homework:. indifference curves definition: for any bundle a and a preference relation over bundles,...

Homework:Problem IV.4Find the maximum for =0=0=0Notice: =0 iff =0Then =0 iff =0 iff iff if Then x,y, or z must be zero. (more than one may be zero, but at least one must be)

Homework:Problem VWe know ab, bc, cd, and da.

Then a b c da, an intransitive cycle.

So preferences are not transitive.

Homework:Problem VITransitivity?If a b and bc, where a,b, and c are holdings of dollars and pesos, then a is worth more than b, and b is worth more than c, since Bill only cares about the value of his money holdings. Then a must be worth more than c. Then a c.YES

Completeness?Every bundle of dollars and pesos has a dollar value, and Bill prefers higher dollar values, so every bundle can be compared.YES

Homework:Problem VILocal Nonsatiation?Consider a bundle a and a ball around a of radius . Let b=a+ /2, /2). Then b is in the ball around a and b is worth more than a, so Bill strictly prefers (by a very small amount) b to a. That is: b a.YES

Convex?Let the exchange rate of pesos per dollar be E>1. Then the value of the bundle (x dollars, y pesos) is x+y/E and Iff Let a==(1,E) and =(2,0). Then . Let c=. Then a and b . Then by the definition of convexity, if preferences are convex a/2+b/2 c=aTherefore (1/2,E/2)+(1,0) (1,E)Therefore (3/2,E/2)(1,E)But 3/2+(E/2)/E=2=1+E/E so a/2+b/2 c, a contradiction

Homework:Problem VIContinuitySuppose a= b =. Then . Let = . Then if we make the radius of the ball around a less than or equal to /2, any c in the ball will be preferred to b.YES

Indifference Curves

• Definition:• For any bundle a and a preference relation over bundles, the indifference curve through a is the set of all bundles that are indifferent to a in that preference relation.• Basically, it’s the curve the traces out which bundles are equally good.

• An indifference map is the set of all indifference curves.• Note that an indifference map will include all possible bundles, as long as preferences are complete.

Indifference Curves

• Remember: we assume preferences are monotonic, complete, and transitive.• Are Upward sloping curves possible?• No!

UNC Football National Championships

Indifference Curves

• Remember: we assume preferences are monotonic, complete, and transitive.• Are Upward sloping curves possible?• No!

Indifference Curves

• Remember: we assume preferences are monotonic, complete, and transitive.• Are thick indifference curves possible?• No.

Indifference Curves

• Remember: we assume preferences are monotonic, complete, and transitive.• Are thick indifference curves possible?• No.

Indifference Curves

• Remember: we assume preferences are monotonic, complete, and transitive.• Can indifference curves cross?• No.

Indifference Curves

• Remember: we assume preferences are monotonic, complete, and transitive.• Can indifference curves cross?• No.

• by monotonicity• But B~C and A~B because they are on the same indifference curves• Then by transitivity, A~C.• Contradiction!

Indifference CurvesG

Good 1

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Preferred

Indifference Curves

• Review: Indifference curves must be:1. Everywhere2. Thin3. Non-increasing4. Non-Crossing

Indifference Curves• Which one of these curves is convex?• Strict Convexity: A preference relation displays strict convexity if for every x, y, and z such that xz and y z, then x+(1- )y z, where 0< <1 and x and y are not the same.

Indifference Curves• Which one of these curves represents con?• Strict Convexity: A preference relation displays strict convexity if for every x, y, and z such that xz and y z, then x+(1- )y z, where 0< <1 and x and y are not the same.

Indifference Curves

• Review: Indifference curves representing convex preferences must be:1. Everywhere2. Thin3. Non-increasing4. Non-Crossing5. Convex (bowed inward)

Indifference Curves

• Indifference curves tell us how people make tradeoffsPiz

Indifference Curves• What does an extremely bowed indifference curve mean?

Good 1

Indifference Curves• This curve represents perfect complements, where goods are consumed only in fixed proportions

Good 1

Indifference Curves• What does a straight indifference curve mean?

Good 1

Indifference Curves• This curve represents perfect substitutes, where goods are interchangeable at some fixed ratio.

Good 1

Indifference Curves

• Intermediately bowed curves are imperfect substitutes or complements. More curved means that the goods are more complementary—the individual wants to consume them together. Less curved means one can be used in place of the other—as a substitute.

Indifference Curves• We’d like to be able to quantify what trades individuals are willing to make between various goods

Good 1

Indifference Curves• Marginal Rate of Substitution

• The MRS between good 1 and good 2 (MRS12) is:1. The maximum amount of Good 2 the individual

would be willing to give up to get one more unit of Good 1

2. The minimum amount of Good 2 the individual would need to receive to give up one unit of Good 1.

Good 1

• Suppose we’re interested in the MRS at a particular bundle, . Let where .• In English: is on the same indifference curve.Then

Good 1

Indifference Curves• Marginal Rate of Substitution:• =1/

• Example: If you’re willing to give up 3 units of for one unit of , you’d be willing give up 1/3 of a unit of for 1 unit of.

Indifference Curves

• Homework:• Finish Petranka Ch. 3• Have a great weekend!

Utility

• Preferences are rational iff a utility representation exists.• If preferences aren’t complete, we couldn’t assign a value to the utility function for some bundles.• If preferences aren’t transitive, we have a b ca. Then U(a)>U(b)>U(c)>U(a).• Then U(a)>U(a), which is impossible.

• Preferences are rational and continuous iff a continuous utility representation exists.

Utility

• Preferences are rational and monotonic iff an increasing utility representation exists.

Utility

• Ordinal interpretation of utility functions:• In this class, we will not be interested in cardinal utility—that is, the exact value of the utility function. Instead, we will be interested in the ordinal properties of utility—that is, which value of the utility function is higher.

Utility

• Ordinal comparisons:• Kate is taller than Jenny.• 5 is more than 3.• Pete likes making $100,000/year more than making $20,000/year.

• Cardinal comparisons• Kate is 3 inches taller than Jenny.• 5 is 2 more than 3.• Pete likes making $100,000/year twice as much as $20,000/year.

Utility

• Monotonic Transformations• Definition: f(x) is monotonic if f(x)>f(y) iff x>y.• In English: if you graph the function, it’s always sloped upward i.e. increasing in all the variables.• Proof: If U(x) represents , then U(x)>U(y) iff x y, and by montonicity, U(x)>U(y) iff f(U(x))>f(U(y)). Then f(U(x))>f(U(y)) iff x y, so f(U(x)) represents

Utility

• Monotonic Transformations• Definition: f(x) is monotonic if f(x)>f(y) iff x>y.• If f’(x)>0 for all x, then f is monotonic.

Utility

• Uses for cardinal utility• In order to decide what allocation of resources is better, we usually need cardinal utility.• Why? Because interpersonal comparisons of utility are impossible with ordinal preferences.

Utility

• Uses for cardinal utility• Examples:• Is a monopoly bad?• CS and PS require that everyone gets the same cardinal utility from money.

• When Bill Gates gives money to impoverished people in Africa, do they gain more welfare than he loses?• Using the concept of decreasing marginal returns to wealth, we’d guess that they gain much more than he loses. But without cardinal utility we have no way to analyze that in an economic model.

Utility

• Why ordinal utility?• There are infinitely many utility functions that can represent a set of preferences, so preferences alone can’t determine cardinal utility. • Specifically, any monotonic transformation of a utility function represents the same preferences.

Utility

• Back to ordinal preferences• Uses of monotonic transformations• Some utility functions can be hard to differentiate, and can be easier after a monotonic transformation.• Example: Cobb-Douglas Utility:

• Define V=ln(U)

Utility

• Back to ordinal preferences• Uses of monotonic transformations• Some utility functions can be hard to differentiate, and can be easier after a monotonic transformation.• Example:• ln is monotonic so we can just maximize• is monotonic so we can just maximize• -

Utility

• Preferences• Transitive +Complete Monotonic Continuous

• Utility function:• Exists Increasing Continuous

Indifference Curves

• We need to get an idea of what utility functions correspond to what type of preferences.

Utility• What sort of utility function represents these preferences?

• The higher a is, the more you want relative to x1.

Good 1

Utility• What sort of utility function represents these preferences?

• The higher a is, the more you like relative to .

Good 1

Utility

• What sort of utility function represents these preferences?

• One possibility: Cobb-Douglas– • The higher a is, the more you prefer good 1.• The more of a good you have, the less attractive it is compared to the

other good.

Utility

• Exercise: come up with utility functions U(x,y) to satisfy the following preferences:1. Your preferences are continuous2. You like x five times more than y, and would always trade 5x for a unit of y.3. You always want to consume twice as many x as y.4. You like having more y than x and you like having some of both more than a lot of one or the other. You always strictly prefer having another unit of either good.You have 10 minutes.

• The MRS between good 1 and good 2 (MRS12) is:1. The maximum amount of Good 2 the individual

would be willing to give up to get one more unit of Good 1

2. The minimum amount of Good 2 the individual would need to receive to give up one unit of Good 1.

Good 1

• Suppose we’re interested in the MRS at a particular bundle, . Let where .• In English: is on the same indifference curve.Then

Good 1

Utility

• A utility function U represents preferences if iff x y.

Utility

• Utility also gives us an easy way to find the marginal rate of substitution (MRS).

• That is, the rate you’d trade A for B at is the marginal utility of A (additional utility of a little more A) divided by the marginal utility of B.

,MRS AA B

Utility

Examples:

,MRS AA B

Utility

• More types of utility functions:Quasilinear: • The tradeoff between and depends on , but not , so

along the x2 axis the indifference curves look the same, just transposed to the left or right.

Good 1

Good 2

Utility

• More types of utility functions:Quasilinear: • The (negated) slope of the indifference curve is

Good 1

Good 2

Constrained Optimization

• We’ve already learned how to optimize a function without constraints:1. Take partial derivatives and set them equal to zero.2. Solve the system of equations.• However, there are almost always constraints on our choices.• Examples:• Choosing the time you spend studying vs leisure each day—you only have 24 hours in a day.• Choosing what to spend your income one—you have a finite budget.

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• The budget set is the set of all possible bundles an individual can choose from.• Usually, it’s the area under a downward sloping line that has all non-negative values for every good in the bundle.

Budget Set

• A budget set is typically as follows: You have w dollars to spend, and each good has a price. If there are two goods, and the quantity of good 1 iswhile the quantity of good 2 is ,• The budget line, which is the upper edge of the budget set, is

Budget Set

• A budget set is typically as follows: You have w dollars to spend, and each good has a price. If there are two goods, and the quantity of good 1 iswhile the quantity of good 2 is ,• The budget line, which is the upper edge of the budget set, is

Budget Set

• Marginal Rate of Transformation• the Marginal Rate of Transformation between Good 1 and Good 2, , is 1. The amount of good 2 the individual must give up to get one more

unit of good 1.2. The amount of good 2 an individual will receive if they give up one

unit of good 1.

• Marginal Rate of Transformation• the Marginal Rate of Transformation between Good 1 and Good 2, , is 1. The amount of good 2 the individual must give up to get one more

unit of good 1.2. The amount of good 2 an individual will receive if they give up one

unit of good 1.3. -1 times the slope of the budget line.4. If prices are fixed and nothing is being given away for free,

• Break time!

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Budget Line

• Can a point below the budget line be optimal if preferences are complete, transitive, convex, and monotonic?• No!• Any optimal bundle MUST be on the budget line.

• Given rational, convex, monotonic preferences and smooth indifference curves, is it possible for an optimal bundle to include both goods and not be where the indifference curve is tangent to the budget line?• No!

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• Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods, the indifference curve must be tangent to the budget line.M

I1I2I3 I4

Constrained Optimization• Equivalently, Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods,.• : The minimum amount of Good 2 the individual would need to receive to give up one unit of

Good 1.• : The greatest amount of good 2 the individual is willing to give up to get one more unit of

good 1• : The amount of good 2 an individual will receive if they give up one unit of good 1.• : The amount of good 2 the individual must give up to get one more unit of good 1.

• Suppose . Then the amount of good 2 the individual must give up to get another unit of good 1 is less than the maximum they’d be willing to give up, so they keep the amount of good 2 that would make them indifferent, plus a little more good 2, so they’d strictly prefer giving up some good 2 for more good 1.• Suppose . Then the amount of good 2 the individual will receive for giving up one unit of good 1 is greater than the minimum they’d be willing to accept. So they get the amount of good 2 that would make them indifferent plus a little more, so they strictly prefer giving up good 1 for more good 2.

• Special cases:• Corner Solutions:

• If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual doesn’t consume one or more goods.• Kinked indifference curves

• If the indifference curve is kinked, then the derivative may not be defined at the optimum.

Constrained Optimization• Special cases:

• Corner cases:• If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual consumes only one good.• If consume only good 1• If consume only good 2

• Corner cases:• If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual consumes only one good.• Generally, you can check corner solutions against interior solutions and other corner solutions directly. Whichever solution has the highest utility is the optimum.

• Special cases:• Kinked indifference curves

• If the indifference curve is kinked, then the derivative may not be defined at the optimum. • Drawing the indifference curves and budget constraint may be helpful.• Check for solutions on the non kinked portion of the curve.

Constrained OptimizationExercise: For each combination of budget constraint and indifference curve, draw the optimum and give the approximate optimal bundle of x and y.Note that the utility functions are quasilinear in x, so every indifference curve looks the same, just transposed to the left or right.

2 4 6 8 10

0 2 4 6 8 100

0 2 4 6 8 1 00

Budget Constraints: Indifference curves:

• Homework:• Homework due Friday• Read Petranka through 5.3

• Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods, the indifference curve must be tangent to the budget line.M

I1I2I3 I4

Constrained Optimization• Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods,.• Given rational, convex, monotonic preferences, a smooth indifference curve, and an optimal bundle with nonzero quantities of all goods, the indifference curve must be tangent to the budget line.

• Corner cases:• If the slope of the indifference curve never equals the slope of the budget constraint, clearly MRS=MRT is not possible. In this case, the individual consumes only one good.• Generally, you can check corner solutions against interior solutions and other corner solutions directly. Whichever solution has the highest utility is the optimum.

• Special cases:• Kinked indifference curves

• If the indifference curve is kinked, then the derivative may not be defined at the optimum. • Drawing the indifference curves and budget constraint may be helpful.• Check for solutions on the non kinked portion of the curve.

Strictly convex preferences display a diminishing marginal rate of substitution. In other words, as an individual gets more of Good A, she needs to receive less of Good B in order to be willing to give up some of Good A

• We know the following:• at the optimal bundle• =

• Then = • That is, the slope of the indifference curve equals the slope of the budget line.

• = • Equivalently, • = • The interior optimum bundle is at the point where the marginal utility per dollar of all goods is equal. • That is, where the additional utility from another dollar’s worth of the good is the same for all goods.• If not, one good would give more utility for another dollar than the other, and we’d want to shift consumption to the more cost effective good.

• Restricted Conditions to use Lagrangian techniques to find a constrained maximum1. The function we want to maximize is continuous.2. The function we want to maximize has a derivative

that is continuous.3. The function we want to maximize is quasi-

concave.4. Our constraint set is defined by an equality.5. Our constraint set is defined by a weakly concave

function.

• Restricted Conditions to use Lagrangian techniques to find a constrained maximum1. The function we want to maximize is

continuous.• If not, we could have very ugly functions

• Restricted Conditions to use Lagrangian techniques to find a constrained maximum2. The function we want to maximize has a

derivative that is continuous.• If not, we have a kinked function, and if the maximum is at the kink we can’t take derivatives to find the optimality condition.

• Restricted Conditions to use Lagrangian techniques to find a constrained maximum3. The function we want to maximize is

quasi-concave.• Definition: f is quasiconcave if for all x and y in the domain (in our case, x and y are bundles) and for any a between 0 and 1.• Quasi-concave utility means that the represented preferences are convex.

• Restricted Conditions to use Lagrangian techniques to find a constrained maximum4. Our constraint set is defined by an

equality.

• As we saw before, monotonicity ensures that the optimal bundle will be on the budget line, so we only need to optimize over the budget line, not the whole budget set.

• Restricted Conditions to use Lagrangian techniques to find a constrained maximum5. Our constraint set is defined by a weakly

concave function.• Linear functions are weakly concave, so the budget line satisfies this condition.

• Procedure for solving a constrained optimization problem: 1. Where x is a bundle and is a number

1. Take first order conditions ,… and set them equal to zero.

2. Solve the system of equations to find the interior optimum.

3. Check the interior optimum against corner solutions.

• Procedure for solving a constrained optimization problem: 1. Where x is a bundle and is a numberWhen we use this procedure, f will usually be U, the utility function, and g will be the budget constraint:

• Example: We want to find the optimal bundle given• U(x1,x2)=x1

1/2x21/2

• w=4• =2• =1

• Exercise: find the optimal bundles for the following problems:

1. U(x1,x2)=x1 + x2

• w=4 , =2, =12. U(x1,x2)=4x1x2

• w=2 , =1, =1• We’ll discuss the answers in 10 minutes

• Homework:• Homework due Friday• Finish Chapter 5 in Petranka

• Example: We want to find the optimal bundle given

1. U(x1,x2)=x1 + ln(x2)• w=6 , =3, =1

• Example: We want to find the optimal bundle given

• U(x1,x2, x3)=x11/3x2

1/3x11/3

• w=6• =1• =2• =3

homework:. indifference curves definition: for any bundle a and a preference relation over bundles,...

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