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WEEK 5 CONSUMER SURPLUS RISK April 4, 2012 Consumer Surplus 2 Valuation The maximum amount the consumer is willing to pay for a product Consumer surplus (CS) for an individual consumer is the difference between the consumer’s valuation for a product and the cost of purchasing the product Aggregate consumer surplus is the sum of all individual consumer surplus EC2101 Semester 1 AY 2012/2013 WEEK 5 Example: Individual Demand for Houses 3 Prefers to buy 0 Prefers not to buy V P Consumer’s demand for the product is ìï 1 Q=í ïî 0 if if P£V P>V EC2101 Semester 1 AY 2012/2013 WEEK 5 Example: Market Demand for Houses 4 Suppose there are 4 consumers with different valuations for houses Consumer 1 values the house at 0.9 million Consumer 2 values the house at 0.7 million Consumer 3 values the house at 0.5 million Consumer 4 values the house at 0.1 million Suppose each consumer buys at most 1 house EC2101 Semester 1 AY 2012/2013 WEEK 5 5 Example: Market Demand for Houses in Graph P 0.9 0.7 0.5 0.1 0 1 2 3 EC2101 Semester 1 AY 2012/2013 4 WEEK 5 Q Example: Calculating Consumer Surplus 6 P Suppose market price is 0.3 million CS= 0.9 - 0.3+ 0.7- 0.3+ 0.5- 0.3 =1.2 0.9 0.7 0.5 P = 0.3 0.1 0 1 2 3 EC2101 Semester 1 AY 2012/2013 4 WEEK 5 Q Notes on Market Demand for Houses 7 Represents consumers’ valuation for the product in decreasing order As the number of consumers with different valuations increase, market demand curve will become “smoother” With a very large number of consumers, market demand can be modeled as a smooth curve EC2101 Semester 1 AY 2012/2013 WEEK 5 8 Consumer Surplus with Smooth Demand Curve P Suppose market demand is Q=100-P 100 The current market price is 20 CS=0.5*80*(100-20)=3,200 20 0 A Total expenditure Q 80 EC2101 Semester 1 AY 2012/2013 WEEK 5 Notes on Consumer Surplus 9 CS measures how much economic value is captured by consumers through their market transactions CS is the area below the demand curve and above the price EC2101 Semester 1 AY 2012/2013 WEEK 5 How does CS change with price? 10 P 100 30 B A 20 0 70 Q 80 EC2101 Semester 1 AY 2012/2013 WEEK 5 Network Externality 11 We have network externality if the amount of good demanded by one consumer depends on other consumers’ demand Positive network externality Quantity demanded by a typical consumer increases as demand from other people grows Negative network externality Quantity demanded by a typical consumer decreases as demand from other people grows EC2101 Semester 1 AY 2012/2013 WEEK 5 Positive Network Externality 12 P 10 8 D200 A B C D 0 200250 Q 400 EC2101 Semester 1 AY 2012/2013 WEEK 5 Negative Network Externality 13 P 1500 1000 A C B D200 0 D 200 260 Q 400 EC2101 Semester 1 AY 2012/2013 WEEK 5 14 Part 1 Uncertainty and Risk EC2101 Semester 1 AY 2012/2013 WEEK 5 What is risk? 15 If consumers do no know the consequences of a decision for sure, we have uncertainty E.g. stock price could go up or down If consequences of a decision are uncertain, we face risk Our goal How to describe/model risk? How to manage risk? EC2101 Semester 1 AY 2012/2013 WEEK 5 How to describe risk? 16 Suppose you have $1000 Option No 1: put it in your savings account risk Option 2: buy a lottery Involves risk All possible outcomes for option 2 Outcome 1: win Outcome 2: lose EC2101 Semester 1 AY 2012/2013 WEEK 5 Probability 17 How likely is each outcome? Outcome 1: 10% Outcome 2: 90% Probability measures the likelihood that each outcome occurs Where does probability come from? Objective: past experience Subjective: personal judgment EC2101 Semester 1 AY 2012/2013 WEEK 5 Expected Value 18 Payoff for each outcome of option 2 Outcome 1 (win): 10000 Outcome 2 (lose): 0 Expected value is the probability-weighted average of the payoffs associated with all possible outcomes Expected value of buying a lottery is 0.9*0+0.1*10000=1000 EC2101 Semester 1 AY 2012/2013 WEEK 5 Calculating Expected Value 19 Suppose there are n possible outcomes Outcome i’s payoff is Xi Outcome i’s probability is Pri Expected value is n E(X) = å Pri Xi = Pr1 X1 + Pr2 X2 + where + Prn Xn i=1 n å Pr = Pr + Pr + i 1 2 + Prn =1 i=1 EC2101 Semester 1 AY 2012/2013 WEEK 5 Degree of Risk 20 Suppose you have option 3: buy a stock Outcomes, probabilities, and payoffs Price increases, probability 50%, payoff $1500 Price decreases, probability 50%, payoff $500 Expected value 0.5*1500+0.5*500=1000 Are option 2 and option 3 equally risky? EC2101 Semester 1 AY 2012/2013 WEEK 5 Deviation 21 Deviation is the difference between actual payoff and expected value Option 2 (lottery) If win, deviation=10000-1000=9000 If lose, deviation=0-1000=-1000 Option 3 (stock) If price increases, deviation=1500-1000=500 If price decreases, deviation=500-1000=-500 EC2101 Semester 1 AY 2012/2013 WEEK 5 How to measure degree of risk? 22 We use standard deviation to measure the degree of risk Option 2 0.1´ (9000)2 + 0.9 ´ (-1000)2 = 3000 Option 3 0.5´ (500)2 + 0.5´ (-500)2 = 500 Option 2 is more risky Same expected value Higher standard deviation EC2101 Semester 1 AY 2012/2013 WEEK 5 Alternative Way: Variance 23 We can also use variance to measure the degree of risk Variance = standard deviation squared Option 2 0.1´(9000)2 + 0.9 ´(-1000)2 = 9000000 Option 3 0.5´(500)2 + 0.5´(-500)2 = 250000 EC2101 Semester 1 AY 2012/2013 WEEK 5 24 Part 2 Preferences towards Risk EC2101 Semester 1 AY 2012/2013 WEEK 5 Which one will you choose? 25 Option 1 (bank) Expected value: 1000 Standard deviation: 0 Option 2 (lottery) Expected value: 1000 Standard deviation: 3000 EC2101 Semester 1 AY 2012/2013 WEEK 5 Evaluating Risky Outcomes Using Utility Function 26 Consider a utility function of income U(I) Consumer’s utility from option 1 is U(1000) Consumer’s utility from option 2 is U(0) if consumer loses U(10000) if consumer wins Consumer’s expected utility from option 2 is 0.9U(0)+0.1U(10000) EC2101 Semester 1 AY 2012/2013 WEEK 5 Expected Utility 27 Expected utility is the probability-weighted sum of all utilities associated with all possible outcomes Suppose there are n possible outcomes Consumer’s utility function is U(X) Outcome i’s payoff is Xi Outcome i’s probability is Pri Expected utility is n E(U) = å Pri U(Xi ) = Pr1 U(X1 ) + Pr2 U(X2 ) + i=1 EC2101 Semester 1 AY 2012/2013 WEEK 5 + Prn U(Xn ) Expected Utility vs. Expected Value 28 Suppose U(I)=I2 Consider a risky income: 50% chance 5, 50% chance 15 Expected value=0.5*5+0.5*15=10 Expected utility 0.5U(5)+0.5U(15)=0.5*25+0.5*225=125 EC2101 Semester 1 AY 2012/2013 WEEK 5 Which one gives you the highest EU? 29 Option 1 (bank) Expected Option 2 (lottery) Expected utility: U(1000) utility: 0.9U(0)+0.1U(10000) It depends on your preference (utility function) EC2101 Semester 1 AY 2012/2013 WEEK 5 Risk Averse 30 A consumer is risk averse if the consumer prefers a certain income to a risky income with the same expected value The consumer’s utility of a certain income is higher than the expected utility of a risky income with the same expected value Risk averse consumer will choose option 1 Both options have the same expected value: 1000 Option 1 has no risk EC2101 Semester 1 AY 2012/2013 WEEK 5 Utility Function of Risk Averse Consumer 31 U(I ) U(1000) > 0.9U(0)+ 0.1U(10000) B U(10000) U(I ) Expected utility A U(1000) 0.9U(0)+ 0.1U(10000) C U(0) 1000 Expected value 10000 EC2101 Semester 1 AY 2012/2013 WEEK 5 I Expected Utility in Graph 32 U(I ) aU(0) + (1- a)U(10000), 0 £ a £1 a= 0 B U(10000) D A 0.5U(0) + 0.5U(10000) a = 0.5 U(1000) 0.9U(0)+ 0.1U(10000) C U(0) 1000 a =1 U(I ) 5000 10000 EC2101 Semester 1 AY 2012/2013 WEEK 5 I What about two options with different expected value? 33 U(I ) A risk averse consumer may or may not prefer a certain income to a risky income with higher expected value B U(10000) E A U(1000) U(I ) 0.5U(0) + 0.5U(10000) D 0.84U(0)+ 0.16U(10000) 0.9U(0)+ 0.1U(10000) C U(0) 1000 1600 5000 10000 EC2101 Semester 1 AY 2012/2013 WEEK 5 I Risk Neutral 34 A consumer is risk neutral if the consumer is indifferent between a certain income and a risky income with the same expected value The consumer’s utility of a certain income is the same as the expected utility of a risky income with the same expected value Risk neutral consumer think both options are the same EC2101 Semester 1 AY 2012/2013 WEEK 5 Utility Function of Risk Neutral Consumer 35 U(I ) U(1000) = 0.9U(0) + 0.1U(10000) U(I ) U(10000) U(1000) U(0) B A 1000 0.9U(0)+ 0.1U(10000) 10000 EC2101 Semester 1 AY 2012/2013 WEEK 5 I Risk Loving 36 A consumer is risk loving if the consumer prefers a risky income to a certain income with the same expected value The consumer’s utility of a certain income is lower than the expected utility of a risky income with the same expected value Risk neutral consumer will choose option 2 EC2101 Semester 1 AY 2012/2013 WEEK 5 Utility Function of Risk Loving Consumer 37 U(I ) U(1000) < 0.9U(0)+ 0.1U(10000) B U(10000) U(I ) C U(1000) U(0) 1000 0.9U(0)+ 0.1U(10000) A I 10000 EC2101 Semester 1 AY 2012/2013 WEEK 5 38 Part 3 How to Manage Risk? EC2101 Semester 1 AY 2012/2013 WEEK 5 What do risk averse consumers do with risk? 39 Most consumers are risk averse Risk averse consumer may still prefer a risky option Consumer will bear risk if there is enough reward to compensate for the risk When to bear risk and when to eliminate risk? How to eliminate risk? EC2101 Semester 1 AY 2012/2013 WEEK 5 Risk Premium 40 Risk premium is the maximum amount of money a risk averse consumer is willing to pay to avoid taking a risk Difference between the expected value of a risky option and a certain income to make the consumer indifferent between the two EC2101 Semester 1 AY 2012/2013 WEEK 5 Risk Premium in Graph 41 U(I ) Risk premium=60-45=15 U(I ) U(80) U(45) U(10) B D C 2 5 U(10) + U(80) 7 7 A 10 45 60 I 80 EC2101 Semester 1 AY 2012/2013 WEEK 5 Calculating Risk Premium 42 Suppose the utility function of a consumer is U(I ) = I Consumer can buy a risky asset Payoff=900 with probability 60% Payoff=400 with probability 40% What is the risk premium associated with this asset? EC2101 Semester 1 AY 2012/2013 WEEK 5 Calculating Risk Premium Cont’ 43 Expected utility of the asset 0.6 900 + 0.4 400 = 26 To get the same utility, consumer needs a certain income of I = 26 2 = 676 Expected value of the asset 0.6 ´ 900 + 0.4´ 400 = 700 Risk premium is 700-676=24 EC2101 Semester 1 AY 2012/2013 WEEK 5 Understanding Risk Premium 44 The consumer is willing to pay 24 to avoid risk Compared to a certain income of 676 Consumer will choose a risky asset only when the expected value of the risky asset is at least 24 higher than the certain income EC2101 Semester 1 AY 2012/2013 WEEK 5 Managing Risk: Insurance 45 Suppose you own a car Your annual income is $50000 You face risk With probability 95% nothing happens to your car, your income is $50000 With probability 5% you have an accident and it will cost you $10000 (your income is $40000) Expected value 0.95*50000+0.05*40000=$49500 EC2101 Semester 1 AY 2012/2013 WEEK 5 Actuarially Fair Insurance 46 Consider the following insurance policy Insurance premium Coverage $500 $10000 Expected payout 0.95*0+0.05*10000=$500 This insurance is actuarially fair because insurance premium is equal to the expected payout EC2101 Semester 1 AY 2012/2013 WEEK 5 Full Insurance 47 If you buy the insurance No accident, income=50000-500=$49500 Accident, income=50000-500-10000+10000=$49500 The insurance provides full coverage You are fully insured Insurance eliminates all risk Any risk averse consumer should buy an actuarially fair insurance that provides full coverage EC2101 Semester 1 AY 2012/2013 WEEK 5