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Chapter 3

Chapter 3. The Utility Function Approach to Consumer Choice. Consumer’s Problem : Choose the BEST BUNDLE she/he can AFFORD. Breaking the problem into 2 stages: Affordability problem – Budget Constraint: M = P S S + P F F

myles-scott
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Chapter 3

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  1. Chapter 3 The Utility Function Approach to Consumer Choice • Consumer’s Problem: Choose the BEST BUNDLE she/he can AFFORD. • Breaking the problem into 2 stages: • Affordability problem – Budget Constraint: M = PSS + PFF • Best Bundle Problem – Utility Functions (older) or Indifference Curves (new) • We begin the analysis with the Affordability problem.

  2. CHAPTER OUTLINE • The opportunity set or budget constraint • Consumer preferences • The best feasible bundle • An application of the rational choice model • BUDGET LIMITATIONS • Abundle: a particular combination of two or more goods. • Budget constraint: the set of all bundles that exactly exhaust the consumer’s income at given prices. • Its slope is the negative of the price ratio of the two goods. Rational Consumer Choice 3-2

  3. Figure 3.2: The Budget Constraint, or Budget Line PSS + PFF = M F = M/PF – PS/PF *S =10 -1/2*S. • Let M=$100/wk; PS=$5/sq. yd, PF=$10/lb • Suppose all M is on F, then F = M/PF= 100/10 = 10 units at L. Suppose all M is spent on S, then S =M/PS= 100/5 =20 units at K • PSS + PFF = M is the budget constraint orsolve for F = M/PF – PS/PF *S =10 -1/2*S. Note that F = L=M/PF shows the value of the intercept L =100/10=10. • Slope = rise/run = -M/PF /M/PS =M/PF * PS/M = - PS/PF= -1/2 3-3

  4. BUDGET SHIFTS DUE TO PRICE OR INCOME CHANGES 1.If the price of ONLY one good changes… • The slope of the budget constraint changes. 2. If the price of both goods change by the same proportion… • The budget constraint shifts parallel to the original one. 3.If income changes …. • The budget constraint shifts parallel to the original one. 3-4

  5. Figure 3.3: The Effect of a Rise in the Price of Shelter (PS from $5 to $10/sq yd) Recall: M = $100 Before PS change: F =10, S =20 After PS change: F =10, S=10 Note that the slope of B2 is larger than that of B1 due to a rise in the price shelter. 3-5

  6. Figure 3.4: The Effect of Cutting Income by Half Old income = M = $100, PF =10, PS =5 New Income = M’ = $50, PF =10, PS =5 • M=$50 so that M/PF =50/10 =5 and M/PS =50/5= 10 • Reducing income by half results in a parallel shift inward of the budget constraint. • Note that changing income does NOT change the slopes, i.e. slope of B2 = slope of B1 = -1/2 3-6

  7. Figure 3.5: The Budget Constraints with the Composite Good = all other goods Y is the composite good with Py =$1 so that the budget constraint is: PYY +PXX = M or Y +PXX=M Slope = -(M/1)/M/PX =- M/1 * PX/M = -PX 3-7

  8. Figure 3.6: A Quantity Discount Gives Rise to a Nonlinear Budget Constraint, i.e. BC with a Kink Slope = (300-400)/(1000-0) = -100/1000 = -1/10 Slope = (0-300)/(7000-1000) = -300/6000 = -3/60 =-1/20 • The idea • Electric company charges $0.10/per kWh for the 1st 1000kWh so the slope = -100/1,000 = -1/10 • Charges $0.05 per kWh for additional amounts • For zero consumption, the consumer has $400 =M to spend on other goods BUT at amounts greater than 1,000 kWh, the slope = -300/6,000 = -1/20. 3-8

  9. Properties of Preference Orderings • Completeness:the consumer is able to rank all possible combinations of goods and services. • More-Is-Better: other things equal, more of a good is preferred to less. • Transitivity: for any three bundles A, B, and C, if he prefers A to B and prefers B to C, then he always prefers A to C. • Convexity: mixtures of goods are preferable to extremes. 3-9

  10. Figure 3.8: Generating Equally Preferred Bundles • IC – a set of bundles among which the consumer is indifferent, ceteris paribus. • Z – Because Z has more of both Food and Shelter, it is preferred to A. • A -Because A has more of both Food and Shelter, it is preferred to W. • The consumer is indifferent between A, B, and C combinations since they lie on the same indifference curve. 3-10

  11. Indifference Curves • Indifference curve: a set of bundles among which the consumer is indifferent. • Indifference map: a representative sample of the set of a consumer’s indifference curves, used as a graphical summary of her preference ordering. Properties of Indifference Curves Indifference curves … • Are Ubiquitous. • Any bundle has an indifference curve passing through it. • Are Downward-sloping. • This comes from the “more-is-better” assumption. • Cannot cross. • Become less steep as we move downward and to the right along them. • This property is implied by the convexity property of preferences. 3-11

  12. Figure 3.11: Why Two Indifference Curves Do not Cross Part of an Indifference Map • 4 properties of preference ordering imply the following about IC and indifference maps. • Any bundle has an IC passing through it due the completeness property • ICs are downward-sloping due to more-is-better property • ICs from the same indifference map cannot cross • Example: E is equally attractive as F and F is preferred to E makes no sense. Due to the crossing of ICs. • Bundles on any IC are less preferred than any bundles on a higher indifference curve • And more preferred than any bundles on any lower IC • That is: I4 > I3 > I2 > I1 based on the property: more is preferred to less (provided the goods are ‘good’ goods) 3-12

  13. Figure 3.12: The Marginal Rates of Substitution • Trade-offs Between Goods • Marginal rate of substitution (MRS): the rate at which the consumer is willing to exchange the good measured along the vertical axis for the good measured along the horizontal axis. • Equal to the absolute value of the slope of the indifference curve. • Enlarged area =slope  around A, if 1 unit of Shelter is given up, we must compensate the consumer with 2 units of Food Figure 3.13: Diminishing Marginal Rate of Substitution • Convexity states that a consumer with more of a product is willing to give up more of it to get the product that she/he has less of. • MRSF,S = rate at which the consumer can give up Food for Shelter without changing total satisfaction or MB of Shelter in terms of Food. • Slope of BC = rate at which one can substitute Food for Shelter without changing total expenditure or the marginal cost (MC) of Shelter in terms of Food. • Note that diminishing MRS  preference for variety; less of what is plenty for more of what is scarce 3-13

  14. Figure 3.14: People with Different Tastes • Tex prefers potatoes to rice while Mohan prefers rice to potatoes • Reasoning: At A, Tex would give up 1lb of potatoes for 1 lb of rice whereas Mohan is willing to give up 2lbs of potatoes for 1 lb of rice • The Best Feasible Bundle • Consumer’s Goal:to choose the best affordable bundle. • -The same as reaching the highest indifference curve she can, given her budget constraint. • - For convex indifference curves.. • the best bundle will always lie at the point of tangency of (a) Budget Constraint and (b) Indifference Curves 3-14

  15. Figure 3.15: The Best Affordable Bundle • Given that M =$100, PF =$10, PS =$5 • Thus, PFF + PSS = M • Given that I3 > I2 > I1 • Best Affordable Bundle – most preferred bundle of those that are affordable • Indifference Map – how the various bundles are ranked in order of preferences • Budget Constraint – which bundles are affordable • Rationality – consumers enter the market thinking about indifference maps and budget constraints and behaves as if they think about these two items only. • At F, the slope of Budget Constraint = slope of the Indifference Curve I2. • Or PS/PF = MRSF,S= -1/2 . This is the Tangency Condition. • Opportunity cost of Shelter in terms of Food = MB of Shelter in terms of Food. • If not at equilibrium, it pays to adjust quantities BUT not prices! • For example, at E (slope of IC, say 1/4 < PS/PF=-1/2) the consumer can be compensated for the loss of 1 sq yd by being given ¼ of Food to get back to F. 3-15

  16. Figure 3.17: Equilibrium with Perfect Substitutes Figure 3.16: A Corner Solution • Case where the consumer does not consumer one of the goods, i.e. there is no interior point of tangency when MRS may be bigger or smaller than PS/PF. • Suppose at A, the MRS =0.25 whereas PS/PF =$5/$10 = -½. It is clear that at A, MRS <PS/PF. • Given market prices, he would have to give up too much Food to get 1 unit of Shelter! Might as well not purchase any Shelter until market prices change. • For some goods, ICs are not convex at all, i.e. fail the convexity assumption. For such goods, a corner solution is the outcome since these are easily substitutable. Note that MRS =-30/15 =- 2 (2 pints of C for 1 pint of J) everywhere but PJ/PC =-4/3 • That is, J has twice the amount of Coke caffeine. • Given these ratios, he consumes at point A since he only cares about caffeine. 3-16

  17. Figure 3.18: Food Stamp Program vs. Cash Grant Program Figure 3.19: Where Food Stamps and Cash Grants Yield Different Outcomes =Composite good =Food • Cash or Food Stamps? • Food Stamp Program • Objective - to alleviate hunger. How does it work? • People whose incomes fall below a certain level are eligible to receive a specified quantity of food stamps. Assume M =$400 and Food Stamps =$100 • Stamps cannot be used to purchase cigarettes, alcohol, and various other items. The government gives food retailers cash for the stamps they accept. • Given PX and M, the initial equilibrium is Point J with maximum Food at $400/PX. Food stamps increase this to $500/PX and equilibrium is at Point K. • Budget constraint with Food Stamps is ADF but ADE with cash equivalent. If he needs more that F of composites, Food Stamps are a drag. However, since K is identical for both, Food stamps and Cash Income are equivalent. • Here the consumer prefers cash since it enables him to be at L which lies on a higher IC than D. • Note that D has the exact value of Food Stamps =$100 while L has less than $100 of Food • .Idea: Food Stamps constraint consumers to spend ALL on Food (often bad food resulting in high obesity). • Congress restricted expenditures on illicit goods – political feasibility 3-17

  18. Appendix: Utility Approach to the Consumer Problem Figure A3.1: Indifference Curves for the Utility Function U=FS Figure A3.2: Utility Along an Indifference Curve Remains Constant • Changes in TU along U=U0 are ∆TU = MUF∆F + MUS∆S • Recall that along the utility curve, ∆TU=0 so that the consumer has the same level of satisfaction at K and L. Thus, • MUF∆F + MUS∆S or MUF∆F = - MUS∆S which can re-arranged to • MUF/MUS = -∆S/∆F and at equilibrium: -MUF/MUS =PF/PS. • MU – is the rate at which TU changes with the consumption of a good. • Note that the slope expression is similar to that under IC analysis: • -MUF/MUS =PF/PS ≈ -MRS =PF/PSOr MUF/PF=MUS /PS • Assume a consumer’s level of satisfaction can be represented by a utility function, U =(F,S) = FS. Need to draw utility curve for U =1 Or FS=1 of S= 1/F. • Thus, for U=2, S = 2/F and so forth. • Note that the utility map looks similar to an indifference map from earlier in the chapter. 3-18

  19. Figure A3.3: A Three-Dimensional Utility Surface Figure A3.4: Indifference Curvesas Projections • Use of ICs only requires people to be able to rank preferences [Ordinal Utility] while the utility approach requires numerical values such as “ Good A gives me 100 times the utility of Good B.”[Cardinal Utility]. That is, U=U(X,Y) which resembles a 3-dimensional graph. • Note that “more-is-better” property implies that the utility mountain has no summit! • Assume U = U0 and slice the mountain at JK parallel to the XY plane, and repeat the exercise with LN at U1 units above. • Slices JK and LN are like ICs or Us that can be projected into X-Y space. • This means that we can derive ICs (two dimensions) from a three-dimensional utility graph. • Given a utility a utility function, U=U(X,Y), we can generate an indifference curve map as shown above. • Although this is possible, it is not necessary since given preference ordering, we can discard the Cardinal Utility Approach and simply use the Ordinal Approach (ICs) 3-19

  20. Figure A3.5: Indifference Curves for the Utility Function U(X,Y)=(2/3)X + 2Y Figure A3.6: The Optimal Bundle when U=XY, Px=4, Py=2, and M=40 • To find an optimal solution that satisfies a budget constraint, given U(X,Y) =XY, M=$40, PX=$4, PY =$2. Budget Constraint is 4X + 2Y =40 • Solve budget constraint for Y= 20 – 2X. • Substitute this into the utility function to get U(X,Y) =X(20 -2X) =20X – 2X2 and take derivative of U(X,Y) with respect to X and equate the result to zero. • ∂U(X,Y)/∂X = 20 -4X=0  X = 20/4 =5. Plug this into the budget constraint to obtain • 4*5 + 2Y =40 or 2Y =40-20 =20 or Y =20/2 =10. Thus, {X=5, Y=10} is the optimal bundle. • Suppose U(X,Y) = (2/3) X + 2Y and that U = U0. • Then we can solve U(X,Y) = U0 = (2/3) X + 2Y for Y =(U0/2) –(1/3)X. • With U =1, 2, 3, the resulting ICs or Us are linear as shown above. 3-20

  21. Worked Problems: Indifference Curves Question: Boris budgets $9/wk for his morning coffee with milk. He likes it only if it is prepared with 4 parts coffee, 1 part milk. Coffee costs $1/oz , milk $0.50/oz. How coffee and how much milk will Boris buy per week? How will your answers change if the price of coffee rises to $3.25/oz? Show your answers graphically. Answer: Let C = coffee (ounces/week) and M = milk (ounces/week). Because of Boris's preferences, C = 4M. At the original prices we have: 4M(l) + M(0.5) = 9 4.5M = 9 So M=2 and C=8 . Represent by Point A below Let M' and C' be the new values of milk and coffee. Again, we know that C'=4M'. With the new prices we have: (4M')(3.25) + M'(.5) = 9 13M' + 0.5M' = 9, 13.5M' = 9, M' = 2/3; C = 8/3.Represented by Point B below The graph represents milk and coffee as Perfect Complements - goods that are consumed together. A B

  22. Worked Problem: Utility Function Approach • Question: Tom spends all his $100 weekly income on two goods, X and Y. His utility function is given by U(X,Y)=XY. If PX =4 and PY =10, how much of each good should he buy? Budget Constraint: PXX + PYY = M  4X + 10Y =$100 • Answer: • Solve the budget constraint, 100 = 4X + l0Y, to get Y = 10  0.4X, then substitute into the utility function to get U = XY = X(10  0.4X ) = 10X  0.4X2. • Equating ∂U/∂X to zero we have 10  0.8X = 0, which solves for X = 12.5. • Substituting back into the budget constraint and solving for Y, we get Y = 5.

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