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1. Chapter 21: Summary and Conclusion. 

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There should be little doubt that you have completed an ambitious program covering
Economics, Mathematics, and Excel.  A fitting conclusion is to review what you have
learned in these three areas.

By repeated demonstration of the core ideas in the Economic Approach, you have a new,
deeper understanding of economics.  Endogenous and exogenous variables, optimal and
equilibrium solutions, initial positions and comparative statics, underlying and
presentation graphs, if-then statements in own units and with elasticity, and interpreting
data through the Economic Approach are all part of your economics vocabulary.

We strongly encourage you to reread the document that started all of this, An
Overall View of the Economic Approach.  You will quickly realize how much you have
learned when you see that the ideas and graphs in that document are what we have been
working on throughout this course.  In addition, you will be able to put all of the material
in this course together into a coherent whole.

The Derivative and Understanding Calculus:

Perhaps the most important mathematical lesson of your work here is a true
understanding of the difference between infinitesimally small (d) and finite size (∆)
changes.  The former are the hallmark of the calculusÑa derivative is nothing more than
the exploration of how a function, Ä(x), changes for infinitesimally small changes in x.

You have seen how economists use the derivative in optimization problems in a
special way.  By setting the first derivative(s) equal to zero, we can find an optimal
solution.

Economists also use the derivative to do comparative statics.  Applying the general
use of a derivative as an exploration of how a function varies, you have learned to
analyze how a reduced form expression varies with an exogenous variable by taking
the derivative with respect to the exogenous variable.

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2. NON-LINEAR REDUCED FORM: When the reduced form expression is
non-linear in an exogenous variable, exploring the derivative of the reduced
form (known as the Method of Reduced Form) will give a different answer
than the Method of Actual Comparison, which is based on a comparison of
two distinctly different values of an exogenous variable.. 

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NON-LINEAR REDUCED FORM: When the reduced form expression is
non-linear in an exogenous variable, exploring the derivative of the reduced
form (known as the Method of Reduced Form) will give a different answer
than the Method of Actual Comparison, which is based on a comparison of
two distinctly different values of an exogenous variable.

LINEAR REDUCED FORM:  A reduced form expression that is linear in a
particular exogenous variable means that the two methods, reduced form
and actual comparison, will yield exactly equivalent results.

Neither method is always right or wrong.  The correct method depends on
the question being asked.   
If you are interested in a qualitative prediction (i.e., the relationship
between two variables), then the sign of the derivative (i.e., > or < 0) is
a quick, easy way to answer the question.  The magnitude of the
derivative at any point is a statement about how the reduced form
expression behaves for small changes around that value of the
exogenous variable.

If you are wondering about the value of the optimal or equilibrium
solution given two separate values of an exogenous variable, then the
Method of Actual Comparison is appropriate.

Our work on constrained optimization enabled us to learn about LagrangeÕs brilliant
strategy of transforming an m-variable, 1-constraint optimization problem into an m+ 1
unconstrained optimization problem by building a composite function of original objective
function + λ(constraint).1

The optimal value of the Lagrangean Multiplier, λ*, reveals the rate of change in the
maximum value function (the optimal value of the objective function) for an infinitesimal
relaxation of the constraint.

In the case of marketable permits, economists have established an interesting
interpretation of λ*.  Assuming an omnipotent, omniscient social planner solves the
constrained optimization problem, λ* reflects the additional cost of reducing pollution by
one unit.  It then turned out that the market's equilibrium price equaled λ*.  Thus,
absent full information by an omnipotent, omniscient social planner, the equilibrium
price in the market for permits reveals valuable information: low Pe means that the firms
find it easy to reduce pollution themselves; high Pe means pollution reduction is difficult.

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This strategy also works with more than one constraint.

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3. Phase Diagrams:. We learned about phase diagrams in our study of the equilibration process. This type of
graph has a powerful featureÑit immediately reveals the equilibrium solution as well as
the type and speed of the equilibration process.. 

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We learned about phase diagrams in our study of the equilibration process.  This type of
graph has a powerful featureÑit immediately reveals the equilibrium solution as well as
the type and speed of the equilibration process.

Creating a new workbook
Opening an existing workbook
Saving an existing workbook elsewhere
Adding, Deleting, Copying, and Moving sheets in a workbook

Scroll bars
Cell addresses
Using the Tools on the various Tool Bars

Copying by clicking on the "fill handle" (little black cross) and dragging with the mouse

Understand the meaning of a relative and an absolute reference
Know the signs for the operations:+,-,*,/,^
Interpreting error values returned by formulas
#DIV/0
#NAME?
#REF!
#VALUE!
Understanding ######

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4. Making charts. Using the Chart Wizard
Editing the chart once it has been created, titles and labels, controlling the axes scale
Power User: Adding labels to the plotted points. 

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Using the Chart Wizard
Editing the chart once it has been created, titles and labels, controlling the axes scale
Power User:  Adding labels to the plotted points

Creating Arrows, Boxes and other Graphics

For Unconstrained Max and Min
For Constrained Optimization
For Equilibrium Models
Using the Solver Dialog Box (Target and Changing Cells, Constraints, and Options)

Loading and installing an add-in
Using the Comparative Statics Wizard to do Comparative Statics
Drawing Presentation Graphs and calculating elasticities

Of course, there's much, much more in Microsoft Excel:

Keep working with Excel because there's much more to learn!

We remind you that we chose Microsoft Excel as the vehicle to deliver the economics and
mathematics content precisely because Excel is powerful and commonly used.  It is the
dominant spreadsheet today and, when bundled with Microsoft Office, offers a full range
of software for business and personal use.  You have learned quite a bit, but there's much
more to Excel than what we have covered.

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5. You should also realize that you've learned more than just narrowly applied Excel skills.
Sit in front of any spreadsheet (for example, Quattro Pro) and you'll be amazed at how
quickly you'll understand and dominate the software!. 

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You should also realize that you've learned more than just narrowly applied Excel skills.
Sit in front of any spreadsheet (for example, Quattro Pro) and you'll be amazed at how
quickly you'll understand and dominate the software!

What You Have Learned about . . . Solving Problems IN GENERAL:

Be organized in your thinking.
Know where you are in the problem.
Finding the initial solution requires underlying graphs, while
comparative statics utilizes reduced forms and presentation graphs.
Know the question and what you are looking for.
Constrained optimization has different underlying graphs than
unconstrained optimization.

Be systematic.
Apply solution recipes one step at a time, in logical order.

Be aware of alternative solution strategies.
You can find optimal solutions in a wide variety of ways, including:
1) A variety of different tables
2) A variety of different graphs
3) Direct Method versus Marginalism
Marginalism based on ∆ versus d
4) Calculus versus ExcelÕs Solver
You can do comparative statics in a variety of different ways, including:
1) Actual Comparison (∆) of alternative solutions versus exploring the
derivative of the Reduced Form (d)
2) Derivative of the Reduced Form versus Comparative Statics Wizard
If a problem has NO CLOSED FORM, all is not lost.  Even if the
reduced form canÕt be found, the Comparative Statics Wizard can be
used to plot out specific points on the analytically intractable reduced
form expression.

You should be proud of what you have learned and accomplished.

We hope your new skills and understanding of the Economic Approach,
mathematics, and Excel will enable you to master the many applications of
economics to the real world and to see the world in a different way.

You have made a valuable investment in yourselfÑone that should pay off
handsomely in other courses, on the job, and in your understanding of the world.

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