Rational Consumer Choice - Texas A&M Universitypeople.tamu.edu/~aglass/econ323/Chapter03Slides.pdf · ... Utility Function Approach to Consumer ... –Reach the highest indifference

Rational Consumer Choice

Chapter 3

Chapter Outline

• Budget Constraints

• Budget Shifts Due to Price or Income Changes

• Consumer Preferences and Indifference Curves

• Best Feasible Bundle

• Appendix: Utility Function Approach to Consumer Choice

©2015 McGraw‐Hill Education. All Rights Reserved. 2

Budget Limitation

• 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 3.1: Two Bundles of Goods


Affordable vs. Unaffordable

• Affordable set, or feasible set: bundles on or below the budget constraint; bundles for which the required expenditure at given prices is less than or equal to the income available.

• Unaffordable set, or unfeasible set: bundles that lie outside the budget constraint


Budget Constraint

• For income M, quantity of food F, quantity of shelter S, price of food PF, and price of shelter PS, the budget constraint is a straight line

PSS + PFF = M or

• The food endpoint is the maximum amount of food affordable M/PF, and the shelter endpoint is the maximum amount of shelter affordable M/PS.


Budget Constraint

• For Figure 3.2, let the price of food be PF = $10/lb, the price of shelter be PS = $5/sq yd. and income be M = $100/week. – What bundles of food and shelter are affordable?

• The budget constraint PSS + PFF = M becomes 5S + 10F = 100 or F = 10 – (1/2)S.

• Maximum food is M/PF = 100/10=10, Maximum shelter is M/PS = 100/5 = 20. Slope of budget constraint –PS/PF = ‐1/2.


Budget Constraint

• Point D containing 4 pounds of food and 5 square yards of shelter per week is feasible.

• F = 4 at PF = 10 will cost $40 and S = 5 at PS = 5 will cost $25 so spend $65 all together, which is less than income per week of $100.

5S + 10 F = $100, 25 + 40 ≤ $100• Point E containing F = 8 and S = 12 is not feasible as spending $80 on food and $60 on shelter would exceed the $100 weekly budget.


Figure 3.2: The Budget Constraint,or Budget Line


• If the price of ONLY one good changes…– The slope of the budget constraint changes.

• If the price of both goods change by the same proportion…– The budget constraint shifts parallel to the original one.

• If income changes...– The budget constraint shifts parallel to the original one.


Budget Shifts Due to Price and Income Changes

Increased Price of Shelter

• For Figure 3.3, suppose the price of shelter doubles from PS = $5 to $10, while income and the price of food remain the same.

• The new budget constraint becomes10S + 10 F = $100 or F = 10 ‐ S

• Maximum shelter falls in half from M/PS = 100/5 = 20 to 100/10 =10. Maximum food M/PF remains unchanged. More steeply downward sloping (slope now ‐1).


Figure 3.3: The Effect of a Risein the Price of Shelter


Income Cut in Half

• For Figure 3.4, suppose income is cut in half from M = $100 to $50, while the prices of food and shelter remain the same as original case.

• The new budget constraint becomes5S + 10 F = $50 or F = 5 – (1/2)S

• Maximum shelter falls in half from M/PS = 100/5 = 20 to 50/5 =10. Maximum food also is halved from M/PF = 100/10 = 10 to 50/10 = 5. Slope unchanged at ‐1/2.


Figure 3.4: The Effect of Cutting Income by Half


Both Prices Double

• If the price of shelter doubles from PS = $5 to $10 at the same time as the price of food also doubles from PF = $10 to $20, has same effect on the budget constraint as if income fell in half.

• The new budget constraint becomes10S + 20 F = $100 or F = 5 – (1/2)S

• Maximum shelter falls in half from M/PS = 100/5 = 20 to 100/10 =10. Maximum food also is halved from M/PF = 100/10 = 10 to 100/20 = 5. Slope unchanged at ‐1/2.


Budgets With More Than Two Goods

• When we have more than 3 goods, the budget constraint becomes a hyperplane, or multidimensional plane.

• In this case, view the consumer’s choice as one between a good, X, and an amalgam of other goods, Y. This amalgam is called the composite good.– The amount of income left after buying good X– The amount the consumer spends on goods other than good X


Composite Good

• For income M, quantity of composite good Y, quantity of good X, price of X PX, and price of composite PY,=1, the budget constraint is a straight line

PXX + Y = M or Y = M – PXX

• The maximum composite is always M, the maximum X is M/PX, and the slope is –PX.


Figure 3.5: The Budget Constraints with the Composite Good


Nonlinear Budget Constraints

• Budget constraints are typically linear due to prices being constant, but budget constraints can be non‐linear when prices vary with quantity such as with quantity discounts.

• For Figure 3.6, M = $400/month. Power costs $0.10/kWh for first 1000 kWh each month, then $0.05 for any additional kWhs.

• Slope is ‐0.1 for first 1000kWh, which will cost $100 of the $400 monthly budget. The remaining $300 can buy 6000kWh more with slope ‐0.05.


Figure 3.6: A Quantity Discount Gives Rise to a Nonlinear Budget Constraint


Preference Ordering

• Preference ordering: a ranking of all possible consumption bundles in order of preference.– Differ widely among consumers

– Four simple properties of preference ordering


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.

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 …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.

Figure 3.9: An Indifference Curve


Figure 3.10: Part of an Indifference Map


Figure 3.11: Why Two Indifference Curves Do Not Cross


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.

Figure 3.12: The Marginal Rates of Substitution


Marginal Rate of Substitution

• Figure 3.13 depicts three different points on an indifference curve with different MRS.

• At S = 3 sq ft of shelter and F = 17 lbs of food each week, MRS = 3 – willing to give up lots of food for an additional sq ft of shelter since eating well but cramped for space.

• At S = 7 and F = 9, MRS = 1• At S = 16 and F = 3, MRS = ¼

– willing to give up little food for more shelter since ample space but hungry.


Figure 3.13: Diminishing Marginal Rate of Substitution


Different Tastes

• A consumer’s willingness to exchange goods is indicated by the slope of the indifference curve.

• Consumers with different tastes (different MRS at a give point) will have indifference curves with different slopes.

• In Figure 3.14, rice lovers (Mohan) are willing to give up more potatoes for a pound of rice than potato lovers (Tex).


Figure 3.14: People with Different Tastes


The Best Feasible Bundle


• Consumer’s Goal: to choose the best affordable bundle.– Reach the highest indifference curve possible, given budget constraint.

– For convex indifference curves, the best bundle will always lie at a point of tangency where the slope of an indifference curve matches the slope of budget constraint (in absolute value).

MRS = PS / PF

Figure 3.15: The Best Affordable Bundle


Corner Solutions

• Corner solution: in a choice between two goods, a case in which the consumer does not consume one of the goods.

• In Figure 3.16, choose maximum food and no shelter.

• More likely when goods are close substitutes, rather than essentials like food and shelter.


Figure 3.16: A Corner Solution


Perfect Substitutes

• In Figure 3.17, suppose MRS is that 2 Cokes are equivalent to one Jolt (for caffeine content).

• Jolt sells for $1/pint and Coke sells for $0.75/pint. For a daily soft drink budget of $15, what is the optimal consumption bundle of Coke and Jolt?

• Budget constraint 0.75C + J = 15 or C = 20 – (4/3)J. Maximum Jolt is 15 pints. Maximum Coke is 15/0.75=20 pints. Slope ‐4/3 (must give up 1.3 Cokes to afford a Jolt).

• Indifference curve is straight line with slope –MRS = ‐2. 15 pints of Jolt (with no Coke) yields same satisfaction as 30 pints of Coke (with no Jolt) and is the best choice.


Figure 3.17: Equilibrium withPerfect Substitutes


Perfect Substitutes

• What is the new optimal bundle if the price of Jolt doubles from $1 to $2/pint?

• New budget constraint 0.75C + 2J = 15 or C = 20 –(8/3)J. Maximum Jolt is 7.5 pints (half as much as before). Maximum Coke remains 15/0.75=20 pints. Steeper slope of budget constraint ‐8/3.

• Reach highest possible indifference curve with 20 Cokes and no Jolt, which is equivalent in caffeine to 10 Jolts and no Coke.


Perfect Substitutes

• How much income is needed to compensate for the increased price of Jolt?

• Was consuming 15 Jolts, equivalent to 30 Cokes.

• Cut back to 20 Cokes, equivalent to 10 Jolts.

• Need $7.50 in additional income to afford 10 more Cokes (cheaper than $10 for 5 more Jolts).


Perfect Complements

• Suppose butter and toast are perfect complements, using one pat of butter for each slice of toast. Butter costs $0.20/pat, toast $0.10/slice and have $12/month to spend. How much toast and butter will be consumed?

• Budget constraint 0.2B + 0.1T = 12 or T = 120 –2B. Maximum butter 12/0.2 = 60, maximum toast 12/ 0.1 = 120. Slope ‐2.

• Preferences are for B = T. Replacing T with B, T = 120 – 2T, 3T = 120, T* = B* = 40 so consume 40 slices of toast with butter each month.


The Utility Function Approach to Consumer Choice

• Finding the highest attainable indifference curve on a budget constraint is just one way to analyze the consumer choice problem

• In this second approach, represent the consumer’s preference not with an indifference map, but with a utility function.


Utility Function and Preerences

• If the utility function is U(F,S) = FS, F = 4 lbs/wkand S = 3 sqft/wk would generate U=12 utils/wk as a measure of satisfaction, F=8 and S=6 lead to utility U=48, etc.

• One bundle B is preferred over another if leads to a higher level of utility.


Using Utility Function to Construct an Indifference Map

• For the utility function U(F,S) = FS, how can we graph the indifference curves that correspond to 1, 2, 3, and 4 utils?

• An indifference curve represents different combinations of food F and shelter S that generate a constant level of utils U0.

• For U0 = 1 = FS, S = 1/F. For U0 = 2 = FS, S = 2/F. In general, S = U0/F. Figure 3A.1 depicts values such as (F,S) = (1,2), (2, 1) for U0 = 2.

•


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


Marginal Utility

• Figure 3A.2 examines a move from point K to point L along indifference curve U0, where point L has more food but less shelter than point K.

• Since both points are on the same indifference curve (representing bundles that generate U0utility), the marginal utility gained from more food must match the marginal utility lost from less shelter.


Marginal Utility

• The slope of an indifference curve at any point is /.

• At the optimal consumption bundle, the slope of the indifference curve must match the slope of the budget constraint tangent to it.

• Equivalently, the marginal utility generated by the last dollar spent on food must match the marginal utility from the last dollar spend on shelter.


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


Generating Indifference Curves for Perfect Substitutes

• For Figure 3A.5, U(X, Y) = (2/3)X + 2Y so 3 units of X gives as much utility (U = 2) as one unit of Y, making X and Y perfect substitutes.

• Setting U0 = (2/3)X + 2Y, indifference curves are straight lines Y = U0/2 – X/3 with slope ‐1/3.

• U0 = 1 can be generated by Y = ½ and no X, X = 1.5 and no Y, or other combinations in between.


Figure A3.5: Indifference Curves for the Utility Function U(X,Y)=(2/3)X + 2Y


Using Calculus to Maximize Utility

• For Figure 3A.6, suppose U(X, Y) = XY, M = 40, PX = 4 and PY =2.

• Budget constraint 4X + 2Y = 40 or Y = 20 – 2X.• To maximize U = XY subject to budget constraint, replace Y in the utility function with expression for Y from the budget constraint so choose X to maximize XY = X(20 – 2X) = 20X – 2X2.

→ X* = 5

Y* = 20 – 2X = 20 – 2(5) = 20 – 10 = 10


Figure A3.6: The Optimal Bundle when U=XY, Px=4, Py=2, and M=40


Problem 1

Martha has $150 of disposable income to spend each week and cannot borrow money. She buys Malted Milk balls and the composite good. Suppose that Malted Milk Balls cost $2.50 per bag and the composite good costs $1 per unit.a. Sketch Martha’s budget constraint.b. What is the opportunity cost, in terms of bags of Malted Milk Balls, of an additional unit of the composite good?


Solution 1

• Budget Constraint (blue line)

2.5X + Y = 150 or Y = 150 – 2.5X

• Maximum composite 150.

• Maximum Malted Milk Balls X 150/2.5 = 60.

• Slope ‐2.5.

• Every $1 spent on composite good is 4/10th of a bag of Malted Milk Balls that cannot afford.


Problem 2

Suppose that in an inflationary period the cost of the composite good increases to $1.50 per unit, but the cost of Malted Milk Balls stays the same.

a. Sketch the new budget constraint.

b. What is the opportunity cost of an additional unit of the composite good?


Solution 2

• Budget Constraint (red line)

2.5X + 1.5Y = 150 or Y = 100 – (5/3)X

• Maximum composite falls from 150 to 100.

• Maximum X remains 150/2.5 = 60.

• Slope ‐2.5/1.5 = ‐5/3.



Solutions 1, 2 & 3 Figure

0

150

0 90Malted Milk Balls (X)

Compo

site (Y

)

Slope = ‐2.5

1

100

60


Problem 3

Suppose Martha demands a pay raise to fight the inflation. Her boss submits and raises her salary so that her disposable income is now $225/wk.

a. Sketch the new budget constraint.

b. What is the opportunity cost of an additional unit of the composite good?


Solution 3

• Budget Constraint (green line)

2.5X + 1.5Y = 225 or Y = 150 – (5/3)X

• Maximum composite returns from 100 to 150.

• Maximum X expands from 150/2.5 = 60 to 225/2.5 = 90.

• Slope stays at ‐2.5/1.5 = ‐5/3.



Problem 4

Picabo, an aggressive skier, spends her entire income on skis and bindings. She wears out one pair of skis for every pair of bindings she wears out.

a. Graph Picabo’s indifference curves for skis and bindings.

b. Now draw her indifference curves on the assumption that she is such an aggressive skier that she wears out two pairs of skis for every pair of bindings.


Solution 4

0

1

2

3

4

5

6

7

8

9

10

0 2 4 6 8 10

S = B

S = 2B

Bindings


Skis

Rational Consumer Choice - Texas A&M Universitypeople.tamu.edu/~aglass/econ323/Chapter03Slides.pdf · ... Utility Function Approach to Consumer ... –Reach the highest indifference

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