Econ 604 Advanced Microeconomics
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 naturally-occurring 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 ernment 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”?
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.
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 |900 |
|2 |800 |
|3 |700 |
|4 |600 |
|5 |500 |
|6 |400 |
|7 |300 |
|8 |200 |
|9 |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 |0 |
|2 |125 |
|3 |250 |
|4 |375 |
|5 |500 |
|6 |625 |
|7 |750 |
|8 |875 |
|9 |1000 |
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 |Qty S |
|100 |9 |1.8 |
|200 |8 |2.6 |
|300 |7 |3.4 |
|400 |6 |4.2 |
|500 |5 |5 |
|600 |4 |5.8 |
|700 |3 |6.6 |
|800 |2 |7.4 |
|900 |1 |8.2 |
Graphically
[pic]
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
QD = QS
1000 – 100P = -125+125P
1125 = 225P
1125/225 = P
5 = 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 |Pd' |Pd |Ps' |
|150 |13 |8.5 |2.2 |
|250 |12 |7.5 |3 |
|350 |11 |6.5 |3.8 |
|450 |10 |5.5 |4.6 |
|550 |9 |4.5 |5.4 |
|650 |8 |3.5 |6.2 |
|750 |7 |2.5 |7 |
|850 |6 |1.5 |7.8 |
|950 |5 |0.5 |8.6 |
Or graphically
[pic]
Again, remembering that these graphical relationships are implicit in the supply and demand equations, we can solve directly
QD = QS
1450 – 100P = -125+125P
1575 = 225P
1575/225 = P
7 = P
1450 – 100(7) = 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
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 |Y |
|10.6066 |0.005518 |
|10 |5 |
|9 |7.937254 |
|8 |9.848858 |
|7 |11.26943 |
|6 |12.36932 |
|5 |13.22876 |
|4 |13.89244 |
|3 |14.38749 |
|2 |14.73092 |
|1 |14.93318 |
|0 |15 |
Graphically
[pic]
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 increments of one good cost society increasing increments of another.
[pic]
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 - 2X2)-1/2(-4X)
= -2X/Y
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 |Y |Opp Cost X for Y |Opp Cost Y for X |
|10.61 |0.01 |-3844.03 |0.00 |
|10.00 |5.00 |-4.00 |-0.25 |
|9.00 |7.94 |-2.27 |-0.44 |
|8.00 |9.85 |-1.62 |-0.62 |
|7.00 |11.27 |-1.24 |-0.80 |
|6.00 |12.37 |-0.97 |-1.03 |
|5.00 |13.23 |-0.76 |-1.32 |
|4.00 |13.89 |-0.58 |-1.74 |
|3.00 |14.39 |-0.42 |-2.40 |
|2.00 |14.73 |-0.27 |-3.68 |
|1.00 |14.93 |-0.13 |-7.47 |
|0.00 |15.00 |0.00 | |
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.
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 |0 |
|1 |19 |
|2 |36 |
|3 |51 |
|4 |64 |
|5 |75 |
|6 |84 |
|7 |91 |
|8 |96 |
|9 |99 |
|10 |100 |
|11 |99 |
|12 |96 |
|13 |91 |
|14 |84 |
|15 |75 |
|16 |64 |
Alternatively, we might approach this graphically
[pic]
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
[pic]
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
[pic]
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 derivative equal to zero implies that you did no “climbing” with the last (infinitesimal) adjustment, so you must be at an optimum
b. 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.
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 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) ................
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