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1 Tour-syl.nb A Brief Tour of Mathematica First, consider the Cobb-Douglass production function x = f(k,l) = k^a l^b where x is units of output, k is units of capital services and l is units of labor Null Y Now let's use Mathematica to graph this function for a = .1 and k = .9 x = k.1 l.9 k0.1 l0.9 Note that the symbol "%" just means the previous expression Tour-syl.nb Plot3D@%, 8k, 0, 25<, 8l, 0, 25<D 2 20 25 10 0 0 5 10 15 20 25 0 5 10 15 20 -SurfaceGraphics- Looks pretty simple doesn't it. To get a fancy looking graph, we would need a more complicated functional form for the production function Y Now lets use Mathematica to find the marginal product of labor, MPl, for our general Cobb-Douglass production function x = ka lb ka lb The Mathematica command for derivative is just "D" so to take the derivative wrt to l we just enter the command l % Tour-syl.nb 3 mpl = l x b ka l-1+b Y Now let's integrate mpl with respect to l and see what we get. The Mathematica command for integration is just "Integrate" impl = mpl l ka lb The answer is the production function. Rather than continuing with our Cobb-Douglass example, let's end this tour of Mathematica with some more complicated math Y First consider some algebra. m = 9 H2 + s L Ht + s L + Hs + t L2 9 H 2 + sL H s + tL + H s + tL 2 Y Now let's raise it to the power of 3 and expand it Expand@%3 D s3 5832 + 9720 s4 + 5400 s5 + 1000 s6 + 17496s2 t + 30132s3 t + 17280s4 t + 3300 s5 t + 17496s t2 + 32076s2 t2 + 19494s3 t2 + 3930 s4 t2 + 5832 t3 + 12636s t3 + 8802 s2 t3 + 1991 s3 t3 + 972 t4 + 1242 s t4 + 393 s2 t4 + 54 t5 + 33 s t5 + t6 Y Now let's factor it Factor@%D Hs + tL3 H18 + 10 s + tL3 Tour-syl.nb 4 Y Now Calculus, partially differentiate wrt s s % 30 Hs + tL3 H18 + 10 s + tL2 + 3 Hs + tL2 H18 + 10 s + tL3 Y maybe this partial derivative can be simplified Simplify@%D 3 H18 + 20 s + 11 tL H18 s + 10 s2 + 18 t + 11 s t + t2L 2 Y Now consider integration of this function wrt t % t 24 s2 H9 + 5 sL2 H9 + 10 sL t + 6 s H2916 + 7533 s + 5760 s2 + 1375 s3L t2 + 3 2 H2916 + 10692s + 9747 s2 + 2620 s3L t3 + H4212 + 5868 s + 1991 s2L t4 + 4 6 6 11 t H207 + 131 sL t5 + 5 2 Y Or integration of the more simple function n y y 1+n y1+n Y Mathematica will also solve equations or systems of equations. For example, y3 - 7 y2 + 3 a y == 0 3 a y - 7 y2 + y3 == 0 Tour-syl.nb Solve@%, yD 5 98y 0<, 9y I7 + 1 2 1 !!!!!! !!!!! !!!!! !!!!!! !!!!! !!!!! ! 49 - 12 a M=, 9y I7 - 49 - 12 a M== 2 Lets end the tour with a 3-D Graphic using trig functions. Don't worry, in Econ 4828 we will use no trig. They do generate neat graphpics. t ParametricPlot3DA9u Sin@tD, u Cos@tD, =, 8t, 0, 15<, 8u, - 1, 1<, Ticks NoneE 3 -Graphics3D-
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Colorado >> ECON >> 4808 (Fall, 2006)
1 Consider higher-order derivatives April 12, 2007 higher.tex We have de.ned the cost function, c = c(x), and the marginal cost function as mc(x) = c0 (x). Since mc(x) is itself a function, one could take its derivative. E.g. dmc(x) dx mc0 (x) = c0...
Colorado >> ECON >> 4808 (Fall, 2006)
1 What is a function? February 13, 2007 1.1 What does it mean to say y = f (x) where x and y are variables A function associates with each value of the variable x a unique value of the variable y. Let\' stick with the variables x and y, but keep ...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Elasticities elasticity.tex April 36, 2007 One of the problems with marginal product functions is that the marginal product is sensitive to the units in which x, l, and k are measured. For example, changing the ouput from pounds of stu to tons o...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Economic applications of total dierentials May 2, 2007 Total dierentials are very useful for getting a better understanding of a lot of the graphs that we are familiar with in economics. Graphs such as isoquants, indierence curves, and isocost li...
Colorado >> ECON >> 4808 (Fall, 2006)
derivative.tex March 12, 2007 1 1.1 Economic Applications of Dierential Calculus The concept of a derivative slopes, continuity, limits and derivatives 1.2 Rules of dierentiation functions of single variables economic applications marginal revenu...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Economic Application of Derivatives deriv-applic.tex and .pdf April 5, 2007 In earlier notes, we have already considered marginal cost as the derivative of the cost function. That is mc(x) = c0 (x) How would you de.ne, in words, the cost function...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Consumer Theory in a Nutshell ct-nutshell.pdf February 6, 2007 You might want to read up on consumer theory in your intermediate micro book. The intent is build a model/theory to explain what an individual will purchase in the market place. Star...
Colorado >> ECON >> 4808 (Fall, 2006)
Cats, lions and zebras: assumptions, predictions, the utility function, and, sufficient but not necessary catfood.tex February 20, 2007 Consider the following theory about \"What\'s for dinner?\": Definitions: 1. C a cat is a furry animal that meows ...
Colorado >> ECON >> 4808 (Fall, 2006)
Economics 4808 - Review questions on economic applications of differentials. Dec 6, 2003 1. 2. 3. What is my maximum willingness to pay for another bottle of scotch in terms of pounds of swiss chocolate if my utility function is u us, c c 2 s us ...
Colorado >> ECON >> 4808 (Fall, 2006)
Review questions Econ 4808 Set 3 April 30, 2007 1. Assume t = m(x) = 4. What is m0 (x)? Answer: 0 2. Consider the function q = q(m) = + m dq dm ; leaving your answer in .ve parts. :5 m 3m :5 em + ln(3m). Find 1 em + m . Note d(ln a) d(ln m) = dm +...
Colorado >> ECON >> 4808 (Fall, 2006)
Basic algebra quiz: Econ 4808 Fall 05 Each of the .rst .ve questions will be graded out of 2 points, so is question 6, but it is extra credit. 1. Factor the expression ax+2 = ax a2 2. Expand and simplify a4 b (a2 b 3 3 )2 = a4 b a4 b 3 6 = b6 b3...
Colorado >> ECON >> 4808 (Fall, 2006)
Econ 4808: review questions for basic calculus quiz January 17, 2007 1. Find 6x + dy dx where 3 x +2 y = f (x) = 3 + 2x + 3x2 + 3 ln x. d(3+2x+3x2 +3 ln x) dx = 2. Assume that h(w) = (w3 w)(5w4 + w2 ). The the derivative of h(w) and simplify it a...
Colorado >> ECON >> 4808 (Fall, 2006)
Economic Examples of Partial Derivatives partialeg.tex April 12, 2004 Let\' start with production functions. A production function is one of the many ways to describe the state of technology for producing some good/product. Assume the firm produces a ...
Colorado >> ECON >> 4808 (Fall, 2006)
Review questions Econ 4808 Set 1 revised February 15, 2007 1. De.ne, in a few sentences, the concept of an economic theory. 2. De.ne a thing or action in words. Refer to this thing as action as A. Then de.ne a condition that is necessary but not su c...
Colorado >> ECON >> 4808 (Fall, 2006)
Brief notes on what is a theory (model) Edward Morey, January 29, 2007 theory.pdf Mathematics greatly helps with the process of building theories What is a theory? It is a thing, specifically a logical construct, with the following three parts: 1. d...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Rules for Dierentiation rules.tex and .pdf April 3, 2007 Deriving the derivative of a function using the basic de.nition of a derivative is revealing but is typically quite tedious when one wants to dierentiate a complicated function. So typicall...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Derivation of the cost function: The production manager\' problem s revised April 20, 04 Let\' consider the cost function in a bit more detail. Where does it come s from? Imagine that you have been hired by a .rm to run its widget factory; that is,...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Optimization optim-start.tex April 20, 2004 1.1 A start I am going to start by solving some problems that we are familiar with. I will add theory after we get a sense of the issues involved in optimization. We currently have enough tools to so...
Colorado >> ECON >> 4808 (Fall, 2006)
0.1 Mathematics is a language of symbols, grammar and logic. It greatly facilitates the process of deduction Jan 18, 2007, language.pdf While reviewing the syllabus, I said mathematics is a language. Languages are ways of conveying information and ...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Economic Models and Static Analysis equilibrium revised March 10, 2007, equilibrium.tex Equilibrium de.ned de.nition static equilibrium and comparative statics dynamic equilibrium some background the continuum from partial to general equilibrium ...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Economic applications of dierentials and total dierentials April 30, 2007 dier.pdf 1.1 The basic theory The dierential dy is a \" measure\" of how much y changes Dierentials are very useful for deriving isoquants, indierence curves, and other bo...
Colorado >> ECON >> 4808 (Fall, 2006)
1 Introduction to Mathematical Economics 4808 Course Syllabus - Spring 2007 2 Econ 4808 Introduction to Mathematical Economics Edward Morey Spring 2007 (January 15, 2007) Course Description: Econ 4808 is a course that will improve your understandi...
Colorado >> ECON >> 4808 (Fall, 2006)
Review questions Economics 4808 Set 2 revised March 10, 2007 1. Wilbur, the evangelist, is in the business of saving souls. If he can get you into his church on Sunday morning (church capacity 46), your soul will be saved and you will donate $47 doll...
Colorado >> ECON >> 6808 (Spring, 2008)
Econ 6808 Introduction to Quantitative Analysis Quiz 2 The quiz is due at the beginning of class on Tuesday September 8. It is a group quiz, groups of three, no more, no less. The group will hand it one quiz. Turn in a hard copy of your pertinent Mat...
Colorado >> ECON >> 6808 (Spring, 2008)
Absolute.nb 1 absolute . nb This notebook addresses the absolute value estimation problem in min&mak.pdf Y Minimizing Sum of Absolute Values of Residuals Suppose a firm produces a single output Y. The quantity produced in period t is Y(t). Y(t) ...
Colorado >> ECON >> 6808 (Spring, 2008)
Page 1 of 6 Consumer Theory in a Nutshell The intent is build a model to explain what an individual will purchase in the market place. Start by defining some terms: xi amount of good i consumed by the individual pi price of good i s y the indivi...
Colorado >> ECON >> 6808 (Spring, 2008)
Feb 20, 2002 Edward R. Morey I. Curvature Properties* *Thanks to Laura Langhoff for initial layout, typing and graphics. 2 I. Curvature Properties What do we mean by curvature properties? A mathematical function (or relation) can be described...
Colorado >> ECON >> 6808 (Spring, 2008)
1 The Compensating Variation: The Basics Two states of the world and . What is the compensation variation associated with the ) from to ?. It is the amount of money that must be subtracted from m in the new state to make one indifference between t...
Colorado >> ECON >> 6808 (Spring, 2008)
Deriving the production function from the cost function March 20, 2001 The intent of these notes is to demonstrate that the production function can be derived from cost function. Let\'s start with a simple graphical example, where production is a func...
Colorado >> ECON >> 6808 (Spring, 2008)
1 The Envelope Theorem: Shephard\'s Lemma, Hotelling\'s Lemma, etc. Suppose where a is a parameter. Choose x to max the function. In general Therefore is the maximum value of Call this a value function . given a. . The profit function is an example...
Colorado >> ECON >> 6808 (Spring, 2008)
Game Theory January 28, 02 Readings The Lost Years of a Nobel laureate, NYT, Nov 13, 1994. Between Genius and Madness, Nyt, June 14, 1998. Review of A Beautiful Mind, NYT, June 11, 1998. Chapter 1 of A Beautiful Mind Varian, Hal R., Intermediate Mic...
Colorado >> ECON >> 6808 (Spring, 2008)
1 Hicksian Demand Functions, Expenditure Functions & Shephard\'s Lemma Consider a world with 2 goods (x and y), where Wilbur has well-defined preferences over bundles of those two goods, and those preferences can be represented by the utility functio...
Colorado >> ECON >> 6808 (Spring, 2008)
1 Introduction to integration with some simple economic examples - Lecture 1 Topics covered: derivatives & antiderivatives (integrals) the notation of integration indeterminacy of integration indefinite vs. definite integration some simple eco...
Colorado >> ECON >> 6808 (Spring, 2008)
7 Introduction to Integration with some simple economic examples - Lecture 2 Reviewing is called an indefinite integral. is called an indefinite integral, rather than a definite integral, because function rather than a specific (definite) number. N...
Colorado >> ECON >> 6808 (Spring, 2008)
17 Applications of Integration: Present Value of Streams of Benefits and Costs - Lecture 3 Consider b is called the base Now consider expodential functions that have the base e = 2.71828. that is, Magically and Now consider where Note that e rt d...
Colorado >> ECON >> 6808 (Spring, 2008)
23 Now let\'s briefly consider multiple integrals. For example, assume the function What is the form of ? (Note that ) That is, what is the form of the function One can think of multiple integrals as such that follows: f ( x, y)dydx = m( x, y)dx...
Colorado >> ECON >> 6808 (Spring, 2008)
Joint Density Functions, Marginal Density Functions,Conditional Density Functions, Expectations and Independence \\ Feb 10, 2002 Define the joint density Consider the joint density function fx, y where fx, y 0 x and y, x, y , and fx, ydxd...
Colorado >> ECON >> 6808 (Spring, 2008)
quiz4_long.nb 1 The University of Colorado at Boulder Deparment of Economics ECON 6808-1998 Y Quiz Four-September 29, 1998 SCENARIO: Wilbur maximizes the linear utility function u=s+b subject to a linear money constraint m=pss+pbb and a linear time...
Colorado >> ECON >> 6808 (Spring, 2008)
maxlikev.nb 1 This notebook plots the likelihood function, and finds the max lik estimate for the one observation Extreme value estimation problem in min % max . pdf h = Exp@2 b - 10D Exp@- Exp@2 b - 10DD E-10+2 b-E -10+2 b 0.35 0.3 Plot@h, 8b, 0...
Colorado >> ECON >> 6808 (Spring, 2008)
Edward R. Morey I. Minimization with constraints) 1. 2. 3. 4. Procedure for Maximizing (or Minimizing) a Function of One Variable Maximizing or Minimizing Functions of Two Variables Maximizing or Minimizing Functions of ...
Colorado >> ECON >> 6808 (Spring, 2008)
An Economic/Statistical Application of Integration: Probability Theory What is a random variable? x is a random variable if it has a known distribution. That is, x is a random variable if oe a and b one can determine the probability that . Note that ...
Colorado >> ECON >> 6808 (Spring, 2008)
1 From the production function to the cost function, including a proof of Shepard\'s Lemma March 10, 2002 Assume a production function x = f (k, l) which is increasing and quasiconcave in k and l 1.1 Derivation of the cost function and conditional...
Colorado >> ECON >> 6808 (Spring, 2008)
The profit function April 2, 2001 Consider a competitive firm with the production function y fk, l Assume proft () maximizing behavior. py wl rk where y fk, l, w is the price of labor, r is the price of captial, and p is the price of y. Substit...
Colorado >> ECON >> 6808 (Spring, 2008)
An Economic/Statistical Application of Integration: Probability Theory rvandprob.tex Feb 22, 2003 What is a random variable? x is a random variable if it has a known distribution. That is, x is a random variable if a and b one can determine the pro...
Colorado >> ECON >> 6808 (Spring, 2008)
Econ 6808 Introduction to Quantitative Analysis Quiz six- in class, groups of one This quiz is worth 20 points. III. Economic Curvature and VI. Economic Applications of Duality Theory 1. (10 points) Define an individual\' compensating variation for ...
Colorado >> ECON >> 6808 (Spring, 2008)
Econ 6808 Introduction to Quantitative Analysis Quiz one I. Constrained Max and Min 1. Assume that the Snerd Corporation produces product X using L and K where X = f(K,L) = L.5K.5 Further assume that the firm buys labor and capital at the parametric ...
Colorado >> ECON >> 6808 (Spring, 2008)
Econ 6808 Introduction to Quantitative Analysis Assignment 1: paper/project report - due Nov 16 This project is worth 10 points. Quoting from the syllabus The Paper: The purpose of the paper will be to show how the mathematical techniques and theor...
Colorado >> ECON >> 6808 (Spring, 2008)
1 Econ 6806, Final, December 17, 1996 Answer the questions to the best of your ability in the allotted time. Make sure you have all nine questions The exam consists of three pages and question two has five parts. You can choose 20 points worth of que...
Colorado >> ECON >> 6808 (Spring, 2008)
1 Econ 6806, Final, December 10, 1999 Answer the questions to the best of your ability in the allotted time. Make sure you have all ten questions The exam consists of two pages. You can choose 20 points worth of questions for which you will be given ...
Colorado >> ECON >> 6808 (Spring, 2008)
6808qz2.nb 1 The University of Colorado at Boulder Department of Economics ECON 6808-1998 Y Quiz Two - September 8, 1998 This defines Wilbur\'s utility function. It is a function of two variables. WilbursUtility@x_, y_D := x.8 y.3 1. This creates ...
Colorado >> ECON >> 6808 (Spring, 2008)
1 Econ 6808 Introduction to Quantitative Analysis Quiz five - Answers This quiz is worth 20 points. III. Economic Curvature and VI. Economic Applications of Duality Theory 1. (5 points) Convince me that the cost function, c = c(y, w) where w = [w1, ...
UCSD >> MATH >> 240 (Spring, 2008)
Exercise L.4. Let g e Cf (lR) and / r lR -\' IR.be an absolutelycontinuousfunction with compact support and p x / be the convoiutionof p axd f ; v * f @ ) , :I p @ - d f ( u ) d y . Jn (a) Show tv * f) (r) : (p\' * f) (r) for all .c IR. # (p\'* f) (\"...
UCSD >> MATH >> 240 (Spring, 2008)
3 (U) (tSp) Showthat the operator is bounded L\'(O,oo),rn)and ll?/ll2 < ? on Crllf llz, whereCz is the constantin (a) with p : 2. ...
UCSD >> MATH >> 240 (Spring, 2008)
2 (u) (t0p) Suppose -> f in Lo (X , p). Show that fn * f in measure. /, (b) (z0p) Suppose -> f in measureand lf\"l < g a.e.with g Lo(X, p). Show fn that f L, (X, p) and f n -> f i n L P ( X , F ) . \'ll 3. Let be a Hilbert space,T:\'11 - ?l a bounded...
UCSD >> MATH >> 240 (Spring, 2008)
2 (u) Show that, for 0 S r S 1, fI k:m axrk j:m (b) Show that \"1111Ianrk:ioo. lc=0 /c:0 Hint: Estimate the left hand side of the formula in (b). 4. (30p) Let {fi} b e a sequence real-valuedfunctionsin L\'(X , p) such that of -) f a.e. with f L\'(...
Colorado >> ECON >> 4545 (Spring, 2008)
Why should there be a special course in environmental economics? whyacourse.pdf September 4, 2007 Environmental and natural resource are first and foremost resources and should be allocated in the same way we want to allocate all resources. That is, ...
Colorado >> ECON >> 4545 (Spring, 2008)
1 How can consumer\' surplus measures be used s to evaluate policies? use-cs.tex October 9, 2007 Assume a state of the world s from the perspective of individual i can be s 0 described by the vector Ss (yi ; Ps ; Cs ) where yi is individual i\' incom...
Colorado >> ECON >> 4545 (Spring, 2008)
1 An Introduction to the travel-cost method for estimating use damages: mining injuries in the Coeur d\'Alene Basin Tc-intro-coeurdalene.doc revised April 1, 2008 Objective: Estimate Damages from Injuries Damage: Willingness to Pay (WTP) for Unin...
Colorado >> ECON >> 4545 (Spring, 2008)
1 Parks and TV: one congestible, one not Edward Morey, April 24, 2008 Our national parks, local parks (e.g. Boulder Mt. Park, Central Park in NYC) and everything in between are often congested. Is the amount of that congestion e cient, and, if not,...
Colorado >> ECON >> 4545 (Spring, 2008)
Ryan Duncan December 16, 2003 Reintroduction Of Wolves: Market Failed or Market Failure? I was recently asked to conduct some research on the reintroduction of wolves into the Colorado area. Wolves used to be a normal part of the Colorado landscape...
Colorado >> ECON >> 4545 (Spring, 2008)
Should I-70 west of Denver be a toll road? Edward Morey, Dec 11, 2007 Kirk Johnson had an article, In Colorado, Traffic on the Interstate is at a Peak, in the New York Times (Sept 19, 2004), The article is about congestion on I-70 west of Denver; as...
Colorado >> ECON >> 4545 (Spring, 2008)
Blake N Melton Professor Morey Environmental Economics 1 April 1999 The Rail Line Quarry Proposal: A Failure to Utilize Available Methods The decision of whether or not to build the proposed Rail Line Quarry in the South Draw tributary to Eldorado is...
UCSD >> MATH >> 240 (Spring, 2008)
Qualifying Exam in Real Analysis, September 7, 2OA5 Instructions: Answer all six problems. No books or notesmay be used in this exam. You rnay cite without proof any result in the text by Folland, in any problem. 1. (45 pts.) Provethe following. Each...
UCSD >> MATH >> 240 (Spring, 2008)
1. Ane,r,vsrs Quar,myrNc Extu, Spnruc 2004 Instructions: Clearly explain and ju^stify your en.qwers. You may cite theoremsfrom tortbools or that were provedin classas long as they are not what the problerr ocplicitly asls you to prove. You may also u...
UCSD >> MATH >> 240 (Spring, 2008)
functionson [0,1J.Suppose that (3) Let AC([0,1]) denotethe absolutely continuous by f c C([0,1],R) is defined rl \'/ f : {f e AC(\';0,11) lf\'(r)l\'d, < 1 andl/(0)lS U. p(f , g) = max{l/(t) - s(t)l : t e [0,1J]. Show F is a precompactsubsetof C([0,1]) ...
UCSD >> MATH >> 240 (Spring, 2008)
2 4. Suppose T^ : X -> Y for n N is a sequence bor:ndedlinear operators between of two Banach spaces,X and Y. Ftrrther assume that lim,-*Trn exists for all r X. Show Tt : limr,*oo 4rr definesa bounded iinear operator from X to Y. 5. Srppose that {...
UCSD >> MATH >> 202 (Spring, 2008)
A P P L IED ALGEBRA QUALIPIER S 2OOO LAB PART Page2 ofSpages lvlay 29, 2OO2 PARTII SYMMETRIC FUNCTIONMETHODS s - l L e t p l - g , 1* r z * r s , pz-r!+x7+13, p s : s i + J \' t r + r a n d P6 :0i f + r! + r$. constmct thepolynomial which...
UCSD >> MATH >> 240 (Spring, 2008)
Qualifying Exam in Real Analysis, June L, 2OO3 No books or notes may be used in this exam. Do Problem 1 Instructions: remaining four problems, 2 through 5. fndicate clearly which any three of the problems you want graded. You may cite without proo...
UCSD >> MATH >> 240 (Spring, 2008)
1. Aw,q.r,ysrs Quer.rryrnc Exeu, Ferr, 2004 Instructions: Clearly explainandjustify your snswers. You may cite theorerns from the text, notes, or class as long as they are not what the probiemoiplicitly asks you to prove. You may also use the results...
UCSD >> MATH >> 202 (Spring, 2008)
APPLIED ALGEBRA QUALIFIER S 2OOO LAB PAR]T Pagel of5pages Llay 29, 2002 APPLIED ALGEBRA QUATIFIER May 29 2002 PART T REPRESENTANON THEORY R.ll-et Gbe a finite group and let {/^}r.rr be a completesetof irreducible unitary representations of...
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