derivative application sums

ECONOMIC APPLICATION OF DERIVATIVES

(Acknowledgement: Sums numbers 1 to 62 have been taken from: ‘Elementary Methods of Mathematical Economics’ written by Dr. (Mrs) Pratibha Borwankar, Seth Publishers

Pvt. Ltd., Mumbai 1995)

1. The demand curve for a commodity is p = 36 – 2d2. The average cost curve is

given as c = 15d. Find the equilibrium output and price under profit

maximization assumption

2. The demand curve for a commodity is p = 50 – 4d. The average cost is zero.

Find the necessary condition for maximum profits.

3. The demand curve for a commodity is p = 10 – 5d. The average cost is c = 3.

Find the TC, MC, TR, MR, maximum profits. Write the necessary condition

and sufficient conditions for maximum profits.

4. The total cost function is C = 2q – 2q2 + q3. Find the minimum AC and verify

that at optimum output AC = MC.

5. The total output varies with the use of labour according to the function which

is given to be Q = 10 + 12L – L2. Find the average product and marginal

product of labour. What should L be to maximize output ? Also find total

output, marginal product and average product of labour at this point.

6. Given C = x3/3 – 3x2 + 9x +16 and R = 21 – x2, does the producer sell in a

monopolistic or a competitive market ? Find the output for maximum revenue.

What is the maximum value of R? Find the output that maximizes his net

revenue.

7. Find the optimum level of production of a firm whose revenue and cost

functions are given as follows: R = 30x – x2 and C = 20 + 4x. (Note: x is the

level of output).

8. Explain the economic significance (in mathematical terms) of the following

values of price demand elasticities: 1, 0, infinity, 4 and ½

9. The demand function is given to be p = 90 – 1/5d. Find the level of output for

which MR will be zero. What will be the price of this output ? If the total cost

function is C = 20 + 2q2 – 20q, find the equilibrium output and price.

10.For a monopolist firm, the cost function is C = 0.004x3 + 20x + 5000 and the

demand function for this firm is p = 450 – 4x. Find the profit maximizing

output. At this level show that MC = MR (Note: x = quantity).

St Xavier’s College, Mumbai - Economics Department. Derivatives & Economic Applications

11. If the monopolist faces a demand curve d = 20 – p/3, at what level would his

total revenue be maximum ? What would be the price of this output level ?

Find out the value of total revenue for this output level.

12.Given p = 30 – 5d, find the elasticity of demand when p = 5. Also find the edp

when d = 4.

13.Prove the mathematical conditions to be satisfied for profit maximization and

also for loss minimization.

14. If AR = 20, the elasticity of demand with reference to price is 2. Find MR and

comment on it.

15.The demand function faced by a firm is p = 500 – 0.2x and its cost function is

given to be C = 25x + 10000. Find the output at which the profits of the firm

are maximum. Also find the price it will charge. (note x = quantity)

16.Given the demand curve p = 10 – 4d, find the TR and MR of this firm. What is

the demand when MR is zero?

17.Given p = 16 – d2/2, find the TR function, MR function, R’(1), R’(5). At what

demand would MR be zero ?

18. If the demand curve for sugar is p = 8 – d, find the respective price elasticity of

demand at P = 1 and P = 3

19.Show that edp is always – 1 for the demand curve (d)(p) = alpha, where alpha

is a constant.

20.Compute price elasticity of supply for the supply function: S = 20 – 0.05p + p1/2

for p = 4, P = 16 and p = 100.

21. If the total cost function is C = f(x), where x is the output, find: elasticity of total

cost (K). Show that the elasticity of average cost is (K – 1). Verify your answer

for the total cost function which given as ax2 + bx.

22.A firm has a total cost function C = (1/10)(x3) – 3x2 + 50x + 100. The firm sells

its product in a competitive market and the fixed market price is Rs 100/3 per

unit of x. Find the out for maximum net revenue.

23.Find the optimum output of a firm whose TR and TC are given as R = 30x – x2

and C = 20 + 4x. Find the value of the MR at the optimum output.

24.The demand curve is p = 12 – 3x. The average cost is c = 2x 2. Find the

necessary and sufficient conditions for determining output for maximum

profits, the price, TR, TC and profit at this point.


25.A firm’s total cost is Rs (0.1x2 + 5x + 100) when x units are produced per

week. The fixed market price is Rs P per unit of x. What is the supply curve

of the firm? At what price will 150 units be produced ?

26.The total cost function is C = 15x – 6x2 + x3, where x is output and C is the

total cost of output. Find the AC function and MC function. At what output is

AC minimum ? What is the minimum AC ? At what output is MC minimum ?

What is the minimum MC ? At what output are average and marginal costs

equal ?

27.The average cost curve of a good is given to be c = 1 + 120x3 – 6x2, where x

is output. Find minimum AC. Verify that minimum AC is equal to MC

28.The production function of a commodity is q = 10x1 + 5x2 – x12 – 2x2

2 + 3x1x2.

Find the marginal productivity of X1 and X2. What is MPX1 and MPX2 if x1 = 1

and x2 = 5 ?

29.The production function of a commodity when three inputs A, B and C are

used in quantities a, b and c is : q = 10a + 20b + 8c – a2 + 2b2 –c2ab. Find the

marginal productivities of A, B and C. Determine these productivities if we are

given that a = 1, b = 2 and c = 3.

30.The employment of ‘a’ man-hours on ‘b’ acres of land gives a wheat

cultivating farmer q = 2(12ab – 5a2 + 4b2) bushels of wheat. Find the AP and

MP curves for labour when 10 acres are cultivated.

31.Find the marginal product and the average product of L and K when we are

given L = 2 and K = 4 for the following function : q = 4L2 + 15LK + K2.

32.Find the MPL and MPK for the following production function: x = 1.01L0.75C0.25,

where x is total putout, L is labour and C is capital.

33.Find the MPL and MPK for the following production function: x = L0.42C0.58,

where x is total putout, L is labour and C is capital.

34. If the utility function u = f(x, y) is given by : u = x2 + xy + y2, find MUx and MUy.

Evaluate them for x = 1 and y = 2.

35. If the demand for a good X1 is given by the demand law x1 = a1 – a11P1 + a12P2,

find the direct price elasticity (e11) and the cross price elasticity (e12).

36.Let the demand function for good x1 be x1 = 63.3 – 1.9P1 + 0.2P2 + 0.5P3, find

the partial elasticities of demand for x1 wrt P1, P2 and P3. Estimate the values

of e11, e12 and e13 given that P1 = 10, P2 = 8 and P3 = 7.


37.Consider four commodities A, B, C and D. The prices of these commodities

are PA, PB, PC and PD respectively. The estimated demand for commodity B is

given by the following function:

DB = 49.07 – 0.02PA – 0.36PB - 0.03PC + 0.03PD, find eAA, eAB, eAC and eAD.

Evaluate these elasticities given PA = 1, PB = 2, PC = 3 and PD = 1.

38.Consider two commodities C and D. The prices of these commodities are PC

and PD respectively. The estimated demand for commodity C is given by the

following function:

DC = 1.30 – 0.05PC + 0.01PD, find eCC and eCD. Evaluate these elasticities

given PC = 2 and PD = 1.

39. If the consumption function for good X is: x = (177.6)(y-0.023)(Px-1.04)(P0.939),

where y is the aggregate real income, PX is the price of good X and P is the

average retail price of all other commodities, find the income elasticity and the

two partial elasticities of demand for X.

40.The demand for bicycles in a country is estimated to be B = 11.2K – 8.6P –

379, where B is the annual total purchase of bicycles, K is an index of

purchasing power ad P is the price of bicycles. Find the purchasing power

elasticity, the price elasticity of demand (purchase) of bicycles. Evaluate these

elasticities if we are given K = 100 and P = 45.

41.The demand functions for two goods X1 and X2 are estimated as follows:

x1=100

P1P212 and x2=

500

P2P113 where P1 and P2 are the prices of good X1 and good

X2 respectively. Determine whether the goods are competitive or

complementary.

42. Find the partial marginal costs functions of good X and good Y for the

following joint cost function: C = 0.1x2 + 5xy + 0.02y2 + 25

43.Consider three commodities X1, X2 and X3 with prices P1, P2 and P3

respectively. If the demand for X3 is X3 = 10.3 + 0.1P1 + 0.1P2 - 0.3P3, find the

partial elasticities of demand for X3 wrt to P1, P2 and P3 and evaluate them if

we are given P1 = 8, P2 = 9 and P3 = 7.

44.The demand curves for products X1 and X2 are P1 = 1 – X1 and P2 = 1 – X2.

The total cost of jointly producing these goods is C = X1X2 and the profit

function is Z = X1 – X12 + X2 - X2

2 – X1X2. Find the necessary and sufficient


conditions for maximum profits and determine the production of these

commodities at this level.

45. If z = xy, then maximize this function subject to the constraint x + 3y = 5.

46.Find the maximum of z = 10x + 20y – x2 – y2, subject to 2x + 5y = 10.

47.Given the utility function of an individual as: u = 4x1 + 17x2 – x12 – x1x2 – 3x2

2

and the budget constraint Y = x1 + 2x2 = 7, find his purchase of the two goods

for maximum utility.

48.Find x and y which maximizes u, where u = xy + 2x, subject to 4x + 2y = 60.

49.Find x and y which maximizes u, given u = xy subject to x + y = 6.

50.Given that the production function is q = f(x, y) and that the total cost

constraint is C = r1x + r2y + b0, where x and y are inputs, r1 and r2 are the input

prices and b0 is the fixed cost of production. Find the necessary and sufficient

conditions for maximum output subject to the cost constraint.

51.Given the cost function C = r1x1 + r2x2 + b0 and a constraint q = f(x1, x2), find

the first order condition for minimum cost for given q at q0.

52.The production function is q = xy and the cost constraint is 50 = 2x + y + 10.

Determine the maximum output q subject to the given cost constraint.

53. If the cost function is C = 5x + 2y + 10, minimize the total cost of output q =

40, given the production function q = xy.

54.Assume a production function of the form q = 10 – 1/x1 – 1/x2. If the price of

the product is P and we are given P = 9 and if r1 and r2 are the input prices of

X1 and X2 respectively and we are given r1 = 1 and r2 = 4, then assuming

perfect competition, find the equilibrium output, input requirement and profit

subject to the given production function.

55.The production function of the firm is q = 5 – x1-1/2 – x2

-1/2 and the price of the

product is P and we are given P = 2, and r1 and r2 are the input price and we

are told r1 = 1 and r2 = 8, if the firm sells in a competitive market, find the input

demands, the quantity produced and the profits earned by the firm, subject to

the given production function.

56. The demand curves for two commodities are: P1 = 28 – 3x1 and P2 = 22 –

2x2. The joint cost function is C = x12 + 3x2

2 + 4x1x2. Find the necessary and

sufficient conditions for maximum profits. Determine the prices, total cost and

profit when profit is maximum.


57.The demand curves for two commodities are: P1 = 7 and P2 = 20. The joint

cost function is as follows: C = x12 + 3x2

2 + x1x2. Find the necessary and

sufficient conditions for maximum profits. Determine the prices, total cost and

profits at the level of maximum profits.

58.Let the utility function be u = xy. Given PX = 1 and PY = 2 and income I = 10,

find the demand for X and Y which maximizes utility.

59.Let the utility function be u = x2y3. Given PX = 1 and PY = 4 and income I = 10,


60.Let the utility function be u = xy. Given PX = 1 and PY = 9 and income I = 10,


61.Given U = (x + 2)(y + 1) and PX = 2 and PY = 5 and income I = 51. Find the

demand for X and Y which maximizes utility.

62.The production function of a firm is : q=12x1 x2−x1−x2

x1 x2. The price of the

product is P = 9 and the input prices are PX1 = 1 and PX2 = 4. Find the input

demands, the total production and the profits made by the firm under the

assumption that the firm wants to make profits by selling under competitive

conditions.


derivative application sums

Documents