Final exam！

Econ 339 - basic consumer model 1

Basic econ model of household

Introduction

• Economic analysis of family looks at individuals making choices, given constraints – as individuals, and collectively

• Will look at both consumption (demand) and production (supply) as well: ultimately households supply labour to market and purchase goods and services with their labour income; also, produce goods and services at home.

• Choices? How to allocate time to best satisfy preferences for goods and services produced at home and purchased in market, given productivity in both sectors.

• Start with basic consumer analysis

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Note: math requirements for course

• Will use both diagrams and simple algebra throughout course

• Two – dimensional diagrams • May need to review:

– Equation of straight line: • is a constant • b is also a constant; gives slope of line; slope = ? • If b > 0, upward sloping line in x-y space • If b < 0, downward sloping line in x-y space

– Will also use curves (non-linear)

• Less concerned with equations

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y a bx= + a

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Consumer problem

Consumer: one of basic agents in economy - purchases goods and services in market (given prices and income) How do we describe/define a “consumer/household” as an economic agent? - given tastes, makes choices to reach highest possible level of satisfaction - goal to “maximize utility subject to constraints” Two components in simplest model: 1. budget set: set of goods and services household can afford, given income and prices; 2. preferences: consumer’s tastes: ranking of bundles of goods and services (indifference curve map; utility function)

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Consumer demand • Budget set frequently easier to understand, so start with

this • Budget constraint:

– Consider two goods: “food”, denoted by f, and “all other goods”, denoted by o

– Let denote a unit of food, and a unit of “other goods”

– Let unit prices be and , and total available income be Y

fq oq

fp op

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Example: suppose food is measured in kilograms (think of this as bulk food), and other goods in $ Then the unit price of “other goods” is 1; Let the price of a kg of food be $2.00 Suppose individual has $30 to spend, so Y=30.

– Household is limited to those bundles of food and “other goods” which cost no more than the income available to be spent.

– Given values above, total expenditure on any combination of kg of

food and other goods is

_ Individual has $30.00 to spend, so feasible bundles satisfy – This is this consumer’s budget constraint, and the bundles satisfying

this inequality are in the consumer’s budget set.

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$2.00 f oq q+

0$2.00 $30.00fq q+ ≤

• Playing with budget set: If purchase no food, have $30 to spend on other goods • If spend all money on food (so ), can purchase ____ kg

• If purchase 2 kg of food, how much is left to spend on other goods?

• If purchase qf of food, how much is left for other goods?

• Maximum amount of other goods:

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30 2o fq q≤ −

30 2o fq q= −

0oq =

From equations to diagram

• Plot budget set, in Cartesian space, with food on horizontal (x) axis and other goods on vertical (y) axis

• Use numbers from example

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Identify budget line, budget set

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Budget sets

f f o op q p q Y+ =

-Diagram: -Budget set is in positive quadrant (non-negative quantities)

- bounded by budget line – points satisfying budget set as strict equality. -In general:

Frequently rearrange equation to write BL as

0

f o f

o

pYq q p p

= − or of o f f

pYq q p p

= −

Given income, prices, quantity of one good: max’m quantity of other good

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Properties of Budget line

• Slope and intercepts of BL? – Formulae?

– Interpretations?

•Effects of changes in income? Prices?

•Consumer’s purchases/consumption restricted to Budget set

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How does consumer evaluate bundles? • Consumer preferences/tastes

– Economists are not interested in any particular consumer, but in patterns

– Assume consumers are “rational”

• “Rationality” assumptions: 1. consumer can always compare bundles: for any two bundles A

and B, can say either “A is preferred to B” or “B is preferred to A” or “A and B are equally preferred” 2. more is preferred to less (ex: food and “other goods” are both

“goods”, not “bads”)

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Preferences

• 3. preferences are consistent (“transitivity”): – Given 3 bundles A, B, and C,

if consumer says “A is preferred to B” and “B is preferred to C”, Then if asked to compare A and C, would say “A is preferred to C”

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Utility function

• Summarize preferences by a Utility function – Mathematical description

– Assigns a real # to bundles of goods

• Simplicity/convenience!

– For two goods, ( , )f oU q q

Utility functions

• How to interpret utility function?

• Bundle with larger number is “better”, according to the consumer in question

• Relative numbers matter, not absolute numbers

• (ordinal rather than cardinal measure - mostly)

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Indifference curves (ICs)

• Graphing preferences: – “indifference curve” – all bundles yielding same level of utility

– Rationality assumptions dictate shapes

• Negative slope (if both are “goods”) • Convex to origin

– Marginal rate of substitution given by slope of IC

– Indifference curve map – one curve through each point

– IMPORTANT: assume preferences, and hence indifference

curves, are fixed/given

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Utility functions and IC’s

• Common functional forms: – Linear (perfect substitutes):

• Representative IC: to find equation, fix level of utility, solve for one of the variables.

• MRS? (slope of IC, times (-1))

1 2U q q= +

2 1q U q= −

Cobb-Douglas utility function

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General functional form: Example: Notice: if quantity of either good = 0, U=0 Properties of indifference curves: -Equation: consider IC through bundle (8, 27)

-then U = 18 (why?) -then equation of IC is

- MRS?

1 ,( )f o f oU q q q q

α α−=

1/3 2/3 ,( )f o f oU q q q q=

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Utility functions and IC’s II

– Perfect complements:

• Representative IC? • MRS?

– Quasi-linear:

• Representative IC? Non-linear, intersects ___axis • MRS? Depends only on one good

1 2min{ , }U q q=

1 2lnU q q= +

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Consumer choice

• Choices depend on both constraints and preferences

• ASSUME: choose bundle to max utility – Diagram: superimpose IC map on budget set

• Goal: highest IC on budget set

• Choice? On BL (spend all available income) tangency of BL and IC – Algebra: on BL: spend all income:

bundle at which MRS = price ratio

Example of solving for optimal bundle numerically

• Consider budget set from example, and diagram with food on horizontal axis (need orientation to define slopes)

• Have:

– Equation of budget line is qo=30-qf • Optimal bundle is on this line, so satisfies this equation

– 2nd condition: MRS = price ratio » psychic rate of trade-off = opportunity cost » slope of IC = slope BL

– Suppose CD preferences, with α=1/3; then MRS = 0.5qo/qf – Slope of BL = -2 – Therefore second equation is 2=0.5qo/qf or qo=4qf Econ 339 - basic consumer model 22

Numerical example continued.

• Now have 2 equations in 2 unknowns

• Budget line equation: qo=30-qf

• MRS = opportunity cost equation: qo=4qf

• Solve these simultaneously to obtain qf=6

• Substitute back into budget line equation to obtain qo=24

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