Econ 604 Advanced Microeconomics
Davis
Spring 2006
Problems:
Reading. Chapter 3 (pp. 66-86) for today
Chapter 4 (pp. 91 to 113) for next
time
To collect today Chapter 2, 2.3, 2.5,2.6, 2.9.
For next time: Chapter 3, 3.2, 3.4, 3.5, 3.7
Lecture #3.
REVIEW
II. The Mathematics of Optimization
A. Maximization of a Function of One Variable.
1. Necessary and sufficient conditions for a maximum.
Recall, our rule is that optimization requires that the first order condition
equal zero, and that the second derivative be negative (for a maximum) or positive
(for a minimum).
Example:
f(x) = 150x – 5x2
f’(x) = 150 – 10x
f’’(x) = -10
B. Functions of Several Variables
1. Partial Derivatives. Given y  f ( x1 , x2 ,...xn ) . The partial derivative of
y f

 f x1  f1 .
y =f(x1, …., xn) with respect to x1 we denote as
x1
x1
(f
)
xi
Second order conditions are
= fij = fji. Young’s Theorem assures
x j
that the cross partial is independent of the order in which the derivatives
were taken.
2. Maximizing Functions of Several Variables.
a. Total differentiation. Given several variables, the total
differential is
dy = f1dx1+ f2dx2+… + fndxn
b. First-Order Condition. Critical points arise when dy=0, or when
f1 = f2 =… = fn =0.
c. Second -Order Condition. The same as the single variable case,
except that there are “cross effects.” For a maximum or a minimum, own
effects must dominate cross effects, or (in the 2 variable case)
f11(f22) –f122 >0
Note: the same second order condition implies a minimum as well as a
maximum (intuitively the own effects must dominate the cross effects).
i. Implicit Functions. Note, we can write functions in “implicit
form” (that is, an equality set equal to zero). If second order
conditions hold, we can solve for one variable in terms of another.
ii. The Envelope Theorem. A major application of the implicit
function theorem that we will use frequently. Consider an implicit
function f(x, y, a) = 0, where x and y and variables and a is a
parameter.
The envelope theorem states that for small changes in a, dy*/da
can be computed by holding x constant at its optimal value x*, and
simply calculating y/a {x = x*(a)}.
C. Constrained Maximization. A problem that arises frequently in practice
(e.g., consumers maximize utility subject to an income constraint.
1. Lagrangian Multiplier Method. In general, we optimize a function f(x1, …, xn)
in light of a series of constraints about those independent variables g(x1, …,
xn).=0. Write: L = f(x1, …, xn) +g(x1, …, xn) and take FONC, a series of n+1
equations in n+1 unknowns.
2. Interpretation of the Lagrangian Multiplier. Each of the first order conditions
in the above system may be solved for . That is f1/g1 = f2/g2 =, …., fn/gn = , or
=
marginal benefit of xi / marginal cost of xi
Observation: One way to better understand the role of  from an intuitive
level would be to consider a constrained function of a single variable. Suppose
f(x) = 10x – x2 and suppose that this was subjected to the constraint that x=3.
Then, L =10x- x2 +(3-x)
From FONC
10 – 2x = 
and
3–x
=
0
When x=3, 10-2(3) = 4. Suppose we relax the constraint, and increased x to 4.
Then
and
10 – 2x =  and
4–x =
0
 = 2. In x, f(x) space, this is just the slope of the line tangent to f(x).
2
Reasoning similarly, when x=5 or more,  =0 and the constraint is no longer binding.
So  is the marginal increase in the objective brought about by relaxing the constraint, as
long as the constraint binds.
3. Duality. Observe that for every constrained maximization problem, there is an
implied constrained minimization problem (this is conceptually no more complicated
than looking at a simple sum, and realizing that there is an implied difference associated
with it. Thus, for my 8 year old daughter, given the problem 24-17 = 7, it was much
easier to solve dual problem 17+7 = 24.) We encounter the same problems in
optimization, albeit they are more complex.
4. Envelope Theorem in Constrained Maximization Problems. The envelop
theorem applies just as it did for unconstrained problems. Find optimal value of
the dependent variable in terms of the parameters at the critical point. Then take
the derivative w.r.t. the constant.
5. Second Order Conditions with Constrained Optimization. A final observation.
Optimize an objective f(x1, x2) subject to a linear constraint, g(x1, x2) =c - b1x1 b2x2=0. Form the Lagrangian expression
L
= f(x1, x2)
+ ( c - b1x1 - b2x2)
Totally differentiating the objective twice, and inserting the constraint, we
find that d2y =
(f11 f22 - 2f12 f1 f2 + f22 f12)(dx12/ f22) <0 as long as the
bolded term is negative, or
(f11 f22 - 2f12 f1 f2 + f22 f12)<0
This is a quasi concavity condition that we will use later. Intuitively, this
indicates the minimum condition for an interior solution to a linearly constrained
optimization problem. In contrast to the simple “mound” condition necessary in
the case of unconstrained optimization, a sufficient condition for an optimum
given a constraint is the weaker condition that the constraint be more convex than
the objective function. As we will see below, this condition will be equivalent to
asserting that any linear combination of two points on a curve be in the interior of
the function.
Note: One way to view quasi concavity is as follows: A three dimensional “mound” is
quasi concave if it can be completely represented in a two dimensional topographical
map
PREVIEW
III. Choice and Demand.
A. Chapter 3. Preference and Utility.
1. Axioms of Rational Choice
2. Utility
a. Nonuniqueness of Utility Measures
b. The Ceteras Paribus Assumption
3
c. Utility form Consumption of Goods
d. Arguments of Utility Functions
e. Economic Goods
3. Trades and Substitution
a. Indifference curves and the Marginal Rate of
Substitution
b. Indifference Curve Map
c. Indifference Curves and Transitivity
d. Convexity of Indifference Curves.
e. Convexity and Balance in Consumption
4. An Alternative Derivation
5. Examples of Utility Functions
a. Cobb-Douglas Utility
b. Perfect Substitutes
c. Perfect Complements
d. CES Utility
6. Generalizations to More than Two Goods
LECTURE_____________________________________________
III. Choice and Demand. The purpose of the next several chapters is to develop
the notion of market demand, starting from primitive assumptions about individual
optimization. In the first chapter (chapter 3= we discuss the assumptions economists
usually use to model individual behavior, and then use those assumptions to develop
indifference curves
A. Chapter 3. Preferences and Utility.
1. Axioms of Rational Choice. One way to begin an analysis of individual
behavior is to state a set of basic “axioms” that characterize rational behavior. We define
3.
a. Completeness: For any two goods, services or situations A and B
exactly one of the following conditions holds
i. A is preferred to B (A  B)
ii. B is preferred to A (B  A)
iii. A and B are equally attractive.
This condition implies that all items can be ranked.
b. Transitivity. If an individual reports A  B and that B  C then A  C .
This transitivity condition implies that choices are internally consistent.
c. Continuity If an individual reports A  B then for a situation A’
sufficiently “close” to A it must be the case that A’  B. This largely technical
assumption rules out “knife’s edge responses to small changes in income and prices.
2. Utility. If individuals follow these three assumptions, all situations must be
rankable. Following Bentham, we label this preference ranking as “utility”. We will say
4
that preferred goods confer higher utility. That is, A  B implies U(A) > U(B) (where
the latter term is an inequality, and the former is a preference ranking.) Formally define
the follow
Utility: Individual preferences are assumed to be represented by a utility function of the
form
utility =
U(X1, X2, …, Xn)
where X1, X2, …, Xn denote quantities of goods 1 to n consumed in a given time frame.
Utility is unique up to an order-preserving constant.
Some comments on this definition:
a. Non-Uniqueness of Utility Measures. Importantly, the utility measure is
only ordinal. We can order utility only up to an order-preserving transformation. For
this reason interpersonal utility comparisons are not possible.
b. The Ceteras Paribus Assumption Utility is a general measure of
satisfaction. Clearly such a measure is affected by general factors, such as one’s outlook
on life, or the opinions of one’s friends. However, economists typically hold these
effects constant when discussing utility, and focus on the utility of quantifiable goods.
c. Utility from Consumption of Goods One implication of the Ceteras
Paribus condition is that we often simplify notation, and omit from models variables we
hold constant. Thus, given a general utility function
utility =
U(X1, X2, …, Xn; other things)
we often write
utility =
U(X1, X2, …, Xn)
or, if we are concerned particularly about just two goods, we might write
utility =
U(X1, X2)
This is done with the understanding that the remaining goods, and preferences exist, but
are held constant.
5
d. Arguments of Utility Functions. Economists often include rather
strange arguments in utility functions. These actually reflect some implicit conventions.
For example, we often write
utility =
U(W)
where W = wealth. Of course individuals do not derive utility directly from wealth, but
rather the wealth spent on consumption. So this expression denotes the utility of
consuming the most satisfying consumption bundle possible with wealth W.
In labor economics, we often express utility as a function of consumption C, and
non work hours, H or leisure
utility =
U(C, H)
Finally, in some contexts were are interested in the intertemporal aspects of
utility. Thus we express utility as a function of consumption today (time period 1) and
tomorrow (time period 2). Or
utility =
U(C1, C2)
Y
e. Economic Goods. We frame the arguments in a utility function as
economic “goods”. That is items that have the property that more is preferred to less.
These goods may include arguments as simple as hot dogs and pizza, or may include
complex composite goods such as leisure. Given that more is preferred to less, we may
view movements away from the origin (e.g., the point of no consumption) as utility
enhancing. In a two-dimensional space, for example.
More preferred than
X', Y'
Y'
Less preferred than X', Y'
X'
X
3. Trades and Substitution. Most economic activity involves trading between one
good and other. If you purchase a soda to bring to class, for example, you exchange
money for something you value more highly, soda. In this section we develop an
apparatus for modeling this trading activity.
6
a. Indifference curves and the Marginal Rate of Substitution. Generally an
individual will be indifferent to several combinations of a particular
combination of goods. Utility from increased consumption of one good
can just offset utility losses from decreased consumption of another. We
gather these points of indifference together as an “indifference curve”
Y
Indifference Curve: A set of consumption bundles among which the
individual is indifferent. That is, the bundles all provide the same level of
utility.
Indifference Curve
Marginal Rates of Substituion
Y1
Y2
U1
X
X1 X2
Notice that the curve is bowed inward. This reflects the assumption that
individuals generally prefer balanced combinations of goods to one or the
other.
One way to analytically characterize the change in substitutability between
goods is with the slope of the line tangent to utility curve U1. This is
termed the Marginal Rate of Substitution
MRS =
-[dY/dX ] |U=U1
Notice that the slope of the line tangent to U1 “flattens” as consumption
bundles move from those comprised mostly of Y to bundles consisting
mostly of X. This flattening reflects that starting from an evenly split
allocation of Y and X, , increasing amounts of X must be given in
exchange for units of Y in order to maintain constant utility. This reflects
the standard assumption that goods substitute imperfectly for each other.
(This, of course, is the opposite of the concavity assumption we impose on
production possibilities curves. There, starting from a center allocation,
decreasing amounts of X must be given up to increase Y and still maintain
constant total production)
7
Of course, for any individual the MRS (and the Indifference curve
in general) is determined by individual choices.
Y
b. Indifference Curve Maps Now in the above illustration only a single
indifference curve, U1 was drawn. More generally, we may conceive of
the X Y quadrant as being packed with indifference curves, each reflecting
a higher level of utility, as it radiates out further from the origin
Increasing Utility
U2
U1
X
Now we explore several properties of these indifference curves.
Y
c. Indifference Curves and Transitivity: One implication of our axioms of
individual choice is that indifference curves cannot intersect. To see this,
consider the illustration below.
Increasing Utility
C
D
E
A
U1
U2
B
X
Is such a relationship possible? In fact, it is not. Notice that A  B and
C  D. However, the consumer is indifferent between B and C. Thus
A  C and C  D. But A and D are on the same indifference curve, a
contradiction.
Result: Indifference curves cannot intersect
8
Y
d. Convexity of Indifference Curves. Another way to illustrate a declining
MRS is to observe that indifference curves are convex. Convexity implies
that any line drawn between two points on an indifference curve will lie
above the curve. Any indifference curve that did not satisfy convexity
would have a ‘wave’ in it, as illustrated below
Y
Nonconvex
e
.
Convex
C
o
n
v
e
X
x
X
i
Convexity and Balance in Consumption: Comparing the above charts, it is
also evident that one can see that convexity implies a preference for
balance in consumption. With convexity (as seen in the left panel, for any
pair of bundles (X1, Y1) and (X2, Y2), a point halfway between ((X1,+X2)/2
, (Y1+ Y2)/2) and (X2, Y2), renders higher utility (with quasi-convexity, it
is possible that a the point falls on a line and that consumers are indifferent
between the three bundles) This is not true if convexity is violated, as
illustrated in the right panel.
Example. Suppose a person’s utility function for hamburgers and softdrinks is given by
Utility =
(XY)1/2
An indifference curve is developed by picking a utility level, and
examining all consumption bundles consistent with that level. For
example, if U=10
10
=
(XY)1/2
The slope of the indifference curve is found by solving for Y
Y=
100/X
Taking the derivative
dY/dX =
-100/X2
9
Thus, at X=5, Y=20, the MRS = 100/25 =4. At Y=5, X=20, MRS = -1/4.
4. An Alternative Derivation. A somewhat more mathematical derivation
proceeds directly from the definition of the utility function. Given
utility =
U(X1, X2, …, Xn)
as defined above, we can define marginal utility of any good as the first
derivative. For example, for X1,
marginal utility
==MU1
=U/X1
Notice that, as before MU1is dependent on the other things held constant,
as well as subjective components. Thus, it is measurable only up to a monotonic
transformation. Taking the total differential of utility
dU
=
(U/X1) dX1 + (U/X2) dX2 + …+ (U/Xn) dXn.
=
MUx1 dX1 + MUx2dX2 + …+ MUxndXn.
To derive the MRS, set the differential to zero, and change only the level
of two goods.
dU
=
(U/X) dX + (U/Y dY
=
0
=
MU
dX
+
MU
dY
=
0
.
x
y
Solving
-dY/dX |U=constant
=
MUx/MUy.
Notice that the absolute measures of utility drop out when constructing the
MRS (e.g., the MRS of, a particular consumption bundles is 4 utils from soda, and
2 utils from pizza. “Utils” cancel out in the ratio). Thus, the MRS is a well
defined concept, even when ‘utils’ are not observable.
Diminishing Marginal Utility and the MRS. Recall in the introduction that
we used the notion of diminish marginal utility to resolve the Diamond/Water
Paradox. Intuitively, it would seem that there is a relationship between
diminishing marginal utility and a diminishing MRS, since both ideas refer to the
idea that a consumer becomes relatively satiated in one good as they consume
more of it. The two concepts, however, turn out to be distinct. The MRS
assumption requires that the utility function be quasi-concave. This is related, in a
rather complex way to the assumption that marginal utility diminishes (fii<0). The
following example illustrates.
Suppose
utility =
(XY)1/2
10
Then marginal utilities for X and Y are
MUx = .5(Y/X)1/2
MUy = .5(X/Y)1/2
Notice in each case, that MUi is inversely related to consumption of i.
Now the MRS =
-dY/dX |U=constant =
.5(Y/X)1/2
= Y/X
1/2
.5(X/Y)
Notice that this illustrates the Standard diminishing MRS relationship.
That is, starting from an allocation of Y=5 and X=10, MRS = .5 (1/2 a unit of Y
trades for a unit of X. But when Y=10 and X=5, MRS = 2, that is, 2 units of Y
trade for a single unit of X.
MUx/MUy
=
Notice also that a monotonic transformation of the utility function does
not affect the MRS. For example, take the natural logarithm of the above utility
function
ln(U) =
.5ln(X)
+
Then MUx
=
.5/X
And MRS
=
MUX/MUy
.5ln(Y)
and MUy
=
=
.5/Y
Y/X
as before. Because utility is insensitive to monotonic transformations, we
will often make such transformations, when convenient.
5. Examples of Utility Functions. Although the form of an individual’s utility
function is not observable, specific functional form are often imposed on utility
functions. We consider some of these here.
a. Cobb-Douglas Utility. This is a general case of the quadratic utility
function used above for purposes of illustration. It is named after a pair of
investigators who used it to investigate production relationships in the
U.S.
utility =
XY
where  and  are positive constants. In general the constants reflect the
importance of the goods to the consumer. Because the function is unique
only to a monotonic transformation, it is often convenient to impose the
restriction that  +  =1.
11
b. Perfect Substitutes. When goods are perfect substitutes the MRS is
constant. That is, a consumer remains on a level curve via a fixed trade-off
between two products
utility =
U(X,Y)
=X+ Y
This relationship might describe consumer preferences between different
brands of the same product. (Exxon gasoline substitutes 1 for 1 for Shell
gasoline)
c. Perfect Complements On the other hand, some goods generate utility
only if consumed in fixed proportion with another. Peanut-butter and
bread, or coffee and cream are examples. Such a utility function is
represented analytically as follows
utility =
U(X,Y)
=min(X, Y)
In this case, the goods are consumed in fixed proportion
Y/X = (/)
Any deviation will result in waste
d. CES Utility The three functions mentioned are special cases of the more
general constant elasticity of substitution function
utility =
U(X,Y)
= X/+
U(X,Y)
= lnX+ lnY
Y/
when >0 and
utility =
when =0.
It can be shown that for such functions, the elasticity of substitution is
 = 1/(1- ) (we will develop this more later)
For the case  = -1, the function becomes
utility =
U(X,Y)
= -X-1 - Y-1
The negative entries make this a curious utility scale. It is, however,
perfectly acceptable, as can be verified by checking first derivatives.
Example. Homothetic preferences.
All of the above utility functions share the property of being “homothetic”
In other words, the MRS for these functions depends only on the ratio of the two
12
goods consumed, and not the absolute levels. For example, in the Cobb-Douglas
function discussed above,
MRS = MUx/MUY = Y/X
a relation which depends only on Y/X.
The analytical advantage of homothetic preferences is that indifference curves are
similar. Indifference curves for higher utility levels are simply substitutes for those at
lower levels of utility.
As a counter example the utility function
utility =
U(X,Y)
= X + ln Y
is nonhomothetic. To see this observe
MUx
Thus
=
1
MUy
=
MRS MUx/MUy
1/Y
=
Y.
So the MRS changes as the level of consumption increases.
6. Generalizations to More than Two Goods
The Cobb-Douglas and CES utility functions can be readily generalized to the
case of n goods.
utility =
U(X1,X2,…Xn)
=
X11 X22… Xnn
or
utility =
U(X1,X2,…Xn)
=
X1/  + X2/  + … + Xn/ 
These functions have the same properties as those developed above for the 2 good
case.
13