Download Mathematics for Economics: Exercise 2

UNIVERSITY OF BRISTOL : DEPARTMENT OF ECONOMICS
MATHEMATICS FOR ECONOMISTS - MODULE 11122
EXERCISE 4
Attempt as many relevant questions as possible before the tutorial class in week 8. The questions
marked with T will be covered in the tutorials whereas the questions marked with EL will be covered
in the exercise lecture in week 7. You are asked to hand in for marking Questions 7 and 9 to your
QM1 tutor by 5pm, Thursday, 29th November. Your mark will count towards the award of your
credit points at the end of the year.
1.
Differentiate the following showing all your steps:
(i) ln 2  3 x 2 
(iii)
T2.
1 3 x
2  5 x
2
 x3 
(ii) x 3 e 4 x
(iv)  7  3 x  x 2 
2
(v)
4 x e2 x  8x 5
2
(a) Find the price elasticity of demand, directly and by using logs, for the demand function
16
Qd  2 when p = 5.
p
(b) Find the price elasticity of supply for the supply function Qs  7  01
. p  0.004 p 2
if the current price is 80. Is supply elastic, inelastic or unit elastic at this price?
3.
Given the demand function p  60  3q where q is the output, find
(i)
the total revenue function Rq  ,
(ii)
the marginal revenue function.
(iii)
4.
the price at which marginal revenue is zero.
1 3
q  15q 2  175q  1000 where q is the output, find
2
the differential of Cq  at q = 11 and use it to estimate the change in cost as
Given the total cost function Cq  
(i) output increases from 11 to 11.4 and (ii) output decreases from 11 to 10.8.
Compare your estimated changes with the actual changes in cost.
5.
For each of the following economic functions, find any points of inflection, maxima
and minima, stating whether the points of inflection are stationary or non-stationary:
T(i) f  x   x 3  6 x 2  9 x  6
1
23
(ii) h x    x 3  2 x 2  4 x 
3
3
2
6.
The demand function for a good is p  1000  4q where q is the quantity demanded
find the value of q which maximises total revenue.
7.
A monopolist has the following total revenue and total cost functions:
Rq  33 q  4 q 2 and Cq  q 3  9 q 2  36 q  6
Find :
T8.
(i)
(ii)
(iii)
(iv)
(v)
his profit -maximising output level
his maximum profit
his AR (average revenue ) function
the price per unit at which this profit-maximising output is sold.
the differential of the revenue function at the profit maximising
output level.
A monopolist faces an (inverse) demand function p  1    
q
.
2
The cost of producing q units is given by C(q) = q. Determine the monopolist’s
optimal price and quantity if he wishes to maximise his profit.
Calculate the elasticity of demand at the optimal price.
9.
A firm has the following short-run production function:
Q  f l   
2l3
 10 l 2
3
where l is the labour input.
(i) Derive the AP function and show that , where AP is a maximum , MP = AP.
( AP is the average product and MP is the marginal product).
(ii) Find the value of l for which total product is a maximum. Also, find the maximum
total product value.
EL10. A firm has the following short-run total cost function:
Cq  q 3  9 q 2  30 q  25
(i) Derive the AVC ( average variable cost) function and show that , when AVC is a
minimum , MC = AVC where MC is marginal cost.
(ii) Derive the ATC (average total cost) function and check that , where q = 5, ATC is
a minimum and MC = ATC.
(iii) Show that the total cost function has a non-stationary point of inflection at the
value
of q for which MC is a minimum.
3
10.(iv)
Graph on the same graph the AFC (average fixed cost), AVC, ATC and MC functions
and compare the minimum values of AVC and ATC.
(For the graph take q = 0,1, 2,.....,7)
Hint: For parts(i) and (iii), you need to find the values of q that minimise AVC and MC
respectively and verify that these are the minima. There are a number of worked
solutions in Chapter 6 of Bradley and Patton that you will find helpful but please use
the notation of your lecture notes rather than the notation of Bradley and Patton where
it is different.
11.
Given the average cost function
AC 
100 e 0.05( q 10)
q 1
(i) Find the minimum AC, verifying it is the minimum.
(ii) Write down the equation of the total cost function.
C.Osborne. November 2001