M.A. (ECONOMICS) PART I (BASIC QUANTITATIVE METHODS)

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1 (BASIC QUANTITATIVE METHDS) ESSN N. 6 AUTHR : DR. VIPA CHPRA ANAYSIS F CNSUMER'S SURPUS Structure: 6. Introduction 6. jectives 6. Definition 6.. Rule to Evaluate Definite Integral 6.. Definite Integral as an Area under a Curve 6.4 Consumer's Surplus 6.4. Consumer's Surplus under Pure Competition 6.4. Consumer's Surplus under Monopoly 6.5 Producer's Surplus 6.6 Application of Integration in Economics 6.7 Summary 6.8 Recommended Books 6.9 ist of Question 6.9. Short Questions 6.9. ong Questions 6. Introduction: In previous lesson we have studied the method of finding out integration more specifically indefinite integration. It is the reverse of differentiation which deals with the situation where the derivative is given and we have to find the primitive function. After dealing with indefinite (general) integrals, we now come to another type of integrals which have unique value and are called definite integrals. The process of integration has very useful application in various fields of economics like in the analysis of Consumer's Surplus Producer's Surplus etc. 6. jectives: After reading this lesson you should e ale to: * define definite integral * eplain the concept of consumer's surplus * distinguish etween consumer's surplus under, state competition and

2 under monopoly * acquaint yourself with the concept of producer's surplus * know some more application of integration in economics 6. Definite Integral The definite integral of a function f() in the closed interval (a, ) denoted y f ( ) d is equal to the difference in the value of the integral of f()d for a two assigned values say a and of the independent variale and is written as a g g g g a f d F c F F a where F() + c = f ( ) d a a is called the lower limit and is called the upper limit of integration (a < ). The interval (a, ) is called the range of integration. 6.. Rule to Evaluate Definite Integral To evaluate a definite integral such fgd as process is as under: a (i) Find the indefinite integral fgd Fg c say. (ii) Put = (upper limit) and get F() + c (iii) Put = a (lower limit) and get F(a) + c (iv) Sutract (iii) from (ii) and get F() F(a). Note: The constant of integration does not play any role in definite integral. Eample : Evaluate the following integrals: Solution: (i) (i) d (ii) M N d t t 5 dt t d. QP M i Q P

3 (ii) 4 QP 4 QP 4 4 QP t t c 5h t dt t 5 dt t 5 t t log tqp. 5log 5 log QP log log 5 log Eample : Evaluate : 6 4 (i) g g d (ii) (iii) 9 Solution: (i) (ii) g g e QP j 4 4 d / / / 5/ / / d 4 id 8 C d d QP 4 QP 6 9 log log log 7 4 QP QP log g d

4 (iii) [ log 6 7 QP log c 6 7 g g RST g log d log. log. } d d d d d o log (+) d q d log.log log d or or Eample : g g g g g log d log log g d log g log g UVW QP Solution: 4 4 Show that d 4 4 d d / N M / 4. 4 Q P 4 / / / d i / QP Definite Integral as an Area under a Curve Every definite integral has a definite value. That value may e interpreted geometrically to e a particular area under a given curve. et y = f () e an increasing function of in a < < and let if PQ e the continuous curve for y = f (). et AC and BD e the ordinates at the points

5 = a and =. 4 The area under the curve y = f() etween the limits a and can written as a definite integral. Area under the curve = a g f d Now we will make use of definite integral to find Consumer's surplus and producers surplus. 6.4 Consumer's Surplus (CS) First we shall study the Concept of Consumer's Surplus. Definition : In the words of Dr. Marshall 'The ecess of price which a person would e willing to pay rather than go without the thing over that which he actually does pay is the economic measure of this surplus of satisfaction. It may e called Consumer's Surplus. It is estimated y the area etween the demand curve and the price which is actually paid. et P = f() e a demand function where y is the price per unit and, the numer of units. et a uyer purchases units at a price p then the total amount paid y the consumer = p. But the total satisfaction derived y the consumer = Area under the demand curve etween = and =. In Fig. area ABC represents the area under the demand curve etween = and =.

6 5 The area under the demand curve from = to = = Consumer's surplus = f d P g g f d CS= what a consumer is prepared to pay what he actually pays. Alternative Method There is an another method to find consumer's surplus y integrating the demand function (as a function of price) with respect to Price. Consumer's Surplus = P demand function as a function of priceg dp, P

7 6 where P is the price at which quantity demanded is ero. It can e otained y putting = in the demand function Consumer's Surplus under Pure Competition Under Pure Competition the point of intersection of demand and supply curves gives the equilirium price and quantity purchased. Step : tain i.e. quantity purchased y solving the demand and supply curve equations. Step : Sustitute this value of in the demand equation the corresponding value of P is otained Consumer's Surplus Under Monopoly Under Monopoly Consumer's Surplus is otained as follows :- Step : Determine Quantity purchased y equating MR and MC. Step : Sustitute the value of in demand equation. tain price Step : tain Consumer's surplus g CS f d P where P = f() is the demand function. Eample : If the market demand curve is p =, where p and are respectively the price and demand of commodity, find the consumer's surplus when p = 4 and p = 8. Solution : Given demand function is p =. When p = 4, 4 = or = 8. When p = 8, 8 = or = 6. Consumer's Surplus = fgd P When p = 4, Consumer's Surplus = d 4 8 d i 8 = 8 8 = 64 When p = 8, Consumer's Surplus = fgd P 6 6 gd 8 6 d i 48 8 g

8 7 = = 6 Eample : Find the customer's surplus for the demand function p = 5 where =. Solution : Customer's surplus = Pd P Given P = 5 when =, P = 5 5 g d [ 5 ] 5 Eample : A monopolist demand function is = P where is the quantity demanded when price is Rs. p per unit with the average cost function AC () = Find the Consumer's Surplus at the price which monopolist would like to fi. Solution : Under monopoly MR = MC = P P = P = Revenue = p = MR AC 6 TC = Total Cost ACg MC = + 6 Equate MR = MC 6 = = 9 = 4 g

9 8 When = 4, 4 = P P = 6 Consumer's Surplus = gd M N Q 4 P Eample 4: The marginal cost function of a firm is given y MC = e. + 5, when is quantity produced. If fied cost is Rs. 8,, find the total cost function of the firm. Solution : Given that MC = e. + 5 and C() = 8, Integrating the marginal cost function, we otain Cg. de 5id k = e k C() = + k = 8 k = 7 Thus total cost function of the firm is given y C() = e Producer's Surplus Sometimes, it so happens that a producer is willing to sell his product at a certain price ut when he actually sells the article he gets more than his epectation. In such cases the difference etween the price at which the producer would e willing to sell his product and the price he actually receives is called Producer's Surplus.

10 9 Suppose producer sells limits at a price P, then the amount which epects is given y the area under the supply curve from = to = and the amount which he actually receives = P. In fig. ABC is the area under the supply curve from = to = Area ABC = g f d Amount actually received y the producer = P = Area DBC Producer's Surplus = P g f d Alternatively Producer's Surplus = Integrate the supply function (as a function of p) with respect to p. P PS Supply function as a function of pg dp. P where p is the price otained y setting = in the supply function. Eample : The supply function of a product is y = + 6. Find the producer's

11 surplus when units are supplied. When =, y = + 6 = 6 Solution: Producer's Surplus = 6 ( 6) d 6 6 = 6 6 = Rs. Eample : Given the demand function P = 5 and the supply function P = 4 +, find the Consumer's Surplus (CS) and Producer's Surplus PS). Solution: Here Demand function is P = 5...(i) Supply function is P = (ii) Solving (i) and (ii), we get 5 = 4 + or = = and P = 5 = 5 = CS ( 5) d and PS ( 4 ) d 6 Eercise :. The su pply cu rve for a commodity is p 9 and the quantity sold is 7 units. Find the Producer's Surplus. Can you find the Consumer's Surplus? If yes find it, if not eplain with the help of diagram why not.. Find the Producer's Surplus for the Supply function p = 9 when = 7. Under pure competition for a commodity the demand and supply laws are P d 8, and P S respectively. Determine the Consumer's Surplus and the Producer's Surplus.

12 6.6 More Application of Integration in Economics. To find Total Revenue for selling units when Marginal revenue is given Total Revenue = M arginal Re venued K Where K is the constant of integration and its value can e found y puttng the initial condition vi., TR = when q = Eample: If the marginal revenue function is MR = 8 6p q, determine the revenue and demand function. Solution : Given M = 8 6q q TR MRdq K 8 6q q dq K 6q TR 8q q K or TR 8q q q K Initial Condition, when q =, TR = K = TR 8q q q. To find the total cost when marginal cost is given Total cost for producing units = c () = MC( ) d K, where k is constant of integration and is determined from the initial conditions given in the question.. To find the maimum profit when Marginal Revenue and Marginal Cost Functions are given Profit = TR TC where TR = MR dq K and TC = MCdq K M Cost d + K i.e. Total Profit = arginal Re venue Marginal To find output that maimises total profits, we equate MR = MC. Eample : The marginal cost and marginal revenue of a firm are given as

13 MC = and MR =. Compute the total profits, given that TC at output ero is. Solution : Given MC = 4 +.8, MR = TC MC ( ) d K ( 4. 8 ) d K.8 TC 4 K TC = K Initial Condition When =, TC = K = = TC = TR MR d K d K TR = + K Initial Condition When =, TR =, K = TR = Profits = TR TC P = 4.4 At equilirium, MR = MC =.8 = 8 = At = Profits = 4.4 () = 4.4 = 8 4 = 4 4. To Find Consumption Function from Marginal Propensity to Consume Consumption function is given y C MPCdy K Here, C is the consumption and Y e the level of income, K is an aritrary constant. The value of K can e determined y putting the initial condition. Eample : Find the consumption function given that income equal dc Consumption when Y =, and that marginal prosperity to consume.7, dy where Y is income and C is consumption.

14 Solution : The marginal propensity to consume is dc.7 dy C.7dy K, where K is constant of integration =.7Y + K (i) Where y =, then Y = C =.7 + K K = from (i) C =.7Y +, which is the required consumption function. 5. To find demand function if elasticity of demand is given Price elasticity of demand = e p = e p e p e p dp d. p logp dp p d logk logp log logk e p e log P p e P p K log logk logk p d. dp Eample : tain the demand function for a commodity whose elasticity of demand is given as a P, where a and are constants and P denotes the e p price per unit of the commodity. Solution : Elasticity of demand d a P dp P a dp P P. d dp a P

15 4 Integrating oth sides we get log = a log P P + C log logp a log e e PC log e e log e = log P a e P+C = P a e P+C, where C is constant of integration. 6. To find capital formation from investment function. 7. In compound interest calculations when compounding is done continuously. 6.7 Summary In the present lesson we have studied the use of definite integration in finding Consumer's Surplus under: (i) Pure competition and (ii) under monopoly. Besides this integration is useful in finding total cost (given marginal cost), total revenue (given marginal revenue), maimum profit (given marginal revenue and marginal cost, and producer's surplus (given supply function) etc. Thus the technique of integraiton has very useful application in the field of Business and Economics. 6.8 Recommended Books :. Dr. S.C. Aggarwal : Basic Mathematics for Economists and Dr. R.K. Rana. Dr. D. Bose : Introduction of Mathematical Economics. Dr. B.M. Aggarwal : Mathematics Economics 4. Dr. Aggarwal and Joshi : Mathematics for Students of Economics 6.9 ist of Questions 6.9. Short Questions. Evaluate (i) ( 7) d 8 (ii) X d. If MC =, find the Total Cost.. If the Marginal Revenue Function MR = 4 Q, find the Total Revenue Function. 4. Define Consumer's Surplus. e

16 5 5. Evaluate 5 d 6. If the marginal revenue is given y MR = 7 + find the total revenue function ong Questions. The demand and supply functions under pure competition are P = 6 and P = + 4 respectively. Find the consumer's and producer's surplus.. Under a monopoly the quantity sold and market price are determined y the demand function. If the demand function for a profit maimiation monopolist is P = 74 and MC = 4 +, find CS.. The demand function for a commodity P = D. The supply function P = D. Find Consumer's Surplus. 4. If the demand function is P = 5 and the demand is, what will e the Consumer's Surplus? 5. Given the Demand function P = 8 and the supply function P = +, find the Consumer's Surplus and the Producer's Surplus. This lesson has een written from the assistance taken from Distance Education Council.