Derive and understand:
1. How Rational People make Choices
2. How this should Guide our own Decisions

• We are now studying the foundations of demand theory.
- Why do demand curves slope downward?
- Why do they shift with changes in prices for substitutes and complements?
- Why do they shift with changes in income?
- What normative significance can we give to demand based on underlying consumer preferences?

• Preferences are complete if for any two consumption points x and x', either x ≤x' (x is at least as good as x') or x' ≥ x (x' is at least as good as x), or both.
• Preferences are reflexive if for all x, x ≥ x (x is at least as good as itself).

• Preferences are transitive if x ≥ x' and x' ≥ x'' implies that x ≥ x''.
• Preferences are strongly monotonic if for any two commodity points x = (x1, x2) and x' = (x'1, x'2) if x1 ≤ x'1, x2 ≤ x'2, and x ≠ x', then x' is preferred to x.
• Preferences are continuous if the set of all choices that are at least as good as a choice x' and the set of all choices that are no better than x' are both closed sets.

Representation Theorem:
If a consumer has a preference relation that is complete, reflexive, transitive, strongly monotonic, and continuous , then these preferences can be represented by a continuous utility function u(x) such that u(x) > u(x') if and only if x > x'.

•
Consider the following (ordinal) utility function for Food (F) and Clothing ( C ) for Emily:
U(F,C) = FC
• Each indifference curve gives the combinations of F and C that yield the same level of satisfaction to Emily (e.g. 25, 50, 100).
•
Think of slicing the utility function at different levels and projecting into F,C space.
• Her marginal rate of substitution of Food for Clothing is given by the slope at any point on an indifference curve
MRS
FC
= - dC/dF|U = a constant
[MRS
FC is the amount of Clothing Emily is willing to give up for one addition unit of food, holding utility constant]

•
Marginal rate of substitution of F for C is equal to the marginal utility of F divided by the marginal utility of C at a point on an indifference curve, i.e holding utility constant (dU = 0) dU(F,C) = U
F dF + U
C dC = 0
Where U
F
= Marginal utility of F
U
C
= Marginal utility of C
U
F
/Uc = -dC/dF|U = constant = MRS
FC
• For U = FC: U
F
= C, U
C
= F -> MRS
FC
= C/F
• Set U = 25: C = 5, F = 5; MRS
FC
= 5/5 = 1
C = 2.5, F= 10; MRS
FC
= 2.5/10 = 0.25
Declining MRS of F for C as F increases holding U constant
Declining marginal utility of Food as F increases U constant

Budget constraints limit an individual’s ability to consume in light of the prices they must pay for various goods and services. This is where scarcity comes in and what helps to make economics the dismal science
• The Budget Line
–
The budget line indicates all combinations of two commodities for which total money spent equals a given fixed level of income (ignore borrowing and saving for now)

Consumers choose a combination of goods that will maximize the satisfaction they can achieve, given the limited budget available to them.
• The maximizing market basket must satisfy two coditions:
1) It must be located on the budget line.
2) Must give the consumer the most preferred combination of goods and services.
Max U(q1, q2, q3,…q n
)
S.T
. Σp i q i
= I

• Let: U = FC
• I = P
F
F + P
C
C
• Write C in terms of F and substitute
• C = I/P
C
– P
F
F/P
C
• Max
F
U = F(I/P
C
– P
F
F/P
C
) = FI/P
C
–
P
F
F^2/Pc
• dU/dF = I/Pc – 2P
F
F/Pc = 0
• F = I/2P
F
• C = I/2P
C
-> MRS = C/F = P
F
/P
C

• A Corner Solution
– If the MRS is, in fact, significantly greater than the price ratio, then a small decrease in the price of frozen yogurt will not alter the consumer’s market basket.
– At point B, the MRS of ice cream for frozen yogurt is greater than the slope of the budget line.
– This suggests that if the consumer could give up more frozen yogurt for ice cream he would do so.
– However, there is no more frozen yogurt to give up since all of her income is going to ice cream

Optimal bundle
MRS = p
1
/p
2
How Rational Consumers Choose: A Graphical
Approach
Good 2
Budget Line : p
1 x
1
+ p
2 x
2
= I
Slope: Ratio of Prices ( p
1
/ p
2
)
Direction of increasing Utility
2 ingredients :
Preferences : for each consumer, tells us tastes of consumers
Budget constraints : tells us prices of goods and a consumer’s income
0
Slope: Marginal Rate of Substitution of Good 1 for Good 2 ( M.R.S
.)
Indifference Curve : tells us between which bundles the consumer is indifferent
Good 1
Budget Set : tells us what the consumer can afford, given his income I and prices p
1
, p
2

Price of Beer: p
B
= 6, Price of Tacos: p
T
= 1, Income: I = 18
Tacos
18
12
If you spend $1 more on beer rather than on tacos:
-you lose -1/ p
T units of Tacos
-you gain 1/ p
B units of beer a / b = M.R.S.
p
B
/ p
T
= price ratio
-1/ p
T
1 /p
B
a b
Indifference curve corresponding to a utility of 5 (say)
Beer
1 3

x = Units of Tacos, y = Units of Beer
M.R.S.
( x , y ) = .5
x / y : MRS of Beer for Tacos
Given formula, say
Price of Beer: 6, Price of Tacos: 1, Income: 18
Need to Solve: .5x/y = 6 and x + 6y = 18
Which gives: x =12, y =1

1.
You have just bought a mansion in Middletown.
Suddenly, after payment, the price of all houses in your new neighborhood increase. Are you better off after the change? Would you be better off if the price had decreased instead (ignore transaction costs, fees…)?
2.
The government decided that potato consumption should be subsidized: for each pound bought, you get
$1 from the government. The government then decides, to balance its budget, to levy the lowest tax per capita- that achieves this goal. Are you overall better off – or worse off?

y = All others goods ($)
B
I
I
’
X
’
X
B
’
(Housing) price increase
X = ( x,y )-bundle chosen before price increase
B = initial budget line; I = initial indifference curve
X = ( x,y )-bundle chosen after price increase
B
’= final budget line;
I = final indifference curve

: All indifference curves to follow are for sake of illustration!
(they are not derived) x = Housing (sq.)

y = All others goods ($)
B
I I’
B’
X
X’
(Housing) price decrease
X= ( x,y )-bundle chosen before price decrease
B= initial budget line; I= initial indifference curve
X= ( x,y )-bundle chosen after price decrease
B’= final budget line; I= final indifference curve x = Housing (sq.)

y = all other goods ($)
150
120
B’’
B
B’
90
70
60
I
X
B : Income: I = $150; (potato) price: $3
X : chosen bundle (initial situation)
B
’ : Income:
I = $150; price: $2 = 3 - 1($1 subsidy)
X ’ : chosen bundle (when subsidy is applied, but not the tax)
B
’’: Income:
I = 150 - Tax; price: $2
(Total Subsidy = Tax)
X
’’: chosen bundle (eventual situation)
I’
X’’
I’’
X’
Initial budget line
Final budget line
20 30 40 50 75 x = potatoes (pds)

• Substitution Effect
• When the price of a good decreases relative to others goods, holding purchasing power constant, consumers substitute toward the good that is relatively cheaper.
• Note that relative prices change, so intuitively, the slope of the budget line will change.
• Note that purchasing power is held constant.
• Thus, we hold the consumer to the same level of utility
(i.e., consumer can only purchase the same level of utility as before).

Movie
• Remember the slope of this budget line is -P
C
/P
M
.
• When P
C falls, the slope becomes less negative.
• First rotate budget line out,
(i.e., from white line to blue line).
• Then parallel shift it back to the green line.
CD

Movie
• Next draw in your original indifference curve.
• The original point is C
1
.
• When the price of CD’s falls, the substitution effect causes consumption to increase to C
2
.
• The substitution effect looks the same for all three cases.
CD
C
1
C
2

• The direction of the income effect depends on whether the good is a normal good (i.e., increased purchasing power increases consumption) or an inferior good (i.e., increased purchasing power decreases consumption).
• The income effects help to distinguish our three cases.

Movie
• Increased purchasing power increases consumption from C
2 to C
3
.
C
2
C
3
U
1
U
2
CD

Movie
• Increased purchasing power increases consumption from C
2 to C
3
.
• Put C
3 to the right of C
2
.
C
1
C
2
C
3
U
1
U
2
CD

Movie
1
3
P
CD
Note: Utility rises as price falls u
1 u
2
CD
P
1
P
2
D
CD
C
1
C
2
C
3
1
C
3
C

Movie
• Derive the Substitution Effect just as before:
•i.e., from C
1 to C
2
C
1
C
2
U
1
CD

Movie
C
1
Be careful that your indiff. curves don’t cross!
U
1
U
2
• Increased purchasing power decreases consumption from
C
2 to C
3
• Put C
3
.
to the left of C
2 but to the right of C
1
.
C
3
C
2
CD

Movie
1
P
CD
3
Note: Again, utility rises as price falls
P
1
Demand still downward sloping.
u
1 u
2
CD
P
2
D
CD
C
1
C
3
C
2
1
C
3
C

Movie
• Derive the Substitution Effect just as before:
•i.e., from C
1 to C
2
C
1
C
2
U
1
CD

Movie
Be careful that your indiff. curves don’t cross!
U
2
• Increased purchasing power decreases consumption from
C
2 to C
3
•Put C
3
.
to the left of
C
2
C
1
.
and to the left of
U
1
C
3
C
1
C
2
CD

Movie
1
P
CD
3
Note: Again, utility rises as price falls.
u
2
P
1
P
2
Here, we get an upward sloping
Demand u
1
CD CD
C
3
C
1
C
2
D
3
C
1
C

• Many public programs to help the poor
“target” subsidies at particular goods and services (food stamps, housing subsidies)
• These programs increase the individual’s real income, but the money must be spent on the targeted goods and services
• If the gov’t transferred an equivalent amount of income without restrictions, the individuals would be better off
• However, these programs are paternalistic and reflect concerns that the funds will be wasted on “beer”

• In the economic model of the utility-maximizing consumer, the consumer’s utility function associates a numerical value to each conceivable choice. Given prices and monetary resources, the consumer chooses the best (utility-maximizing) bundle from among all those she can afford.
– This optimal choice occurs when, given his choice , what an additional unit of a commodity is worth to him equals the price of the commodity
• The model of the utility-maximizing consumer is rationalized by economists as an as if model. No one believes that consumers actually maximize a utility function. But if the consumer’s choice behavior conforms to relatively simple rules, the consumer acts as if she maximizes utility. And important consequences follow.
– Unhappily, systematic violations of these simple rules can be observed in real life.
Consumer marketers and advertising executives are well compensated for their skills in manipulating how consumers frame their choice
– Economists continue to use the model of the utility-maximizing consumer, in the belief that the violations are usually insignificant or in the hope that the conclusions drawn from models so constructed are not grossly affected by violations