(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 43123, 1082] NotebookOptionsPosition[ 40035, 973] NotebookOutlinePosition[ 40378, 988] CellTagsIndexPosition[ 40335, 985] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Math 256 course info", "Title", CellChangeTimes->{{3.440417834109375*^9, 3.440417839265625*^9}}], Cell["Steven Tschantz", "Subtitle", CellChangeTimes->{{3.440417844421875*^9, 3.440417846703125*^9}}], Cell["11/3/09", "Subsubtitle", CellChangeTimes->{{3.4404178526875*^9, 3.440417854953125*^9}, { 3.466270500578125*^9, 3.4662705100625*^9}}], Cell[CellGroupData[{ Cell["Basic data", "Section", CellChangeTimes->{{3.44041789003125*^9, 3.44041789321875*^9}, { 3.440419016296875*^9, 3.440419018578125*^9}}], Cell[TextData[{ "This is course information for\n\nMath 256 - Spring 2010\nMathematical \ Models in Economics\nTTh 2:35-3:50\nSC2200 (computer lab)\n\nProf. Steven \ Tschantz\nOffice: SC1507\nOffice hours: (Spring) TTh 1-2 and by appt.\nOffice \ phone: (32)2-6664\nEmail: tschantz@math.vanderbilt.edu\n\nCourse homepage: ", ButtonBox["http://www.math.vanderbilt.edu/~tschantz/m256/", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.math.vanderbilt.edu/~tschantz/m256/"], None}] }], "Text", CellChangeTimes->{{3.44041789953125*^9, 3.440418065796875*^9}, { 3.4404181529375*^9, 3.4404181529375*^9}, {3.46627049584375*^9, 3.466270496640625*^9}, {3.466275768421875*^9, 3.46627578671875*^9}, { 3.466275898515625*^9, 3.46627594278125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Policies", "Section", CellChangeTimes->{{3.44041818159375*^9, 3.44041820609375*^9}}], Cell[TextData[{ "This course is about applying mathematics, defining and experimenting with \ mathematical models, with particular emphasis on applications in economics. \ I will assume some basic economics and calculus. Some statistics experience \ would also be helpful. We will be using the computer program/language ", StyleBox["Mathematica", FontSlant->"Italic"], " to document the models, program the experiments, and evaluate the \ results. No background in ", StyleBox["Mathematica", FontSlant->"Italic"], " is assumed, but an interest in programming is valuable. Initially, we \ will be spending a good deal of time working through the basics of ", StyleBox["Mathematica", FontSlant->"Italic"], ", considering simple economic problems and mathematical models" }], "Text", CellChangeTimes->{{3.440418210890625*^9, 3.44041825928125*^9}, { 3.440418866109375*^9, 3.44041890778125*^9}}], Cell[TextData[{ "Assignments will be weekly ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks to complete and submit by e-mail. There will be a term project \ where you may work in small teams to research a question, formulate and \ program a model, presenting your results to the class during the last one or \ two days of class. There will be no final exam. The term project will count \ for approximately 25% of your final grade. Class will include lectures on \ the math and econ, instruction in ", StyleBox["Mathematica", FontSlant->"Italic"], ", and a significant amount of lab time where you will practice ", StyleBox["Mathematica", FontSlant->"Italic"], ", working on the assignments, with immediate assistance available. \ Assignments will be posted Tuesdays, generally to be due the following \ Tuesday. We will start out slowly, but assignments should soon include more \ challenging and open ended questions." }], "Text", CellChangeTimes->{{3.440418210890625*^9, 3.440418344171875*^9}, { 3.440418841*^9, 3.4404188781875*^9}, {3.440418947171875*^9, 3.44041898696875*^9}}], Cell[TextData[{ "I have been involved in part-time economics consulting with ", ButtonBox["Luke Froeb", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.owen.vanderbilt.edu/vanderbilt/About/faculty-research/f_\ profile.cfm?id=102"], None}], " of the Owen graduate school of management here at Vanderbilt, devising and \ programming mathematical models to help understand various economic \ questions. This course reflects the kind of work I have been doing. Treat \ this course as a part-time job. It is important not only that you keep up \ with posted assignments, but you should go beyond, applying the tools you \ will learn to better your understanding of questions that interest you." }], "Text", CellChangeTimes->{{3.440418210890625*^9, 3.440418344171875*^9}, { 3.440418446953125*^9, 3.440418513734375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Sample problem", "Section", CellChangeTimes->{{3.466270589359375*^9, 3.46627059271875*^9}}], Cell["\<\ Suppose a firm has a product it will sell to consumers. How should the firm \ set the price it will charge for its product? We want a simple model that \ illustrates the relationship between the price of a product, how much \ consumers will buy, how much it will cost the firm to produce that much of \ the product, and what profit the firm will make. Intuitively, the more the \ firm charges, the less it will be able to sell. It costs more to produce \ more of the product, though perhaps the added cost of increasing production \ by some amount may depend on the level of production. In any case, the firm \ faces a basic tradeoff: increasing the price increases the amount of profit \ on each unit, but decreases the number units that will be sold.\ \>", "Text", CellChangeTimes->{{3.466270610734375*^9, 3.466270774328125*^9}, { 3.466270831203125*^9, 3.466270990375*^9}, {3.466271036*^9, 3.4662710525*^9}, {3.466271231296875*^9, 3.466271238625*^9}, { 3.46627136003125*^9, 3.466271360171875*^9}, {3.466271421984375*^9, 3.466271678078125*^9}}], Cell[TextData[{ "Suppose the firm sets the price at ", Cell[BoxData[ FormBox["p", TraditionalForm]], FormatType->"TraditionalForm"], ". We describe the demand as a function of this variable, ", Cell[BoxData[ FormBox[ RowBox[{"q", "(", "p", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ", the amount that can be sold at price ", Cell[BoxData[ FormBox["p", TraditionalForm]], FormatType->"TraditionalForm"], " each month, say. The firm should know how much raw material and labor it \ will take to produce a given quantity ", Cell[BoxData[ FormBox["q", TraditionalForm]], FormatType->"TraditionalForm"], ", and so will have some idea of the cost ", Cell[BoxData[ FormBox[ RowBox[{"C", "(", "q", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " of producing this much product. The profit of the firm is then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[CapitalPi]", "(", "p", ")"}], "=", RowBox[{ RowBox[{"p", "\[VeryThinSpace]", RowBox[{"q", "(", "p", ")"}]}], "-", RowBox[{"C", "(", RowBox[{"q", "(", "p", ")"}], ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " each month. We assume that the firm will seek to maximize profit, \ concluding that the price charged will be determined by the demand and cost \ functions, ", Cell[BoxData[ FormBox[ RowBox[{"q", "(", "p", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"C", "(", "q", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ", and so predicting how price will be affected by changes in demand and \ cost." }], "Text", CellChangeTimes->{{3.466271707578125*^9, 3.46627212775*^9}, { 3.46627217034375*^9, 3.466272219671875*^9}, {3.466272265828125*^9, 3.466272279890625*^9}}], Cell[TextData[{ "We have a simple model of how firms set prices, expressed as a simple \ profit maximization problem. Mathematically, we solve for the profit \ maximizing price using simple calculus. At a critical point,\n", Cell[BoxData[ FormBox[ RowBox[{"0", "=", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{"\[CapitalPi]", "(", "p", ")"}]}], RowBox[{"\[PartialD]", "p"}]], "=", RowBox[{ RowBox[{"q", "(", "p", ")"}], "+", RowBox[{"p", FractionBox[ RowBox[{"\[PartialD]", RowBox[{"q", "(", "p", ")"}]}], RowBox[{"\[PartialD]", "p"}]]}], "-", RowBox[{ RowBox[{"C", "'"}], RowBox[{"(", RowBox[{"q", "(", "p", ")"}], ")"}], FractionBox[ RowBox[{"\[PartialD]", RowBox[{"q", "(", "p", ")"}]}], RowBox[{"\[PartialD]", "p"}]]}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\nWe can identify important factors in determining the best price. For \ example, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"C", "'"}], RowBox[{"(", "q", ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " is the change in cost per unit change in quantity, what economists call \ the marginal cost. We might rearrange the terms in this condition to \ conclude\n", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"p", "-", RowBox[{ RowBox[{"C", "'"}], RowBox[{"(", RowBox[{"q", "(", "p", ")"}], ")"}]}]}], "p"], "=", RowBox[{"-", SuperscriptBox[ RowBox[{"(", RowBox[{ FractionBox[ RowBox[{"\[PartialD]", RowBox[{"q", "(", "p", ")"}]}], RowBox[{"\[PartialD]", "p"}]], FractionBox["p", RowBox[{"q", "(", "p", ")"}]]}], ")"}], RowBox[{"-", "1"}]]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\nthe quantity on the right being the profit margin, the profit for each \ additional unit as a fraction of the price, and the quantity in parentheses \ on the right being the own price elasticity of demand, the fractional change \ in demand as a multiple of the fractional change in price." }], "Text", CellChangeTimes->{{3.4662722979375*^9, 3.46627302059375*^9}, { 3.466273089390625*^9, 3.466273091078125*^9}}], Cell[TextData[{ "To solve for the profit maximizing price we would need the functions ", Cell[BoxData[ FormBox[ RowBox[{"q", "(", "p", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"C", "(", "q", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], ". To get a practical answer for a real world problem we would need to \ estimate at least the elasticity of demand and marginal cost. To understand \ how demand and cost determine the optimum price, we can assume simple forms \ for the demand and cost functions, with parameters that can be adjusted to \ fit various scenarios. The accuracy of the conclusions drawn will depend on \ whether the assumed forms are flexible enough and whether the parameters can \ be estimated with any precision. By stating the assumptions of the model, \ making explicit the functional forms and parameter values, and solving \ various scenarios, we can better evaluate the implications of the model in \ the real world. " }], "Text", CellChangeTimes->{{3.466273030703125*^9, 3.4662736926875*^9}, { 3.466273724484375*^9, 3.46627373325*^9}, {3.466273785484375*^9, 3.46627390421875*^9}}], Cell[TextData[{ "To illustrate this problem we can start by assuming a simple linear demand \ form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"q", "(", "p", ")"}], "=", RowBox[{"b", "+", RowBox[{"m", "\[VeryThinSpace]", "p"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " with demand decreasing with increasing price so ", Cell[BoxData[ FormBox[ RowBox[{"m", "<", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ". We imagine a linear form for cost ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"C", "(", "q", ")"}], "=", RowBox[{"fc", "+", RowBox[{"mc", "\[VeryThinSpace]", "q"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", with some overhead fixed cost ", Cell[BoxData[ FormBox["fc", TraditionalForm]], FormatType->"TraditionalForm"], " and constant marginal cost ", Cell[BoxData[ FormBox["mc", TraditionalForm]], FormatType->"TraditionalForm"], ". We define these functions and the profit formula in ", StyleBox["Mathematica", FontSlant->"Italic"], " and solve the corresponding first order conditions. Assuming values for \ the parameters, we can plot the demand, revenue, cost, and profit as \ functions of price to help understand how these are determined by the price. 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