Econ 1011A: Section 3 Notes1
TA: Zhenyu Lai (zlai@fas.harvard.edu), 9/22/2011
Preferences and Utility
Convex Sets and Quasi-Concave Functions
Indirect Utility
1. How to Build a Model of Individual Decision Making, Starting from Zilch
Microeconomic theory provides a structured framework in order to think about why individual
decision agents behave the way they do in the world. Thus far in the course, the goal was to
become familiar with the tools of calculus and the language of optimization.
We can now actually begin to investigate the theory of choice, or how agents make decisions.
Start with a set of possible alternative bundles, S. For example,
S = fchocolate, strawberry, [java chip]g
Now, given this set S, introduce the idea that individual i has a (weak) preference relation %i
over item bundles in S. A preference relation consists of statements which can take 3 forms:
x % y for x; y 2 S :
x y for x; y 2 S :
x y for x; y 2 S :
\x is weakly preferred to y”
\x is strictly preferred to y”, which means x % y and y 6% x
\indi¤erent between x and y”, which means x % y and y % x
Specify a utility function to represent these preferences. More precisely, a function u : X !
< is a utility function representing preference relation % if, for all x; y 2 X,
x % y , u (x)
u (y)
Why do we want to bother doing this? It makes things alot less cumbersome. Suppose you
have 10 alternatives. Think about what you are doing under the ‘preference’ framework.
When an individual makes a choice, you would then have to analyze this by listing 10 C2 = 45
pairwise comparisons. Under the ‘utility’ framework, you would calculate a representative
‘utility value’ for each of the 10 alternatives. The most preferred element of S is just the
solution to a familiar looking problem, which we are very good at dealing with
max u (x)
x2S
Because of the way we de…ned this, utility functions are unique only up to an order-preserving
(“monotonic”) transformation, i.e. if u (x) = 7 and u (y) = 6, 7 6 can only imply x % y.2
Axioms of rational choice: A preference relation is ‘rational’if and only if it satis…es
1. Completeness. For all x; y 2 X, either x % y or y % x. [Give example of incomplete %]
2. Transitivity. For all x; y; z 2 X, if x % y and y % x, then y % z. [Give counter-example]
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These notes draw heavily on the notes from previous years, and in particularly from last year’s TF, Matt Fiedler.
Just as if u (x) = 70 and u (y) = 60, 70 60 can only imply x % y.
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Why do we require this ‘rationality’ given that it may be controversial? The ability to
represent preferences by a utility function is closely linked to these assumptions of rationality.
Proposition 1 A preference relation % can be represented by a utility function only if it is rational.
Proof. We show that if utility function represents %, then % must be complete and transitive.3
1. Completeness. Because u ( ) is a real-valued function de…ned on X, it must be that for
any x; y 2 X, either u (x)
u (y) or u (y)
u (x). But because u ( ) is a utility function
representing %, this implies either that x % y or that y % x. Hence % must be complete.
2. Transitivity. Suppose x % y and y % x. Becaues u ( ) represents %, we must have u (x) u (y)
and u (y)
u (z). Therefore u (x)
u (z). Because u ( ) represents %, this implies x % z.
Thus, we have shown that x % y and y % z imply x % z, and so transitivity is established.
When dealing with utility functions, we generally assume that u (x1 ; :::; xn ) is strictly increasing in each xi . People prefer to have more of everything. But having more of one good at the
expense of having less of another requires a trade-o¤.
Fig 1: Indi¤erence curves
Fig 2: Why indi¤erence curves can’t intersect
Consider 2 inputs x and y, an indi¤erence curve shows a set of consumption bundles,
which represents all possible alternative combinations of x and y, among which the individual
is equally well-o¤, i.e. all bundles providing the same level of utility.
Exercise. Why are intersecting indi¤erence curves inconsistent with rational preferences?
Answer. From Fig 2: If two indi¤erence curves intersect, then c a and a b imply c b
(by transitivity). However, bundle c dominates b, and hence c b. This is a contradiction.
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It is not required that you know any proofs for this course, but this is meant to give you a ‡avor of what is
required for rigorous economic theory, or economics grad school. Currently, in the fall graduate version of this course,
Econ 2010A, you would already be required to understand these assertions in detail.
This short proof is taken from Prop 1.B.2 of Mas-Colell, Whinston and Green. There is a more important statement,
the Utility Representation Theorem (Prop 3.C.1) that follows from this, which states “Suppose % is rational and
continuous, then there is a continuous utility function u (x) that represents %”. The mathematical proof of this is
longer and more involved, and I defer its exposition to the reference.
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The slope of the indi¤erence curve must be negative. This represents the tradeo¤: if an
individual is forced to give up some y, he or she must be compensated by an additional
amount of x to remain indi¤erent between the two bundles of goods.
The marginal rate of subsitution is the negative of the slope of the indi¤erence curve.
This represents the rate at which the consumer is willing to substitute one good for another
so as to remain on the same level of utility.
Mathematical Representation: If the utility a person receives from two goods is represented
by u (x; y), we can write the total di¤erential of this function as
du = ux dx + uy dy
Along the indi¤ erence curve, the change in utility is zero, i.e. du = 0. Thus,
Marginal Rate of Substitution, M RS =
dy
dx
=
U =constant
ux
uy
2. Don’t be Confused: Convex Preferences vs. Quasi-Concave Utility Function
** The key takeaway from this section is this: Mathematically, a convex set of preferences is
used to state the economic principle of diminishing marginal rate of substituion.
A convex combination x0 of two points x and y is a weighted average, x0 = x + (1
)y
Convex Set: A set of points is said to be convex if any two points within the set can be
joined by a straight line that is contained completely within the set. Formally, all convex
combinations of any two points in the set must remain in the set.
Convex Function: We say that a function f is convex if the weighted average of the function
values at the points x and y is greater than or equal to the function value at the weighted
average of the points.
f (x) + (1
) f (y)
f ( x + (1
) y)
For any two points on a convex function, the line representing the set of convex combinations
of these two points must lie entirely above the graph of the function.
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Concave Function: Similarly, we say that a function f is concave if the weighted average
of the function values at the points x and y is less than or equal to the function value at the
weighted average of the points.
f (x) + (1
) f (y)
f ( x + (1
) y)
For any two points on a concave function, the line representing the set of convex combinations
of these two points must lie entirely below the graph of the function.
In utility theory, the appropriate concepts that we typically require for our functions are
quasi-concavity and quasi-convexity. This is because any strictly increasing transformation of a utility function represents the same preferences, i.e. invariance to monotone
transformation. Yet, a concave function can become convex under a permissable transforp
mation. For example, raising the concave function u (x) = x to the fourth power yields
u (x) = x2 , which is convex.
Quasi-convex Utility Function: A function f is quasiconvex if the lower contour set
fxjf (x) cg is a convex set for all values of c, i.e.
f (x)
f (y)
f (x )
f (x )
imply f ( x + (1
) y)
f (x )
Quasi-concave Utility Function: A function f is quasiconcave if the upper contour set
fxjf (x) cg is a convex set for all values of c, i.e.
f (x)
f (y)
f (x )
f (x )
imply f ( x + (1
) y)
f (x )
Note that any monotone transformation of a quasi-concave (quasi-convex) function is also
quasi-concave (quasi-convex).
Assuming diminishing MRS is equivalent to a Quasi-concave utility function.
In words, all combinations of any two bundles x and y, which are preferred to or indi¤erent
to another bundle x , will create a bundle that is also preferred to x
Notice that this is exactly the de…nition of a convex set of preferences
x%x
y%x
imply x + (1
)y % x
and hence we have established that a quasi-concave utility function represents convex preferences and gives a convex upper contour set. It is useful to see a diagram here.
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Strictly convex preferences and strict quasi-concavity follow from making inequalities strict.
Note that concavity of a utility function is a stricter assumption that quasi-concavity. A
concave (convex) function is quasi-concave (quasi-convex) but most quasi-concave (quasiconvex) functions are not concave (convex).
Don’t be confused. A convex utility function does not imply convex preferences. Rather, a
quasi-concave utility function implies convex preferences.
By using the notion of convexity, we can show that individuals prefer some balance in their
consumption. Suppose that an individual is indi¤erent between the combinations (x1 ; y1 ) and
2 y1 +y2
(x2 ; y2 ). If preferences are convex, then the combination x1 +x
will be preferred to
2 ;
2
either of the initial combinations.
3. Indirect Utility: Its all about the Envelope Theorem!
In consumer theory, we often care about how much a person’s happiness depends on the price
of whatever he consumes. Yet the e¤ect of price on happiness is indirect: It is actually the
consumption of the good that provides the agent with utility rather than how much it costs.
Consider the problem:
max u (x1 ; :::; xn )
s.t. p1 x1 + ::: + pn xn = Y
x1 ;:::;xn
,
max L (x1 ; :::; xn ; ; p1 ; :::; pn ; Y ) =
x1 ;:::;xn ;
max [u (x1 ; :::; xn ) + (Y
x1 ;:::;xn ;
p1 x1
:::
pn xn )]
We know how to solve this constrained maximization. At optimium, the individual will choose
x1 = x1 (p1 ; p2 ; :::; pn ; Y )
..
.
xn = xn (p1 ; p2 ; :::; pn ; Y )
By applying the implicit function theorem, we made (x1 ; x2 ; :::; xn ) a function of parameters (p1 ; p2 ; :::; pn ) and Y . These demand functions specify how the individual’s optimal
quantity of x demanded depend on the vector of prices (p1 ; p2 ; :::; pn ) and his budget Y .
The optimal choice of inputs x determine the individual’s utility stated as an implicit function
U (x1 ; x2 ; :::; xn )
= U (x1 (p1 ; p2 ; :::; pn ; Y ) ; x2 (p1 ; p2 ; :::; pn ; Y ) ; :::; xn (p1 ; p2 ; :::; pn ; Y ))
which we are going to rede…ne as a value function on parameters
= V (p1 ; p2 ; :::; pn ; Y )
=
max L (x1 ; :::; xn ; ; p1 ; :::; pn ; Y ) = L (x1 ; :::; xn ;
x1 ;:::;xn ;
; p1 ; :::; pn ; Y )
This value function represents the solution to the constrained maximization problem, over
which we can now apply the envelope theorem.
Vpi = Lpi (x1 ; :::; xn ;
; p1 ; :::; pn ; Y ) ,
VY = LY (x1 ; :::; xn ;
; p1 ; :::; pn ; Y )
In words, if prices or income were to change, we only need to consider how this a¤ects the
consumer’s constrained maximization of his …nal utility, rather than having to consider all
the intermediate changes to the optimal inputs as long as the consumer is always optimizing.
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