Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 2 Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets
IV. Market failure Outline of the course
I. Introduction
II. Individual choice
Budget constraint (Ch. 2)
Preferences (Ch. 3)
Utility (Ch. 4)
Consumer problem (Ch. 5)
III. Competitive markets
IV. Market failure Consumer Theory
Consumers choose the best consumption bundle they can afford.
– this is virtually the entire theory in a nutshell
– but this theory has many surprising consequences
Two parts to theory:
– “can afford”  budget constraint
– “best”  according to consumers’ preferences 3 Consumption bundle
L goods in the economy, labeled ` D 1; 2; : : : ; L
.x1; x2; : : : ; xL/  how much of each good is consumed
.p1; p2; : : : ; pL/  prices of each good
m  money the consumer has to spend
Budget constraint: p1x1 C p2x2 C : : : C pLxL Ä m
all nonnegative .x1; x2; : : : ; xL/ that satisfy this constraint make up the
budget set of the consumer. See ﬁgure on blackboard. 5 Two goods
theory works with more than two goods, but can’t draw pictures.
two goods enough to understand conceptual issues
often think of good 2 (say) as a composite good, representing money to
spend on other goods.
budget constraint becomes: p1x1 C p2x2 Ä m :
money spent on good 1 (p1x1) plus the money spent on good 2 (p_2x_2)
must be less than or equal to the amount of money available (m). 6 Budget line
The set of consumption bundles that satisfy the budget constraint with
equality: p1x1 C p2x2 D m ;
which can also be written as x2 D
budget line has slope of p1
p2 m
p2 p1
x1
p2 and vertical intercept of m
p2 m
set x1 D 0 to ﬁnd vertical intercept ( p ); set x2 D 0 to ﬁnd horizontal
2
m
intercept ( p )
1 slope of budget line measures opportunity cost of good 1  how much of
good 2 you must give up in order to consume more of good 1 7 Changes in budget line
increasing m makes parallel shift out
increasing p1 makes budget line steeper
increasing p2 makes budget line ﬂatter
just see how intercepts change
multiplying all prices by t same as dividing income by t
multiplying all prices and income by t does not change consumption
possibilities 8 The numeraire
can arbitrarily assign one price a value of 1 and measure other price
(and income) relative to that
useful when measuring relative prices; e.g., British pounds per dollar,
1987 dollars versus 1974 dollars, etc.
the good whose price is normalized to one is called the numeraire good 9 Taxes, subsidies and rationing
quantity tax  tax levied on units bought: p1 C t
value tax  tax levied on dollars spent: p1 C p1. Also known as ad
valorem tax
subsidies  opposite of a tax p1 s or .1 /p1 lump sum tax or subsidy  amount of tax or subsidy is independent of the
consumer’s choices. Also called a head tax or a poll tax. 10 Example  food stamps
before 1979 was an ad valorem subsidy on food
– paid a certain amount of money to get food stamps which were worth
more than they cost
– some rationing component  could only buy a maximum amount of
food stamps
after 1979 got a straight lumpsum grant on food coupons. Not the same
as a pure lumpsum grant since could only spend the coupons on food. 11 Preferences
Preferences are relationships between bundles
Binary relations: heavier than, taller than, etc.
If a consumer would choose bundle .x1; x2/ when .y1; y2/ is available it
is natural to say that bundle .x1; x2/ is preferred to .y1; y2/ by this
consumer
Preferences have to do with the entire bundle of goods, not with
individual goods. 12 Notation
.x1; x2/ .y1; y2/ means the x bundle is strictly preferred to the
y bundle
.x1; x2/ .y1; y2/ means the x bundle is regarded as indifferent to the
y bundle
.x1; x2/ .y1; y2/ means the x bundle is at least as good as the
y bundle 13 Assumptions about preferences
Complete  any two bundles can be compared:
for all bundles x and y , either x y or y x. Reﬂexive  any bundle is at least as good as itself:
for all x , x x. Transitive  if x y and y z , then x z. 14 Indifference curves
graph the set of bundles that are indifferent to some bundle. See ﬁgure
on blackboard.
boundary of the upper contour set
Note that indifference curves describing two distinct preference levels
cannot cross. 15 Examples of preferences
perfect substitutes
– red pencils and blue pencils; quarts and gallons
– constant rate of tradeoff between the two goods
perfect complements
– always consumed together
– right shoes and left shoes; coffee and cream
bads
neutrals
satiation or bliss point 16 Wellbehaved preferences
monotonicity  more of either good is better.
Implies indifference curve has negative slope.
convexity  averages are preferred to extremes.
– slope gets ﬂatter as you move further to the right
– example of nonconvex preferences. See ﬁgure on blackboard. 17 Marginal Rate of Substitution
slope of the indifference curve
changes as we move along the curve
Let x2 D f .x1/ denote a certain indifference. Then, x2
MRS at .x1; x2/ D f .x1/ D
x1
0 along an indifference curve natural sign is negative, since indifferent curves have negative slope
measures how the consumer is willing to trade off consumption of good
1 for consumption of good 2.
measures marginal willingness to pay (give up)
– not the same as how much you have to pay
– but how much you would be willing to pay
18 Utility Function
summarizes preferences
a utility function assigns a number to each bundle of goods so that more
preferred goods get higher numbers
that is, u.x1; x2 / > u.y1; y2/ if and only if .x1; x2/ .y1; y2/ only the ordering of bundles counts, so this is a theory of ordinal utility Utility functions are not unique
If u.x1; x2 / is a utility that represents some preferences, and f . / is any
increasing function, then f .u.x1; x2// represents the same preferences.
Why? Constructing a utility function
Easy task when preferences are monotonic. Examples
from utility to indifference curves
from indifference curves to utility
examples
– perfect substitutes  all that matters is total number of pencils, not
their color, so u.x1; x2/ D x1 C x2 does the trick
– can use any monotonic transformation as well, such as log.x1 C x2/
– perfect complements  all that matters is the minimum of the left and
right shoes you have, so u.x1; x2/ D minfx1; x2g works
– quasilinear preferences  u.x1; x2/ D v.x1/ C x2, that is,
indifference curves are vertically parallel. See ﬁgure on blackboard.
– CobbDouglas preferences
bc
utility has form u.x1; x2/ D x1 x2
convenient to take transformation f .u/ D u
a1
or x1 x2 a , where a D b=.b C c/ 1
b Cc b
b Cc and write x1 c
b Cc x2 Marginal utility
extra utility from extra consumption consumption of one of the goods,
holding the other good ﬁxed
this is a derivative, but a special kind of derivative  a partial derivative
just means you look at the derivative of u.x1; x2/ keeping x2 ﬁxed treating it like a constant
examples
– if u.x1; x2/ D x1 C x2, then M U1 D @u
D 1:
@x1 a1
– if u.x1; x2/ D x1 x2 a , then @u
a
1
D ax1 1x2
M U1 D
@x1 a : Marginal utility
@u
.x1; x2/
M U1 .x1; x2/ D
@x1
note that MU depends on the choice of the utility function  not an ordinal
concept
however, MU is closely related to MRS, as we will see, which is ordinal Relationship between MU and MRS
Fix a utility level k and consider the correspondent indifference curve: u.x1; x2/ D k
This equation implicitly deﬁnes a function f .x1/ such that u.x1; x2/ D k if and only if x2 D f .x1/
Recall the deﬁnition of MRS: MRS.x1 ; x2/ D f 0.x1/ whenever u.x1; x2/ D k
In principle, need to know f , which is only implicitly deﬁned. Relationship between MU and MRS
We have u.x1; x2/ D k
along the indifference curve x2 D f .x1/
Thus, u.x1; f .x1// D k for all x1: By the chain rule of calculus, @u
@u
df
d
.x1 ; f .x1// C
.x1; f .x1//
.x1/ D
u.x1; f .x1// D 0:
@x1
@x2
dx1
dx1
Hence, df
MRS D
D
dx1 @u
@x1
@u
@x2 D M U1
M U2 Optimal choice
move along the budget line until upper contour set doesn’t cross the
budget set
note that tangency occurs at optimal point. In symbols: MRS.x1; x2 / D p1
p2 Exceptions: kinks and corner solutions.
Warning: tangency is not a sufﬁcient condition, unless indifference
curves are convex. Optimal choice
If we know the utility function, the prices and the income we can
calculate the consumer’s optimal bundle.
Two equations, two unknowns: p1
p2
Dm MRS.x1; x2/ D
p1x2 C p2x2 This always works when preferences are convex and optimum is interior.
Optimal choice depends only on:
– preferences (not utility)
– price ratio MRS is an ordinal concept
Why?
Suppose v.x1 ; x2/ and u.x1; x2/ represent the same preference.
Hence, v.x1; x2/ D f .u.x1; x2// where f is an increasing function.
By the chain rule of calculus, MRS under v D @v
@x1
@v
@x2 D @u
f 0.u.x1; x2// @x1 f 0 .u.x ; x / @u
1 2 @x2 D MRS under u Examples
perfect substitutes: x1 D m=p1 if p1 < p2, and 0 otherwise
perfect complements: x1 D m=.p1 C p2/
neutrals and bads: x1 D m=p1
discrete goods: good is either consumed or not; compare
.1; .m p1/=p2/ with .0; m=p2/
concave preferences: similar to perfect substitutes. Note that tangency
doesn’t work.
CobbDouglas preferences (Homework): x1 D am=p1 . Note that
parameter a gives the budget share. Which is better, a commodity or an income tax?
original budget constraint: p1x1 C p2x2 D m
with tax: .p1 C t/x1 C p2x2 D m
optimal choice with tax: .p1 C t/x1 C p2x2 D m
revenue raised is tx1
income tax that raises same amount of revenue yields the budget
constraint: p1x1 C p2x2 D m t x1
this budget line passes through .x1 ; x2 /, so....
Income tax is at least as good! caveats
only applies for one consumer
income is exogenous  if income responds to taxes, problems
no supply response, only looked at demand side Demand function
optimal choice depends on prices and income
when we hold income ﬁxed, and examine dependence of optimal
consumption on prices this is called the demand function
when we hold prices ﬁxed, and examine dependence of optimal
consumption on income this is called the Engel function ...
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 Spring '09
 SAMUELSON
 Microeconomics, Utility

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