Quản trị kinh doanh - Chapter 4: Consumer choice

1Consumer ChoiceChapter 4Copyright (c)2014 John Wiley & Sons, Inc.2Chapter Four Overview1. The Budget Constraint2. Consumer Choice3. Duality4. Some Applications5. Revealed PreferenceChapter FourCopyright (c)2014 John Wiley & Sons, Inc.3Budget Set: The set of baskets that are affordableBudget Constraint: The set of baskets that the consumer may purchase given the limits of the available income. Budget Line: The set of baskets that one can purchase when spending all available income.Chapter FourKey DefinitionsCopyright (c)2014 John Wiley & Sons, Inc.4 Price of x: Px ; Price of y: Py Income: ITotal expenditure on basket (X,Y): PxX + PyY Assume only two goods available: X and YThe Basket is Affordable if total expenditure does not exceed total Income:PXX + PYY ≤ IChapter FourThe Budget ConstraintCopyright (c)2014 John Wiley & Sons, Inc.5Two goods available: X and YI = $10Px = $1Py = $2All income spent on X → I/Px units of X boughtAll income spent on Y → I/Py units of X boughtChapter FourA Budget Constraint ExampleBudget Line 1: 1X + 2Y = 10OrY = 5 – X/2Slope of Budget Line = -Px/Py = -1/2Copyright (c)2014 John Wiley & Sons, Inc.6I/PX = 10YX••AC B•I/PY= 5Budget line = BL1-PX/PY = -1/2Chapter FourA Budget Constraint ExampleCopyright (c)2014 John Wiley & Sons, Inc.7Chapter FourBudget ConstraintLocation of budget line shows what the income level is.Increase in Income will shift the budget line to the right.More of each product becomes affordableDecrease in Income will shift the budget line to the left.less of each product becomes affordableCopyright (c)2014 John Wiley & Sons, Inc.8 YX 10 5BL1612BL2I = $12PX = $1PY = $2Y = 6 - X/2 . BL2If income rises, the budget line shifts parallel to the right (shifts out)If income falls, the budget line shifts parallel to the left (shifts in)Shift of a budget lineChapter FourA Budget Constraint ExampleCopyright (c)2014 John Wiley & Sons, Inc.9Chapter FourBudget ConstraintLocation of budget line shows what the income level is.Increase in Income will shift the budget line to the right.More of each product becomes affordableDecrease in Income will shift the budget line to the left.less of each product becomes affordableCopyright (c)2014 John Wiley & Sons, Inc.10 YX 56I = $10PX = $1PY = $3Y = 3.33 - X/3 . BL2 BL1BL23.3310If the price of X rises, the budget line gets steeper and the horizontal intercept shifts inIf the price of X falls, the budget line gets flatter and the horizontal intercept shifts outChapter FourRotation of a budget lineA Budget Constraint ExampleCopyright (c)2014 John Wiley & Sons, Inc.11Two goods available: X and YI = $800Px = $20Py = $40All income spent on X → I/Px units of X boughtAll income spent on Y → I/Py units of X boughtChapter FourA Budget Constraint ExampleBudget Line 1: 20X + 40Y = 800OrY = 20 – X/2Slope of Budget Line = -Px/Py = -1/2Copyright (c)2014 John Wiley & Sons, Inc.12Chapter FourA Budget Constraint ExampleCopyright (c)2014 John Wiley & Sons, Inc.13Consumer’s Problem:Max U(X,Y) Subject to: PxX + PyY < IChapter FourConsumer ChoiceAssume:  Only non-negative quantities  "Rational” choice: The consumer chooses the basket that maximizes his satisfaction given the constraint that his budget imposes.Copyright (c)2014 John Wiley & Sons, Inc.14Interior Optimum: The optimal consumption basket is at a point where the indifference curve is just tangent to the budget line.A tangent: to a function is a straight line that has the same slope as the functiontherefore.“The rate at which the consumer would be willing to exchange X for Y is the same as the rate at which they are exchanged in the marketplace.”Chapter FourInterior OptimumMRSx,y = MUx/MUy = Px/PyCopyright (c)2014 John Wiley & Sons, Inc.15YX •Optimal Choice (interior solution)ICBL0C••BPreference DirectionChapter FourInterior Consumer OptimumCopyright (c)2014 John Wiley & Sons, Inc.16 Chapter FourInterior Consumer OptimumCopyright (c)2014 John Wiley & Sons, Inc.17Chapter FourBasket A:MRSx,y = MUx/MUy = Y/X = 4/4 = 1Slope of budget line = -Px/Py = -1/4Basket B:MRSx,y = MUx/MUy = Y/X = 1/4Interior Consumer OptimumAssumptions U (X,Y) = XY and MUx = Y while MUy = X I = $1,000 P = $50 and P = $200 Basket A contains (X=4, Y=4) Basket B contains (X=10, Y=2.5) Question: Is either basket the optimal choice for the consumer?XYCopyright (c)2014 John Wiley & Sons, Inc.18YX •U = 2502.51050X + 200Y = IChapter FourInterior Consumer OptimumExampleCopyright (c)2014 John Wiley & Sons, Inc.19“At the optimal basket, each good gives equal bang for the buck”1. MUx/Px = MUY/PY2. PxX + PyY = INow, we have two equations to solve for two unknowns (quantities of X and Y in the optimal basket): Chapter FourEqual Slope ConditionMUx/Px = MUy/PyCopyright (c)2014 John Wiley & Sons, Inc.20Chapter FourContained OptimizationWhat are the equations that the optimal consumption basket must fulfill if we want to represent the consumer’s choice among three goods? MU / P = MU / P is an example of “marginal reasoning” to maximize P X + P Y = I reflects the “constraint”XXYXYYCopyright (c)2014 John Wiley & Sons, Inc.21Chapter FourContained OptimizationU(F,C) = FCPF = $1/unitPC = $2/unitI = $12Solve for optimal choice of food and clothingCopyright (c)2014 John Wiley & Sons, Inc.22Composite Goods: A good that represents the collective expenditure on every other good except the commodity being consideredChapter FourSome ConceptsCorner Points: One good is not being consumed at all – Optimal basket lies on the axisCopyright (c)2014 John Wiley & Sons, Inc.23Chapter FourSome ConceptsCopyright (c)2014 John Wiley & Sons, Inc.24Chapter FourSome ConceptsCopyright (c)2014 John Wiley & Sons, Inc.25Chapter FourSome ConceptsCopyright (c)2014 John Wiley & Sons, Inc.26Chapter FourSome ConceptsCopyright (c)2014 John Wiley & Sons, Inc.27Chapter FourSome ConceptsCopyright (c)2014 John Wiley & Sons, Inc.28The mirror image of the original (primal) constrained optimization problem is called the dual problem. Min PxX + PyY (X,Y) subject to: U(X,Y) = U* where: U* is a target level of utility.Chapter FourDualityIf U* is the level of utility that solves the primal problem, then an interior optimum, if it exists, of the dual problem also solves the primal problem.Copyright (c)2014 John Wiley & Sons, Inc.29YX •Optimal Choice (interior solution)U = U*PXX + PYY = E*0Decreases inexpenditure levelExample: Expenditure MinimizationChapter FourOptimal ChoiceCopyright (c)2014 John Wiley & Sons, Inc.30YX•U = 2502.51050X + 200Y = EExample: Expenditure MinimizationChapter FourOptimal Choice25 = XY (constraint)Y/X = 1/4 (tangency condition)Copyright (c)2014 John Wiley & Sons, Inc.31Suppose that preferences are not known. Can we infer them from purchasing behavior? If A purchased, it must be preferred to all other affordable bundlesChapter FourRevealed PreferenceCopyright (c)2014 John Wiley & Sons, Inc.32Suppose that preferences are “standard” – then:All baskets to the Northeast of A must be preferred to A.This gives us a narrower range over which indifference curve must lieThis type of analysis is called revealed preference analysis. Chapter FourRevealed PreferenceCopyright (c)2014 John Wiley & Sons, Inc.