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 Page 1 13. R’(5). where alpha is a constant. Show that the elasticity of average cost is (K – 1). If the demand curve for sugar is p = 8 – d. MR function. What is the demand when MR is zero? 17. (note x = quantity) 16. Find the out for maximum net revenue.Economics Department. Given the demand curve p = 10 – 4d. Mumbai . Given p = 16 – d2/2. The demand curve is p = 12 – 3x.2x and its cost function is given to be C = 25x + 10000. 20. Prove the mathematical conditions to be satisfied for profit maximization and also for loss minimization. Also find the edp when d = 4.05p + p1/2 for p = 4. If the total cost function is C = f(x). Compute price elasticity of supply for the supply function: S = 20 – 0. St Xavier’s College. TR. The firm sells its product in a competitive market and the fixed market price is Rs 100/3 per unit of x. Given p = 30 – 5d. If AR = 20. Find the necessary and sufficient conditions for determining output for maximum profits. R’(1). 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. 23. Verify your answer for the total cost function which given as ax2 + bx. 22. the price. find the TR and MR of this firm. TC and profit at this point.11. A firm has a total cost function C = (1/10)(x3) – 3x2 + 50x + 100. Show that edp is always – 1 for the demand curve (d)(p) = alpha. find the TR function. 15. 12. 21. find the respective price elasticity of demand at P = 1 and P = 3 19. Also find the price it will charge. the elasticity of demand with reference to price is 2. The average cost is c = 2x2. Find the value of the MR at the optimum output. P = 16 and p = 100. At what demand would MR be zero ? 18. Find MR and comment on it. 14. 24. find: elasticity of total cost (K). Derivatives & Economic Applications Page 2 . where x is the output. If the monopolist faces a demand curve d = 20 – p/3. find the elasticity of demand when p = 5. Find the output at which the profits of the firm are maximum. The demand function faced by a firm is p = 500 – 0. Find the optimum output of a firm whose TR and TC are given as R = 30x – x2 and C = 20 + 4x. find the partial elasticities of demand for x1 wrt P1.25. where x is total putout. Find minimum AC. Derivatives & Economic Applications Page 3 . 32. where x is output and C is the total cost of output. B and C. find the direct price elasticity (e11) and the cross price elasticity (e12).3 – 1. Find the MPL and MPK for the following production function: x = L 0. 34. The production function of a commodity is q = 10x1 + 5x2 – x12 – 2x22 + 3x1x2. 30.Economics Department. The production function of a commodity when three inputs A.1x2 + 5x + 100) when x units are produced per week. The total cost function is C = 15x – 6x2 + x3.5P3. y) is given by : u = x2 + xy + y2.42C0. b = 2 and c = 3. Find the AP and MP curves for labour when 10 acres are cultivated.2P2 + 0. Find the marginal productivity of X1 and X2. 31. Determine these productivities if we are given that a = 1. 36. where x is output. Evaluate them for x = 1 and y = 2. Find the marginal productivities of A. where x is total putout. e12 and e13 given that P1 = 10. Mumbai . A firm’s total cost is Rs (0. L is labour and C is capital. P2 = 8 and P3 = 7. B and C are used in quantities a. 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. St Xavier’s College. Verify that minimum AC is equal to MC 28.25. Estimate the values of e11. The average cost curve of a good is given to be c = 1 + 120x3 – 6x2. Find the AC function and MC function.01L 0. 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. 35.75C0. 33. P2 and P3. What is the supply curve of the firm? At what price will 150 units be produced ? 26. Let the demand function for good x1 be x1 = 63. Find the MPL and MPK for the following production function: x = 1.58. The fixed market price is Rs P per unit of x. b and c is : q = 10a + 20b + 8c – a2 + 2b2 –c2ab. If the utility function u = f(x. The employment of ‘a’ man-hours on ‘b’ acres of land gives a wheat cultivating farmer q = 2(12ab – 5a2 + 4b2) bushels of wheat. What is MPX1 and MPX2 if x1 = 1 and x2 = 5 ? 29.9P1 + 0. L is labour and C is capital. find MUx and MUy. If the demand for a good X1 is given by the demand law x1 = a1 – a11P1 + a12P2. Mumbai . Evaluate these elasticities if we are given K = 100 and P = 45. If the demand for X3 is X3 = 10. The estimated demand for commodity C is given by the following function: DC = 1. 44. eAC and eAD. PC = 3 and PD = 1. The demand curves for products X1 and X2 are P1 = 1 – X1 and P2 = 1 – X2. P2 = 9 and P3 = 7.023)(Px-1. If the consumption function for good X is: x = (177. PB = 2. PC and PD respectively. 41.03PD. The prices of these commodities are P C and PD respectively. find eCC and eCD. Consider two commodities C and D. P2 and P3 and evaluate them if we are given P1 = 8.6)(y-0.03PC + 0.Economics Department. 40. K is an index of purchasing power ad P is the price of bicycles. Consider four commodities A. PX is the price of good X and P is the average retail price of all other commodities. Evaluate these elasticities given PA = 1.939). The demand functions for two goods X1 and X2 are estimated as follows: and X2 respectively. 38. PB.2K – 8.1P1 + 0. 39. X2 and X3 with prices P1.0. where y is the aggregate real income. find the income elasticity and the two partial elasticities of demand for X.6P – 379. Find the purchasing power elasticity.3P3.1x2 + 5xy + 0.02y2 + 25 43.04)(P0. Evaluate these elasticities given PC = 2 and PD = 1. Find the partial marginal costs functions of good X and good Y for the following joint cost function: C = 0.36PB . The estimated demand for commodity B is given by the following function: DB = 49. C and D. find the partial elasticities of demand for X3 wrt to P1. where B is the annual total purchase of bicycles. where P1 and P2 are the prices of good X1 and good Determine whether the goods are competitive or complementary. P2 and P3 respectively. find eAA. The demand for bicycles in a country is estimated to be B = 11. Derivatives & Economic Applications Page 4 .30 – 0. 42. eAB. Find the necessary and sufficient St Xavier’s College.3 + 0.1P2 .X22 – X1X2.07 – 0.37.0. The prices of these commodities are PA. B.02PA – 0.05PC + 0.01PD. the price elasticity of demand (purchase) of bicycles. The total cost of jointly producing these goods is C = X1X2 and the profit function is Z = X1 – X12 + X2 . Consider three commodities X1. Find the necessary and sufficient conditions for maximum profits. Determine the maximum output q subject to the given cost constraint. 53. Find x and y which maximizes u. subject to the given production function. Assume a production function of the form q = 10 – 1/x1 – 1/x2. Mumbai . St Xavier’s College. Given the utility function of an individual as: u = 4x 1 + 17x2 – x12 – x1x2 – 3x22 and the budget constraint Y = x1 + 2x2 = 7. Given that the production function is q = f(x. r1 and r2 are the input prices and b0 is the fixed cost of production. find his purchase of the two goods for maximum utility. then assuming perfect competition. If the cost function is C = 5x + 2y + 10. 48. total cost and profit when profit is maximum. 50. given u = xy subject to x + y = 6. find the input demands. 52. Find x and y which maximizes u. input requirement and profit subject to the given production function. y) and that the total cost constraint is C = r1x + r2y + b0.conditions for maximum profits and determine the production of these commodities at this level. The demand curves for two commodities are: P1 = 28 – 3x1 and P2 = 22 – 2x2. Find the maximum of z = 10x + 20y – x2 – y2. Derivatives & Economic Applications Page 5 . 51. If z = xy. Given the cost function C = r1x1 + r2x2 + b0 and a constraint q = f(x1. if the firm sells in a competitive market. given the production function q = xy. where x and y are inputs. and r1 and r2 are the input price and we are told r1 = 1 and r2 = 8. subject to 4x + 2y = 60. find the first order condition for minimum cost for given q at q0. 46. 54. The joint cost function is C = x12 + 3x22 + 4x1x2. the quantity produced and the profits earned by the firm. 56. 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. 49. If the price of the product is P and we are given P = 9 and if r 1 and r2 are the input prices of X1 and X2 respectively and we are given r1 = 1 and r2 = 4. then maximize this function subject to the constraint x + 3y = 5. x2). Determine the prices. minimize the total cost of output q = 40. 45. where u = xy + 2x. 55.Economics Department. 47. The production function is q = xy and the cost constraint is 50 = 2x + y + 10. subject to 2x + 5y = 10. Find the necessary and sufficient conditions for maximum output subject to the cost constraint. find the equilibrium output. 62. The joint cost function is as follows: C = x12 + 3x22 + x1x2. 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. Given U = (x + 2)(y + 1) and PX = 2 and PY = 5 and income I = 51. The price of the product is P = 9 and the input prices are PX1 = 1 and PX2 = 4. Let the utility function be u = x2y3. Mumbai . The demand curves for two commodities are: P1 = 7 and P2 = 20. Let the utility function be u = xy.57. 59. 61. Given PX = 1 and PY = 4 and income I = 10. Find the input demands. Given PX = 1 and PY = 9 and income I = 10. Find the necessary and sufficient conditions for maximum profits. total cost and profits at the level of maximum profits. Let the utility function be u = xy. 60. find the demand for X and Y which maximizes utility.Economics Department. find the demand for X and Y which maximizes utility. 58. Given PX = 1 and PY = 2 and income I = 10. Find the demand for X and Y which maximizes utility. Determine the prices. find the demand for X and Y which maximizes utility. The production function of a firm is : . St Xavier’s College. Derivatives & Economic Applications Page 6 .