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:1.Affordability problem – Budget Constraint: M = PSS + PFF2.Best Bundle Problem – Utility Functions (older) or Indifference Curves (new)3.We begin the analysis with the Affordability problem.

Rational Consumer Choice

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CHAPTER OUTLINE

The opportunity set or budget constraintConsumer preferencesThe best feasible bundleAn application of the rational choice model

BUDGET LIMITATIONS

A bundle: 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.

Figure 3.2: The Budget Constraint, or Budget Line

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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 or solve 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

PSS + PFF = MF = M/PF – PS/PF *S =10 -1/2*S.

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.

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Figure 3.3: The Effect of a Rise in the Price of Shelter (PS from $5 to $10/sq yd)

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Note that the slope of B2 is larger than that of B1 due to a rise in the price shelter.

Recall: M = $100Before PS change: F =10, S =20After PS change: F =10, S=10

Figure 3.4: The Effect of Cutting Income by Half

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M=$50 so that M/PF =50/10 =5 and M/PS =50/5= 10Reducing 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

Old income = M = $100, PF =10, PS =5New Income = M’ = $50, PF =10, PS =5

Figure 3.5: The Budget Constraints with the Composite Good

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Y is the composite good with Py =$1 so that the budget constraint is:PYY +PXX = M or Y +PXX=MSlope = -(M/1)/M/PX =- M/1 * PX/M = -PX

= all other goods

Figure 3.6: A Quantity Discount Gives Rise to a Nonlinear Budget Constraint, i.e. BC with a Kink

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The idea1.Electric company charges $0.10/per kWh for the 1st 1000kWh so the slope = -100/1,000 = -1/102.Charges $0.05 per kWh for additional amounts3.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.

Slope = (300-400)/(1000-0) = -100/1000 = -1/10

Slope = (0-300)/(7000-1000) = -300/6000 = -3/60 =-1/20

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.

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Figure 3.8: Generating Equally Preferred Bundles

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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.

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 CurvesIndifference curves …1. Are Ubiquitous.

Any bundle has an indifference curve passing through it. 2. Are Downward-sloping.

This comes from the “more-is-better” assumption.3. Cannot cross.4. Become less steep as we move downward and to the right

along them. This property is implied by the convexity property of preferences.

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Figure 3.11: Why Two Indifference Curves Do not Cross

Part of an Indifference Map

Bundles on any IC are less preferred than any bundles on a higher indifference curveAnd more preferred than any bundles on any lower ICThat is: I4 > I3 > I2 > I1 based on the property: more is preferred to less (provided the goods are ‘good’ goods)

4 properties of preference ordering imply the following about IC and indifference maps.a.Any bundle has an IC passing through it due the completeness propertyb.ICs are downward-sloping due to more-is-better propertyc.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.

Figure 3.12: The Marginal Rates of Substitution

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Trade-offs Between GoodsMarginal 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

Figure 3.14: People with Different Tastes

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The Best Feasible BundleConsumer’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

1. Tex prefers potatoes to rice while Mohan prefers rice to potatoes

2. 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

Figure 3.15: The Best Affordable Bundle

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Given that M =$100, PF =$10, PS =$5Thus, PFF + PSS = MGiven that I3 > I2 > I1

Best Affordable Bundle – most preferred bundle of those that are affordableIndifference Map – how the various bundles are ranked in order of preferencesBudget Constraint – which bundles are affordableRationality – 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.

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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.

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Figure 3.18: Food Stamp Program vs. Cash Grant Program

Figure 3.19: Where Food Stamps and Cash Grants Yield Different Outcomes

Cash or Food Stamps?Food Stamp ProgramObjective - 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 =$100Stamps 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.

=Composite good

=Food

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

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Figure A3.1: Indifference Curves for the Utility Function U=FS

Figure A3.2: Utility Along an Indifference Curve Remains Constant

Appendix: Utility Approach to the Consumer Problem

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.

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 toMUF/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/PS Or MUF/PF=MUS /PS

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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)

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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

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.

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 =401.Solve budget constraint for Y= 20 – 2X.2.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.3.∂U(X,Y)/∂X = 20 -4X=0 X = 20/4 =5. Plug this into the budget constraint to obtain4.4*5 + 2Y =40 or 2Y =40-20 =20 or Y =20/2 =10. Thus, {X=5, Y=10} is the optimal bundle.

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) = 913M' + 0.5M' = 9, 13.5M' = 9, M' = 2/3; C = 8/3. Represented by Point B belowThe graph represents milk and coffee as Perfect Complements - goods that are consumed together.

A

B

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.