Download TRENT UNIVERSITY DEPARTMENT OF ECONOMICS Dr. M. Arvin

TRENT UNIVERSITY
DEPARTMENT OF ECONOMICS
Dr. M. Arvin
Economics-Administration 2250H
2010-11
Assignment* #3
Due Date:
Beginning of class on Thursday, December 2, 2010. Penalty for late
assignments is 20% for each day or part thereof – as per course outline, page 2.
. Please make sure you read your course outline and understand this clearly. (For
example, if an assignment is late by ½ hour, it still receives a 20% penalty.)
Notes:
Please make sure that your answers are neat and legible. Also ensure that your
assignment is securely held together and contains your name and student number.
The numbers in the margin refer to the mark for each part. Please type your answers
or use a pen. Do not use a pencil.
(15) 1.
Royal Cruises sails in the Caribbean. The number of cabins that can be filled is
determined by the demand function:
Q = 20(1600 – 0.2P)½
(20)
2.
, where P < $8,000
i)
Find an expression for the price elasticity of demand. Is the demand elastic or
inelastic when the price of a cabin is $2,000?
ii)
Use differentiation to determine whether this demand function is (strictly)
concave/convex. Draw a rough diagram to indicate what your curve looks
like.
Suppose the demand and supply sides of the market are represented by the following
two equations:
Qd = b0 + b1 P
Qs = a0 + a1 P
*
As.sign.ment (ă-sin-mĕnt) n. A thing or task that is assigned to a person (Oxford American Dictionary, 1980).
where Qd and Qs represent quantities demanded and supplied, P is price, and it
is assumed that b0 >0, b1<0, b0 > a0, a1>0, and b0 a1 > b1 a0 . Suppose the
government imposes a constant per unit tax of t on the supplier on each unit it sells.
i)
Assuming there is equilibrium in the market (after the tax), find the reduced
form solution for equilibrium quantity. (Note: Obviously, the tax raises the
equilibrium price. You may assume that the tax is not set so high to result in
a zero equilibrium quantity.)
ii)
Use partial differentiation to comment on the effect of an increase in the tax
rate on the market equilibrium quantity. (If you do not use partial
differentiation, you will receive little credit for this part.)
iii)
Suppose the government is interested in maximizing its total tax revenue.
Continue to assume that equilibrium holds. What is its optimal tax rate (in
terms of the parameters of the model? (Be sure to use calculus and check
your second-order condition.)
(10) 3.
A builder wants to fence in 135,000 square feet of land in a rectangular shape.
Because of security reasons, the fence in the front will cost $2 per foot, while the
fence for the other three sides will cost $1 per foot. How much of each type of fence
will be needed in order to minimize the cost? What is the minimum cost? [Be sure
to use calculus to work out your answer and check your second-order condition.]
(15) 4.
Baumol (Quarterly Journal of Economics, 1952) and Tobin (Review of Economics
and Statistics, 1956) developed a model of money demand based on an individual’s
decisions on the trade-off between holding bonds, which pay interest, and holding
money, which is used to purchase goods and services. At the beginning of each
month, the individual gets an income of Y and converts this costlessly to bonds. Over
the course of the month, the individual makes n equal-size withdrawals, each in the
amount of Y/n. The cost of each withdrawal is c. This means that the average cash
balance held during the month is Y/2n and the total cost of managing the portfolio,
TC, is
TC = (n c) + (rY/2n),
where the first term in parentheses is the transaction cost and the second term is
interest forgone (the interest rate in effect is 100r%). Suppose the individual’s
choice variable is the number of withdrawals she makes. Treat everything else as
a parametric constant. Find the value of n that minimizes total cost (be sure to
also check your second-order condition), and calculate the average cash holdings
consistent with this value of n.
(30)
5. A monopolist producing a patented pharmaceutical product is faced with the demand
function
p=a–bq
where p is price, q is quantity, and a and b are parametric constants (both positive).
The company’s total costs are
TC =  q +  q 2
where  and  are parametric constants (both positive). Assume a > .
(10)
6.
i)
Use calculus to find the profit maximizing output of the medicine and
maximum profit to this company. (Your answer will be in terms of the
parameters of the model.) Be sure to check your second-order condition.
ii)
To reduce excessive economic profits of the monopolist from this essential
drug, the government decides to tax the company: It imposes a constant per
unit tax of t on each unit it sells. Use calculus to determine how the optimal
output and profit change with the imposition of this tax. (Note: Your answer
will be in terms of the parameters of the model; you may assume a >  + t .)
Be sure to check your second-order condition.
iii)
Use the expression you obtained for optimal output in part ii) to write down an
expression for the government total tax revenue from this firm. If the
government picks t to maximize this tax revenue, find the value of t which
maximizes this revenue. (Note that the optimal value of t will be in terms of
the parameters of the model.) Be sure to check your second-order condition.
iv)
Continuing from part iii), prove that if the government uses the optimal value
of t, it appropriates half of the protected monopolists profit prior to taxation.
A company charges $200 for each box of tools on orders of 150 or fewer boxes.
The cost to the buyer on every box is reduced by $1 for each ordered in excess of
150. Use calculus to determine for what size order the revenue is maximized. Be
sure to check your second-order condition. Calculate the maximum revenue.