λ. It is usually positive; if it is zero then the constraint is not binding.

Transcription

1 hater 4 Utilit Maimization and hoice rational consmer chooses his most referred bndle of commodities from the set of feasible choices to consme The rocess of obtaining this otimal bndle is called tilit-maimization in microeconomics To choose sch a tilit-maimizing bndle, a consmer needs to have two ieces of information: his references or indifference crves and the set of all bndles that he can afford the set is determined b his bdget constraint ssme that the consmer has dollars to allocate between goods and with constant rices, resectivel The consmer is constrained b Sloe = / +, That is, no more than can be sent + = on the two goods in qestion Grahicall, the line the bdget line + = divides the entire commodit sace into two areas: the affordable area and naffordable area h4 Formerl, the mathematical roblem of tilit maimization is ma (, ) s t + {, } B the nonsatiation (monotonicit) assmtion, we can write the bdget constraint + as + = and the maimization roblem becomes ma (, ) s t + = {, } To solve this linear rogramming roblem, let s first review the basics of otimization Otimization (maimization, minimization) of an one-variable fnction: fnction = f ( ) achieves otimm at * if it satisfies both First-order condition: f'( *) =, and Second-order condition: f''( *) < (ma) and f''( *) > (min) Eamles: = (ma); = (min); = (inflection) h4 Otimization of a two-variable fnction withot constraints: fnction z = f(, ) achieves otimm at ( *, *) if it satisfies both First-order condition: f( *, *) = and f(*, *) =, and Second-order condition: f( *, *) <, f(*, *) < and ff ( f) > (ma); and f( *, *) >, f(*, *) > and ff ( f) < (min) n eamle: z = with ( *, *) = (,) and ma = Otimization of a two-variable fnction with a constraint: z = f(, ) with constraint g (, ) = c; the Lagrangian mltilier method Write L (,, λ) = f(, ) + λ[ c g(, )], then L (,, λ) achieves otimm at ( *, *, λ *) if it satisfies both First-order condition: L( *, *, λ*) = L( *, *, λ*) = L λ( *, *, λ*) =, and Second-order condition: H = ggl gl gl > (ma) and H = ggl gl gl < (min) n the case of a linear constraint, ie, g (, ) = a+ b, the second-order condition becomes ff fff+ ff < (ma) and f f f f f + f f > (min) This is becase from the first-order condition, a = f / λ and b = f / λ, and ths g = a = f / λ and g = b = f / λ Sbstitting both g and g into H = g g L g L g L, we have H = = λ ggl gl gl [ fff ff ff ]/ n eamle: z = with + = Foc ields (*, *, λ *) = (,,), and soc indicates that it is a ma with z* = 8 The meaning of the Lagrangian mltilier (λ): it measres the change in the otimal vale of z when the constraint c is increased b a small amont: z * λ t is sall ositive; if it is zero then the constraint is not binding c Often it is referred to as the shadow rice it measres the ratio between otential (marginal) benefit otential (marginal) cost The tilit maimization roblem { ma (, ) s t + = } is siml {, } the last case discssed above Ths, to solve the roblem, we write L (,, λ) = (, ) + λ( ) h4 3 h4 4

2 and the first-order conditions are L = λ = L = λ = L = = λ f this eqation sstem is solvable, we obtain an interior eqilibrim soltion (*, *, λ ) Kee in mind that we need to check the second-order condition to be sre that the soltion indeed maimizes tilit nterretations of first-order conditions From the first two eqations of the first-order conditions, we obtain MRS = = and λ = = n words, the first condition states a necessar condition for tilit maimization is that the marginal rate of sbstittion eqals the rice ratio between the two goods The second condition sas that, eqivalentl, tilit- maimization reqires the marginal tilities er dollar sent across commodities to be eqalized and eqal to the shadow rice λ which can also be interreted as the marginal tilit of income B frther rewriting the first-order conditions as = =, we t λ λ tilit-maimization roblem into the general framework of marginal analsis: marginal cost=marginal benefit n or sitation, the marginal cost of consming one more nit is its rice and the marginal benefit is / λ This set of eqations also elains the water-diamond arado that zzled dam Smith long time ago The second-order condition of tilit-maimization: in order for the soltion from the first-order conditions to maimize tilit, we mst have at choice (*, *), H = > or + < That is, the tilit fnction (, ) is qasi-concave the indifference crve is conve Soc is satisfied b all eamles of tilit fnctions so far h4 5 h4 6 Utilit-maimization conditions Since we assme the law of diminishing MRS, the second-order condition is assmed to be satisfied in this class The roblem of tilit-maimization is siml to solve the following set of eqations (, ) = (, ) = + and there is no need to set the Lagrangian and start over again Solving the eqation, we have * =, * = 4 and the maimm tilit is * = ( *, *) = The shadow rice: λ = (,4) / = (,4) / 5 = grahical illstration of tilit-maimization Since λ *, it follows that λ = means: if is increased from $ to $, or = $, then total tilit is increased b λ = = f λ =, then the increase in * wold be n eamle Sose a consmer has $ for a lnch of izza () and soda () The rices of the two items are = $ er slice = $5 er / / c The consmer s tilit fnction is (, ) = = What is the consmer s otimm consmtion? Soltion: From the tilit-maimization condition, we obtain = and = + Fill in the nmbers, = and = * B * D h4 7 h4 8

3 The imortance of the second-order conditions orner soltions 3 B n eamle: ma( + ) s t + = {, } h4 9 h4 n alication: birthda gifts -- $ cash or $ s tos? O G Tos () $ s Gifts Other alications: food stams and rationing in general B Bdget line with $ cash Utilit-maimization and the Marshallian demand fnctions When the arameters of, and are not secified, the tilitmaimization roblem { ma (, ) s t + = } does not rovide a {, } single nmerical soltion ( *, *) nstead it rovides relationshis between (*, *) and (,, ): * = (,, ) * = (,, ) which are referred to as the Marshallian demand fnctions (also called the Walrasian demand fnctions) The Marshallian demand fnctions for for tilit fnctions o The obb-doglas tilit fnction (, ) α = with α, > α * = ( α + ) * = For α = = /, * = * = ( α + ) t is interesting to notice that for a consmer with sch a tilit fnction, he alwas sends fied roortion of income on each commodit -- α α + h4 h4

4 on and α + on t can be roved that the obb-doglas fnction is the onl fnction that can have this roert o The erfect sbstittes tilit fnction (, ) = α + Most likel we will have corner soltions since foc ma not be satisfied The general soltions are * = / and * = if α / > / ; * = and * = / if α / < / ; mltile soltions if α / = / o The erfect comlements tilit fnction (, ) = min{ α, } * = + α α * = + α / o The ES tilit fnction (, ) = ( α + ) with γ γ γ γ γ * = ( ) /( + α ) with γ = γ γ γ γ γ * = ( α) /( + α ) ndirect tilit fnction: given the Marshallian demand fnctions, we can establish a link between the maimm tilit and (,, ): v(,,) [(,,), (,,)] which is referred to as the indirect tilit The indirect tilit fnctions of the for tilit fnctions o The obb-doglas tilit fnction (, ) α = with α, > α α α,, ) = = α = = /,,, ) = α α + α ( α + ) ( α + ) ( α + ) [ ] o The erfect sbstittes tilit fnction (, ) = α + α,, ) = ma{, } o The erfect comlements tilit fnction (, ) = min{ α, } α,, ) = + α / o The ES tilit fnction (, ) = ( α + ) with For h4 3 h4 4 γ γ γ γ / γ,, ) = ( α + ) with γ = Proerties of the indirect tilit fnction f (, ) satisfies all the assmtions in hater 3, then,, ) is ontinos in, and, Homogenos of degree zero in (,, ), 3 Strictl increasing in, 4 Decreasing in, 5 onve in,, ) 6 Ro s identit: if,, ) is differentiable and, then,, )/,, )/ * and *,, )/,, )/ n eamle: for the obb-doglas tilit fnction (, ) =, we know that * = /, * = / and,, ) = learl, conditions 5 are satisfied b the indirect tilit fnction lso 3/ v/ v/ = /(4 ) and v/ = /( ), ths = v/ Ro s identit is also satisfied ondition 5 is of articlar interest: f rice flctates between with robabilit π with robabilit π and the average (eected) rice is, then πv(,, ) + ( π) v(,, ) > v(,, ) This elains wh consmers like to see the sales sign to go on and off it imroves their tilit! nmerical illstration: in the revios obb-doglas tilit fnction, let = $, = $ be 5 and 75 with eqal robabilit Then = 5 5 v(,5,) + 5 v(,75,) = 5( + 55) = 577 > 44 = v(, 5, ) n alication: the lm-sm rincile: a general income ta redces tilit to a smaller etent than does a single commodit ta that ields the same revene First a nmerical eamle: sing the revios obb-doglas tilit fnction and let = $, = $ = 5 Sose government weights between an income ta and a sales ta on to collect $5 revene With -- h4 5 h4 6

5 the income ta, income falls to ' = 5 and the tilit level becomes v ' = 5 With a sales ta of 5 cents on, the rice of becomes = 5 and the qantit demanded of becomes The ta revene in either case is 5 cents Bt the tilit level with the sales ta is v " = 44 which is smaller than v ' = 5 + = * ( + t ) + = 3 Grahical illstration of the lm-sm rincile + = t The eenditre fnction: cost-minimization One side of consmer behavior is tilit-maimization, and the other side is cost-minimization These two sides are dal to each other: the are internall consistent ntitivel seaking, a tilit maimizer mst also be an eenditre minimizer: for an given level tilit level that he is at, he wants to send as little resorces as ossible o Mathematicall, the eenditre minimization roblem is min( + ) s t (, ) = {, } The eenditre-minimization conditions are (, ) = (, ) (, ) = The soltions to this eqation sstem are the so-called Hicksian demand fnction c ** = (,, ) c ** = (,, ) o The eenditre fnction shows the minimal eenditres necessar to achieve a given tilit level for a articlar set of rices That is h4 7 h4 8 E (,, ) = ** + ** Eamles of Hicksian demand fnctions and eenditre fnctions o The obb-doglas tilit fnction (, ) = with α, > The eenditre-minimization conditions are α α and The Hicksian demand fnctions are α ** = ** = For α = = /, α ** = ** = α The eenditre fnction is α α α α + α + α + α + E (,, ) = [( ) + ( ) ] For α = = /, α E (,, ) = o The erfect sbstittes tilit fnction (, ) = α + The Hicksian demand fnctions are ** = / α and ** = if α / > / ; ** = and ** = / if α / < / ; mltile soltions if α / = / The eenditre fnction is E (,, ) = min{, } α o The erfect comlements tilit fnction (, ) = min{ α, } ** = α, and E (,, ) = ( + ) α ** = / o The ES tilit fnction (, ) = ( α + ) with ** = ( ) [ α( ) + ( α) ] and ** = ( α) [ α( ) + ( α) ] / E (,, ) ( γ γ = α γ + γ ) γ with γ = h4 9 h4

6 The roerties of the eenditre fnction f (, ) satisfies all the assmtions in hater 3, then E (,, ) is ontinos in, and, Homogenos of degree in (, ), 3 Strictl increasing in for strictl ositive rices, 4 ncreasing in, 5 oncave in 6 Shehard s lemma: if E (,, ) is differentiable with ositive rices, then E (,, ) E (,, ) ** = and ** = n eamle: for the obb-doglas tilit fnction (, ) =, we know that ** = /, ** = / and E (,, ) = learl, conditions 5 are satisfied b the eenditre fnction lso it is eas to verif that Shehard s lemma is also satisfied ondition 5 also has an interesting interretation similar to condition 5 of indirect tilit fnctions Relations between tilit-maimization and eenditre-minimization a E [,,,, )] = f $5 can b a maimm of b v [,, E (,, )] = nits of tilit, then nits of tilit will cost at least $5 c **[,, v(,, )] = *(,, ) *[,, E(,, )] = **(,, ) d These reslts iml that one does not need to derive the indirect tilit fnction and the eenditre fnction searatel: deriving one and the other one is derived throgh the relations n eamle: The eenditre fnction for (, ) = α + is E (,, ) = min{, } α hange to v and E to, then solve for v,,, ) = α /min{, } ma{, } α = h4 h4 Eercises Joanne has $3 to send to bild a small wine cellar She enjos two vintages in articlar: an eensive 987 French Bordea ( w F ) at $ er bottle and a less eensive 993 alifornia varietal wine ( w ) riced at $4 Her tilit is characterized b the following fnction: /3 /3 w ( F, w) = wf w a How mch of each wine shold she rchase? b f the rice of the 987 French Bordea had fallen to $ a bottle and the alifornia wine remains stable at $4 er bottle, how mch of each wine shold she rchase? c nswer the above qestions for nne whose tilit fnction haens to be w (, w ) = w + w F The ES tilit fnction is sometimes given b (, ) = +, a Derive the Marshallian demand fnctions and its indirect tilit fnction b Derive the Hicksian demand fnctions and the eenditre fnction F c Verif Ro s identit, Shehard s lemma and the relations between the indirect tilit fnction and the eenditre fnction 3 Let,, ) = + a Verif that this satisfies all of the roerties of an indirect tilit fnction b Find the associated eenditre fnction c Find the associated Marshallian demand fnctions 4 consmer of two goods faces ositive rices and has a ositive income His tilit fnction is (, ) = ma{ a, a} + min{, } with < a < Derive the Marshallian demand fnctions and the Hicksian demand fnctions 5 Sose individals reqire a certain level of food () to remain alive Let this amont be given b Once is rchased, individals obtain tilit from food and other goods () of the form (, ) ( ) α = with α + = h4 3 h4 4

7 a Show that if > the individal will maimize tilit b sending α( ) + on good and ( ) on good nterret or reslt b How do the ratios / / change as income increases in this roblem? 6 t a given wage rate an individal wold choose to work si hors er da, bt instittional constraints force the erson to work eight hors or not at all Show that the nemloment benefit necessar to indce the erson to qit is less than if he were allowed to work si hors Hint: label the -ais as all other goods (OG) and the -ais as leisre with a maimm of 4 hors and the sl labor = 4 leisre consmed 7 Prove that a erson with a homothetic tilit fnction will alwas refer cash to tos (ie, additional ) h4 5