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foreground="[0,0,255]" name="Warning" readonly="true" size="10"/><Font background="[0,0,0]" italic="true" name="_cstyle263"/><Font background="[0,0,0]" name="Normal"/><Font background="[0,0,0]" italic="true" name="_cstyle262"/><Font background="[0,0,0]" bold="true" executable="false" family="Times New Roman" foreground="[0,0,0]" italic="true" name="Normal256" readonly="false" size="12" underline="false"/><Font background="[0,0,0]" italic="true" name="_cstyle261"/><Font background="[0,0,0]" italic="true" name="_cstyle260"/><Font background="[0,0,0]" italic="true" name="_cstyle259"/><Font background="[0,0,0]" italic="true" name="_cstyle258"/></Styles><Page-Numbers enabled="false" first-number="1" first-numbered-page="1" horizontal-location="right" style="Page Number" vertical-location="bottom"/><Section><Title><Text-field layout="Heading 2" style="Heading 2"><Font encoding="ISO8859-1" family="Times New Roman"> 1.2 El Modelo B\341sico</Font></Text-field></Title><Section collapsed="true"><Title><Text-field layout="Heading 3" style="Heading 3"><Font encoding="ISO8859-1" family="Times New Roman">1.2.1. Los Agentes Econ\363micos.</Font></Text-field></Title><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">En esta secci\363n hablaremos del ejemplo m\341s sencillo de un modelo de equilibrio general en el que a\372n no aparece el gobierno. 
Como a lo largo del libro vamos a referirnos a temas Macroecon\363micos,  cuando hablemos de bienes, factores o agentes econ\363micos lo estaremos haciendo en forma agregada.  Es decir,  hablaremos de un "grupo de individuos" en vez de a un solo individuo. Igualmente, estaremos pensando en un </Font><Font family="Times New Roman" style="_cstyle261">grupo de empresas</Font><Font family="Times New Roman">, en lugar de una sola empresa. Asimismo, tendremos <Font style="_cstyle262">un solo bien</Font>, un producto y/o servicio que bien puede identificarse en abstracto con el Producto Interno Bruto o con una canasta de bienes.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Por lo general, una vez que tenemos al grupo de consumidores, la Macroeconom\355a Neocl\341sica los tratar\341 como la juxtaposici\363n de <Font style="_cstyle258"> individuos id\351nticos.</Font>Cabe se\361alar que esta agregaci\363n no presupone <Font style="_cstyle259">colusi\363n.</Font> En efecto, estaremos sumando las decisiones de individuos que concurren a los mercados sin tener ning\372n tipo de poder monop\363lico o monops\363nico. As\355 pues, primero se analiza la respuesta de un individuo representativo y atom\355sticamente peque\361o,  y despu\351s se construye la oferta o demanda agregada correspondiente.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman"> Lo mismo sucede del lado de las empresas. Es pr\341ctica com\372n en la formalizaci\363n del Modelo Neocl\341sico, se agregan muchas peque\361as empresas para llegar a las funciones de producci\363n, costos y demanda de factores agregadas as\355 como una oferta agregada de bienes. En cualquier caso, tanto productores como consumidores se comportar\341n como </Font><Font family="Times New Roman" style="_cstyle260">tomadores de precios</Font><Font family="Times New Roman">.</Font></Text-field><Text-field layout="Normal" style="Normal"/></Section><Section collapsed="true"><Title><Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman">1.2.2. Comportamiento de las familias</Font></Text-field></Title><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Iniciemos la construcci\363n del modelo b\341sico con la descripci\363n de lo que sucede con el sector de las familias en una econom\355a dada (pa\355s, regi\363n, etc). Las familias est\341n compuestas  por personas que no s\363lo consumen, sino que ofrecen su mano de obra a cambio de la cual obtienen un ingreso. M\341s a\372n, tambi\351n como parte del sector familias se encuentran las personas que son due\361as de las empresas y cuyos ingresos provienen directamente de las ganancias que tales empresas generan.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">Supongamos que el sector familias ( <Font style="_cstyle263">familia representativa</Font><Font encoding="ISO8859-1">) tiene una funci\363n de utilidad que depende, positivamente, de la cantidad de bienes que consume, </Font><Font style="_cstyle264">C</Font>, es decir, del consumo agregado o consumo nacional y negativamente, del trabajo realizado, <Font style="_cstyle265">T</Font><Font encoding="ISO8859-1">. En otras palabras,  el bienestar de las familias se incrementa al aumentar el tiempo dedicado al ocio y disminuye en la medida en que m\341s tiempo se destine a trabajar. 
Podemos ilustrar lo anterior mediante una expresi\363n como la que aparece enseguida. Se trata de una funci\363n de utilidad se conoce como Cobb-Douglass donde </Font></Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman">  tomar\341 valores entre cero y uno a lo largo del ejercicio.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">restart;</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">U:=(C,T)-&gt; ln((C^alpha)*(H-T)^(1-alpha));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSJVR2YqNiQlIkNHJSJURzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHRiktJSNsbkc2IyomKTkkJSZhbHBoYUciIiIpLCYlIkhHRjQ5JSEiIiwmRjRGNEYzRjlGNEYpRilGKQ==</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">La gr\341fica que aparece a continuaci\363n nos muestra lo que sucede con el nivel de bienestar de las familias para distintos valores del consumo (entre 0 y 100 unidades), del trabajo (entre 0 y 24 unidades) y del par\341metro de preferencias </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman"> (entre 0.1 y 0.9).  Ciertamente no es posible ver el la relaci\363n entre estas  cuatro variables (</Font><Font family="Times New Roman" style="_cstyle266">U, C, T</Font><Font family="Times New Roman"> y </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman">) en una sola gr\341fica. Por esta raz\363n se opt\363 por  hacer 10 gr\341ficas que muestran la relaci\363n entre </Font><Font family="Times New Roman" style="_cstyle267">U,C</Font><Font family="Times New Roman"> y <Font style="_cstyle268">T</Font> para 10 distintos valores de </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman">. M\341s a\372n, si estas diez gr\341ficas se presentan una despu\351s de otra tendremos algo parecido a lo que ocurre cuando presentamos varios dibujos en secuencia: es decir,  una animaci\363n. Dicha animaci\363n puede activarse mediante el posicionamiento del cursor sobre la gr\341fica</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">with(plots):
H:=24:
animate3d(U(C,T),C=0..100,T=0..24,alpha=0.1..0.9,
axes=normal,
frames=10,
style=patchcontour,
shading=zhue,
orientation=[-150,40],
style=patchcontour,
title=`Funcion de Utilidad (animada) `,titlefont=[TIMES,BOLDITALIC,12], axesfont=[TIMES,ITALIC,8],labelfont=[TIMES,BOLDITALIC,9]); 
H:='H':</Text-field></Input><Output><Text-field layout="Warning" style="Warning">Warning, the name changecoords has been redefined
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layout="Normal" style="Normal"><Font family="Times New Roman"> </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Conviene mencionar que  las curvas de nivel son c\363ncavas hacia abajo. Esto es debido a que decidimos graficar trabajo, </Font><Font family="Times New Roman" style="_cstyle269">T</Font><Font family="Times New Roman">, contra consumo <Font style="_cstyle270">C</Font> . Si en lugar de <Font style="_cstyle271">T</Font><Font encoding="ISO8859-1">  hubi\351ramos graficado el ocio </Font><Font style="_cstyle272">(H-T) </Font>contra consumo<Font style="_cstyle275"> C, </Font><Font encoding="ISO8859-1">observar\355amos la forma convencional de las curvas de indiferencia. Por otro lado, y como es de esperarse, a medida que </Font></Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman"> toma valores m\341s cercanos a la unidad, las curvas de nivel toman la forma de l\355neas rectas paralelas al eje de la variable </Font><Font family="Times New Roman" style="_cstyle273">T</Font><Font family="Times New Roman">. </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Supongamos ahora que los individuos que nos ocupan son racionales. Lo anterior significa que buscan siempre alcanzar el m\341ximo bienestar, dadas sus restricciones. En este caso su restricci\363n est\341 dada por el hecho de que para poder comprar bienes, deben contar con un ingreso. Este puede provenir ya sea de su trabajo o de las ganancias que obtengan las empresas en que tengan alguna participaci\363n accionaria. En este modelo el ingreso familiar, pues, depender\341 de tres variables: del salario por hora, del n\372mero de horas trabajadas y de las ganancias que reciban las familias.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">En virtud de que supusimos que las  familias eran "tomadoras de precios", el nivel de salario por hora de trabajo, <Font style="_cstyle274">w, </Font><Font encoding="ISO8859-1"> es visto como una variable ex\363gena, es decir, sobre la cual no tienen control alguno. Lo mismo sucede con la variable ganancias, </Font></Font><Equation input-equation="pi;" style="2D Comment">NiMlI3BpRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman">. De esta forma, la \372nica manera en que las familias pueden controlar por s\355 mismas el nivel de ingreso que reciben es mediante la determinaci\363n del n\372mero de horas que est\351n dispuestas a trabajar. En s\355ntesis, dados los salarios y las ganancias, mientras m\341s horas se trabajen m\341s ingreso puede alcanzarse y m\341s puede consumirse, y viceversa.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">R:=(w,pi,T)-&gt;w*T+pi;</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSJSR2YqNiUlIndHJSNwaUclIlRHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKiwmKiY5JCIiIjkmRjFGMTklRjFGKkYqRio=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"><Font family="Times New Roman"> </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Para encontrar la soluci\363n al problema de optimizaci\363n de las familias, resultar\341 conveniente definir la siguiente funci\363n Lagrangiana:</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">Lambda:=(C,T,lambda)-&gt;U(C,T)+lambda*(C-R(w,pi,T));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSdMYW1iZGFHZio2JSUiQ0clIlRHJSdsYW1iZGFHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKiwmLSUiVUc2JDkkOSUiIiIqJjkmRjQsJkYyRjQtJSJSRzYlJSJ3RyUjcGlHRjMhIiJGNEY0RipGKkYq</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">    </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Es sabido que las condiciones de primer orden de Kuhn-Tucker para un extremo (m\341ximo o m\355nimo)  se cumplen  cuando, simult\341neamente, las derivadas parciales de la funci\363n Lagrangiana respecto de las variables de control (</Font><Font family="Times New Roman" style="_cstyle277">C</Font><Font family="Times New Roman"> y <Font style="_cstyle278">T</Font>) y respecto del multiplicador de Lagrange (</Font><Equation input-equation="lambda;" style="2D Comment">NiMlJ2xhbWJkYUc=</Equation><Font family="Times New Roman">) se igualan a cero:</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">lc:=(C,T,lambda)-&gt;diff(Lambda(C,T,lambda),C):
lt:=(C,T,lambda)-&gt;diff(Lambda(C,T,lambda),T):
ll:=(C,T,lambda)-&gt;diff(Lambda(C,T,lambda),lambda):</Text-field></Input></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">A continuaci\363n, y s\363lo con fines informativos, se presentan las expresiones correspondientes a las referidas derivadas parciales.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">lc(C,T,lambda);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMsJiomJSZhbHBoYUciIiIlIkNHISIiRiYlJ2xhbWJkYUdGJg==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">lt(C,T,lambda);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMsJiomLCYiIiJGJiUmYWxwaGFHISIiRiYsJiUiSEdGJiUiVEdGKEYoRigqJiUnbGFtYmRhR0YmJSJ3R0YmRig=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">ll(C,T,lambda);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMsKCUiQ0ciIiIqJiUid0dGJSUiVEdGJSEiIiUjcGlHRik=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">solucion_familias:=solve({lc(C,T,lambda)=0,
lt(C,T,lambda)=0,
ll(C,T,lambda)=0},
{C,T,lambda}):
assign(solucion_familias);</Text-field></Input></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">Antes de seguir adelante conviene recordar que estamos tratando de encontrar la forma en que los individuos determinan su consumo y la cantidad de trabajo que desean ofrecer en el mercado de trabajo, dados los salarios vigentes en el mercado y las ganancias que esperan recibir. Es decir, queremos obtener una <Font style="_cstyle279">demanda de bienes</Font> y una <Font style="_cstyle280">oferta de trabajo</Font> a partir del comportamiento optimizador de las familias.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Las soluci\363n simult\341nea de las  </Font><Font family="Times New Roman" style="_cstyle281">condiciones de primer orden</Font><Font encoding="ISO8859-1" family="Times New Roman"> del ejercicio de optimizaci\363n,  resulta en las expresiones correspondientes a la funci\363n consumo, </Font><Font family="Times New Roman" style="_cstyle282">c[d]</Font><Font family="Times New Roman">, y oferta de trabajo, <Font style="_cstyle283">t[s]</Font>.<Font encoding="ISO8859-1">
Faltar\355a complementar estas expresiones con las  restricciones de no-negatividad (es decir, precios y cantidades positivas) y otras condiciones relacionadas con la curvatura de la funci\363n objetivo,  para poder asegurar que estamos efectivamente hablando de la funci\363n consumo y oferta de trabajo.  Nos referiremos a ellas m\341s adelante. Asumamos por ahora que tales condiciones adicionales se cumplen. </Font></Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">c[d]:=unapply(C,w,pi);
</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JiUiY0c2IyUiZEdmKjYkJSJ3RyUjcGlHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGLCwmKiglJmFscGhhRyIiIjkkRjMlIkhHRjNGMyomRjJGMzklRjNGM0YsRixGLA==</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">t[s]:=unapply(T,w,pi);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JiUidEc2IyUic0dmKjYkJSJ3RyUjcGlHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGLComLCgqKCUmYWxwaGFHIiIiOSRGNCUiSEdGNEY0KiZGM0Y0OSVGNEY0RjghIiJGNEY1RjlGLEYsRiw=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">lambda[o]:=unapply(lambda,w,pi);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JiUnbGFtYmRhRzYjJSJvR2YqNiQlIndHJSNwaUc2IjYkJSlvcGVyYXRvckclJmFycm93R0YsLCQqJiIiIkYyLCYqJjkkRjIlIkhHRjJGMjklRjIhIiJGOEYsRixGLA==</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">En virtud de la especificaci\363n que hicimos de la funci\363n de utilidad, </Font><Font family="Times New Roman" style="_cstyle284">U</Font><Font encoding="ISO8859-1" family="Times New Roman">,  la funci\363n consumo, </Font><Font family="Times New Roman" style="_cstyle285">c[d]</Font><Font encoding="ISO8859-1" family="Times New Roman">,  indica que las familias buscar\355an asignar a la compra de bienes una proporci\363n igual a </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font family="Times New Roman"> del ingreso potencial:  </Font><Equation input-equation="w*H+pi;" style="2D Comment">NiMsJiomJSJ3RyIiIiUiSEdGJkYmJSNwaUdGJg==</Equation><Font encoding="ISO8859-1" family="Times New Roman">.  Por su parte,  los consumidores estar\341n dispuestos a trabajar m\341s horas en la medida en que los salarios, </Font><Font family="Times New Roman" style="_cstyle286">w</Font><Font family="Times New Roman">, sean mayores y los ingresos provenientes de las utilidades de las empresas, </Font><Equation input-equation="pi;" style="2D Comment">NiMlI3BpRw==</Equation><Font family="Times New Roman">, sean menores, siempre y cuando </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font family="Times New Roman"> sea positiva pero menor que la unidad.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">S\363lo falta verificar el cumplimiento de las condiciones  Kuhn-Tucker de segundo orden. Estas condiciones son suficientes para afirmar que las funciones de demanda de bienes y oferta de trabajo efectivamente correspondan a un punto en que las familias est\341n </Font><Font family="Times New Roman" style="_cstyle287">maximizando</Font><Font encoding="ISO8859-1" family="Times New Roman"> su bienestar dada la restriccion presupuestal.  Lo que estamos buscando es un valor positivo para el determinante del Hessiano orlado correspondiente a la funci\363n Lagrangiana con que se ha venido trabajando.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">with(linalg):
A:=hessian(Lambda(x,y,z), [x,y,z]);</Text-field></Input><Output><Text-field layout="Warning" style="Warning">Warning, the protected names norm and trace have been redefined and unprotected
</Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSJBRy0lJ21hdHJpeEc2IzclNyUsJComJSZhbHBoYUciIiIlInhHISIjISIiIiIhRi03JUYxLCQqJiwmRi1GLUYsRjBGLSwmJSJIR0YtJSJ5R0YwRi9GMCwkJSJ3R0YwNyVGLUY5RjE=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">
Tal Hessiano orlado se obtiene al substituir la demanda de bienes, <Font style="_cstyle288">c[d]</Font>, la oferta de trabajo, <Font style="_cstyle289">t[s], </Font>en la matriz <Font style="_cstyle291">A </Font><Font encoding="ISO8859-1">en la forma en que se indica a continuaci\363n:</Font></Font></Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">x:=c[d](w,pi):
y:=t[s](w,pi):</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">Delta:=factor(collect(det(A),alpha));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSZEZWx0YUcsJCoqJSJ3RyIiIywmISIiIiIiJSZhbHBoYUdGK0YqRixGKiwmKiZGJ0YrJSJIR0YrRislI3BpR0YrISIjRio=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">El numerador del determinante </Font><Equation input-equation="Delta;" style="2D Comment">NiMlJkRlbHRhRw==</Equation><Font family="Times New Roman">, </Font><Equation input-equation="-w^2;" style="2D Comment">NiMsJCokJSJ3RyIiIyEiIg==</Equation><Font family="Times New Roman">, es obviamente negativo . Por su parte, el denominador  es negativo siempre y cuando </Font><Equation input-equation="1-alpha &lt; 0;" style="2D Comment">NiMyLCYiIiJGJSUmYWxwaGFHISIiIiIh</Equation><Font family="Times New Roman">, lo cual hemos venido suponiendo a lo largo del ejercicio.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Text-field layout="Maple Output" style="Maple Output"/></Section><Section collapsed="true"><Title><Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman">1.2.3 Comportamiento de las Empresas</Font></Text-field></Title><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">El tratamiento que se da a las empresas es muy similar al de los consumidores. De hecho, suponemos que los empresarios son racionales y que participan en mercados competitivos.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Las empresas tienen como objetivo maximizar sus ganancias mediante la producci\363n y venta de bienes y/o servicios. Para lograr lo anterior concurren al mercado de factores de la producci\363n, en este caso muy simple es el mercado de mano de obra. De ah\355  se allegan los recursos necesarios para llevar a cabo la producci\363n de bienes, ello en un entorno determinado por una cierta tecnolog\355a o funci\363n producci\363n como la que se define a continuaci\363n. </Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">T:='T':</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">y:=(T)-&gt;T^beta;</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSJ5R2YqNiMlIlRHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCk5JCUlYmV0YUdGKEYoRig=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Para visualizar las caracter\355sticas de esta ecuaci\363n podemos ayudarnos otra vez de una gr\341fica animada.. En ella puede verse que cuando </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font encoding="ISO8859-1" family="Times New Roman"> se aproxima a la unidad, esta toma la forma de una l\355nea con pendiente de 45 grados que pasa por el origen. Este es el caso en que la tecnolog\355a exhibe rendimientos constantes a escala. Por su parte, los rendimientos decrecientes e hacen m\341s pronunciados a medida que los valores del par\341metro </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font encoding="ISO8859-1" family="Times New Roman"> son m\341s cercanos a cero.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">with(plots):
animate(y(T),T=0..24,beta=0.5..0.9,color=blue,frames=50,title=`Funcion Produccion (animada)`,titlefont=[TIMES,BOLDITALIC,12], axesfont=[TIMES,ITALIC,8],labelfont=[TIMES,BOLDITALIC,9]); </Text-field></Input><Output><Text-field layout="Maple Plot" style="Maple Plot"><Plot height="300" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" type="two-dimensional" 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layout="Normal" style="Normal"><Font family="Times New Roman">    </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">El siguiente paso es modelar el comportamiento optimizador de las empresas. Para ello es necesario especificar la funci\363n de beneficios econ\363micos de la empresa representativa como la diferencia entre ingresos y costos. Los ingresos provienen de las ventas de la producci\363n en el mercado y los costos del pago de la n\363mina, seg\372n aparece a continuaci\363n.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">B:=(T,w)-&gt;y(T)-w*T;
b:=unapply(diff(B(T,w),T),T,w);

</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSJCR2YqNiQlIlRHJSJ3RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHRiksJi0lInlHNiM5JCIiIiomOSVGMkYxRjIhIiJGKUYpRik=</Equation></Text-field><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JSJiR2YqNiQlIlRHJSJ3RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHRiksJiooKTkkJSViZXRhRyIiIkYxRjJGMCEiIkYyOSVGM0YpRilGKQ==</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Las empresas buscar\341n maximizar las ganancias mediante la determinaci\363n del n\372mero de trabajadores que estar\355an dispuestos a contratar. As\355, la variable de control es </Font><Font family="Times New Roman" style="_cstyle292">T</Font><Font encoding="ISO8859-1" family="Times New Roman"> . Por cierto, en esta ocasi\363n la optimizaci\363n no presenta otra restricci\363n que la propia especificaci\363n de la funci\363n producci\363n, de manera que las condici\363n de primer orden es la anulaci\363n de la derivada parcial de la funci\363n beneficios respecto del nivel de empleo.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">solucion_empresa:=solve(b(T,w)=0,{T}):
assign(solucion_empresa);
t[d]:=unapply(T,w);
</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiM+JiUidEc2IyUiZEdmKjYjJSJ3RzYiNiQlKW9wZXJhdG9yRyUmYXJyb3dHRistJSRleHBHNiMsJComLSUjbG5HNiMqJiUlYmV0YUciIiI5JCEiIkY5LCZGOEY5RjlGO0Y7RjtGK0YrRis=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Ahora bien, hemos dicho que aunque las empresas contratan trabajadores, ellas no hacen como un fin en s\355 mismo. En efecto, su prop\363sito es contratar trabajadores para producir bienes. Es por ello que la demanda de trabajo, </Font><Font family="Times New Roman" style="_cstyle293">t[d]</Font><Font family="Times New Roman">, y la oferta de bienes, <Font style="_cstyle294">c[s]</Font>, son resultado del mismo proceso optimizador.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Para obtener la oferta de bienes, bastar\341 con sustituir la demanda de trabajo,  </Font><Font family="Times New Roman" style="_cstyle295">t[d]</Font><Font encoding="ISO8859-1" family="Times New Roman">,  en la funci\363n producci\363n,  </Font><Font family="Times New Roman" style="_cstyle296">y[T]</Font><Font encoding="ISO8859-1" family="Times New Roman">, como se muestra a continuaci\363n:</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">c[s]:=(w)-&gt;y(t[d](w)):
c[s](w);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMpLSUkZXhwRzYjLCQqJi0lI2xuRzYjKiYlJWJldGFHIiIiJSJ3RyEiIkYuLCZGLUYuRi5GMEYwRjBGLQ==</Equation></Text-field></Output></Group><Text-field layout="Maple Output" style="Maple Output"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Las condiciones de segundo orden para este ejercicio de optimizaci\363n ser\341n satisfechas si  </Font><Equation input-equation="diff(B(T,w),T,T) &lt; 0;" style="2D Comment">NiMyLSUlZGlmZkc2JS0lIkJHNiQlIlRHJSJ3R0YqRioiIiE=</Equation><Font encoding="ISO8859-1" family="Times New Roman">. Ello ocurrir\341 mientras la funci\363n producci\363n muestre rendimientos decrecientes, es decir, si </Font><Equation input-equation="beta &lt; 1;" style="2D Comment">NiMyJSViZXRhRyIiIg==</Equation><Font family="Times New Roman">.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">T:='T':</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">simplify(factor(diff(B(T,w),T,T)));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMqKCklIlRHLCYlJWJldGFHIiIiIiIjISIiRihGJ0YoLCZGJ0YoRihGKkYo</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Hyperlink"/><Text-field layout="Normal" style="Normal"/></Section><Section collapsed="true"><Title><Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman">1.2.4 El Equilibrio General: Ley de Walras</Font></Text-field></Title><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">De los ejercicios resueltos en las dos secciones previas obtuvimos las ecuaciones necesarias para determinar el equilibrio simult\341neo en los mercados de bienes y mano de obra, es decir,  a lo que nos referiremos como el equilibrio general. </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Una condicion que impondremos en este equilibrio provendr\341 de sustituir la funci\363n de beneficios que resulta de la optimizaci\363n de las empresas en el nivel de ganancias que utilizan los trabajadores para definir su demanda de bienes. Esta condici\363n presupone que los consumidores cuentan con muy amplia informaci\363n, y que procesan dicha informaci\363n de la manera en que mejor responde a sus objetivos. </Font></Text-field><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">Hecho lo anterior,  determinaremos  el punto donde se cruzan las ofertas y las demandas respectivas. Es decir, donde el exceso de demanda en el mercado de bienes y el exceso de demanda en el mercado de trabajo se hacen iguales a cero. </Font></Text-field><Text-field layout="Normal" style="Normal"><Font family="Times New Roman"> Las funciones de exceso de demanda en el mercado de bienes <Font style="_cstyle297">, E[c]</Font>, y en el mercado de trabajo,  <Font style="_cstyle298">E[t], </Font>son las que enseguida aparecen: </Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">E[c]:= w-&gt;c[d](w,B(t[d](w),w))-c[s](w):</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">E[c](w);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMsKCooJSZhbHBoYUciIiIlIndHRiYlIkhHRiZGJiomRiVGJiwmKS0lJGV4cEc2IywkKiYtJSNsbkc2IyomJSViZXRhR0YmRichIiJGJiwmRjVGJkYmRjZGNkY2RjVGJiomRidGJkYsRiZGNkYmRiZGK0Y2</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">E[t]:=w-&gt;t[d](w)-t[s](w,B(t[d](w),w)):</Text-field></Input></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">E[t](w);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMsJi0lJGV4cEc2IywkKiYtJSNsbkc2IyomJSViZXRhRyIiIiUid0chIiJGLiwmRi1GLkYuRjBGMEYwRi4qJiwqKiglJmFscGhhR0YuRi9GLiUiSEdGLkYuKiZGNUYuLCYpRiRGLUYuKiZGL0YuRiRGLkYwRi5GLkY5RjBGOkYuRi5GL0YwRjA=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Aunque no es evidente a primera vista,  bastar\341 con que se tenga equilibrio en uno de los mercados para obtener el equilibrio en el otro mercado. Para probar esta afirmaci\363n ser\341 necesario  sumar el </Font><Font family="Times New Roman" style="_cstyle299">valor de los excesos de demanda </Font><Font family="Times New Roman"> en ambos mercados: <Font style="_cstyle300">E[c]+wE[t]</Font> . Una vez realizadas las simplificaciones necesarias resulta que dicha suma es igual a cero. Eso quiere decir, que si en un mercado el exceso de demanda es cero, tambi[en lo ser[a en el otro.  Esta propiedad del modelo con que estamos trabajando se conoce como la <Font style="_cstyle301">Ley de Walras</Font>.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">simplify(E[c](w)+w*E[t](w));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMiIiE=</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman"> En su forma m\341s general, la </Font><Font family="Times New Roman" style="_cstyle302">Ley de Walras</Font><Font family="Times New Roman"> establece que cuando se tienen </Font><Equation input-equation="N" style="2D Comment">NiMlIk5H</Equation><Font encoding="ISO8859-1" family="Times New Roman"> mercados en los que participan agentes optimizadores, basta con que se\341 calcule el equilibrio en </Font><Equation input-equation="N-1" style="2D Comment">NiMsJiUiTkciIiJGJSEiIg==</Equation><Font encoding="ISO8859-1" family="Times New Roman"> de ellos para garantizar que el \372ltimo tambi\351n est\351 en equilibrio.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">Tomemos ahora una de  las funciones de exceso de demanda, por ejemplo, el exceso de demanda en el mercado debienes, <Font style="_cstyle303">E[c]</Font>, y encontremos el nivel de salario real  que garantice que tal exceso de demanda es igual a cero.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">solve(E[c](w)=0,w);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMqKi0lJGV4cEc2IyomLSUjbG5HNiMqKiUlYmV0YUciIiIsKCUmYWxwaGFHISIiKiZGL0YtRixGLUYtRi1GLUYwRi9GLSUiSEdGLUYtRixGLUYtRi5GLUYvRjBGMkYw</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">S\363lo con fines ilustrativos, comprobamos que el resultado es el mismo si en vez de resolver el equilibrio en el mercado de bienes lo hacemos en el mercado de trabajo.</Font></Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">solve(E[t](w)=0,w);</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMqKi0lJGV4cEc2IyomLSUjbG5HNiMqKiUlYmV0YUciIiIsKCUmYWxwaGFHISIiKiZGL0YtRixGLUYtRi1GLUYwRi9GLSUiSEdGLUYtRixGLUYtRi5GLUYvRjBGMkYw</Equation></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">El equilibrio puede visualizarse de varias formas. Un primer ejemplo aparece en las siguientes dos gr\341ficas, las cuales corresponden al caso hipot\351tico en el que </Font><Equation input-equation="alpha = .5;" style="2D Comment">NiMvJSZhbHBoYUctJSZGbG9hdEc2JCIiJiEiIg==</Equation><Font family="Times New Roman"> y </Font><Equation input-equation="beta = .4;" style="2D Comment">NiMvJSViZXRhRy0lJkZsb2F0RzYkIiIlISIi</Equation><Font family="Times New Roman">. Cada curva muestra respectivamente el exceso de demanda en el mercado de bienes y en el mercado de trabajo para distintos niveles del salario real, <Font style="_cstyle304">w. </Font> Resalta  la forma que toman ambas curvas, donde una de ellas es aproximadamente  la imagen de espejo de la otra, por virtud de la <Font style="_cstyle305">Ley de Walras</Font><Font encoding="ISO8859-1">. El lector podr\341 modificar el valor de los par\341metros aludidos para hacer la est\341tica comparativa del equilibrio general ante cambios en las formas de la funci\363n de utilidad y la funci\363n producci\363n.</Font></Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field bookmark="grafica1" layout="Normal" prompt="&gt; " style="Maple Input">with(plots):
alpha:=0.5:
beta:=0.4:
H:=24:
grafica1:=plot(E[c](w),w=0.1..0.15,title=`Mercado de Bienes`,titlefont=[TIMES,BOLDITALIC,12], axesfont=[TIMES,ITALIC,8],labelfont=[TIMES,BOLDITALIC,10]):

grafica2:=plot(w*E[t](w),w=0.1..0.15,title=`Mercado de Trabajo`,titlefont=[TIMES,BOLDITALIC,12], axesfont=[TIMES,ITALIC,8],labelfont=[TIMES,BOLDITALIC,10]):
display(grafica1);
display(grafica2);
alpha:='alpha':
beta:='beta':
H:='H':</Text-field></Input><Output><Text-field layout="Maple Plot" style="Maple Plot"><Plot height="300" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" type="two-dimensional" width="400">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</Plot></Text-field><Text-field layout="Maple Plot" style="Maple Plot"><Plot height="300" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" type="two-dimensional" width="400">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</Plot></Text-field></Output></Group><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">Los niveles de empleo en el equilibrio, <Font style="_cstyle306">t[e]</Font><Font encoding="ISO8859-1"> , y de actividad econ\363mica, y=</Font><Font style="_cstyle307">c[e],</Font><Font encoding="ISO8859-1">  se determinan, respectivamente,  sustituyendo la soluci\363n del salario de equilibrio, </Font><Font style="_cstyle308">w[e]</Font>, en la demanda,<Font style="_cstyle309"> t[d] </Font>(u oferta, <Font style="_cstyle310">t[s]</Font><Font encoding="ISO8859-1">) de trabajo,  y este resultado a su vez en la funci\363n producci\363n,</Font><Font style="_cstyle311"> y[T]</Font>.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font family="Times New Roman">   </Font></Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">W:=solve(E[c](w)=0,w):
w[e]:=unapply(W,alpha,beta,H):
simplify(w[e](alpha,beta,H));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMqKikqKiUlYmV0YUciIiIsKCUmYWxwaGFHISIiKiZGKUYnRiZGJ0YnRidGJ0YqRilGJyUiSEdGJ0YmRidGKEYnRilGKkYsRio=</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">T:=t[d](w[e](alpha,beta,H)):
t[e]:=unapply(T,alpha,beta,H):
simplify(t[e](alpha,beta,H));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMpKiwlJWJldGFHIiIiKSoqRiVGJiwoJSZhbHBoYUchIiIqJkYqRiZGJUYmRiZGJkYmRitGKkYmJSJIR0YmLCRGJUYrRiZGKUYrRipGJkYtRiYsJComRiZGJiwmRiVGJkYmRitGK0Yr</Equation></Text-field></Output></Group><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">C:=c[s](w[e](alpha,beta,H)):
c[e]:=unapply(C,alpha,beta,H):
simplify(c[e](alpha,beta,H));</Text-field></Input><Output><Text-field layout="Maple Output" style="2D Output"><Equation style="2D Output">NiMpKSosJSViZXRhRyIiIikqKkYmRicsKCUmYWxwaGFHISIiKiZGK0YnRiZGJ0YnRidGJ0YsRitGJyUiSEdGJywkRiZGLEYnRipGLEYrRidGLkYnLCQqJkYnRicsJkYmRidGJ0YsRixGLEYm</Equation></Text-field></Output></Group><Text-field layout="Fixed Width" style="Fixed Width"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Aunque las expresiones que aparecen arriba pueden parecer un tanto cuanto ilegibles, una r\341pida inspecci\363n nos permite llegar a un par de conclusiones interesantes:</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font family="Times New Roman">El equilibrio general depende sola y precisamente de  </Font><Equation input-equation="alpha,H;" style="2D Comment">NiQlJmFscGhhRyUiSEc=</Equation><Font family="Times New Roman">  y de </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font encoding="ISO8859-1" family="Times New Roman">. Es decir, cualquier cambio en la asignaci\363n de recursos y, por lo tanto, cualquier variaci\363n en los niveles de actividad y empleo tendr\355an que darse como resultado de variaciones en las preferencias y/o en la tecnolog\355a</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font encoding="ISO8859-1" family="Times New Roman">S\363lo hay un precio en la econom\355a: el salario real, </Font><Font family="Times New Roman" style="_cstyle312">w.</Font><Font encoding="ISO8859-1" family="Times New Roman"> Sin embargo, hay dos mercados, uno en el que se intercambian bienes y otro en el que se compra y vende el factor de la producci\363n</Font><Font family="Times New Roman" style="_cstyle313"> T. </Font><Font family="Times New Roman"> Esto es el resultado de que escogimos como <Font style="_cstyle314">numerario</Font><Font encoding="ISO8859-1"> a los bienes de consumo cuyo precio, en t\351rminos de s\355 mismo es igual a la unidad, por definici\363n. En otras palabras, en un entorno como el modelado en esta secci\363n del curso, lo que importa en la determinaci\363n del equilibrio y la distribuci\363n del ingreso son los precios relativos y no sus niveles absolutos. </Font></Font></Text-field><Text-field layout="Normal" style="Normal"/></Section><Section collapsed="true"><Title><Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman">1.2.5 El Equilibrio General y el bienestar de la sociedad.</Font></Text-field></Title><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">La gr\341fica que aparece a continuaci\363n, muestra la posici\363n de la funci\363n de utilidad, </Font><Font family="Times New Roman" style="_cstyle315">U, </Font><Font encoding="ISO8859-1" family="Times New Roman">y de la frontera de posibilidades de producci\363n, alrededor del equilibrio general. </Font><Font family="Times New Roman">
</Font></Text-field><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">with(plots):
alpha:=0.4:
beta:=0.5:
H:=24:
indiferencia:=plot([c[d](w,B(t[d](w),w)),24-t[s](w,B(t[d](w),w)),
w=.65*w[e](alpha,beta,H)..1.35*w[e](alpha,beta,H)],
title=`Equilibrio General`,
titlefont=[TIMES,BOLDITALIC,12], 
labels=[`consumo`,`ocio`],
labeldirections=[HORIZONTAL,VERTICAL],
axesfont=[TIMES,ITALIC,8],labelfont=[TIMES,BOLDITALIC,10],color=black):

produccion:=plot([c[s](w,B(t[d](w),w)),24-t[d](w,B(t[d](w),w)),
w=.65*w[e](alpha,beta,H)..1.55*w[e](alpha,beta,H)],color=blue):

display(indiferencia,produccion);
alpha:='alpha':
beta:='beta':
H:='H':</Text-field></Input><Output><Text-field layout="Maple Plot" style="Maple Plot"><Plot height="300" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" type="two-dimensional" width="400">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</Plot></Text-field></Output></Group><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">
La curva c\363ncava hacia abajo y de color azul corresponde a las combinaciones de producci\363n </Font><Font family="Times New Roman" style="_cstyle317">c[s]</Font><Font family="Times New Roman">,  y ocio <Font style="_cstyle316">H-t[s]</Font><Font encoding="ISO8859-1">  que resultan del ejercicio de optimizaci\363n de las empresas. Por su parte, la l\355nea convexa y de color negro es la curva de indiferencia para un nivel de utilidad equivalente al que se obtiene cuando el consumidor representativo determina su demanda de bienes, </Font><Font style="_cstyle319">c[d],</Font>  y oferta de trabajo, <Font style="_cstyle320">t[s]</Font>,  al nivel del salario real de equilibrio <Font style="_cstyle318">w[e]</Font>.  El resultado puede resumirse como sigue:
</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font encoding="ISO8859-1" family="Times New Roman">En el equilibrio, la tasa marginal de sustituci\363n de los consumidores es igual a la tasa marginal de transformaci\363n de las empresas. Dicha tasa corresponde al nivel del salario real, ' </Font><Font family="Times New Roman" style="_cstyle321">w</Font><Font family="Times New Roman">, y se desprende del hecho de que ambas curvas son tangentes entre si en dicho equilibrio </Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font encoding="ISO8859-1" family="Times New Roman">Dadas las decisiones de los productores, las familias no pueden alcanzar una combinaci\363n m\341s satisfactoria de consumo y ocio que la que tienen en el equilbrio. Toda vez que las curvas de indiferencia no se cruzan entre si, como vimos al principio de este cap\355tulo, cualquier otro punto sobre la frontera de producci\363n corresponder\355a a curvas de indiferencia con un menor nivel de bienestar.</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font encoding="ISO8859-1" family="Times New Roman">Toda vez que la frontera de posibilidades de producci\363n proviene del ejericicio de optimizaci\363n de las empresas, tenemos que concluir que los productores est\341n asignando optimamente los recursos en cada punto.</Font></Text-field><Text-field layout="Normal256" style="Normal256"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Por lo tanto, en esta sociedad hipot\351tica de agentes optimizadores, el resultado de decisiones individuales enfocadas a maximizar los objetivos de cada persona, se traducen a trav\351s de su correncia en el mercado, en resultados de producci\363n y asignaci\363n que maximizan el bienestar colectivo.</Font></Text-field><Text-field layout="Normal" style="Normal"/></Section><Section><Title><Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman">1.2.6 Usando el Modelo</Font></Text-field></Title><Section collapsed="true"><Title><Text-field layout="Heading 4" style="Heading 4"><Font encoding="ISO8859-1" family="Times New Roman">Pregunta 1. \277Qu\351 tan diferentes pueden ser los pa\355ses?</Font></Text-field></Title><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">De acuerdo con los resultados que se presentaron anteriormente, las \372nicas razones  por las cuales la econom\355a de un pa\355s puede ser diferente a la de otro son diferencias en tecnolog\355a o en preferencias.  La gr\341fica siguiente presenta los niveles que mostrar\355a el Producto Nacional Bruto per capita  de un pa\355s para distintos valores de </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font family="Times New Roman"> y </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman"> dentro de un rango entre 0.1 y 0.9.  <Font encoding="ISO8859-1">
Dicha gr\341fica sugiere que es posible observar grandes diferencias en el ingreso per capita en un mundo con individuos optimizadores. Por ejemplo, la econom\355a m\341s peque\361a registra un PNB per c\341pita de aproximadamente 1, mientras que la m\341s grande toma un valor de alrededor de 16.</Font></Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Claro esta que las diferencias se deben, en parte a las preferencias y en parte a la tecnolog\355a.  Si rotamos la gr\341fica, colocando el cursor sobre ella, de manera que desaparezca el eje </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman"> , veremos exclusivamente el efecto de cambios en </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font family="Times New Roman">. En ese caso, y aun si el valor de </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman"> se mantiene en 0.9, veremos importantes diferencias en el PNB per capita, con rangos que van desde alrededor de 4 hasta 16. <Font encoding="ISO8859-1">
En consecuencia, seg\372n la teor\355a que subyace a este modelo, no son solamente las diferencias en tecnolog\355a las que explican la brecha entre naciones m\341s ricas y las menos pr\363speras, sino que las preferencias pueden explicar una parte importante de estas discrepacias. </Font></Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">with(plots):
H:=24:
plot3d(c[e](alpha,beta,24),alpha=0.1..0.90,beta=0.1..0.90,
axes=boxed,
title=`Valores Observables del PIB `,
labels=[a,b,``],
titlefont=[TIMES,BOLDITALIC,12], 
axesfont=[TIMES,BOLDITALIC,8],labelfont=[SYMBOL,9]); 
H:='H':</Text-field></Input><Output><Text-field layout="Maple Plot" style="Maple Plot"><Plot height="300" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" type="three-dimensional" 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collapsed="true"><Title><Text-field layout="Heading 4" style="Heading 4"><Font encoding="ISO8859-1" family="Times New Roman">Pregunta 2. \277C\363mo se distribuye el ingreso entre trabajadores y empresarios?</Font></Text-field></Title><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">En este modelo, el ingreso se distribuye entre trabajadores y propietarios de las empresas. Por eso, la pregunta que podemos contestar es qu\351 proporci\363n del Producto Nacional Bruto va a la n\363mina, y obviamente por diferencias, qu\351 parte toma la forma de ganancias.
La gr\341fica siguiente reporta los valores que toma el cociente  </Font><Font family="Times New Roman" style="_cstyle322">n[e]/c[e]</Font><Font family="Times New Roman">, donde <Font style="_cstyle323">n[e]</Font><Font encoding="ISO8859-1">, la n\363mina total que resulta de multiplicar los salarios en el equilibrio por el nivel de empleo en equilibrio, </Font><Font style="_cstyle324">w[e] t[e]</Font>. Por su parte, <Font style="_cstyle325">c[e]</Font>, es el producto nacional bruto.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Se observa que la distribuci\363n del ingreso depende exclusivamente de la tecnolog\355a. Es decir, para un nivel dado de </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman">, cambios en el valor de </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman"> no se traducen en diferencias en la distribuci\363n del ingreso.  </Font></Text-field><Text-field layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">nomina:=w[e](alpha,beta,H)*t[e](alpha,beta,H):
n:=unapply(nomina,alpha,beta,H):
H:=24:
plot3d(n(alpha,beta,24)/c[e](alpha,beta,24),alpha=0.1..0.90,beta=0.1..0.90,
axes=boxed,
labels=[`a`,`b`,``],
title=`Participacion de la Nomina en el PNB`,
orientation=[20,50],
titlefont=[TIMES,BOLDITALIC,12], axesfont=[TIMES,ITALIC,8],
labelfont=[SYMBOL,9]); 
H:='H':</Text-field></Input><Output><Text-field layout="Maple Plot" style="Maple Plot"><Plot height="300" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" type="three-dimensional" 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layout="Normal" style="Normal"/><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Se observa que la distribuci\363n del ingreso depende exclusivamente de la tecnolog\355a. Es decir, para un nivel dado de </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman">, cambios en el valor de </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman"> no se traducen en diferencias en la distribuci\363n del ingreso.  Adicionalmente, la participaci\363n de la n\363mina en el PNB es igual al par\341metro </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman">.</Font></Text-field><Text-field layout="Normal" style="Normal"/></Section><Section collapsed="true"><Title><Text-field layout="Heading 4" style="Heading 4"><Font encoding="ISO8859-1" family="Times New Roman">Pregunta 3. \277Es correcto equiparar mayores niveles de producci\363n con mayores niveles de bienestar?</Font></Text-field></Title><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Antes de contestar la pregunta es necesario hacer algunas precisiones. En las secciones anteriores vimos que el equilibrio general se alcanzaba en el punto donde la frontera de posibilidades de producci\363n y cierta curva de indiferencia eran tangentes entre s\355. En dicho caso, era posible ver niveles de producci\363n mayores, sin embargo el bienestar que dicha producci\363n implicaba era menor puesto que se lograba a costa de un menor ocio. En consecuencia, el modelo reconoce que </Font><Font family="Times New Roman" style="_cstyle327">fuera del equilibrio</Font><Font encoding="ISO8859-1" family="Times New Roman"> es posible ver mayor producci\363n y menor bienestar.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">Sin embargo, si refraseamos la pregunta para saber si mayores niveles de producci\363n </Font><Font family="Times New Roman" style="_cstyle326">de equilibrio</Font><Font encoding="ISO8859-1" family="Times New Roman"> corresponden a mayores niveles de bienestar, la respuesta ser\341 diferente.. </Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">La gr\341fica que aparece enseguida muestra varias curvas. Cada una de ellas describe la relaci\363n entre el nivel de producci\363n de equilibrio, el cual cambia como resultado de variaciones en el valor del par\341metro </Font><Equation input-equation="beta;" style="2D Comment">NiMlJWJldGFH</Equation><Font family="Times New Roman">, para distintos niveles </Font><Equation input-equation="alpha;" style="2D Comment">NiMlJmFscGhhRw==</Equation><Font encoding="ISO8859-1" family="Times New Roman">. En todos los casos, la relaci\363n es positiva, si bien se refleja el hecho de que la utilidad marginal del consumo es decreciente, puesto que a niveles de bienestar relativamente altos, se necesita de aumentos muy importantes en la producci\363n para observar incrementos sucesivos en la satisfacci\363n del p\372blico.</Font></Text-field><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">En s\355ntesis, de acuerdo con este modelo, una mayor capacidad de producci\363n es condici\363n indispensable para elevar el bienestar de la sociedad.</Font></Text-field><Text-field layout="Normal" style="Normal"/><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input">H:=24:
for j from 1 by 2 while j &lt; 10 do
  alpha:=evalf(j/10):

grafica[j]:=plot([U(c[e](alpha,beta,H),24-t[e](alpha,beta,H)),c[e](alpha,beta,H),beta=0.1..0.90],
axes=boxed,
labels=[Bienestar,PNB],
title=`Combinaciones PNB-bienestar`,
titlefont=[TIMES,BOLDITALIC,12], 
labeldirections=[HORIZONTAL,VERTICAL],
color=black,
axesfont=[TIMES,ITALIC,8],labelfont=[TIMES,BOLDITALIC,9]):

end do:

display(grafica[1],grafica[3],grafica[5],grafica[7],grafica[9]);
H:='H': 
alpha:='alpha':
beta:='beta':
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layout="Normal" style="Normal"/></Section></Section><Section collapsed="true"><Title><Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman"> Lecturas Recomendadas</Font></Text-field></Title><Text-field layout="Normal" style="Normal"><Font encoding="ISO8859-1" family="Times New Roman">El lector puede profundizar en las ideas presentadas en este cap\355tulo consultando las secciones referentes a teoremas de bienestar y equilibrio general en los siguientes textos.</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font family="Times New Roman">McCallum, Bennett. <Font style="_cstyle350">Monetary Economics.</Font> Mcmillan, New York, 1989</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font family="Times New Roman">Parkin, Michael. <Font encoding="ISO8859-1" style="_cstyle353">Microeconom\355a.</Font><Font encoding="ISO8859-1"> M\351xico, Pearson, 2001</Font></Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font family="Times New Roman">Romer, David. <Font style="_cstyle357">Advanced Macroeconomics.</Font>  McGraw Hill, New York, 1996</Font></Text-field><Text-field layout="Bullet Item" style="Bullet Item"><Font family="Times New Roman">Varian, Hal. <Font style="_cstyle360">Intermediate Microeconomics.</Font> WW Norton, Cincinnati, 1999</Font></Text-field></Section></Section><Group><Input><Text-field layout="Normal" prompt="&gt; " style="Maple Input"/></Input></Group><Text-field/></Worksheet>