qdm intro complete

8/20/2019 QDM Intro Complete

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An Introduction to Mathematica (complete)

(Quantitative Dynamic Macroeconomics, Lecture Notes, Thomas Steger, University of Leipzig)

This chapter provides a concise introduction to Mathematica. Instead of giving a rigorous discussion or a comprehen-

sive summary of commands, the basic principles are demonstrated by going through some instructive examples. For a

systematic introduction see, for instance, Gräbe and Kofler (2007) and Jankowski (1998) and the references cited

therein. For a concise summary of notation and basic calculation see the supplement (Section 7 below).

1. First steps

Basics 1

We set up the equation "y=a+bx" and label it "equ"

equ = y a + b x

y a + b x

Next solve this equation for x by applying the command "Solve" and label the result "sol "

sol = Solve@equ, xD

::x → −a + yb

>>

One can extract the RHS of this solution as follows

sol@@1, 1, 2DD−a + y

b

Basics 2

Here we define a function f [ x, y]


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f@x_, y_D := x0.3 y0.7

and then ask Mathematica for the value of f [ x, y] at x=10 and y=20

f@10, 20D16. 245

Notice the two different assignments which have been used so far: "=" is an immediate assignment and ": =" denotes

a delayed assignment. The difference becomes clear using an example

a = Random @D; b := Random @D;

88a, a, a

<,

8 b, b, b

2 QDM_Intro_complete.nb


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We visualize the solution by plotting the quadratic equation. To plot the expression x2 + a x+ b over x, we must

specify the parameters a and b numerically

Clear@a, bD; a = 1; b = −1;

Plot@exp, 8x, −2, 2 8x, exp


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System of equations

à Analytical solution

Consider the following non-linear system a1 x1a1-1 x2

a2 = a and a2 x1a1 x2

a2-1 = b. Notice that this system represents the

first-order conditions of the problem maxI x1, x2M

9 x1a1 x2

a2 - a x1 - b x2=.

Clear@a, bD; sys = 9α1 x1α1−1 x2α2 a, α2 x1α1 x2α2−1 b=;

The system is solved by applying "Sol ve"

sol = Solve@sys, 8x1, x2>

Mathematica gives the solution in a somewhat unusual form. A manual derivation (see supplement below) and

numerical evaluation indicates that the result is correct.

Here we evaluate the solution for a baseline set of parameters

8α1, α2, a, b< = 80.3, 0.4, 0.1, 0.3

813. 2077, 5. 87009<

Graphically, the solution is the intersection of the curves, given by the two first-order conditions, in the ( x1, x2)-plane.

To illustrate we plot the two curves

curve1 = SolveAα1 x1α1−1 x2α2 a, x2E@@1, 1, 2DD;curve2 = Solve

Aα2 x1α1 x2α2−1 b, x2

E@@1, 1, 2

DD;



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Clear@α1, α2, a, bD; 8α1, α2, a, b< = 80.3, 0.4, 0.1, 0.3


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Integrate@diff, xD

a x + x2

There are several distinct ways to form a partial derivative

8D@Sin@xD, xD, ∂x Sin@xD, Sin'@xD<

8Cos@xD, Cos@xD, Cos@xD<

The total derivative can be formed using the command "Dt "

Dt@f@x, y, zDD

Dt @zD f H0, 0, 1L@x, y, zD + Dt @yD f H0, 1, 0L@x, y, zD + Dt @xD f H1, 0, 0L@x, y, zD

Note the notation: Dt [ z] means dz and f H0, 0, 1L@x, y, zD signifies ∑ f H.L∑ z

.

Some Cobb-Douglas algebra

At first, we define Y to denote output according to a standard Cobb-Douglas production function

Clear@Y, A, L, K, αD; Y = A Lα K1−α;The competitive wage and rental price of capital then is

8w = D@Y, LD, r = D@Y, KD<

9A K 1−α L−1+α α, A K −α Lα H1 − αL=

The production elasticity of labor and capital are defined as hYL :=dYêY dLê L

=dYêdL

Y ê L and hYK :=

dYêY dK êK

=dYêdK Y êK

: D@Y, LDY ê L ,

D@Y, KDY ê K >

8α, 1 − α<

Euler's theorem



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w L + r K êê Simplify

A K 1−α Lα

A CES example

Clear@Y, w, rD; Y = Ia Lα + H1 − aL KαM1êα;

8w = D@Y, LD, r = D@Y, KD<

:a L−1+α HH1 − aL K α + a LαL−1+1α , H1 − aL K −1+α HH1 − aL K α + a LαL−1+

1α >

w L + r K Y êê Simplify

Tr ue

Euler's theorem holds, of course, also true in the general CES case.

4. Lists, functions, and some functional programming

Lists

A list is a collection of objects. In Mathematica, a list is the fundamental data structure used to group objects together.

As a first and simple example consider

a = 82, 4, 6, 8, 10


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8a@@1DD, b@@1DD, b@@1, 1DD<

82, 81, 2


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sampleMean =1

iLength ‚

i=1

iLength

listRandom @@iDD

sampleDeviation =

1

iLength − 1 ‚i=1iLength

HlistRandom @@iDD − sampleMeanL2

1ê2

0. 501975

0. 313544

The same can be done using the built-in functions "Mean" and "Standar dDevi at i on"

8 Mean@listRandom D, StandardDeviation@listRandom D<

80. 501975, 0. 313544<

Functions

Mathematica contains a number of built-in-functions. In addition, one can easily define new functions. The general

form of a function definition is f [ x_] : = body. Note that the function argument x is followed by an underscore.

The combination x_ is called a pattern. Also note the "special" definition symbol : =.

Built-in functions. Here is a plot of f H xL = e x together with its first-order Taylor series approximation about x = 0

series1 = Series@Exp@xD, 8x, 0, 1


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Plot@8Exp@xD, series2


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Plot@f@xD, 8x, −6, 6


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mean@x_D := 1Length@xD ‚

i=1

Length@xDx@@iDD

Here we test the function

mean@Table@Random @D, 8100 000>

Observe that Mathematica names the arbitrary constant of integration C[ 1] .

We visualize the result by using an example. First, we extract the solution from the above output expression

toplot1 = sol@@1, 1, 2DD;Here we replace the constant of integration (C[ 1] ) by the numerical value "10"

toplot = toplot1 ê. C@1D → 10

−

a

b+ 10 b t

Specifying parameters a and b numerically, we can use Pl ot command to plot the result



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Clear@a, bD; a = 2; b = 0.2;Plot@toplot, 8t, 0, 30


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list = Table@sol@@1, 1, 2DD, 8t, 0, 20

Clear@a, bD; a = −0.75; b = 2;ListPlot@list, PlotJoined → True, AxesLabel → 8t, xt


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6. Graphics

An isoquant plot of CES functions

Especially for teaching it might be helpful to plot isoquants of a CES technologies, like Y = I x1s+ x2

sM1ês. There are

two basic possibilities to do this.

Possibility 1

QDM_Intro_complete.nb 15


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ManipulateA9Plot3DAIx1σ + x2σ M1êσ , 8x1, 0.001, 50


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Clear@α, p1, p2D; α = .35; p1 =

PlotAEvaluateATableASolveAIx1α + x2αM1êα Y, x2E@@1, 1, 2DD,8Y, 50, 200, 10


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ListPlot@dataList, AxesLabel → 8t, "Sin@tD"


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list =

Import@"http:êêwww.wifa.uni−leipzig.deêfileadminêuser_upload êitvwl−vwlê

makroêLehreêqdm êGDP_Per_Capita_USA.xls"D; p1 = ListPlot@list, PlotStyle → [email protected], Black


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Notation

à Arithmetic operators

^ stands for exponentiation, as in 2^4, or in StandardForm 24

/ stands for division

* stands for multiplication, however is frequently omitted since space between two operands is understood as a

multiplication operation:

2 ∗ 3

2.1 π

. is the matrix multiplication operator (dot product)

à The four kinds of braces

( a+b) c parantheses are for grouping

(round parentheses) are to be used only to indicate order of evaluation: Ha + bL c is the quantity a + b multiplied by c. Note that space indicates multiplication. In other cases space means nothing at all.

f [ x] , Si n[ x] square brackets for functions

(square brackets) are used with functions and commands: Sin@ xD is the sine of x. SinH xL does not work.

{a, b, c, 5. 4, "Hel l o" } curly braces for lists

v[ [ i ] ] double brackets for indexing lists

repeated square brackets are used to refer to a specific part of a list. Given the five element list

v = 8a, b, c, 5.4, Hellois a transformation rule, rules are frequently employed to define options in functions, as in

Pl ot [ Cos[ x] , {x, 0, 2π}, Pl ot Range- >{- 2, 2}]

/ / is the postfix form of the expression f [ x ] , for example Si n[x] may be written x/ / Si n

Many Mathematica functions have associated symbolic representations. Some of the most commonly encountered

are:

f @@ expr is equivalent to Repl aceAl l [ f , expr]



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f / @expr is equivalent to Map[ f , expr ]

st r i nga st r i ngb is equivalent to St r i ngJ oi n[ st r i nga, st r i ngb]

à Punctuation

As in C and other programming languages the semicolon (;) is a command terminating character. Additionally, it is

used to block kernel output to screen. The comma (,) is used as a simple separator. Mathematica allows multiple

statements in a cell or even a single line. The characters "(*" and "*)" denote beginning and end of comments.

à Function names

Mathematica is case sensitive.

If you know what you want to do but you don't know the name of the Mathematica function to use, some rules can

help you guess. (Of course, you could always look it up in the Help Browser, but who wants to read documentation

when you can guess instead?)è All function names start with capital letters, and multi-word names use internal capitalization (the way

German should but doesn't). There is an exception for widely accepted abbreviations of common functions

such as Si n, Log, Exp, etc.

è Nothing is ever abbr. (except N and D). Integer is Integer not Int. Integrate is Integrate, not Int.

è Functions named after a person are the person's name plus the commonly used symbol. Examples:

Bessel J

Legendr eP

ChebyshevT

Exception:

Zet a (should have been Ri emannZ)

è When in doubt, think long. Examples:

Fact or I nteger

Li near Pr ogr ammi ng

Mat hi euChar act er i st i cExponent

A Mathematica variable or symbol may consist of any number of contiguous letters and numbers. The name should

not start with a number. Some special characters should not be used: underscore character, ampersand, period. Hereare a few examples of BAD variable names:

8x_y, x.y, x & y<

à Constants

Built-in constants adhere to this convention: I, E, Pi, Infinity (or in the newer forms: Â, ‰, p, ¶ ). The Traditional-

Forms are all obtained via the Escape key.

I I

QDM_Intro_complete.nb 21


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N@ED

N@π D

1

∞

0 ∞

Basic symbolic and numeric calculations

We begin with a discussion of symbolic vs. numeric calculations based on material in Glynn/Gray, Chapters 5 and 23.

à Simple calculations

Even at this early stage, you will encounter some interesting examples since most of you are familiar with numerical

calculations only.

1 + π

N@%D

N@%%, 50D

1

3

N@%D Note the difference between the following two results. You need to be aware of the fact, that when using real numbers

or N, all calculations are approximate (within the precision of computers floating-point representation).

SinB π 3F

SinB π 3.

F

The following example will give another example of the difference between Mathematica's exact and approximate

number representations.



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M = :: 1116

,1

2>, : 1

3, −

11

5>>;

M.Inverse@ M D

N@ M D.Inverse@ N@ M DDSome more symbolics using complex numbers.

−4

ExpB I π 2

F

:ReBExpB I π 3

FF, Im BExpB I π 3

FF>

ConjugateBExpB I π 3

FF

ExpToTrigBExpB I π 3

FF

The value of a variable need not be a number. Here is another example.

y = 2 x + 1

y2 + 2 x

Here is a method of checking your basic math skills. The == symbol is a relational operator, it tests for equality.

Log@10, π D == Log@π DLog@10D

Cos@xD2 + Sin@xD2 == 1The last example shown what happens when Mathematica does NOT understand the input - it simply returns the input

unevaluated (see below for the proper way of testing this well known trigonometric identity).

TrigToExpACos@xD2 + Sin@xD2E == 1

Other relational operators are: >, =,


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à Algebra and calculus

Two result in the preceding section did not yield the desired results. Here we will try again.

Expand Ay2E − 4 x

SimplifyACos@xD2 + Sin@xD2E == 1

Expand and Simplify are two examples of a large family of functions useful in manipulating algebraic expressions.

We move on to examples from calculus. Note that you can place comments on Mathematica input lines.

∂x xn H∗ this is a derivative ∗L

∂8x,2< xn

‡ % x H∗ % stands for the last output ∗L

Here is a "favorite" of electrical engineering students. It gives the inverse Fourier transform of the ideal lipase filter of

bandwidth 1rad

sec.

sinc =1

2 π

‡ −11

Exp@I ω tD ω

Follow with some algebraic manipulation and we get

ExpToTrig@%DThere are however many integrals for which no explicit formula can be given. These can be solved numerically.

‡ Sin@Sin@3 xDD x

Definite integrals have the same general form. However, the second argument is now an iterator. It has the generalform {x, xmin, xmax}.

NIntegrate@Sin@Sin@3 xDD, 8x, 0, 1

qdm intro complete

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