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The brief \ reference to zoning laws at the end of Chapter 1 opens the door for the more \ involved discussion of how patterns of land use are affected by regulation. \ This chapter examines land use from the standpoint of the community. If one \ finds that the bid rent curve in a particular area, rather than having a \ smooth downward sloping shape, is a series of jagged lines not necessarily \ pointing in any direction, it may be that market participants are constrained \ by regulators who have decided what is best for land users regardless of \ economic considerations. Indeed, one of the harshest criticisms of \ government planning is that the motives of policymakers are political rather \ than economic. Thus, land use often proceeds not on the basis of its most \ efficient use but on the basis of the size and level of protest of vocal \ groups who have the power to elect or re-elect officials who do their \ bidding.\nThe landscape is littered with spectacular government-inspired land \ use failures such as Federal housing projects and rent control but one also \ observes the occasional successful urban renewal. No conclusion is likely to \ be reached here, nor is it our purpose to advocate for a specific position. \ Rather, the goal of this chapter is to provide the reader with (1) a way of \ thinking about land use regulation and (2) a model to rationally resolve a \ conflict with a regulatory agency. The chapter will propose a theoretical \ model that permits one to optimize the conditions of regulation in a general \ sense. Following that, an actual municipal decision is illustrated with a \ case study based on real data.\nThe theory of rent determination advanced in \ Chapter 1 was developed in a simpler time. Urbanization on a large scale to \ accommodate a burgeoning population introduces complexities. Observe a \ transaction between two economic agents, in our case landlord and tenant. Do \ their choices affect only them? Perhaps they do not. Economists have a name \ for the effects transactions have on third parties: Externalities. When I buy \ a car from a dealer I get a car and the dealer gets my money. A trade has \ been completed. But when I drive the car I emit pollutants into the air that \ you breathe. You have been affected by the decision of a car buyer and \ seller to engage in a transaction to which you were not a party. The \ transaction imposed a cost on you in the form of soiling the air you breathe. \ This is known as The Problem of Social Cost. This chapter addresses the \ social cost issues affecting real estate and how land use is determined in \ the presence of social costs.\nAn advanced civilization is a society of \ rules. To deal with competing interests, cultural differences and the \ occasional rogue operator we come together as a community to establish what \ constitutes socially acceptable behavior. The business aspect of society has \ a set of norms reached through negotiation over many years. The study of this \ is an active area of research called \"Institutional Economics\" or \"Law and \ Economics\". Academics in this field study the economic consequences of \ passing laws to regulate human economic behavior. Among the more interesting \ findings are the unintended consequences of placing barriers in the way of \ those who would otherwise seek what is best for their own self-interest.\nThe \ underlying conflict may be simplified as one in which we must choose between \ what is good for the individual versus what is good for the community. Part \ of the debate is: Who shall decide? In economics, institutional factors are \ constraints on freedom of choice; and the choice we are interested in here is \ the choice of how land may be used. The unanswered question is: Shall the \ choice be made by the landowner or the community in which the land is \ located.\nTariffs and trade agreements govern how commerce crosses \ international boundaries. Laws prohibiting collusive and coercive activities \ govern domestic trade at a national level. Our interest lies in local \ government. For the private real estate investor local land use regulation is \ a significant aspect of the decision making process. In urban settings it is \ no overstatement to say that real estate investment success is in large part \ dependent on an understanding of the regulatory environment in which the \ local real estate market exists. Whether it is zoning or rent control, real \ estate investors ignore local politics at their peril.\nSeveral general ideas \ make this subject important. \nFirst, the unique fixed-in-location aspect \ that makes real estate different from financial assets provides both \ stability for investors and a fixed target for policymakers. Businesses that \ can easily move out of an oppressive jurisdiction retrain policymakers who \ might otherwise enact ruinous legislation. But the fact that structures are \ not on wheels and their owners cannot merely roll their buildings across the \ county line, taking their businesses with them, represents a temptation to \ local government.\nSecond, directly affecting residential investment, housing \ is a politically charged topic. Economists consider housing a \"merit good\" \ meaning that part of society has decided that all its members \"deserve\" a \ minimum standard of housing regardless of their economic status or ability to \ pay for it. Out of that mentality arises a host of subsidies, programs, \ controls and standards designed to shape the market into something that fits \ the will of a few elected officials, not necessarily market participants.\n\ Third, and often working against the housing issues just mentioned, are the \ parochial views of the community's established citizenry. Popularized as \ \"NIMBYism\", this manifests itself in the form of local planning groups \ populated by activists who profess a heightened environmental sensitivity, \ concern for preservation of \"the neighborhood\" and who often merely oppose \ everything that represents change. The unintended consequences of this \ activity are interesting to study. They can be as benign as imposing a brief \ delay in obtaining a building permit to extreme outcomes such as litigation \ that bankrupts a developer pursuing a politically unpopular project.\nIn a \ modern city the list of development constraints and regulations is a long \ one. A builder must comply with the general plan, zoning, minimum lot size, \ open space requirements, minimum setbacks from lot lines, maximum floor area \ ratios, building height limitations, grading limitations on slopes, minimum \ landscaped area, view corridors, off street parking, curb cuts, building \ codes, fire prevention and suppression regulations, and traffic counts just \ to name a few. In areas designated as special districts they may also have to \ deal with architectural and design requirements. Some property owners must \ get permission to change the color of their building when they repaint it. \ Charles M. Tiebout (1956) saw a market concept at work for cities. He \ proposed a model for residential homeowners that views the universe of \ potential locations as a group of municipalities competing for \ citizen-taxpayers who \"vote with their feet\" by moving into communities \ offering the best (most efficient) mix of services and taxes (benefits and \ costs) and out of those communities offering less efficient combinations. \ Thus, under the Tiebout hypothesis, communities that fail to provide services \ demanded at a market price (reasonable taxes) are punished by an exodus of \ tax-paying citizens. On the positive, communities that provide high quality \ services at or below market prices attract tax-paying citizens.\nThese \ dynamics influence the choices of commercial land users as well. The recent \ past has seen a rise in the interest of state and local jurisdictions in \ being competitive in the regulatory arena. These range from as little as \ advertising their communities as \"business friendly\" to as much as offering \ major tax concessions for many years after construction of a facility.\nThere \ is no particular reason to choose for our study one form of land use \ regulation over another. Zoning, environmental protection or rent control, \ each has compelling arguments for and against. The method of thinking \ proposed here is a classical microeconomics approach that leads to the \ conclusion that the best answer is the one that accomplishes the most good \ for the most people. One should recognize, however, that the implementation \ of a ", StyleBox["rational", FontSlant->"Italic"], " model in a political environment represents a daunting challenge. People \ are often not rational. Does that mean we should abandon all use of rational \ models? No, often there is an opportunity to present a well-formed argument \ to cooler heads. Such an argument can not only be well received, it can carry \ the day when it is time to vote a project up or down.\nThere are hundreds if \ not thousands of examples from the residential field to draw from. Rather \ than take one of those and its somewhat straightforward analysis, the setting \ for the analysis here comes from the commercial area. This presents \ additional challenges that deserve attention, and at the same time \ illustrates how a somewhat esoteric land use conflict can be modeled." }], "Text", FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["The idea of \[OpenCurlyDoubleQuote]utility\"", CharacterEncoding->"WindowsANSI"], "" }], "Section"], Cell[TextData[{ "Central to the development of a theoretical model of this type is the use \ of an abstraction known as \"utility\", a term economists employ to describe \ a more general form of happiness or betterment. Our model needs a yardstick \ that describes the gratification that comes with success and that yardstick \ is \"utility\". We can quantify this and with further analysis describe \ situations that are better or worse in terms of increased or diminished \ utility. The utility abstraction may seem foreign to non-economists thus the \ analogy to happiness or betterment. While perhaps ill defined, most of us \ know when we are more or less happy or satisfied. Utility is just the word \ economists use to describe that feeling, nothing more. As we wish to \ mathematically model this result, \"disutility\", meaning negative or smaller \ amounts of utility, translates roughly to unhappiness or less happiness, of \ course something to be avoided. Clearly, unhappiness is inferior to happiness \ and thus any mathematical result having a lower value represents a tendency \ toward unhappiness. Utility is ordinal, not cardinal. That is, the actual \ number we produce in any calculation has no meaning by itself (unless one \ believes there is a unit of measure known as \"utils\"). This frustrates \ those who have labored to \"get the numbers right\" in other investment \ settings by calculating the \"right\" answer in the form of some specific \ number. What matters where any number is concerned is ", StyleBox["the ranking", FontSlant->"Italic"], " of various values of utility computed under differing conditions. Thus, \ I may know that I am happier than my brother-in-law but I probably would not \ say that he has a happiness value of 80 unless I was convinced I have a \ happiness value of, say, 95. (The \"happiness\" metaphor tends to be \ stretched rather thin at about this point).\nOnce we accept the utility \ abstraction the next step is to construct a way in which utility is achieved. \ This leads to a \"production function\" which is nothing more than a \"rule\" \ by which people \"manufacture\" utility. Returning to our happiness \ metaphor, most readers have heard someone say that our success or happiness \ is the sum of all of our choices. In such a case the production function or \ rule we use is merely to add up all the choices (implicitly subtracting the \ bad choices that may be seen as adding negative numbers) we have made and \ that determine our happiness. \nSuch a rule becomes more complex in a real \ estate setting but nonetheless is still just some sort of rule. The rule we \ often use for economic choices has two essential properties, both of which \ are fairly intuitive. First, we assume we are always interested in more \ happiness thus the utility function is always rising. This is formally known \ as the property of non-satiation. Second, despite its constant increase,", StyleBox[" the rate at which it increases", FontSlant->"Italic"], " slows as utility increases. This is formally referred to as diminishing \ marginal returns, meaning that while we are happier with each new increment \ of utility we are not as much happier with the next increment as we were with \ the increment last received. \nA silly example may help here. Suppose I love \ bananas to the point of craving. If, like Groucho, I have no bananas I may be \ willing to pay quite a tidy sum for a single banana. I would trade money for \ the utility I receive from eating a banana. Suppose that tomorrow I inherit \ from my deceased rich uncle a large productive banana plantation providing me \ an ample supply of bananas. I still have the craving love of bananas but what \ has changed is what I am willing to trade for yet another banana. Because my \ utility function for bananas exhibits diminishing marginal returns with \ increased ownership of bananas, the amount I am willing to pay for another \ banana when I already have millions of bananas is, although a positive amount \ due to the non-satiation principle, very small.\nHowever you approach an \ understanding of it, utility is a useful abstraction for considering the cost \ and benefits of different choices we face. The reader is encouraged to find a \ comfort level with this abstraction as it is one we will return to again in \ this book.\nWith the free market lessons of Chapter 1 in mind we proceed with \ the counter example: Political land use determination." }], "Text", FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["The Model", "Section"], Cell[TextData[{ "Suppose a community wishes to protect the environment (", StyleBox["Env", "Input"], "), specifically the ", StyleBox["visual", FontSlant->"Italic"], " environment, by regulating the commercial advertising (", StyleBox["A", "Input"], ") of local businesses. We assume that community retail merchants \ advertise via outdoor signage. Regulation comes in the form of restricting \ the height, size, mass, design, shape, illumination, position, color, copy, \ etc. of signs. Resources the community spends on aesthetic regulation reduce \ scarce resources in the form of tax revenue available for other services the \ municipality must furnish such as police and fire protection (", StyleBox["M", "Input"], "). One characterization of the latter would be \"hard benefits\" rendered \ by the city to its residents. On the other hand, regulation of the \ aesthetics of the local visual environment may be termed \"soft benefits\".\n\ Citizens derive utility (", StyleBox["U", "Input"], ") from having visually uncluttered or appealing commercial vistas (", StyleBox["Env", "Input"], ") and from the receipt of municipal services (", StyleBox["M", "Input"], "). A conflict exists between merchants who wish to maximize advertising \ to saturation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", StyleBox[\(A\_0\), "Input"], ")"}], " "}], TraditionalForm]]], "and residents who wish to regulate signage as close to zero as possible. \ A trade-off exists because the reduction of advertising brings about the \ related but not exactly equivalent reduction of municipal income from taxes. \ Tax revenue is a function of (1) sales which, in turn, are a function of \ advertising; and (2) property values. As property values are, through rent, \ an indirect a function of sales, we impound all tax effects into the sales \ tax and ignore for simplicity the dual source of municipal revenue. Thus, the \ city's tax revenue must be allocated between paying for the soft benefits \ afforded by aesthetic regulation and the hard benefits of \ non-aesthetic-regulation municipal services. \nThe city derives its income \ from taxes levied on sales (", StyleBox["S", "Input"], ") at a tax rate (", StyleBox["q", "Input"], ") set exogenously by the State. Merchants who employ signs to advertise \ their businesses to passing consumers generate sales, in part, on the basis \ of the productivity (", StyleBox["\[Gamma]", "Input"], ") of their advertising, which is related to characteristics of the \ individual signs such as size, height, etc. One of the ways the city may \ regulate advertising (", StyleBox["A", "Input"], ") is by reducing the efficiency (", StyleBox["\[Gamma]", "Input"], ") of signs by restricting their characteristics.\nThe city must maximize \ utility (", StyleBox["U", "Input"], ") by choosing the ", StyleBox["correct", FontSlant->"Italic"], " amount of allowable advertising (", StyleBox["A*", "Input"], "). All other variables are exogenous." }], "Text"], Cell[BoxData[{\(CleanSlate[];\), "\[IndentingNewLine]", \(<< Utilities`Notation`\), "\n", \(UpdateNotationsInNotebook[]\), "\n", \(<< Graphics`Graphics3D`\), "\n", \(<< Graphics`Arrow`\), "\n", \(<< Graphics`Legend`\), "\[IndentingNewLine]", \(<< Graphics`Animation`\), "\n", StyleBox[\(Off[General::spell1];\), FontFamily->"Courier New"], "\n", StyleBox[\(Off[General::spell];\), FontFamily->"Courier New"], "\[IndentingNewLine]", StyleBox[\($DefaultFont = {"\", 12. };\), FontFamily->"Courier New"]}], "Input", CellLabel->"In[1]:=", FontSize->12], Cell[BoxData[ RowBox[{"Symbolize", "[", TagBox[ TagBox[\(A\_0\), NotationBoxTag, TagStyle->"NotationTemplateStyle", Editable->True], NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}]], "Input", CellLabel->"In[11]:=", FontSize->12], Cell[TextData[{ "Our notation guide is as follows:\n", StyleBox["A", "Input"], "\t=\tAdvertising \n", StyleBox["q", "Input"], "\t=\ttax rate \n", StyleBox["U", "Input"], "\t=\tUtility \n", StyleBox["M", "Input"], "\t=\tMunicipal services\n", StyleBox["Env", "Input"], "\t=\tEnvironmental protection \n", StyleBox["S", "Input"], "\t=\tSales volume\n", StyleBox["\[Alpha]", "Input"], "\t=\tproportion of Utility arising from citizens' preference for \n\t\t\ environmental regulation, 0 < \[Alpha] < 1\n1-", StyleBox["\[Alpha]", "Input"], "\t=\tproportion of Utility arising from citizens' preference for non-\n\t\t\ environmental \tregulation community services\n", StyleBox["\[Beta]", "Input"], "\t=\tcitizens' negative utility from the appearance of advertising \n", StyleBox["\[Gamma]", "Input"], "\t=\tmerchants' productivity of advertising \[Gamma] > 0\n", Cell[BoxData[ FormBox[ StyleBox[\(A\_0\), "Input"], TraditionalForm]]], " \t= \tthe maximum imaginable amount of advertising possible - full \n\t\t\ saturation, full \tcoverage by any measure, an amount beyond \n\t\twhich it \ is impossible to go." }], "Text", FontSize->12], Cell[TextData[{ "The city derives revenue from sales taxes levied on sales generated by \ businesses. Businesses depend on advertising to promote sales. The following \ expression describes sales, ", StyleBox["S", "Input"], ", as a function of advertising, ", StyleBox["A", "Input"], ", where ", StyleBox["\[Gamma]", "Input"], " represents the productivity of advertising:" }], "Text", FontSize->12], Cell[BoxData[ \(\(S = A\^\[Gamma];\)\)], "Input", CellLabel->"In[12]:=", FontSize->12], Cell[TextData[{ "The following expression describes Municipal Services (", StyleBox["M", "Input"], ") in the form of an annual budget wherein revenue is derived from taxing \ sales (in the interests of simplicity property taxes are not considered here \ even though increases in sales increases property values and therefore \ property taxes):" }], "Text", FontSize->12], Cell[BoxData[ \(\(M = q\ S;\)\)], "Input", CellLabel->"In[13]:=", FontSize->12], Cell["\<\ Citizens find advertising objectionable and have a production function (rule) \ for Environmental protection based on their disutility of Advertising:\ \>", "Text", FontSize->12], Cell[BoxData[ \(\(Env = \((A0 - A)\)\^\[Beta];\)\)], "Input", CellLabel->"In[14]:=", FontSize->12], Cell[TextData[{ "The disutility is subtle. The term \"", StyleBox["A", "Input"], "\" must be viewed as \"Allowed Advertising\". The controversy surrounds \ the difference between the maximum amount of advertising, ", Cell[BoxData[ FormBox[ RowBox[{"(", StyleBox[\(A\_0\), "Input"], ")"}], TraditionalForm]]], ", and that which is allowed, ", StyleBox["A", "Input"], ". Merchants want A to be as high as possible, as close to full \ saturation, ", Cell[BoxData[ FormBox[ RowBox[{"(", StyleBox[\(A\_0\), "Input"], ")"}], TraditionalForm]]], ", as they can get. This makes the term ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ StyleBox[\(A\_0\), "Input"], "-", StyleBox["A", "Input"]}], ")"}], TraditionalForm]]], " approach zero. Residents want ", StyleBox["A", "Input"], " to be as low as possible, making the difference between the maximum and \ the allowed advertising ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ StyleBox[\(A\_0\), "Input"], "-", StyleBox["A", "Input"]}], ")"}], TraditionalForm]]], " as large as possible. The condition ", StyleBox["A", "Input"], " = 0 may be viewed as \"full regulation\", the case of no advertising \ allowed. Plotting ", StyleBox["Env", "Input"], ", the term ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ StyleBox[\(A\_0\), "Input"], "-", StyleBox["A", "Input"]}], ")"}], TraditionalForm]]], ", for an arbitrary value of ", Cell[BoxData[ FormBox[ RowBox[{"(", StyleBox[\(A\_0\), "Input"], ")"}], TraditionalForm]]], "and two different values of ", StyleBox["\[Beta]", "Input"], " against ", StyleBox["A", "Input"], " shows that the amount of environmental protection (", StyleBox["Env", "Input"], ") residents achieve falls with the increase of ", StyleBox["A", "Input"], ". The exponent \[Beta] indicates the intensity with which residents \ derive utility from the ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ StyleBox[\(A\_0\), "Input"], "-", StyleBox["A", "Input"]}], ")"}], TraditionalForm]]], " term thus determines ", StyleBox["the rate", FontSlant->"Italic"], " at which Env falls with the rise in ", StyleBox["A", "Input"], "." }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ShowLegend", "[", "\n", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Env", "/.", RowBox[{"{", RowBox[{\(A0 \[Rule] 1000\), ",", StyleBox[\(\[Beta] -> .7\), FontColor->RGBColor[0, 0, 1]]}], "}"}]}], ",", RowBox[{"Env", "/.", RowBox[{"{", RowBox[{\(A0 \[Rule] 1000\), ",", StyleBox[\(\[Beta] -> .8\), FontColor->RGBColor[1, 0, 0]]}], "}"}]}]}], "}"}], ",", \({A, 0, 999}\), ",", \(PlotStyle \[Rule] {{Dashing[{ .02, .02}], Hue[ .6]}, {Hue[ .9]}}\), ",", \(AxesLabel \[Rule] {"\", "\"}\), ",", \(DisplayFunction \[Rule] Identity\)}], 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The choice of range for ", StyleBox["\[Beta]", "Input"], " is arbitrary, unlike ", StyleBox["\[Alpha]", "Input"], " it need not be constrained [0,1]" }], "Text"], Cell[BoxData[ \(\(Plot3D[Env /. A0 \[Rule] 1000, {A, 0, 999}, {\[Beta], .1, .9}, AxesLabel \[Rule] {"\", "\< \[Beta]\>", "\"}, PlotRange \[Rule] All];\)\)], "Input", CellLabel->"In[16]:="], Cell[TextData[{ "The utility function describes the total utility that citizens receive \ from (1) municipal services and (2) environmental protection in the form of \ aesthetic regulation. The following expression describes that utility, ", StyleBox["U", "Input"], ", as a variant of a Stone-Geary production function where \[Alpha] is the \ citizens' preference for Environmental Regulation (0 < ", StyleBox["\[Alpha]", "Input"], " < 1):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(U = Env\^\[Alpha]\ M\^\((1 - \[Alpha])\)\)], "Input", CellLabel->"In[17]:=", FontSize->12], Cell[BoxData[ \(\((\((\(-A\) + A0)\)\^\[Beta])\)\^\[Alpha]\ \((A\^\[Gamma]\ q)\)\^\(1 - \ \[Alpha]\)\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell[TextData[{ "Note in the output above that the first term is Environmental Protection \ (", StyleBox["Env", "Input"], ")and the second term is Municipal Services (", StyleBox["M", "Input"], "). We wish to maximize this function. It is mathematically helpful and \ common practice to take the Log of both sides of the utility function. \ Maximizing the Log of the function also maximizes the function because the \ Log is monotonic and convex for all positive log bases:" }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(logu = PowerExpand[Log[U]]\)], "Input", CellLabel->"In[18]:=", FontSize->12], Cell[BoxData[ \(\[Alpha]\ \[Beta]\ Log[\(-A\) + A0] + \((1 - \[Alpha])\)\ \((\[Gamma]\ Log[A] + Log[q])\)\)], "Output", CellLabel->"Out[18]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Optimization and Comparative Statics", "Section"], Cell[TextData[{ "Comparative statics allows us to examine how the model output changes with \ changes in the inputs. This is accomplished by taking the partial derivative \ of the Logged function. Because Log[Utility] is monotonically increasing in \ Utility we will sometimes discuss the change in Utility even though it is the \ Log of Utility that we actually differentiate.\nThe partial derivative of \ utility ", StyleBox["with respect to tax rate", FontSlant->"Italic"], " describes how utility changes with changes in tax rate. Taking the \ partial derivative of Log[", StyleBox["U", "Input"], "] w.r.t. ", StyleBox["q", "Input"], " produces a positive sign, indicating that as tax rate rises, utility \ rises. This ignores the interplay between taxes and the level of sales (which \ is not our story. Therefore, we shall pursue it no further):" }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(solv0 = \[PartialD]\_q\ logu\)], "Input", CellLabel->"In[19]:=", FontSize->12], Cell[BoxData[ \(\(1 - \[Alpha]\)\/q\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell[TextData[{ "The ", "partial derivative of Utility ", StyleBox["with respect to Advertising", FontSlant->"Italic"], " is our real interest. This describes how utility changes with changes \ in advertising. Taking the partial derivative of Log[", StyleBox["U", "Input"], "] w.r.t. ", StyleBox["A", "Input"], " produces:" }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(\[PartialD]\_A\ logu\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(\(-\(\(\[Alpha]\ \[Beta]\)\/\(\(-A\) + A0\)\)\) + \(\((1 - \[Alpha])\)\ \[Gamma]\)\/A\)], "Output", CellLabel->"Out[20]="] }, Open ]], Cell["This expression can be made simpler for our later use", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(solv1 = FullSimplify[D[logu, A]]\)], "Input", CellLabel->"In[21]:=", FontSize->12], Cell[BoxData[ \(\(\[Alpha]\ \[Beta]\)\/\(A - A0\) + \(\[Gamma] - \[Alpha]\ \ \[Gamma]\)\/A\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell[TextData[{ "We want to know the value of ", StyleBox["A", "Input"], " at which the community achieves the optimal (most) utility. Setting the \ above equal to zero creates an implicit equation\n", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\[Beta]\/\(A - A\_0\) + \(\[Gamma] - \ \[Alpha]\[Gamma]\)\/A = 0\)], "Input"], "\nTransferring the second term on the left to the right hand side of the \ equation sets marginal cost equal to marginal benefit. We know that \ optimality is achieved in economic settings such as this when marginal cost \ equals marginal benefit.\n", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\[Beta]\/\(A - A\_0\) = \(\[Alpha]\[Gamma] - \ \[Gamma]\)\/A\)], "Input"], "\nReturning to the expression we named \"", StyleBox["solv1", "Input"], "\", setting it equal to zero and solving for optimum ", StyleBox["A", "Input"], " results in an unambiguous solution for ", Cell[BoxData[ FormBox[ StyleBox[\(A\^*\), "Input"], TraditionalForm]]], ", the optimal amount of allowed advertising:" }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(solv2 = Solve[solv1 == 0, A]; \(A\^*\) = A /. solv2[\([1]\)] // Simplify\)], "Input", CellLabel->"In[22]:=", FontSize->12], Cell[BoxData[ \(\(-\(\(A0\ \((\(-1\) + \[Alpha])\)\ \[Gamma]\)\/\(\[Alpha]\ \((\[Beta] \ - \[Gamma])\) + \[Gamma]\)\)\)\)], "Output", CellLabel->"Out[22]="] }, Open ]], Cell[TextData[{ "Remember the ordinal nature of utility. If we achieve an optimum this \ represents a \"peak\" utility, the highest possible. All change from that \ point must be in a direction resulting in diminished utility. Thus, to test \ the model for optimality, we are only interested in the sign of certain \ derivatives.\nComparative statics performed on this expression below have the \ correct sign. Increases in ", StyleBox["\[Gamma]", "Input"], ", the productivity of advertising, shifts the marginal benefit curve \ outward and results in increases in optimal advertising, as expected. \ Increases ", StyleBox["\[Beta]", "Input"], ", the disutility for advertising, shifts the marginal cost curve inward \ and results in decreases in optimal advertising (recall that 0 < ", StyleBox["\[Alpha]", "Input"], " < 1 is what makes the numerator in both expressions negative). These \ expressions constitute the dilemma the city finds itself in when it reduces \ the efficiency of advertising or experiences an increase in disutility for \ advertising." }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[\[PartialD]\_\[Gamma]\( A\^*\)]\)], "Input", CellLabel->"In[23]:=", FontSize->12], Cell[BoxData[ \(\(-\(\(A0\ \((\(-1\) + \[Alpha])\)\ \[Alpha]\ \[Beta]\)\/\((\[Alpha]\ \ \((\[Beta] - \[Gamma])\) + \[Gamma])\)\^2\)\)\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[\[PartialD]\_\[Beta]\( A\^*\)]\)], "Input", CellLabel->"In[24]:=", FontSize->12], Cell[BoxData[ \(\(A0\ \((\(-1\) + \[Alpha])\)\ \[Alpha]\ \[Gamma]\)\/\((\[Alpha]\ \((\ \[Beta] - \[Gamma])\) + \[Gamma])\)\^2\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell[TextData[{ "We also have an intuitive result for the derivative of ", Cell[BoxData[ FormBox[ StyleBox[\(A\^*\), "Input"], TraditionalForm]]], " w.r.t ", StyleBox["\[Alpha]", "Input"], ". Increases in ", StyleBox["\[Alpha]", "Input"], " represent an increase in the community's ", StyleBox["preference", FontSlant->"Italic"], " for environmental regulation over the other forms of municipal services. \ The negative derivative indicates that an increase in ", StyleBox["\[Alpha]", "Input"], " decreases advertising from the optimal." }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[D[\(A\^*\), \[Alpha]]]\)], "Input", CellLabel->"In[25]:=", FontSize->12], Cell[BoxData[ \(\(-\(\(A0\ \[Beta]\ \[Gamma]\)\/\((\[Alpha]\ \((\[Beta] - \[Gamma])\) + \ \[Gamma])\)\^2\)\)\)], "Output", CellLabel->"Out[25]="] }, Open ]], Cell["\<\ Below is the second derivative of Utility with respect to Advertising. As \ both numerators are negative, the second derivative is negative, indicating \ that we have a global optimum. (This is not surprising because all functions \ are strictly convex).\ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[{ \(\(solv3 = FullSimplify[D[logu, {A, 2}]];\)\), "\n", \(solv3 // TraditionalForm\)}], "Input", CellLabel->"In[26]:=", FontSize->12], Cell[BoxData[ \(TraditionalForm\`\(\((\[Alpha] - 1)\)\ \[Gamma]\)\/A\^2 - \(\[Alpha]\ \ \[Beta]\)\/\((A - A0)\)\^2\)], "Output", CellLabel->"Out[27]//TraditionalForm="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["A Graphic Illustration", "Section"], Cell["\<\ To create graphics that illustrate this process we define the marginal \ benefit and marginal cost as functions for plotting the marginal benefit and \ marginal cost curves for Advertising. We also provide some functions for \ formatting:\ \>", "Text", FontSize->12], Cell[BoxData[{ \(\(mb[ A_, \[Alpha]_, \[Gamma]_] = \((1 - \[Alpha])\)\ \[Gamma]/ A;\)\), "\n", \(\(mc[A_, \[Alpha]_, \[Beta]_, A0 : _] = \[Alpha]*\[Beta]/\((A0 - A)\);\)\), "\n", \(\(h[h_] := Hue[h*Pi/360];\)\), "\n", \(\(t[t_] := Thickness[t/100];\)\), "\n", \(\(dash[d_] := Dashing[{d, d}];\)\)}], "Input", CellLabel->"In[28]:=", FontSize->12], Cell["\<\ We can insert arbitrary values for dependent variables to produce numeric \ answers for marginal benefit and marginal cost. Keep in mind that these \ numbers don't mean anything until we have others for comparison.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(mb[ .8, .5, .9]\), "\n", \(\tmc[1, .5, 1\ , 100]\)}], "Input", CellLabel->"In[33]:=", FontSize->12], Cell[BoxData[ \(0.5625`\)], "Output", CellLabel->"Out[33]="], Cell[BoxData[ \(0.005050505050505051`\)], "Output", CellLabel->"Out[34]="] }, Open ]], Cell["\<\ Plots of marginal benefit and marginal cost against advertising indicate the \ functions have the expected classical shape\ \>", "Text"], Cell[BoxData[ \(\(Plot[mb[A, .5, 5], {A, 1, 100}, PlotRange -> {0, .1}, AxesLabel \[Rule] {"\", "\"}];\)\)], \ "Input", CellLabel->"In[35]:=", FontSize->12], Cell[BoxData[ \(\(Plot[mc[A, .5, 5, 100], {A, 10, 70}, AxesLabel \[Rule] {"\", "\"}];\)\)], \ "Input", CellLabel->"In[36]:=", FontSize->12], Cell[BoxData[{ \(mb1 = mb[A, .5, 20]; mc1 = mc[A, .5, 25, 100];\), "\n", \(\(p1 = Plot[{mb1, mc1}, {A, 1, 99.9}, PlotStyle -> {{h[90], t[ .5]}, {h[225], t[1.1]}}, FrameLabel -> {Advertising, "\"}, FrameTicks -> Automatic, PlotRange -> {0, 1}, Frame \[Rule] {True, True, False, False}, PlotStyle -> AbsoluteThickness[4], DisplayFunction \[Rule] Identity];\)\)}], "Input", CellLabel->"In[37]:=", FontSize->12], Cell["\<\ The intersection of marginal cost and marginal benefits curves marks the \ optimal Advertising, which in turn maximizes utility for the community.\ \>", "Text", FontSize->12], Cell[BoxData[{ \(\(line1 = Solve[mb1 == mc1, A];\)\), "\n", \(\(L1 = A /. line1[\([1]\)];\)\), "\n", \(\t\(L2 = mc1 /. A -> L1;\)\), "\n", \(p2 = Graphics[{dash[ .015], Line[{{L1, 0}, {L1, L2}}]}]; p3 = Show[{Graphics[p1], Graphics[p2]}, FrameTicks -> {{{L1, "\"}}, None}, PlotLabel -> "\", FrameLabel -> {None, "\"}, DisplayFunction \[Rule] $DisplayFunction];\)}], "Input", CellLabel->"In[39]:=", FontSize->12], Cell[TextData[{ "The following shows how an increase in ", StyleBox["\[Gamma]", "Input", FontColor->RGBColor[1, 0, 0]], " moves the marginal benefit function out and an increase in ", StyleBox["\[Beta]", "Input", FontColor->RGBColor[0, 0, 1]], " moves the marginal cost function in (dotted lines represent the new \ functions). " }], "Text", FontSize->12], Cell[BoxData[{ RowBox[{ RowBox[{"mb2", "=", RowBox[{"mb", "[", RowBox[{"A", ",", ".5", ",", StyleBox["20", FontColor->RGBColor[1, 0, 0]]}], "]"}]}], ";", RowBox[{"mb3", "=", RowBox[{"mb", "[", RowBox[{"A", ",", ".5", ",", StyleBox["10", FontColor->RGBColor[1, 0, 0]]}], "]"}]}], ";"}], "\n", RowBox[{ RowBox[{"mc2", "=", RowBox[{"mc", "[", RowBox[{"A", ",", ".5", ",", StyleBox["25", FontColor->RGBColor[0, 0, 1]], ",", "100"}], "]"}]}], ";", RowBox[{"mc3", "=", RowBox[{"mc", "[", RowBox[{"A", ",", ".5", ",", StyleBox["35", FontColor->RGBColor[0, 0, 1]], ",", "100"}], "]"}]}], ";"}], "\n", \(p4 = Plot[{mb2, mb3, mc2, mc3}, {A, 1, 90}, PlotStyle -> {{h[90], t[ .4], dash[1]}, {h[90], t[ .1], dash[ .009]}, {h[225], t[ .9]}, {h[225], t[ .1], dash[ .009]}}, FrameLabel -> {Advertising, "\"}, FrameTicks -> None, PlotRange -> {0, 1}, Frame \[Rule] {True, True, False, False}, DisplayFunction \[Rule] Identity];\)}], "Input", CellLabel->"In[43]:=", FontSize->12], Cell["\<\ The cells below contain the calculations which set the endpoints of the \ dotted lines on the following graphic\ \>", "Text", FontSize->12], Cell[BoxData[{ \(\(line2 = Solve[mb2 == mc2, A];\)\), "\n", \(\(L3 = A /. line2[\([1]\)];\)\), "\n", \(\t\(L4 = mc2 /. A -> L3;\)\)}], "Input", CellLabel->"In[46]:=", FontSize->12], Cell[BoxData[{ \(\(line5 = Solve[mb3 == mc3, A];\)\), "\n", \(\(L6 = A /. line5[\([1]\)];\)\), "\n", \(\t\(L7 = mc3 /. A -> L6;\)\)}], "Input", CellLabel->"In[49]:=", FontSize->12], Cell[BoxData[{ \(\(line6 = Solve[mb2 == mc3, A];\)\), "\n", \(\(L8 = A /. line6[\([1]\)];\)\), "\n", \(\t\(L9 = mc3 /. A -> L8;\)\)}], "Input", CellLabel->"In[52]:=", FontSize->12], Cell[BoxData[{ \(\(line10 = Solve[mb3 == mc2, A];\)\), "\n", \(\(L10 = A /. line10[\([1]\)];\)\), "\n", \(\t\(L11 = mc2 /. A -> L10;\)\)}], "Input", CellLabel->"In[55]:=", FontSize->12], Cell[BoxData[{ \(p5 = Graphics[{dash[ .015], Line[{{L3, 0}, {L3, L4}}]}]; p6 = Graphics[{dash[ .015], Line[{{L6, 0}, {L6, L7}}]}]; p7 = Graphics[{dash[ .015], Line[{{L8, 0}, {L8, L9}}]}]; p8 = Graphics[{dash[ .015], Line[{{L10, 0}, {L10, L11}}]}];\), "\n", \(\(p9 = Show[{Graphics[p4], Graphics[p5], Graphics[p7], Graphics[p6], Graphics[p8]}, PlotLabel -> "\", FrameTicks -> {{{L8, "\"}, {L10, "\"}, {L3, "\"}, {L6, \ "\"}}, None}, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 350];\)\)}], "Input", CellLabel->"In[58]:=", FontSize->12], Cell[TextData[{ StyleBox["A*", "Input"], " is optimal advertising where Utility is maximized. B is the result of \ increases in ", StyleBox["\[Beta] ", "Input"], "(such as election of a city council member hostile to business) moving the \ marginal cost curve inward while marginal benefit curve does not change. C \ is a reduction of ", StyleBox["\[Gamma]", "Input"], " (resulting from a vote of the city council to increase regulations by \ reducing sign size, height, etc.) leading to a downward shift of the marginal \ benefit curve with the marginal cost curve unchanged. D is the most drastic \ result (the new city council member influences an even more draconian level \ of regulation) where \[Beta] is increased ", StyleBox["and", FontSlant->"Italic"], " ", StyleBox["\[Gamma]", "Input"], " is lowered at the same time. At D allowed advertising is the farthest \ from optimal, thus utility is the lowest of the four." }], "Text", FontSize->12], Cell[TextData[{ "Below this cell we show the changes separately. First show the effect \ when the city lowers ", StyleBox["\[Gamma]", "Input"], " and therefore the marginal benefit of advertising falls." }], "Text", FontSize->12], Cell[BoxData[{ \(\(p4a = Plot[{mb2, mb3, mc2}, {A, 1, 90}, PlotStyle -> {{h[90], t[ .4], dash[1]}, {h[90], t[ .1], dash[ .009]}, {h[225], t[ .9]}}, FrameLabel -> {None, "\"}, FrameTicks -> None, PlotRange -> {0, 1}, Frame \[Rule] {True, True, False, False}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p10 = Show[{Graphics[p4a], Graphics[p5], Graphics[p8]}, PlotLabel -> "\ Sub-Optimal Conditions\>", FrameTicks -> {{{L10, "\"}, {L3, "\"}}, None}, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 350];\)\)}], "Input", CellLabel->"In[60]:=", FontSize->12], Cell[TextData[{ "Now to show the effect of raising ", StyleBox["\[Beta]", "Input"], ":" }], "Text", FontSize->12], Cell[BoxData[{ \(\(p4b = Plot[{mb2, mc2, mc3}, {A, 1, 90}, PlotStyle -> {{h[90], t[ .4], dash[1]}, {h[225], t[ .9]}, {h[225], t[ .1], dash[ .009]}}, FrameLabel -> {None, "\"}, FrameTicks -> None, PlotRange -> {0, 1}, Frame \[Rule] {True, True, False, False}, DisplayFunction \[Rule] Identity];\)\), "\n", \(\(p11 = Show[{Graphics[p4b], Graphics[p5], Graphics[p7]}, PlotLabel -> "\ Sub-Optimal Conditions\>", FrameTicks -> {{{L8, "\"}, {L3, "\"}}, None}, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 350];\)\)}], "Input", CellLabel->"In[62]:=", FontSize->12], Cell["\<\ Recalling that the optimum is the \"peak\", we illustrate this by making \ Utility a function of Advertising.\ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(u[A_, \[Alpha]_, \[Beta]_, \[Gamma]_, A0 : _, q_] = U\)], "Input", CellLabel->"In[64]:=", FontSize->12], Cell[BoxData[ \(\((\((\(-A\) + A0)\)\^\[Beta])\)\^\[Alpha]\ \((A\^\[Gamma]\ q)\)\^\(1 - \ \[Alpha]\)\)], "Output", CellLabel->"Out[64]="] }, Open ]], Cell[TextData[{ "As mentioned previously, regulation of commercial signage usually comes in \ the form of reducing some physical aspect of it. We can view that as \ modifying the sign ordinance so as to improve public vistas (", StyleBox["Env", "Input"], ") at the expense of the efficiency (", StyleBox["\[Gamma]", "Input"], ") of advertising. Thus, advertising is implicitly limited by reducing ", StyleBox["\[Gamma]", "Input"], ". Here is a set of arbitrary values for all variables that produce a \ numeric ", StyleBox["A*", "Input"], ". Note the value for ", StyleBox["\[Gamma]", "Input"], "." }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sampleA", "=", RowBox[{\(A\^*\), "/.", RowBox[{"{", RowBox[{\(\[Alpha] -> .5\), ",", \(\[Beta] -> 4.1\), ",", StyleBox[\(\[Gamma] -> 3.1\), FontColor->RGBColor[1, 0, 0]], ",", \(A0 -> 100\)}], "}"}]}]}]], "Input", CellLabel->"In[65]:=", FontSize->12], Cell[BoxData[ \(43.05555555555556`\)], "Output", CellLabel->"Out[65]="] }, Open ]], Cell["\<\ Using those same variables in the Utility Function, produces the following \ value for Utility, a number that will gain meaning when compared with an \ alternative.\ \>", "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L12", "=", RowBox[{"u", "[", RowBox[{"sampleA", ",", ".5", ",", "4.1", ",", StyleBox["3.1", FontColor->RGBColor[1, 0, 0]], ",", "100", ",", ".07"}], "]"}]}]], "Input", CellLabel->"In[66]:=", FontSize->12], Cell[BoxData[ \(358071.3425262896`\)], "Output", CellLabel->"Out[66]="] }, Open ]], Cell["\<\ Locating this number on the plot shows that indeed utility peaks at that \ value. Notice the importance of domain and range values with changes in \ parameters. Remember also that the actual values have no meaning except in \ reference to other values calculated the same way. The importance of the \ general model is that it achieves an optimum for all combinations of \ numerical values given the parameters. What we are interested in is what \ happens when equilibrium is disturbed. Assume you are considering a certain \ community for locating your business. You find the present condition \ (equilibrium for our purposes) of sign regulation as plotted above. How does \ a change in the political landscape change your decision to locate? How does \ it change the fortunes of market participants? How does that change of \ fortune affect other business owners' decision to locate in the community? \ Taking aesthetic regulation as just one example of the restrictions on \ freedom of choice imposed by government, what would you expect the aggregate \ effect of numerous restrictions to be?\ \>", "Text", FontSize->12], Cell[BoxData[{\(p9a = Graphics[{dash[ .015], Line[{{sampleA, 0}, {sampleA, L12}}]}];\), "\n", \(p9b = Graphics[{dash[ .015], Line[{{0, L12}, {sampleA, L12}}]}];\), "\n", RowBox[{ RowBox[{"p12", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"A", ",", ".5", ",", "4.1", ",", StyleBox["3.1", FontColor->RGBColor[1, 0, 0]], ",", "100", ",", ".07"}], "]"}], ",", \({A, 1, 100}\), ",", \(FrameLabel -> {Advertising, "\"}\), ",", \(PlotRange -> {0, 400000}\), ",", \(PlotStyle -> {h[150], t[ .4]}\), ",", \(Ticks \[Rule] {{sampleA}, {L12}}\), ",", \(AxesLabel \[Rule] {"\", "\"}\), ",", \(DisplayFunction \[Rule] Identity\)}], "]"}]}], ";", \(Show[{Graphics[p12], Graphics[p9a], Graphics[p9b]}, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 450]\), ";"}]}], "Input", CellLabel->"In[67]:="], Cell[TextData[{ "A reduction in the value of ", StyleBox["\[Gamma]", "Input"], " results in a reduction in both advertising and, as expected, utility." }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"subA", "=", RowBox[{\(A\^*\), "/.", RowBox[{"{", RowBox[{\(\[Alpha] -> .5\), ",", \(\[Beta] -> 4.1\), ",", StyleBox[\(\[Gamma] -> 2.3\), FontColor->RGBColor[1, 0, 0]], ",", \(A0 -> 100\)}], "}"}]}]}]], "Input", CellLabel->"In[70]:=", FontSize->12], Cell[BoxData[ \(35.93749999999999`\)], "Output", CellLabel->"Out[70]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"L13", "=", RowBox[{"u", "[", RowBox[{"subA", ",", ".5", ",", "4.1", ",", StyleBox["2.3", FontColor->RGBColor[1, 0, 0]], ",", "100", ",", ".07"}], "]"}]}]], "Input", CellLabel->"In[71]:=", FontSize->12], Cell[BoxData[ \(82218.02136292354`\)], "Output", CellLabel->"Out[71]="] }, Open ]], Cell[BoxData[{\(p12a = Graphics[{dash[ .015], Line[{{subA, 0}, {subA, L13}}]}];\), "\n", \(p12b = Graphics[{dash[ .015], Line[{{0, L13}, {subA, L13}}]}];\), "\n", RowBox[{ RowBox[{"p13", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"A", ",", ".5", ",", "4.1", ",", StyleBox["2.3", FontColor->RGBColor[1, 0, 0]], ",", "100", ",", ".07"}], "]"}], ",", \({A, 1, 100}\), ",", \(FrameLabel -> {Advertising, "\"}\), ",", \(PlotRange -> {0, 400000}\), ",", \(PlotStyle -> {h[220], t[ .9]}\), ",", \(Ticks \[Rule] {{subA}, {L13}}\), ",", \(DisplayFunction \[Rule] Identity\)}], "]"}]}], ";", \(Show[{Graphics[p13], Graphics[p12a], Graphics[p12b]}, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 450]\), ";"}]}], "Input", CellLabel->"In[72]:="], Cell[TextData[{ "Combining the last two plots shows the cost, in terms of lost utility, of \ reducing the effectiveness of advertising,", StyleBox[" \[Gamma]", "Input"], ". It is this argument that may persuade the one vote an investor needs \ from the local council. If the vote is close and swing vote is rational this \ argument may only need to ring true with that one member. " }], "Text"], Cell[BoxData[{ \(\($TextStyle\ = \ {FontFamily\ -> \ "\", \ FontSize\ -> \ 12};\)\), "\n", \(\(p14 = Show[{Graphics[p12], Graphics[p13]}, {Graphics[p9a], Graphics[p12a]}, FrameLabel -> {"\< \ Advertising\>", "\"}, PlotLabel -> "\", Frame \[Rule] {True, True, False, False}, FrameTicks -> {{{sampleA, "\"}, {subA, "\"}}, None}, \n\ \ \ \ Epilog\ -> \ {Arrow[{80, \ 210000}, \ {61, 210000}, HeadCenter -> .5], \n\ \ \ \ \ Text["\", \ \ {90, \ 230000}]}, DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 350];\)\)}], "Input", CellLabel->"In[75]:=", FontSize->12], Cell["\<\ Double clicking on the next graphic below will demonstrate how utility falls \ continuously with reduction in permitted advertising from optimal \ advertising.\ \>", "Text"], Cell[BoxData[{ \(\(pix = Animate[Show[{Plot[{u[A, .5, 4.1, \[Gamma], 100, .07], u[A, .5, 4.1, 3.1, 100, .07]}, {A, 1, 100}, FrameLabel -> {"\< Advertising\>", \ "\"}, PlotRange -> {0, 400000}, PlotStyle -> {{h[220], t[ .9]}, {h[150], t[ .9]}}, Frame -> {True, True, False, False}, PlotLabel -> "\", FrameTicks -> {{{sampleA, "\"}, {\(A\^*\) /. {\[Alpha] -> \ .5, \[Beta] -> 4.1, \[Gamma] -> \[Gamma], A0 -> 100}, "\"}}, None}]}, {Graphics[p9a], Graphics[{dash[ .015], Line[{{\(A\^*\) /. {\[Alpha] -> .5, \[Beta] -> 4.1, \[Gamma] -> \[Gamma], A0 -> 100}, 0}, {\(A\^*\) /. {\[Alpha] -> .5, \[Beta] -> 4.1, \[Gamma] -> \[Gamma], A0 -> 100}, u[\(A\^*\) /. {\[Alpha] -> .5, \[Beta] -> 4.1, \[Gamma] -> \[Gamma], A0 -> 100}, .5, 4.1, \[Gamma], 100, .07]}}]}]}, Frame \[Rule] {True, True, False, False}], {\[Gamma], 3.1, .00001}];\)\), "\[IndentingNewLine]", \(\(SelectionMove[EvaluationNotebook[], After, EvaluationCell];\)\), "\[IndentingNewLine]", \(\(SelectionMove[EvaluationNotebook[], Next, CellGroup];\)\), "\[IndentingNewLine]", \(FrontEndTokenExecute["\"]\)}], "Input", CellLabel->"In[77]:=", FontSize->12], Cell[TextData[StyleBox["The animation below is a simplified version of the \ one above.", "Text"]], "Text", FontSize->12], Cell[BoxData[{ \(\(Animate[ Plot[{u[A, .5, 4.1, \[Gamma], 100, .07], u[A, .5, 4.1, 3.1, 100, .07]}, {A, 1, 100}, FrameLabel -> {Advertising, "\"}, PlotRange -> {0, 400000}, PlotStyle -> {{h[220], t[ .9]}, {h[150], t[ .9]}}, Frame -> {True, True, False, False}, FrameTicks -> {None, None}], {\[Gamma], 3.1, 2}];\)\), "\[IndentingNewLine]", \(\(SelectionMove[EvaluationNotebook[], After, EvaluationCell];\)\), "\[IndentingNewLine]", \(\(SelectionMove[EvaluationNotebook[], Next, CellGroup];\)\), "\[IndentingNewLine]", \(FrontEndTokenExecute["\"]\)}], "Input", CellLabel->"In[81]:=", FontSize->12], Cell[TextData[{ "Utility, ", StyleBox["U", "Input"], ", changes with the change in allowed advertising, ", StyleBox["A", "Input"], ". But the efficiency, \n", StyleBox["\[Gamma]", "Input"], ", of advertising also affects utility. If those were the only variables, \ ", StyleBox["Mathematica", FontSlant->"Italic"], " does a good job illustrating the problem in three dimensions. But allowed \ advertising is affected by ", StyleBox["community preference", FontSlant->"Italic"], " for ", StyleBox["Env", "Input"], ", ", StyleBox["\[Beta]", "Input"], ", which introduces a fourth dimension. As we cannot illustrate more than \ three dimensions, we break them up into two separate 3-D graphics.\nBelow is \ a 3D graphic displaying the change in Utility with the changes in ", StyleBox["A", "Input"], " and \n", StyleBox["\[Gamma]", "Input"], ".We see that the effect on utility of a change in allowed advertising is \ greatest when the efficiency is highest. This is reasonable as the merchants \ lose more and tax revenue falls more." }], "Text", FontSize->12], Cell[BoxData[ \(\(p19 = Plot3D[u[A, .5, 5, \[Gamma], 100, .07], {A, 0, 100}, {\[Gamma], .0001, 5}, AxesLabel -> {A, \[Gamma], "\"}, Ticks \[Rule] {Automatic, Automatic, None}, Mesh \[Rule] True, MeshStyle \[Rule] Automatic, ViewPoint \[Rule] {1.3, \(-2.4\), 2. }, ImageSize \[Rule] 350];\)\)], "Input", CellLabel->"In[85]:=", FontSize->12], Cell["\<\ We can rotate our plot (double click on the graphic below) to get a better \ view of the changes from different viewpoints.\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"p20", "=", StyleBox[\(Table[ Plot3D[u[A, .5, 5, \[Gamma], 100, .07], {A, 0, 100}, {\[Gamma], .0001, 5}, \ PlotPoints\ -> \ 25, \ {ViewPoint\ -> \ {5\ Cos[turn], \ 5\ \ Sin[turn], \ 2. }}, \ \ SphericalRegion\ -> \ True, \ \ Axes\ -> \ False\ ]\ , {turn, \ 0.0, \ 2\ Pi, \ .1}]\), FormatType->StandardForm, FontFamily->"Courier New", FontSize->10, CharacterEncoding->"WindowsANSI"]}], StyleBox[";", FormatType->StandardForm, FontFamily->"Courier New", FontSize->10, CharacterEncoding->"WindowsANSI"]}], "\n", \(SelectionMove[ EvaluationNotebook[], After, EvaluationCell];\), "\[IndentingNewLine]", \(SelectionMove[ EvaluationNotebook[], Next, CellGroup];\), "\[IndentingNewLine]", \ \(FrontEndTokenExecute["\"]\), "\[IndentingNewLine]", "Null"}], "Input", CellLabel->"In[86]:="], Cell["\<\ We now show the change in Utility with the changes in A and \[Beta]. We note \ that when \[Beta] is small there is little change in utility with changes in \ A. \ \>", "Text", FontSize->12], Cell[BoxData[ \(\(p21 = Plot3D[u[A, .5, \[Beta], 3.1, 100, .07], {A, 0, 100}, {\[Beta], .0001, 5}, AxesLabel -> {A, \[Beta], "\"}, Ticks \[Rule] {Automatic, Automatic, None}, Mesh \[Rule] True, MeshStyle \[Rule] Automatic, ViewPoint \[Rule] {1.3, \(-2.4\), 2. }, ImageSize \[Rule] 350];\)\)], "Input", CellLabel->"In[91]:=", FontSize->12], Cell["Rotating this plot:", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"p22", "=", StyleBox[\(Table[ Plot3D[u[A, .5, \[Beta], 3.1, 100, .07], {A, 0, 100}, {\[Beta], .0001, 5}, \ PlotPoints\ -> \ 25, \ {ViewPoint\ -> \ {5\ Cos[turn], \ 5\ \ Sin[turn], \ 2. }}, \ \ SphericalRegion\ -> \ True, \ \ Axes\ -> \ False\ ]\ , {turn, \ 0.0, \ 2\ Pi, \ .1}]\), FormatType->StandardForm, FontFamily->"Courier New", FontSize->10, CharacterEncoding->"WindowsANSI"]}], StyleBox[";", FormatType->StandardForm, FontFamily->"Courier New", FontSize->10, CharacterEncoding->"WindowsANSI"]}], "\n", \(SelectionMove[ EvaluationNotebook[], After, EvaluationCell];\), "\[IndentingNewLine]", \(SelectionMove[ EvaluationNotebook[], Next, CellGroup];\), "\[IndentingNewLine]", \ \(FrontEndTokenExecute["\"]\), "\[IndentingNewLine]", "Null"}], "Input", CellLabel->"In[92]:="] }, Open ]], Cell[CellGroupData[{ Cell["Conclusion", "Section"], Cell["The implications of this exercise should be clear:", "Text"], Cell[TextData[{ " ", StyleBox[ CounterBox["ItemizedText"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " People make decisions on the margins. Marginal analysis is a very \ powerful tool for measuring the net effect of a trade-off between two \ alternatives. Many, if not all, economic choices between two alternatives may \ be modeled on a cost-benefit basis provided one makes plausible assumptions \ about how people generate well-being, happiness or utility." }], "ItemizedText"], Cell[TextData[{ " ", StyleBox[ CounterBox["ItemizedText"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Any item on the list of development constraints mentioned in the \ introduction to this chapter could be substituted for the one illustrated \ here. The aggregate of all such constraints, if applied by a heavy handed \ legislative body, can operate as a strong disincentive to entrepreneurial \ activity in a community." }], "ItemizedText"], Cell[TextData[{ " ", StyleBox[ CounterBox["ItemizedText"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Arguments for change and arguments for preservation are often equally \ persuasive, especially when couched in an emotional framework. 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