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# Consumer behavior1

## by Susantha Wanasinghe on Dec 31, 2013

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## Consumer behavior1Document Transcript

• Consumer Behavior MA (Economics), University of Kelaniya 2012 Athula Ranasinghe
• Three Approaches  Cardinal Approach  Ordinal Approach  Revealed Preference Approach
• 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)
• Illustrative Example MU 1 = 1000 - 20 X1 MU 2 = 500 - 5X 2 M = 20, P1 = 2 , P2 = 1
• 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.
• Illustrative example 1MU = 2MU 1P 2P 1000 1 2 20X- = 500 1 5X- 2 X 2 = 12X
• 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.
• 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
• 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
• Demand function p 2 1 Q 5 7
• 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
• 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
• What we learnt  Negative relationship between demand and price.  Positive cross-price effect?  Positive income effect.  Homogenous of degree zero.
• 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
• Utility function  Index to map commodity space to utility space.  Indifference curve (Mathematical derivation andDiscussion)  Consumer choice under unlimited options
• 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
• 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,
• X1 Commodity Space and Indifference Map X2 O
• X1 Commodity Space and Budget Constraint X2 O
• X1 Commodity Space and Consumer Equilibrium: Desire, Ability match X2 O
• 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
• M = p X + p X 11 22 dM = p dX + p X 11 22 dX1 p2 = dX 2 p1
• 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.
• Demand curves Let , U = X1 a X 2 b be utility function . U MU1 =a X1 U MU2 = b X 2
• 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
• 1 ö . ÷÷ ø ö . ÷÷ ø b b + + ß a a a æ . çç è æ . çç è = = p
• 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.
• 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?)
• 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
• Illustrative example  Calculate price and income elasticities of the above demand functions.  What the degree of homogeneity ofthese demand functions. Explain reasons for that.
• 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
• 2 p ö . ÷÷ . ß a ß 0.4375 + æ . çç . êëù úû . . çç . ÷÷ . 0.45 . ö . ÷÷ . a . ß a
• 0.5625 + . ççè= æ . çç . ê . = 0.45 0.35 p p 1 2 U M 0.80 0.57796 =
• 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.
• 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?
• 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.
• 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
• 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
• 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
• 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
• = p p
• 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?