1 General Equilibrium in the Robinson Crusoe Economy.

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Stewart Harrell
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1 General Equilibrium in the Robinson Crusoe Economy. Robinson Crusoe is a fictional shipwrecked person from a novel by Daniel Defoe. InthenovelhelivedaloneonanIslandfor8years,producingeverything heconsumedbyhimself. Thistermhasbeenadoptedbyeconomiststodescribe a very simple economy, where there is one consumer and one production possibilitiesfrontierppf).onecanassumethattheppfiscreatedbythevarious ways the one consumer can divide his labor and other resources. In this handout IwillassumepreferencesareCobb-DouglassF,C)=F α C β andthattheproductionpossibilitiesfrontierhasthefollowingsimpleform: a f F γ f +ac C γ c =T, where a f,a c,t > 0 and γ f,γ c >. 3 The following graph shows us such an economy. C F The dark line is the production possibilities frontier, and the light lines are various indifference curves. Now obviously maximizing the utility over this set will be very similar to maximizing it over a budget set, and so the solution will be at a tangency. The slope of the PPF we call the Marginal Rate of Transformation, orthemrt.wefindboththemrsandthemrtbytaking He did have the assistance of "Friday," a local man. But his man Friday was a native American inanovel written by aeuropean,so he didn tcount. 3 In mostapplicationsγ f =γ c =γ.

2 derivatives of an indifference curve and the PPF respectively. F,C) = Ū M f +M c C F = 0 likewisewecanfindthemrt: MRS = C F = M f M c M f = α F,M c= β α C,MRS= F β C a f γ f F γ f +a c γ c C γ c C F = 0 and the tangency condition is: = C F MRT = C F = a fγ f F γf a c γ c C γ c αc βf = a fγ f F γf a c γ c C γ c C γ c αγc a c = F γ f βγf a f ) βγf a f C = F γ γ c f. αγ c a c Tofindouthowmuchclothingthisconsumershouldconsumeweplugthisback into the PPF: a f F γ f +ac C γ c = T βγf ) ) γc a f F γ a f f +ac F γ γ c f = T αγ c a c a f F γ βγ f f a f +ac F γ f αγ c a c + βγ ) f a f F γ f αγ c F = = T = T αγ c αγ c +βγ f T a f α β ) γ f you should recognize the close similarity between this solution and the standard Cobb-Douglassdemands,InthestandarddemandF = β I α+β hereαγ c replaces β, βγ f replaces α and the entire thing is raised to a power. Overall it is not really that different. I will let you solve for the general value for C on your own.

3 The important thing about this equilibrium to an economist is that Robinson s decision process can be separated into tility maximizing and Revenue maximizing. The revenue maximizer will solve the following problem: R=max F,C F+p c C λa f F γ f +ac C γ c T) the utility maximizer will solve the problem max F,C Fα C β λ F+p c C I) andthemodelwillbeclosedbyhavingi =R,ortherevenuethefirmmakes. Ishouldnotethatinordertosolvesuchaproblemwereallyneedγ f =γ c,we also need this if the utility function is not a Cobb-Douglass. In general we will assume this. TogiveanexampleassumethatF,C)=FC 4 andf +C =5. Then the Pareto efficient outcome is: max F,C min λ FC4 λ C +F 5 ) C 4 Fλ = 0 4C 3 F Cλ = 0 C +F 5 = 0 λ = C 4 F =C F F = C ) C + C = C = 5 C = 0 F = 0)=5 and we can decentralize this by finding the appropriate prices. MRS= M f = C4 M c 4FC 3 = 4 C F = = let us just let p c = and = for the fun of it. Then we can solve the revenue maximization problem first: max min F,C λ F+C λ C +F 5 ) 3 p c

4 λf = 0 λc = 0 C +F 5 ) = 0 λ = 4F = C F = C andatthispointfromaboveyoucaneasilyseethatthesolutionisc=0and F = 5. This means that Robinson s Income is I = 5)+0 = 5, and his utility maximization problem is: max min F,C λ FC4 λ pf+c 5 ) C 4 λp = 0 4FC 3 λ = 0 F+C 5 = 0 λ = C 4 =4FC 3 F = C ) C +C 5 = 0 C = 0 andasyoucanseethesolutionisthesame.. Robinson Crusoe and International Trade. Now what would happen if Robinson Crusoe could trade at a different price vectorfromp= asabove? Whatwouldhappenifhecouldtradeatp=? Obviously now he would choose a different bundle to produce as a revenue maximizer) and consume as a utility maximizer.) In the example above 4

5 F,C)=FC 4 andf +C =5)thiswouldresultinthefollowinggraph: y x TherevenuemaximizingpointisF,C)=0,5)theutilitymaximizingpointis F,C)= 5,0) soobviouslythiswillnotwork. ThereisnowaythatRobinson can get the extra 5 units of C that he needs and nothing to do with the 5 units of leftover food. Or is there? What if he could trade with other islands? This is what happens in a model of international trade. Now say that Robinson has just discovered that the next island over has a very productive economy run by highly civilized Native Americans that do notrunaroundhalfnakedkillingeveryeuropeantheycomeacross. 4 Should Robinson open up trade? Should the Native Americans? After all they can produce everything Robinson can, and do it more efficiently. The answer is yes, andthereisaverysimplewaytofigureoutifbotheconomieswillbenefitfrom trade. Are theimplicit) price vectors different in the two economies? p to ascaling factor ofcourse, westill canonly identifym prices inageneral Equilibrium model.) If yes, then they will strictly benefit from trade. Pretty amazing isn t it? What could cause these different price vectors? Well many different things, theseareafewintheorderofrelativeimportance.. Difference in Resource Endowments. Differences in Technology 3. Differences in Demand 4. Economies of Scale 4 Though,cometothinkofit,itprobablywouldhavebeenbetterforallNativeAmericans if they had been a little more... blood thirsty... when meeting the first Europeans. 5

6 IshouldpointoutthatthelastonecouldmeanthatthePPFwasnotconcave, if this happens it could cause problems but it generally doesn t. Notice aswellthatthetermsoftradewillchange,almostsurelyinputpriceswillshift whentradeisopenedup. Thismeansthatsomepeoplewillgainandsomewill loose. WhenIsaythattheyshouldopenuptotradeImeanthatthewinners could pay off the losers. nfortunately in general this does not happen this isaseriousproblembutnotoneiwanttodiscusshere.. Robinson Crusoe and the Dead Weight Loss from Taxes. Now I want to explore the idea of dead weight loss caused by taxes. To be precisewhatweare goingtodoislook atarobinsoncrusoe economy where firstthegovernmenttaxeseachunitoflaborafixedamounttandthenreturns thefundstohimasatransferτ =tl,wherel istheoptimalamountoflabor Robinson supplies. Since we are trying to mimic an economy where there are millions of laborers we will assume that Robinson does not think about the fact that τ will change if L does, or we will assume Robinson acts as if he is a competitiveworker. Noticeaswellthatthisgovernmentisawasteoftime. If theyhadanybrainstheywouldn tcollectataxjusttogiveitback,ingeneral therearemuchbetterthingstodowiththemoney wewilldiscussthislater in the semester. But to continue with this problem, assume the production function for food andclothingisf = L f andc= L c androbinsondecideshowmuchlabor tosupplybymaximizinghisutilityfunction: F,C,L)=FC4 L)subject tothebudgetconstraint F+p c C wl. maxfc4 L) λ F+p c C wl) F λ = 0 C λp c = 0 4 L +λw = 0 F p c C wl = 0 C p c F C = λ λp c F = C p c ) +p c C wl = 0 6

7 4 L C = λw λp c L = C w p c+4 C p c ) +p c C w C ) w p c+4 = 0 C = 8 w p c F = C p c =8 w p c =8 w p c ) 8 wpc L = C w p c+4= w p c+4=6 Notice the rather shocking fact that the quantity of Labor supplied is independent of all prices. To understand this we should think about him consuming leisureinsteadofsellinglabor. Leisureisthetimethisguyusestodoanything besides work watch TV, sleep, have dinner, etceteras) so l = 4 L. If we substitute for L in the budget constraint we get: F+p c C w4 l) F+p c C+wl w4 soyoucanseethathistotalpotential incomeis4wandthepriceofaunitof α leisure is also w, since in Cobb-Douglass demands the general form is m I Σ N m= αn p m andinthiscasep m =w,i=4wwecanseewhythelaborsupplyisconstant. Nowwewanttosolveforthesupplyoffoodandclothing,thisiseasiesttodo ifwewritetheprofitfunctionintermsoff,inordertodothissincef =L f wecanseethatl f =F max L f F wf wf = 0 F s = w F s = w =8 w =F d = 4w,F s =F d = w 4w)= and obviously by symmetry p c = 4w, and at this point let us normalize the wageratetoone. SoourequilibriumisF =C=,L=6and =p c =4. Nowwiththetaxournewproblemis: maxfc4 L) λ F+p c C L τ) 7

8 obviously the only difference is that the income of the consumer has increased byτ,sothesolutionwillstillbe F =C p c,l= Cp c +4 when we put this into the budget constraint: F+p c C L τ = 0 C p ) c +p c C Cp c +4) τ = 0 C = 8 + τ p c 3p c F = 8 + τ 3p c 8 L = + τ p c 3 p c = 3 τ 8+4 = 6 3 τ butnowlet splugbackinthefactthatτ =tl,andweget: L = 6 3 tl ) p c +4 L = 48 t+3 = 48) 3 t+3) 3 =6 andobviouslyilikethissolutionbecauseift=0, allovertheplaceweget: C = 8 + τ = t6 p c 3p c 3p c F = 8 ) +t andtosolvetheothersideoftheproblem: ) +4 = 3p c +t)f = 0 F s = +t) =. Pluggingthisin 6t+4 + t 3 max L f F +t)f )) = 8 ) +t p c 8

9 F s = +t) = 8 p f = 6+t) = 4+t) ) +t =F d F = +t) = +t) 4+t) = Not elegant, I agree, but heh. The final solution is F = C =,L = and = p c = 4+t). So how do we compare the two 6 situations? If we wanted to be precise we should use compensating variation how much more income would we have to give the consumer to make him indifferent)butiamnotgoingtorequirethatthisyear. Aneasyruleofthumb istojustlookatthetwoutilities. Theutilityafterthetaxis: Ft),Ct),Lt)) = FC4 L) ) = = = 4 4 = 3 t+ + t 3 ) 4 ) ) 6 ) ) 8t+8 ) ) 4 6 ) 9

10 Thecasewithoutataxist=0, andutilityisequalto3atthislevel. Now theroetically this should be monotonically decreasing in t, but it is not, it only decreasesaftert, themaximumisactuallyatt=asyoucanseeinthe graph t Ifindthisratherpeculiar,anddon treallyunderstandwhatisgoingon. The best I can understand is that increasing the tax in this general equilibrium model hastwoeffects. Firstthereistheincreaseinallprices,secondistheincrease 0

11 in income. Apparently in this model the second effect outweighs the first for low levels of taxes.

Lecture Notes October 18, Reading assignment for this lecture: Syllabus, section I.

Lecture Notes October 18, Reading assignment for this lecture: Syllabus, section I. Lecture Notes October 18, 2012 Reading assignment for this lecture: Syllabus, section I. Economic General Equilibrium Partial and General Economic Equilibrium PARTIAL EQUILIBRIUM S k (p o ) = D k k (po