Download Dr. M. Arvin 2009/2010

TRENT UNIVERSITY
DEPARTMENT OF ECONOMICS
ECON 425H
ASSIGNMENT #1
Dr. M. Arvin
2009/2010
DUE DATE: Beginning of class on Tuesday, February 9, 2010. As page 6 of your course
outline states, penalty for late assignments is 20 percentage points for each day or part
thereof for which the assignments is late. Please type your assignment or use a pen.
(15) 1.
(5)
2.
Suppose a utility function u = u(x,y) with a marginal rate of substitution equal to
r is replaced by ln u.
i)
Is r replaced by ln r? Illustrate why or why not. [Hint: Does your answer
accord with your class notes on taking a transformation of a utility
function?]
ii)
Will this transformation leave  (the Lagrangian multiplier of the Classical
utility maximization problem subject to an income constraint) unchanged?
[Hint: Write the optimization problem for each utility function and see if
there is a change in the multiplier.]
Find the eigenvalue and eigenvector for matrix A where
 3  1

A = 
1 1 
Normalize your solution for the eigenvector as usual.
(25)
3.
A utility maximizing consumer has a generalized Cobb-Douglas utility function
n
n
1
2
 i  1
,
u(Q)  [Q1 ] [Q2 ] [Qn ]
i 1
and a budget constraint
n
Y   Pi Qi .
i 1
i)
Prove that the share of income spent on good i depends on the size of i
relative to the sum of i’s.
ii)
Prove the following statements regarding the various elasticities of demand:
a)
Own price elasticity of demand for good i is equal to “-1”.
b)
Income elasticity of demand for good i is equal to 1.
c)
Cross price elasticity of demand for good i is equal to 0.
[Hint: find an expression for Qi and work out the various elasticities.]
(10)
4.
Find an expression for output as a function of time given that
3
d3y
d2y
dy

5
2
 et ,
3
2
dt
dt
dt
where y is output and t is time.
(25)
5.
An aircraft manufacturing company can operate manufacturing plants in two
countries. In country A, its cost as a function of output x  0 is
C A ( x)  n(1  3x / 100) .
In country B, its cost as a function of output y  0 is
CB ( y)  2n(1  y / 100) .
The company allocates production between the two plants in order to minimize the
total cost of producing at least q units of output worldwide, where q is some
positive value.
i)
Write this information as a non-linear programming problem, specifying
your Kuhn-Tucker conditions.
ii)
Suppose you are now told that a local restriction prevents you from not
producing in either country (for example, local governments get upset and
this precludes you from having a zero output in country A or in country B).
Using your Kuhn-Tucker conditions find the optimal amount of output for
country A and for country B. (Hint: Your answer for x and y will be in
terms of q .)
iii)
(20)
6.
Given part ii) , what numerical restriction must we impose on q for the
answer in ii) to be valid?
Consider the dynamics of price adjustments in a model of a competitive market.
Suppose that price adjusts in response to the gap between quantity demanded (qd)
and quantity supplied (qs) as follows:
p   (q d  q s )
;
 0 .
However, this model fails to consider the inventory of unsold merchandise that
arises when there is excess supply. How is the dynamics of price adjustment
affected if we take into account the possibility of existence of such inventory? The
idea may be expressed by the following differential equation which hints that there
is a downward pressure on price not only when there is excess supply being
produced at the current price, but also when there is an inventory of unsold
merchandise.


t
p   q d  q s    [q s ( )  q d ( )]d
0
;
  0,   0
The second term in equation above is the integral of (accumulated stock of) past
differences between quantity supplied and demanded. As such, it is the inventory of
unsold merchandise at time t . With   0 , this term causes price to adjust
downward when the inventory is greater than zero – just as with   0 price adjusts
upwards when there is excess demand and downward when there is excess supply.
(To simplify the analysis, assume that the inventory of unsold merchandise is always
non-negative.) By solving a differential equation, find an expression for price as a
function of time t if the demand curve is given by
q d  A  BP
and the supply curve is given by
q s  F  GP
with G  B . (Note: To simplify the analysis further assume that the roots of your
characteristic equation are real.)