Eco 410/510 exercise 1, Spring 2019 K. K. Yun
Justify your answers. precisely, due on Tuesday February.26
- Suppose a coupon pays pays a person 1000 dollars each year starting
immediately this year for n periods (only). If the interest rate is 4 per cent
(which is used as a discount rate), its present value is V = 1000+ 1000
(1+0:04)+
1000
(1+0:04)2 ++
1000
(1+0:04)n1 : Compute the present value of the coupon. Find
the limit of the present value when n increases to 1 (without bound). - Suppose production function for output q is given by q = K
1
2 L
1
2 ; where
K and L are quantities of capital and labor respectively. Suppose time
changes of K and L are given by: K = (t + 1); L = t; where t 1:
Compute d
dt q(t): at t = 1: - Consider f(x) = x
2
x where x > 0:
(a) compute f
0
(x): And obtain the range over which f
0
(x) > 0 and the
range over which f
0
(x) < 0:
(b) Using your answer to b., Önd x that maximizes f: - A monopolist faces a demand function x =
1
2
(11 p) where p is the price
of the product and x is the quantity demanded. The monopolist has a
cost function 2x; x 0:
(a) Writing the inverse demand function as p = p(x) where x is the
quantity demanded at p; derive the revenue function R(x) = p(x)x:
(b) Suppose that the government impose a tax of t per unit of output
sold. Derive the proÖt function (x) = R(x) c(x) tx and obtain
proÖt maximizing x as a function of t:
(c) Calculate the value of t that maximizes governmentís tax revenue.
Prove that the value maximizes the government tax revenue.
(d) What is the proÖt of the monopolist at the tax revenue maximizing
tax rate?. - Suppose you are driving south on the interstate 87 toward New York City
past Albany. You drove for exactly 2 hours after you passed Albany and
you want to estimate how many miles you have covered after Albany. All
you know is that the speed was between 65 miles per hour and 70 miles
per hour. Using the mean value theorem, give the range of miles you have given.
Sample Solution
