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# Consumer behavior1

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### Consumer behavior1

1. 1. Consumer Behavior MA (Economics), University of Kelaniya 2012 Athula Ranasinghe
2. 2. Three Approaches  Cardinal Approach  Ordinal Approach  Revealed Preference Approach
3. 3. Cardinal Approach  Economic rationality  Utility can be numerically measured  Marginal utility of money is constant  Law of diminishing marginal utility  Independent utility  Introspection method (based on own feelings)
4. 4. Illustrative Example MU 1 = 1000 - 20 X1 MU 2 = 500 - 5X 2 M = 20, P1 = 2 , P2 = 1
5. 5. Illustrative example  Problem:  To allocate given income between two commodities to maximize total utility.  Solution:  Allocate income between two commodities such that per-rupeeMarginal Utility from two commodity willequal.
6. 6. Illustrative example 1MU = 2MU 1P 2P 1000 1 2 20X- = 500 1 5X- 2 X 2 = 12X
7. 7. Illustrative example  Any commodity combination satisfying the above condition will maximize utility.  However, his freedom of choice according to the solution given aboveis limited by income constraint.  Therefore, we have to find the commodity combination satisfying both conditions.
8. 8. Illustrative example: Budget constraint M = p X + p X 11 22 20 = 2X1 + X 2 20 = 2X1 + X 2 20 = 2X1 + 2X1 20 = 4X1 X1 = 5 , X 2 = 10
9. 9. Ceteris-paribus price change  Now assume that price of commodity1 drops to Rs. 1.  This affects the budget constraint. 20 = X1 + X 2 Increases the 20 = X1 + 2X1 demand of both 20 = 3X1 commodities. Why? X1 = 7 , X 2 = 14
10. 10. Demand function p 2 1 Q 5 7
11. 11. Increase income  Suppose all other factors remain constant but income increases.  This affects demand through budgetconstraint.  The new budget constraint after income doubles is given below. 40 = 2X1 + X 2 Substituting this to first condition 40 = 2X1 + 2X1 40 = 4X1 X1 = 10
12. 12. Doubling all factors  Suppose now that all factors determining demand are doubled. (Income and all prices).  Then, the budget constraint is 40 = 4X1 + 2X 2  Note that this will not affect the demand
13. 13. What we learnt  Negative relationship between demand and price.  Positive cross-price effect?  Positive income effect.  Homogenous of degree zero.
14. 14. Ordinal Analysis  Utility cannot be quantified.  Commodity baskets can be ranked based onpreference.  Preferences are  Complete  Reflexive  Transitive  Continue  Strong monotonicity  Law of diminishing rate  Diminishing marginal rate of substitution
15. 15. Utility function  Index to map commodity space to utility space.  Indifference curve (Mathematical derivation andDiscussion)  Consumer choice under unlimited options
16. 16. Budget constraint  Consumers ability to purchase (graphical)  Equilibrium: (ability and willingness)  Mathematical approach  Marshallian Demand curve  Hicksian demand curve  Indirect utility  Expenditure function  Welfare change
17. 17. Graphical Method  Indifference map represents whatconsumer wants to do.  If he/she can have a free choice go tothe highest indifference curve.  Budget constraint represents whatthe consumer can do.  In the equilibrium, consumer choosesthe best commodity combinationwithin his/her ability,
18. 18. X1 Commodity Space and Indifference Map X2 O
19. 19. X1 Commodity Space and Budget Constraint X2 O
20. 20. X1 Commodity Space and Consumer Equilibrium: Desire, Ability match X2 O
21. 21. Mathematical Derivation of Equilibrium At equilibrium, slope of the indifference curve is equal to the slope of budget constraint. Slope of indifference curve; U = (X1 , X 2 ) dU = MU dX + MU dX = 0 11 22 dX MU 1 = - 2 = MRS dX 2 MU1 1,2
22. 22. M = p X + p X 11 22 dM = p dX + p X 11 22 dX1 p2 = dX 2 p1
23. 23. Equilibrium 1 2 MU MU = 1 2 p p Re-arranging the terms 2 2 p MU = 2 1 p MU Same condition derived from cardinal analysis.
24. 24. Demand curves Let , U = X1 a X 2 b be utility function . U MU1 =a X1 U MU2 = b X 2
25. 25. Slope of indifference curve Equilibrium condition X X b a X1 b a X 2 p 2 p 2 1 = X1 Demand Functions M X 2 p 1 M
26. 26. 1 ö . ÷÷ ø ö . ÷÷ ø b b + + ß a a a æ . çç è æ . çç è = = p
27. 27. Indirect Utility: a ß éæ a ö M ù éæ b ö M ù U =.. .. ÷÷çç÷÷çç êúê ú a+ b p a+ b p ëè ø 1 û ëè ø 2 û Solve this for M (for given U to obtain expenditure function. Expenditure function measures the minimum income required to attain a given level of utility. Given the utility level, the minimum expenditure is a function of prices. It is a homogenous of degree one function of prices. When all the prices are doubled, the minimum expenditure required to attain the given level of utility will be double the initial income.
28. 28. Expenditure function and Hicksian Demand function. First derivativeof the expenditure function will be the Hicksian demand function. Two measures of welfare change: -Compensating Variation (CV): This measures the minimum incomecompensation required for a consumer to be unaffected after pricechange (policy implemented). -Equivalent Variation (EV): This measures the minimum income shouldbe withdrawn from the consumer to be indifferent between before and after price change (policy change). Price Indices and Welfare Changes: -Two price indices, Laspreyer (base year quantity) and Paasche(current year quantity). -Laspreyer measures the minimum income compensation required fora consumer to consume the same commodity basket that he/sheconsumed before price change. Conceptually this is consistent with CV. However, this over estimates CV. (Why?) -Paasche measures the minimum income required for a consumer to consume the commodity basket that he/she would consume afterprice change. Consistent with EV. Paasche index underestimates the EV (Why?)
29. 29. Illustrative numerical example  Assume a = 0.45 and ß = 0.35.  Questions.  Identify the degree of homogeneity of thisfunction?  What is the meaning of it?  Can the utility function be homogeneous ofdegree one or above? Give reasons for your answer.  Resulted demand functions are; 0.45 ö M æ 0.35 ö M M æ M X1 = ç ÷ = 0.5625 and X 2 = ç ÷ = 0.4375 è 0.45 + 0.35 ø p1 p1 è 0.45 + 0.35 ø p2 p2
30. 30. Illustrative example  Calculate price and income elasticities of the above demand functions.  What the degree of homogeneity ofthese demand functions. Explain reasons for that.
31. 31. Illustrative example  Substitute them into the utilityfunction to derive indirect utility. 0.35 M p 1 M p 1 U U ß a ú . 2 p . . M ÷÷ . M
32. 32. 2 p ö . ÷÷ . ß a ß 0.4375 + æ . çç . êëù úû . . çç . ÷÷ . 0.45 . ö . ÷÷ . a . ß a
33. 33. 0.5625 + . ççè= æ . çç . ê . = 0.45 0.35 p p 1 2 U M 0.80 0.57796 =
34. 34. Illustrative example: indirect utility  Indirect utility function is homogeneous of degree zero. Why isthat?  When the indirect utility function issolved for M (for given level of utility), expenditure function is derived.
35. 35. Illustrative example: expenditure function U 0.80 0 0.45 0.35 0.45 0.35 M = p p = 1.73 p pU 12 120 0.57796 0.56 0.44 1.25 M = 1.98 p pU 1 2 0  Expenditure function is homogeneous ofdegree 1 with respect to prices. What is the meaning of it?  First derivative of expenditure function with respect to own price is Hicksian Compensated demand curve. Why?
36. 36. Illustrative example: Compensated demand 0.56 0.44 1.25 M = 1.98 p pU 1 2 0 ¶M -0.44 0.44 1.25 = (1.98)(0.56) p pU 1'2 0 ¶p1  Calculate own price-elasticity of the Compensated demand curve and comparethat with the own price elasticity of Marshallian demand curve.
37. 37. Numerical example  Assume that M= 1000, p1 = 1 and p2 = 1.  Use the utility and demand functions derived in previous slides.  Inserting these into the two demandfunctions; 1,000 X = 0.5625 = 563 1 1 1,000 X 2 = 0.4375 = 437 1
38. 38. Numerical example  Insert X1 and X2 to the utilityfunction; 0.45 0.35 U = ( 563 ) ( 437 ) = 145 0  Insert U0 and prices to calculate expenditure function 0.56 0.35 1.25 M = 1.98(1) (1) (145) = (1.98)(504) = 1,000
39. 39. Numerical example  Now assume that all other factors remain constant but p1 increases from 1 to 2. Using the expenditure function derived above, expenditure requires to attain theinitial level of satisfaction U0 can be 0.56 0.44 1.25 calculated. M = 1.98 p pU 1 2 0 p = 2, p = 1, U = 145 12 0 Then , 0.56 0.44 1.25 M = 1.98(2) (1) (145) = Rs .1,474
40. 40. Measuring welfare change  When price of commodity increasesfrom 1 to 2, consumer needs additional Rs. 474 to enjoy the initial level of utility. 0 , 0 2 . 0 1  This is called Compensating M p ) M ( p Variation: CV. 0 , 0 2 . 1 1 ( ) CV U U
41. 41. = p p
42. 42. CV and Laspreper Price Index  LPI (Base year basket).  How much a consumer needs to purchase the commodity basket that he/she purchased before price change.  In this example, consumer needs 2(563) + 1(437) = Rs. 1,563 to buy the initialcommodity basket after price changed. This is LPI.  Note that LPI is an over estimate of CV. Why?