Econ 604 Advanced Microeconomics
Davis
Spring 2004
Reading.
Chapters 1 and 2 (pp. 4 to 56) for today
Chapter 3 (pp. 66-86) for next time
Lecture #1.
I. Introduction. To introduce students to the study of microeconomics, it is useful to
reflect on what it is economists attempt to do with their models, and how they proceed.
This introductory material consists of four parts. First we discuss the logical status of
economic models. Then we review some components common to all economic models.
In a third subsection we overview the development of economic thought by reviewing the
intellectual history of the Diamond/Water Paradox. A fourth subsection illustrates the
interplay between assumptions, analytical tools and predictions
A. The logical status of theoretical models. The world is a complicated place. Like
scientists in other fields, we attempt to organize behavior with the use of very simple
models. The models, by necessity are “oversimplifications” of a complex reality.
1. Evaluating Models Models are evaluated with 2 approaches
a) Direct: Testing the truth of the underlying assumptions
b) Indirect: Observing the extent to which the model’s predictions organize
behavior.
The indirect or instrumentalist approach is most frequently used.
Example: Do firms maximize profits. To evaluate this directly, you might ask
responsible corporate officials about their primary objectives. Unfortunately, for
a variety of reasons you are unlikely to get a clean response. Firms “do the best
they can,” or “move to corner the market” or “identify and focus on core
competencies to provide value to the consumers.” Indirectly, however, we might
develop predictions about how a profit-maximizing firm would behave in a
particular context. Conglomerates, for example, may divest unrelated units.
Milton Friedman famously argues that this “indirect” approach is the best to
which we can aspire. An economic agent can no more tell you how he or she
optimizes than a baseball player can describe the series of differential equations
that must be solved in order to catch a baseball.
2. Importance of Empirical Analysis. If we are evaluating models on the basis of
outcomes, then developing empirically testable predictions becomes a paramount
consideration in theoretical analysis. Thus, while a lot of what we do will focus on
the Mathematics of Optimization, it is important to understand both the reasoning
underlying the mathematics, and that our ultimate objective is to create testable
implications
B. General Features of Economic Models: Economists construct a host of models,
constructed at varying levels of “aggregation” or abstraction, ranging from the responses
of rats to changes in the number of level pushes necessary to get food pellets, to the
global effects of international trade agreements. Despite their heterogeneity, these
models share a number of central features.
1. The Ceteris Paribus Assumption: Any modeling exercise requires that some
elements of complex reality be held constant. Thus, for example, when we
predict wheat prices with a simple partial equilibrium supply and demand model,
we must “hold constant” a host of potentially relevant factors, including the
following
The weather
The technology for growing wheat
Consumers’ attitudes for bread
The Population
The prices of potatoes, corn and other starches.
But, of course, some of these variables do, in fact change. Scientists in other
fields similarly invoke ceteras paribus conditions when theorizing. However,
unlike scientific endeavor in many other fields, economists find testing economic
propositions under laboratory conditions relatively difficult. Tests in naturallyoccurring circumstances must control for changes in ceteras paribus conditions.
Although statistical techniques for making such adjustments, the ceteras paribus
assumptions are more problematic.
2. Optimization Assumptions: Economists typically assume that agents optimize.
Firms, for example maximize profits, while households maximize utility
and.Government Regulators maximize public welfare.
Critics frequently challenge the relevance this approach. Individuals, after
all, are clearly not calculating machines. Nevertheless, the assumption of
optimizing behavior is useful for two reasons. First, it provides a framework that
allows for predictions. Second, despite their imprecision, optimizing assumptions
often organize behavior quite well. (Individuals, after all, can optimize subject to
a number of constraints)
3. Positive/Normative Distinction. Economists are typically careful to distinguish
between “positive” and “normative” questions. “Positive” questions pertain to
“what is”. For example, how are corn prices affected by the development of a
new hybid seed? Again, how will unemployment rates respond to a change in the
minimum wage? These are distinct from “normative” statements, which involve a
value judgments, such as “low corn prices are good for consumers and “we should
seek to reduce unemployment”?
2
Many economists argue that economists can only analyze positive
questions. Much of the text attempts to focus on positive questions. However, it
is important to realize that normative goals color the judgment of even the coldest
of economists.
C. Some Historical Perspective: The Development of the Economic Theory of
Value. Here at the outset, it is perhaps useful to provide some brief historical context for
economic thought. Although individuals have pursued economic activities (have
attempted to profit from those activities) for millennia, the study of economics as a field
of inquiry is relatively new, starting in the 18th century. Consider, as an example the
theory of value. Today, we regard “value” as synonymous with “price”. This was not
always true.
1. Early Economic Thought. “Value” and “price” were separate concepts. “Value
was an inherent, perhaps determined characteristic. “Price” was something set by
man. “Unjust” prices could arise if price for an item differed from its value.
2. Early Economists: Adam Smith, who laid out the foundation of modern
economics, continued with this distinction. Items had “value in use” and “value
in exchange” These two values, however need not equal. Water, for example is
essential to survival and has a very high value in use. Diamonds, on the other
hand are completely inessential, yet have a high value in exchange. This
difference between value in use and value in exchange was termed the Diamond
Water Paradox.
3. Labor Theory of Exchange Value. One obvious explanation for the different
exchange values regards the differing production costs of items. Potable water, for
example, can be made with relatively little labor. On the other hand, diamonds
require an immense amount of human effort. This notion is a lynch pin for
Marxist Thought
4. The Marginalist Revolution. From the 1850’s to the 1880’s economists became
increasingly aware that to create a substitute for a labor theory of value, the
difference between value in use and value in exchange needed resolution. The
answer eventually came in the form of marginal analysis. Value, either in use or
exchange is determined not by the total usefulness of a product, but by the
usefulness and costs of the last unit produced and consumed. Alfred Marshall
formalized this notion in his Principles of Economics (1890). Thus, it is not
supply conditions that determine value, nor any perceived usefulness of a product.
Rather value is determined by the marginal cost and the marginal value of the last
unit exchanged.
3
D. Basic Models and Analytical Contexts. In this course, we will delve deeply into
Alfred Marshal’s insights. However, in closing this chapter we set out the basic structure
of this model, to highlight the assumptions underlying this very standard analysis, as well
as the benefits of characterizing a problem analytically. We consider two models very
basic to introductory economics: Supply and Demand Analysis and the Production
Possibilities Frontier
1 .Supply Demand Equilibrium. Suppose we wish to identify equilibrium price
and quantity predictions for peanuts.
a) Market Demand. Suppose first that on the basis of the statistical
analysis of historical data we determined that peanuts are a function of
price. More specifically the relationship was the following linear relation:
quantity demanded = QD = 1000 – 100P
Where QD refers to bushels of peanuts, and P is the price per bushel.
Tabularly, this expression is
Price
Quantity Demanded
1
2
3
4
5
6
7
8
9
900
800
700
600
500
400
300
200
100
b. Market Supply. Similarly, statistical analysis of historical data reveals
that a supply relationship is determined by the expression
quantity supplied = QS = -125+125P
Tabularly,
Price
Quantity Supplied
1
2
3
4
5
6
7
8
9
0
125
250
375
500
625
750
875
1000
4
c. Equilibrium. Now, comparing the supply and demand tables reveals that
QD = QS at a price of $5.
To graph these relationships in the usual way (e.g., with price on the
vertiacal axis), we need to express both quantity supplied and quantity
demanded with Price as the dependent variable. This is termed indirect
supply and indirect demand. Or
Inverse demand
PD
=
10 – QD/100
Inverse supply
PS
=
1+ QS/125
Tabularly
Price
Qty D
100
200
300
400
500
600
700
800
900
Qty S
9
8
7
6
5
4
3
2
1
1.8
2.6
3.4
4.2
5
5.8
6.6
7.4
8.2
Graphically
14
12
10
S
8
6
4
D
2
0
0
200
400
600
800
Of course, looking at the tables, or the graph, it is obvious that the
equilibrium price/quantity combinations are $5 and 500 units,
respectively. However, acknowledging that the relationship graphed in the
above figure is implied by our quantity supplied and quantity demanded
relationships, we can succinctly solved for the underlying equilibrium
5
QD
1000 – 100P
1125
1125/225
5
=
=
=
=
=
QS
-125+125P
225P
P
P
1000 – 100(5) =
500.
d. Comparative Statics Effects. Suppose that one of the ceteras paribus
conditions changes. For example suppose that new proponents of the
Atkins Diet find that peanuts are the only truly healthy acceptable protein.
As a consequence the demand for peanuts becomes
QD’ =
1450 – 100P
Solving for inverse demand yields
P
=
14.5 – QD`/100
Expressed tabularly, the new supply and demand relationship becomes
Qty
150
250
350
450
550
650
750
850
950
Pd
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
Pd'
13
12
11
10
9
8
7
6
5
Ps'
2.2
3
3.8
4.6
5.4
6.2
7
7.8
8.6
Or graphically
14
12
10
S
8
6
4
D
2
0
0
200
400
600
6
800
Again, remembering that these graphical relationships are implicit in the
supply and demand equations, we can solve directly
QD
1450 – 100P
1575
1575/225
7
=
=
=
=
=
1450 – 100(7) =
QS
-125+125P
225P
P
P
750.
e. Summary. This simple exercise illustrates two important points
i. The relationships between algebraic analyses, tables and graphs
ii. That the combination of supply and demand conditions combine to
determine the price.
Question: What would happen here if the Government regulated price at
$5. What would be quantity supplied after the demand shift? What would
be quantity demanded?
2. General Equilibrium Analysis and the Production Possibilities Frontier. The
above analysis is a very simple example of a partial equilibrium analysis. That is,
evaluate how supply and demand combine to determine an equilibrium. We can
also do comparative statics exercises, where we assess the effects of changes in
underlying conditions on an equilibrium.
General Equilibrium analysis, pioneered by Leon Walrus asks the broader
question: What are the repercussions of a change in one market on other market.
Walrus approached this problem by considering a system of equations
characterizing each market. This is an interesting, and sometimes useful exercise.
Nevertheless, to address broader questions, it is frequently more useful to make a
further abstraction. This is typical, for example, in Macroeconomics, where labor
markets are combined into a single entity, as are investment and output markets.
We illustrate this here by considering a production possibilities frontier.
a. The Production Possibilities Frontier: Illustrates the different
combinations of two goods that can be produced form a certain amount of
scarce resources. It also illustrates the opportunity cost of shifting between
possible resource allocations.
Consider a society that produces and consumes X clothes (nonperishable
items) and Y food (perishable items). Suppose that the production
possibilities frontier may be characterized as
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2X2 + Y2 = 225.
Graphically, one might illustrate this by solving for one of the variables.
Y
=
(225
2X2)1/2
-
Tabularly, this may be expressed as
X
10.6066
10
9
8
7
6
5
4
3
2
1
0
Y
0.005518
5
7.937254
9.848858
11.26943
12.36932
13.22876
13.89244
14.38749
14.73092
14.93318
15
Food
Graphically
16
14
12
10
8
6
4
2
0
unattainable
opportunity
cost
0
Inefficient
5
10
15
Clothes
b. Observations about the PPF.
i) The boundary illustrates the maximum combinations of Food
and Clothes available to the society. Quantities within the
boundary are inefficient. Quantities outside the boundary are
unattainable.
ii) The marginal opportunity cost can be expressed as the slope of
the line tangent to the curve at a point.
iiI) The outward bow of the productions possibilities frontier
illustrates increasing marginal opportunity costs. That is, constant
8
Food
increments of one good cost society increasing increments of
another.
16
14
12
10
8
6
4
2
0
0
5
10
15
Clothes
c. Analytical Expression of the PPF. We can also express this relationship
analytically. Recall
=
(225
-
2X2)1/2
Taking the derivative of Y with respect to X yields
dY/dX =
=
.5(225 -2X/Y
2X2)-1/2(-4X)
Thus, when X = 5 (Y=13.22) the opportunity cost of producing one more unit of
X is
-10/13.22 = -.76. To produce one more unit of X society must forego ¾ a unit of
Y.
X
10.61
10.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
Y
0.01
5.00
7.94
9.85
11.27
12.37
13.23
13.89
14.39
14.73
14.93
15.00
Opp Cost X for Y
-3844.03
-4.00
-2.27
-1.62
-1.24
-0.97
-0.76
-0.58
-0.42
-0.27
-0.13
0.00
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Opp Cost Y for X
0.00
-0.25
-0.44
-0.62
-0.80
-1.03
-1.32
-1.74
-2.40
-3.68
-7.47
Note finally from the rightmost column above that marginal opportunity costs for
Y also increase in terms of X.
This notion of increasing marginal opportunity costs is attributable to gains from
specialization and the division of labor.
E. Modern Developments
The primary purpose of this course is to improve the tool set you have as
economists. That involves taking the largely graphical and tabular expressions you
learned in your undergraduate courses and converting them into analytical expressions,
expressions that are both succinct, and facilitation testing. The foundation of this
analytical framework is attributable to Paul Samuelson, the first American winner of the
Nobel Prize for Economics, but it has been vastly expanded in the decades since World
War II.
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II. Chapter 2. The Mathematics of Optimization. The material in this second chapter
should be a review for all of you. However, given that the intuition underlying the use of
these tools is of primary importance in this course, I will spend some class time to
reviewing some of the principles of optimization
A. Maximization of a Function of One Variable.
1. Intuition of Optimization. Suppose you were the CEO of a firm that made a
single product, x. You are, of course, interested in maximizing profits. Intuitively, the
best way to approach this problem would be to consider the profits associated with all
levels of output, x.

=
f(x)
For example, were the function

=
20x – x2
We might simply plot out values in a table, and identify that profits are maximized at x =
10.
x

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
19
36
51
64
75
84
91
96
99
100
99
96
91
84
75
64
11
Alternatively, we might approach this graphically
120
100

80
60
x
40
20
0
0
5
10
15
Notice that in either case, it is obvious that a quantity of 10 maximizes profits. Looking
at the figure suggests an alternative way to find the maximum. You might proceed
marginally from a low level of output, and then consider the marginal effects of
incremental output changes on profits. You would be at a maximum when incremental
profits no longer increased with you last increment in production.
That is, you might use the following rule: Consider the effects of a change from x1
to x2. Increase output further if marginal profits increased, that is if 2 -1 >0.
Otherwise stop.
Another way to say the same thing is to consider the ratio
 2  1
x2  x1


x
You would be at an optimum when this ratio equals zero.
2. Derviatives defined. Now you probably know that a derivative is nothing more
than this slope evaluated over an infinitesimally small range of x. Tthat is, consider a
point x1, and a very small increment to x2 = x1+ h. Then
f ( x1  h)  f ( x1 )
d
 lim
dx
( x1  h)  x1
h o
Thus, when you take the derivative of a function, you could find the slope at any
point xi by inserting xi into the derivative.
3. Optimization:
a.
Necessary conditions. A necessary condition for an optimum is
that the derivate equals zero at point xi. This follows from the
logic of the graph shown on the middle of this page. A
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derivative equal to zero implies that you did no “climbing” with
the last (infinitesimal) adjustment, so you must be at an optimum
Sufficient Conditions. This first order condition, however, does
not necessarily define a maximum. To see this consider the two
figures shown below. In the chart on the left, the first derivative
equals zero, but the function is minimized. In the function on
the right the derivative equals zero, but only because the
function is at a point of inflection.
b.
35
50
30
40
25
30
20
20
15
10
10
0
5
-10
0
5
10
0
0
5
10
-20
Minimum
Point of Inflection
To identify a local optimum as a maximum or a minimum, we take
the second derivative.
Rule: At a point x1,
f’(x1) = 0 and f’’(x1) <0 implies a maximum
f’(x1) = 0 and f’’(x1) >0 implies a minimum
Intuition: Think of climbing a mountain. If you’re at a flat place
(e.g., f’(x)=0, and you were ascending at a slower rate f’’(x)<0, you must
be at a the top of the mountain. On the other hand, if you find yourself at
a flat place and you’re rate of ascent is increasing (that is, you’re
descending less negatively) then you’re at a minimum.
3. Some rules for derivatives. We have well known rules for taking derivatives,
including the following
1. f(x) =
a
f’(x)
=
0
2. f(x) =
ax
f’(x)
=
a
3. f(x) =
axb
f’(x)
=
abxb-1
4. f(x) =
g(x) + h(x)
f’(x)
=
g’(x) + h’(x)
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5. f(x) =
g(x)h(x)
f’(x)
=
g’(x) h(x) + g(x) h’(x)
6. f(x) =
g(x)/h(x)
f’(x)
=
[g’(x)h(x) – h’(x)g(x)]/h(x)2
7. f(x) =
ln(x)
f’(x)
=
1/x
8. f(x) =
g(h(x))
f’(x)
= g’(h(x))h’(x)
This last rule, called the chain rule is a very convenient way to find how one variable
affects another through some intermediate variable z.
B. Functions of Several Variables. Economic problems rarely involve only single
variables. A firm’s production function, for example, usually involves input
combinations. Similarly, consumers maximize utility by selecting optimally among
multiple items. Thus we must consider optimization conditions for function of the form
y = f(x1, x2, … , xn)
1.
Partial Derivatives. As with the single variable case, we are interested in
finding a “flat place “ However, now the “mountain” has multiple
dimensions. On a North/South, and East/West Grid, for example, we
can find flat places in either direction. However, we can only be at a
mountain top if we find a flat place in all dimensions.
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