ECON 410 Consumer Theory

budget line

set of all bundles which exhaust income

budget set

set of all bundles under and including the budget line

model of consumer theory

• budget constraints

• tastes or preferences

• maximizes well-being

marginal rate of transformation (MRT)

absolute value of the slope of the budget line

MRT formula

Px/Py

3 ways to interpret MRT

• how much y a consumer must trade for one more unit of x

• the opportunity cost of buying x in terms of y

• the relative price of x in terms of y

bigger MRT means?

steeper slope

smaller MRT means?

flatter slope

properties of preferences

• completeness

• transitivity

• more is better

completeness

when facing a choice, a consumer can rank them so that either a>b, b>a, or a - b

transitivity

consumers rankings are logically consistent in the sense that if a>b, b>a, then a>c

more is better

all else the same, more of a commodity is better than less

rational preferences

preferences are rational if they are complete and transitive

indifference curve

graphical set of all bundles of goods that an individual views as equally desireable

indifference map

a complete set of indifference curves that summarize a consumer’s tastes

if preferences are rational and monotonic, 4 properties must hold

• every bundle lies on an IC curve

• bundles on IC’s farther from the origin are preferred to those on IC’s closer to origin

• IC’s cannot cross

• IC’s cannot slope upward

what does the curvature of the indifference curve signify?

the rate at which an individual is willing to tradeoff between two goods

utility

a set of numerical values that reflect the relative rankings of various bundles of goods

utility function

the mathematical relationship between utility measurers and every possible bundle of goods

what measure is utility?

an ordinal measure

examples of monotonic transformations

• adding a constant

• multiplying by a positive constant

• assuming each good is positive, raising to a positive power

• natural log/exponential

perfect substitutes utility function

• used to represent two goods a consumer views as perfect goods for one another

• consumer always willing to substitute one good for another at the same rate

perfect complements utility function

• used to represent preferences for two goods that must be consumed in a fixed proportion

• a consumer is completely unwilling to substitute one good for another

• more of one good without more of another does not change utility

cobb douglas utility function

represents complete transitive, monotonic, and convex preferences

quasilinear utility function

• can describe preferences for a consumer who purchases the same amount of a good regardless of their income

• the MRS is a function of only one of the goods

marginal rate of substitution (MRS)

• absolute value of the slope of an IC

• MRS of x for y measures the amount of y a consumer is wiling to trade for one more unit of x to remain as well off

MRS formula

= ΔY/ΔX OR = MUx/MUy

marginal utility

extra utility a consumer gets from consuming the last unit of a good

marginal utility of x formula

= MUx = du/dx

marginal utility of y formula

= MUy = du/dy

if MUx/y > 0

x/y is a “good” good

if MUx/y = 0

x/y is a “neutral” good

if MUx/y < 0

x/y is a “bad” good

explanation of downward sloping IC

to keep utility constant, you must consume less of good Y as you consume more of good X

explanation of upward sloping IC

to keep utility constant, you must consume more of good Y as you consume more of good X

explanation of vertical sloping IC

utility does not change as you consume less or consume more of good Y

explanation of horizontal sloping IC

utility does not change as you consume less or consume more of good X

perfect substitute general utility function

U(x,y) = ax + by

perfect complement general utility function

U = (x,y) = min (ax,by)

cobb douglas general utility function

U(x,y) = cx^ay^b

quasilinear general utility function

• U(x,y) = f(x) + ay

• U(x,y) = ax + f(y)

constrained optimization conditions

• if preferences are rational and monotonic, the optimal point will be on the budget line

• if preferences are rational, monotonic, and convex, then the optimal point will be at a point where MRS = MRT

why must the MRS = MRT @ the utility maximizing bundle

the rate at which I am willing to trade goods must equal the rate at which the market dictates I trade them

Bang per Buck formula

MUx/Px = MUy/Py

Lagrangian Method Formula

L = U(x,y) + ƛ (I - PxX - PyY)

first order conditions

• dL/dx = MUx - ƛPx = 0

• dL/dy = MUy - ƛPy = 0

• dL/d = I - PxX PyY = 0

Lagrangian steps

• create Lagrange

• take first order conditions

• solve system of equations

Corner Solution

• an optimal bundle in which one of the goods is not consumed

• the BL/IC tangency condition need not apply

what happens when MRS does not equal MRT for perfect substitutes

the optimal bundle will occur at a corner solution

where does the optimal bundle lie for perfect complements

on the budget line and at the vertex of an IC

when utility is quasilinear, one of the goods’ MU is?

constant

will there be a a corner solution with downward and convex IC’s

• Cobb Douglas - Never

• Quasilinear - Could be

• Other - Depends

will there be a corner solution for perfect complements?

never

will there be a corner solution for downward and linear IC’s

• MRS=MRT?

  • Yes - Could be

  • No - Always

will there be a corner solution for downward concave IC’s?

always

demand function

the mathematical relationship between the quantity demanded of a good, its price, the price of related goods, and a consumer’s income

steps to solve for demand functions

• create lagrange with specified utility function leaving I, Px, and Py as parameters

• take FOC

• solve for x and y. these are the demand functions: x = f(Px,Py,I) and y = (Py, Px, I)

OR

• MRS = MRT

• plug into budget constraint

Income Consumption Curve

shows what happens to consumption of both goods as income changes

Engel Curve

shows what happens to consumption of only one good when income changes

normal good

goods for which there is a positive relationship between income and demand

inferior good

goods for which there is a negative relationship between income and demand

direction of the shift in demand due to a change in income depends on

if the good is normal or inferior

what does elasticity equal for all Cobb Douglas functions

+1

cross-price elasticity of demand

the percent change in quantity demanded of good x that results from a percent change in another good’s price

for all Cobb Douglas will the demand functions be independent or dependent of the price of the other good

independent

income elasticity of demand

the percent change in quantity demanded of good x that results from a percent change in income

for all Cobb Douglas functions what does income elasticity equal

+1

what derivative is needed when graphing a demand curve for good x

dx/dPx

what derivative is needed when graphing an Engel Curve for good x

dx/di

what derivatives are needed when graphing a PCC

dx/dPx and dy/dPx

what derivatives are needed when graphing an ICC

dx/di and dy/di

a change in price of a good has two effects on an individual’s demand

• substitution effect

• income effect

substitution effect (SE)

the change in quantity demanded when the good’s relative price changes, holding consumer’s utility (purchasing power) constant

income effect (IE)

the change in quantity demanded when purchasing power changes, holding relative prices constant

total effect (TE)

the sum effect of substitution and income effects on quantity demanded

for IC’s downward sloping and convex, what ALWAYS happens to the SE

the SE shows the consumer substituting away from the good whose relative price increased and towards the good whose relative price decreased

properties of Giffen Good’s

• inferior good by definition

• IE > SE leading to the TE breaking the law of demand

what to list when asked to interpret elasticity

• elastic/inelastic/unit elastic

• whether the good follows the LOD

• a 1% change in price causes X% change in quantity demanded

steps to solving for '“will a consumer ever choose a corner solution?”

• find demand functions

• ask: will either of these functions produce a non-positive number

• the good who’s demand function gives a non-positive number, the consumer would not consume

• set function that gives non-positive ≤ 0 to find condition

steps to graph IE and SE

• construct chart

• draw BLs

• add bundles A,B, and C to graph

• add IC1 and IC2

Income and Substitution: Point A

• original optimal bundle

• tangency between IC1 and BL1

Income and Substitution: Point B

• hypothetical optimal bundle between IC1 and BLc

Income and Substitution: Point C

• new optimal bundle

• tangency between IC2 and BL2

Substitution Effect Points

• A → B

• Xb - Xa

Income Effect Points

• B → C

• Xc - Xb

Total Effect Points

• A → C

• Xc - Xa