate of depreciation on capital. Therefore, the
.
evolution of capital stock is given by: K  sY   K  s K    K  K ( s K  1   ) .
The second equation assumes that a capital stock K generates a flow of pollution
K  (i.e., this is the pollution ‘production function’), and that the stock of pollution
decays (i.e., it is naturally absorbed by the environment – by oceans and vegetation)
.
at rate γ. Thus, P is the net rate of change in the stock of pollution, which is given by
the right- hand-side of the second equation. It is reasonable to assume 0 < α < 1 , 
> 0 , γ > 0 , and β > 1.]
i)
Find the two possible steady state solutions (the two intertemporal equilibria)
in the non-negative orthant of the (K,P) plane. I am looking for an algebraic
answer. No diagram is necessary.
ii)
By using the algebraic method, find the type of equilibrium we have at our
interior solution. Again: no diagram is required: use the algebraic approach
that I taught you. (Hint: Do not forget that this is a non-linear system of
equations.)
2.
There are three parts to this question. Answer all parts. (Each part is an independent
question.)
a)
A primitive tribe gathers one unit of berries at each unit of time. They can
consume berries or store them without cost or deterioration. Ancient tribal custom
enjoins them to maximize

T
e  rt ln C (t )dt ,
0
where C is consumption, r>0 is a constant discount rate, and T is the planning
period. Henceforth suppress the time script t.
i) Explain why the appropriate state equation for this tribe is:
W  1  C ,
where W is stock of berries.
ii) Assuming W (0)  0, W (T )  WT , find an expression for optimal C and for
optimal W for this tribe. [Note: Your final answer must involve the parameter
WT .]
iii) Is the second-order condition for this problem satisfied? Explain.
b)
Consider a utility function for a risk-averse individual
u( y) 
1


y

where y is income and α is a constant.
i) Explain what set of restrictions we need on α .
ii) Prove that this utility function exhibits a constant degree of relative risk
aversion.
c)
Prove if the individual has a constant absolute risk aversion, then u''' (y)>0 .