Exercise 8.

3
-C. Prove that the demand law is
ab
1. The marginal revenue function of afirm is
(r- b)²
a
p=,-c.
b-x
dR = 100 0.004x. Find the
2. The marginal revenue function of a monopolist is given as dx
demand function.
functions for each of the
3. Find the total revenue functions and the market demand
following marginal revenue functions:
i) MR = 50 - 2x (ii) MR= 125-2x2
(i) MR =100 4x-3x (iv) MR = 15e-*(1 -x)
30 25
(v) MR = -6 (vi) MR= - 4
I+2 (r+1)
 (vii) MR = 1 (vii) AMR = 10e
20
2/*
30 +0.03r (x) MR
MR =
(ix)
Wr+400 (*+2)2
ab
marginal revenue function is MR = -C, show that the demand law is
4. If the (r+b
C.

5. A firnm has a marginal revenue function given by MR= -c, where x is output and a,

b,c are constants. Show that the demand law is given by p = log -c
6. The elasticity of demand of a commodity is e= 5(p+ 6)' where p denotes price. Find the
demand equation if x= 25 when p= 10.
Hint: Take log cas the constant of integration.
7. The elasticity of demand of a commodity is e=
2(r +4) , where x denotes
quantity
demanded. Find the demand equation ifx=12 whenp= 6.
p
8. The elasticity of demand of a commodity is where x is quantity demanded. Find the
demand equation if x= 8 when p = 5.
9. The elasticity of demand of a commodity is e= where O <p< 16. Find the
289- p
demand equation if x = 600 when p = 15.
10. Determine the demand function of a commodity which has a constant price elasticity of
demand a.
11. The rate of change of demand x with respect to price p is inversely proportional to
square of the price. Given that x= 20 when p = 4 and x = 19 when p =5, find the demand
equation.
dx dx k.
Hint: where k is constant of proportionality and the minus sign indi
dp p dp p
cates that x and p vary in opposite directions.
12. The marginal cost in lacs of rupees is given by MC= 4+ 5x*+ewhere x is quantity
produced. Find the total cost of production when x =2 and fixed cost is Rs 6 lacs.
13. The marginal cost function of a firm is given by MC =3000:03* +50, where x is the
quantity produced. If fixed cost is Rs 80,000, find the total cost function of the firm.
14. The marginal cost function of a firm is given by MC =0.03r² -8x+ 1500, where xdenotes
the levelof output. If fixed cost is Rs 6,500, find the total cost of the firm.
4
15.If the marginal cost function is given by MC= /3x + 36 and fixed cost is Rs 20, find
average cost of producing 15 units of output.
Integra

16.The marginal cost function of producing x units is given by MC= If the cost of
Jax +b
zero output is zero, find the totalcost function.
.The marginal cost function of a firm is MC -(logx. Find its total cost function when
total cost of producing 1unit is Rs 20
18.If MC==AC for all values of x, show that average cost is always equal to aconstant.
dC C dC dx
Hint: or
dx C
19.1f the marginal cost of a commodity is equal to one third of its average cost, show that
fixed cost is equal to zero. If the cost of producing 8 units is Rs 50, find the total cost
function.
20.1f the marginal cost of a commodity is equal to its average cost, show that fixed cost is
equal to zero. Deduce that the total cost of production is directly proportional tó the level
of output.
21.The marginal cost C(*) and marginal revenue R(«) are given by C(*) = 20+ 20
and

R(*)= 30 respectively. The fixed cost is Rs 200. Determine the maximum profit and the
number of items produced at this profit.
22. The ABCCo. Ltd. has approximated the marginal revenue function for one of its products
by MR = 20x -2x. The marginal cost is approximated by MC = 81-16x +x. If the fixed
cost is positive constant, determine the profit maximising level of output and total profit
at the optimal output.
23.A company suffers a loss of Bs 1,000 if its product does not sells at all. The marginal
+40
revenue and marginal cost of the firm are given as MR = 40- 2x and MC = x*- 22x
respectively, where x denotes the level of output. Find the equilibrium output and the
maximum profits.
24. The marginal cost and marginal revenue functions of a monopolist are MC = 26+x 2

and MR = 40-x
2 respectively, where c denotes the level of output. If the fixed cost of
production is Rs 24, find:
(1) The optimal level of output and the associated profits.
(ii) The break-even points and the total profits between them.
Hint: For total profit between the break-even points, integrate marginal profit function.
25. If the marginal revenue of a firm equals its average revenue for alllevels of output, show
that average revenue is always equal to a constant. What type of market situation
corresponds to this situation?
26. Given the following demand functions,calculate the amount of money that the consumer
is willing to pay for purchasing the given units of commodity.
(i) p=200 - 4x,when consumer purchases 25 units.
(ü) p= 350
-when
3 consumer purchases 10 units.
(ii) p= 0.01x - x+ 35, whern consumer purchases 40 units.
27.If the demand function is x 3p- 40p + 93, find the consumer's surplus when the
consumer purchases 25 units of the commodity.
8.32 Mathematics for Economics and Business
28.If the demand function is p =10e where x denotes thousand units, find consumer's
surplus when 2,000 units are purchased.
29.The supply function of a producer is given by p=** +25x +100, where x denoles
hundred units. Find producer's surplus when 600 units are sold.
30.The supply equation of a commodity is 10p - 2yx+300 = 0. Find producer's surplus
when the market price is Rs 8.
31.The supply function of a producer is given by p= 0.4e, where x denotes thousand
units. Find his surplus when sales are 2,000 units.
32. Determine the consumer's surplus and producer's surplus under pure
competition for the
demand function p = 36- x and the supply function p=6+ where p is price and x
is quantity.
33.The demand and supply laws for a commodity are pa =18- 2x x and p, = 2x -3
respectively. Find the consumer's and producer's surplus at the equilibrium.
34. Amonopolist's demand function is p=16-x, where p is price and x is quantity
demanded. If his total cost function is C=6x+ find the consumer's surplus.
2
35. Determine the consumer's surplus and the producer's surplus at equilibrium, when the
demand function is p, = 28-5x +.x? and the supply function is p, =5x+- Assume a
4 4
monopoly situation.
36.The demand and supply functions under perfect competition are p=16-x and
p=2x*+4 respectively. Find the market price and the consumer's as well as the
producer's surplus.
37.The demand law for a monopolist is p= 50 *² and the marginal cost is MC =1+x'.
Determine the consumer's surplus at a price which the monopolist will like to fix. What
willbe the consumer's surplus if the market becomes perfectly competitive?
38. Find the consumer's and the producer's surplus for the demand law p= 20-* and the
supply law p = 12+2x under conditions of
(a) monopoly and (b) pure competition.
39. The demand function for a monopolist is 3x =60- 10p and his average cost function is
20
+1+0.2x, where p and x refer to the price and quantity of the commodity
respectively. Determine the effect on consumer's surplus if the monopolist decides to
maximise sales instead of profits.
40.If the marginalpropensity to save is 0.25, find the consumption function. Given that the
level of consumption is Rs 900 at zero income.
dS = 0.35-0.2,-0.5Y here
41.If the marginal propensity to save, in an economy, is given by dY
Ydenotes income in thousand crores of rupees. If the level of consumption at an income
level of Rs 10,000 crores is Rs 8,000 crores, find the consumption function of the economy.
Integration 8.33

42. An individual's preference scale for twogoods Xand Y is defined by the marginal rate of
substitution of Yfor X, MRSyx = ay Show that s= xy can be taken as one form of the
utility function.
43.If R = aly+b)
Bx+a) is the marginal rate of substitution of Xfor Y, show that one form of the
individual's utility function is U= (r+a)(y +b.
44. The rate of salesof the product of amanufacturer is given by s't) = 1,500
450e,where
tdenotes the number of months the product is on the market. Find total sales
during the
fir_t year.
45. The current level of sales of a company is Rs 2,00,000 per year. It
has been
sales increase by 15% per year. Assuming that this rate of growth continues,estimated
find
that
(i) Total sales in3 years.
(ii) Average sales per year for the period of three years.
(iüi) Sales in the 4h year.
Hint: Average sales = 1
b-a a
46. The sales of a firm is promoted by an
ds = 100e-0.25+ where S
advertisement campaign according to the rule
denotes the sales and t denotes the number of years since the
close of campaign. Calculate the sales promoted during the 5th year and
result of advertisement. total sales as a
Hint: Total sales = 100e-0.25d

47.Suppose that the rate of sales of a new product is given by f(t) = 200- 90e,where tis
the number of daysthe product is on the market. Find total sales
during the first 4 days.
48. After an advertising campaign a product has sales rate f(t) = 1,000e ,
where t is the
number of months since the close of the campaign. Find (i) total cumulative sales after 3
months, (ii)the sales during 4h month and (i) the total sales as a result of the
49.The rate of maintenance cost of the television of a company is given by campaign.
where t is the number of years the contract is in force. What
C=100 +0.5e,
amount the company must
charge per year for an 8 years contract?
50. M/sSuper Sales Corporation is considering the opening of a new sales
Rs 4,90,000 on the belief that after a certain number of centre at a cost of
years the
offset by the savings resulting from increased sales. The rate ofincurred cost will be
savings per year is
estimated to be S=75,000+ 2,400 , where tdenotes years and S denotes rupees saved.
Find the total savings in 4 years and the number of years needed to offset the cost.
51.ABC manufacturing company wants to purchase a relatively more efficient machine. This
machine, if installed, will reduce the cost of production according to the
function
ft) = 13,500Jt where t denotes years and f(t) the reduction in cost of
price of the machine is Rs 72,000, determine the number of years
production. If the
required to offset the
cost of purchase.
 Mathematics for Economics and Business
8.34
dV 15
52. The value of a machine depreciates over time according to the relation dt where V

is the value of the machine in ten thousand rupees and t is years. Find the depreciation in
a period of4 years.
53, An investment function is given by I(0) = 150e,where t denotes time. If the initial stok
of capital is given to be 400, find (i) the time path of the capital stock and (ii) capital
formation during first four years.
54.A firm is currently manufacturing 10,000 items per day. It has been estimated that the
marginal product of the workers is given by the function MP, = 200-3/L. If the firm
employs 25 additionalworkers, determine the daily production of workers.
55. The production manager of an electric company obtained the following function:
lr)= 1,500x, where g(x) is the rate of labour hours required to assemble xth unit of the
product. This function was estimated on the basis of the information obtained from
assembling of first 150 units. If the company has received an additional order of 50 units,
determine the labour hours required to complete the job.
56.The equation ofa learning curve is y= ax', where y is the number of hours required to
produce the xth unit. Find the values of the constant a and b for an 80% learning curve if
15 hours of labour are required to produce the first unit.
Hint: 80% learning curve implies that 80% hours are required for the production of the next unit.
57.Pareto's hypothesis concerning income distribution is given by y= Axb+1), A and b
being positive constants, andy represents the number of persons with an income of Rs x.
Find the number of persons whose income exceeds p. Also find the average income
received by these persons.
Hint: No. of persons with income 2p=|
p
Ar*dx. Average income of these recipients
1 00
A
N xydx, where N=p.
b
Exercise 8.3
2. p= 100 0.0013x 3. (i) p= 50- x (ii) p=
125-x
(ii) p = 100- 2x-x (iv) p= 15e (v) p=0log(1 +)-6
25
(vi) p=-4 (vi) p=+ (vii) p= 10e%o
(ix) p=(r+ 400) + 0.01x² 120 (*) p= e+
6. X = 100 7.
P= 8. p=37-r9. x= 75/289-p 10. xp" =1 )X= 43-s
11. x=+15 12. Rs 28.63 lakh 13. 10000e0.3* +50x +70000
14. 0.01x -4x² +1500x +6500 15. 1.87 16. 2Jax +b-B)
(P+s)s
17. x(logx-2logx +2)+18e 19. 25x3 21. 800,,20022. 9, loss equal to fixed cost 23. 20,
333.33 24. (i) 7, 25 (ii) 2, 12, 0
26.(i)1250 (ii) 222.22 (ii) 813.33 27. 12 28. 101 -) 29. 594 30. 2560
31.0.6e+0.2 32. 83.33, 16.67 33. 27, 6.75 34.3.09 35. 8.67, 27.33
36.12, 5.33, 10.67 37. 28.58, 84.21 38. (a) 1.58, 6.52 (b) 5.33,4.00 39.3.75, 15
40. C = 900+ 0.75Y 41. C=0.65Y-0:4e +1.503 44. 13050.03
45.(i) 757749.58 (ii) 252583.19 (iii) 338408.82 46. 32.55,400 47.711.65
48.(1) 1553.75 (ii) 175.59 (iii) 2000 49. 286.25 50. 351200, 6.13 51, 4
 DUSNess

52.134800 53. (i) K(t) = 100 +300e05t (iü) 1916.72 54. 13750 55. 431.58
56.15, -0.322 58. p " p