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RATIONAL CONSUMER CHOICE

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RATIONAL CONSUMER CHOICE

Chapter Outline

- The Opportunity Set or Budget Constraint
- Consumer Preferences
- The Best Feasible Bundle
- The Utility Function Approach to Consumer Choice
- Cardinal Versus Ordinal Utility
- Using Calculus to Maximize Utility: The Method of Lagrangian Multipliers

Figure 3-1: Two Bundles of Goods

- A bundle is a specific combination of goods.
- eg. Bundle A = 5 unit of shelter & 7 unit of food

Figure 3-2: The Budget Constraint,or Budget Line

- Line B describes the set of all bundles the consumer can purchase of given income & prices.
- Its slope is the opportunity cost of an additional unit of shelter
- the number of units of food that must be sacrificed in order to purchase 1 additional unit of shelter

Budget Constraint or Budget Line

- Suppose consumer’s income M = $100, Price of shelter PS = $5 & Price of food PF = $10
- If the consumer spent all her income on shelter,
- she could buy M/PS = 100/5 = 20
- K (20,0)
- Horizontal intercept

- If the consumer spent all her income on food,
- she could buy M/PF = 100/10 = 10
- L (0,10)
- Vertical intercept

- Budget Constraint - straight line that joins points K & L (Line B)

Budget Constraint or Budget Line

- Budget triangle– bounded by budget constraint & the 2 axes.
- Feasible setoraffordable set- the bundle on or within the budget triangle
- eg. Bundle D (5,4) costs $65

- Bundles that lie outside the budget triangle are infeasible or unaffordable, eg. Bundle E

Budget Constraint or Budget Line

- The consumer’s weekly expenditure on shelter & food must add up to her weekly income.
- If S & F denote the quantities of shelter & food, the constraint must satisfy:
PSS + PFF = M (3.1)

- Solve Eq 3.1 for F in terms of S:

Figure 3-3: The Effect of a Risein the Price of Shelter

- Price of shelter increased from PS1 = $5 to PS2 = $10
- Vertical intercept remains
- Rotates the budget constraint inward

Figure 3-4: The Effect of CuttingIncome by Half

- Income M is cut by half from $100 to $50
- horizontal intercept M/PS falls (from 20 to 10)
- vertical intercept M/PF falls (10 to 5)
- Same slope = - PS/PF = -1/2
- B2 is parallel to the B1

Budgets Involving More Than 2 Goods

- Consumer’s choice as between a particular good X & the composite good Y
- Composite good the amount of money the consumer spends on numerous goods other than X

Figure 3-5: The Budget Constraints with the Composite Good

- For simplicity, the price of a unit of composite good = 1
- if the consumer devotes none of his income to X, he will be able to buy M units of the composite good

Figure 3-6: A Quantity Discount Gives Rise to a Nonlinear Budget Constraint

- Budget constraint straight line when relative prices are constant - the opportunity cost of one good in term of any other is the same
- Budget constraint kinked lines - eg. quantity discounts

Figure 3-7: Budget Constraints Following Theft of Gasoline, Loss of Cash

- A theft of $40 worth of gasoline has the same effect on the budget constraint as the loss of $40 in cash.
- The bundle chosen should be the same, irrespective of the source of the loss.

Consumer Preferences Loss of Cash

- Preference ordering a scheme whereby the consumer ranks all possible consumption bundles in order of preference.
- Assume world with only 2 goods, the consumer is able to make 1 of the 3 possible statements:
- A is preferred to B
- B is preferred to A
- A & B are equally attractive.

Consumer Preferences Loss of Cash

- 4 properties of preference orderings:
- Completeness
- A preference ordering is complete if it enables the consumer to rank all possible combinations of goods

- More-Is-Better
- Other things equal, more of a good is preferred

- Transitivity
- For any 3 bundles, if he prefers A to B & prefers B to C, then he always prefers A to C.

- Convexity
- Mixtures of goods are preferable to extremes

Figure 3-8: Generating Equally Loss of CashPreferred Bundles

- Z > A > W
- A = B = C

- Z is preferred to A because it has more of each good than A has.
- For the same reason, A is preferred to W.

- It follows that on the line joining W & Z there must be a bundle B that is equally attractive as A.
- In similar fashion, we can find a bundle C that is equally attractive as B.

Figure 3-9: An Indifference Curve has.

- An indifference curve (I) is a set of bundles that the consumer considers equally attractive
- L > I > K

Figure 3-10: Part of an Indifference Map has.

- I4 > I3 > I2 > I1

Properties of Indifference Curves has.

- Indifferences curves are ubiquitous.
- Any bundle has an indifference curve passing through.
- This is assured by the completeness property of preferences.

- Indifferences curves are downward-sloping
- An upward-sloping I would violate the more-is-better property

- Indifferences curves cannot cross
- Indifferences curves become less steep as we move downward and to the right along them
- This property is implied by the convexity property of preferences

Figure 3-11: Why Two Indifference Curves Do not Cross has.

- E = D (because they lie on the same indifference curve).
- D = F (same I)
- By transitivity assumption, E = F. But we know that F > E

Figure 3-12: The Marginal Rates has. of Substitution

- Marginal Rates 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
- = the absolute value of the slope of the indifference curve (ΔFA/ΔSA)

Figure 3-13: Diminishing Marginal has. Rate of Substitution

- The more food the consumer has, the more she is willing to give up to obtain an additional unit of shelter
- The convexity property - the consumers like variety

Figure 3-14: People with Different Tastes has.

- Tex is a potato lover; Mohan, rice lover
- Tex’s MRS of potatoes for rice is smaller than Mohan’s
- Tex willing to exchange 1 potatoes for 1 rice at A
- Mohan would trade 2 potatoes for only 1 rice

The Best Feasible Bundle has.

- Best affordable bundle - the most preferred bundle of those that are affordable
- WHERE is the best affordablebundle located?
- the bundle on the budget constraint that lies on the highest attainable indifferent curve
- the bundle that lies at tangency between indifference curve & budget constraint

Figure 3-16: A Corner Solution has.

- When the MRS of food for shelter is always < the slope of the budget constraint, the best the consumer can do is to spend all his income on food

Corner Solutions has.

- Corner solution - a case the consumer does not consume one of the goods
- Corner solutions occur when goods are perfect substitutes
- With perfect substitutes, indifference curves are straight lines, MRS does not diminishing
- If MRS steeper than budget constraint, we get a corner solution on the horizontal axis
- If MRS less steep, a corner solution on the vertical axis

Figure 3-17: Equilibrium with has. Perfect Substitutes

B = budget constraint

The Utility Function Approach to Consumer Choice has.

- This approach represent the consumer’s preferences with a utility function
- A utility function is a formula that yields a number representing the satisfaction provided by a bundle of goods
- Eg: U(F,S) = FS, where F = food, S = shelter
- if he consume 4 F & 3 S, his utility = 12
- if he consume 3 F & 4 S, his utility = 12

- Eg: U(F,S) = FS, where F = food, S = shelter

Figure A3-1: Indifference Curves for has. the Utility Function U=Fs

- To get the indifference curve that corresponds to all bundles that yield a utility level of U0, set FS = U0 and solve for S to get S = U0/F

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

- Marginal utility (MU) is the rate at which total utility changes as the consumption of goods change

- K & L lie on the same indifference curve (same utility)
- K L: The loss in utility from having less shelter, MUSΔS = the gain in utility from having more food, MUFΔF

Optimal bundle Constant

Cardinal Versus Ordinal Utility Constant

- Ordinal utility approach
- Assumed people are able to rank each possible bundle in order of preference
- Does not require that people be able to make quantitative statements about how much they like various bundle
- Eg: Able to say prefers A to B, not able to make statement as “A is 6.43 times as good as B”

- Cardinal utility approach
- The satisfaction provided by any bundle can be assigned a numerical, or cardinal value by a utility function of the form
U = U(X,Y)

- The satisfaction provided by any bundle can be assigned a numerical, or cardinal value by a utility function of the form

The Method of Lagrangian Multipliers Constant

- We want to find the values of X & Y that produce the highest value of U subject to the constraint that the consumer spend only as much as his income.
Maximize £ = U(X,Y) - (PXX + PYY = M)

X, Y,

The Method of Lagrangian Multipliers Constant

- Taking 1st partial derivatives of £ wrt X, Y & and setting them = 0

- Divide Eq (A.3.9) by Eq (A.3.10)

Example: The Optimal Bundle when Constant U=XY, Px=4, Py=2, and M=40

- Budget constraint is PXX + PYY = M
4X + 2Y = 40

Y = 20 - 2X

- U(X,Y) = X(20 - 2X) = 20X - 2X2

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

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