3 The Utility Maximization Problem


 June Conley
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1 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best ordble bundle, cn then be described s n optiiztion proble. The proble is to nd bundle (x 1; x ); which is in the budget set, ening tht x 1 + x nd x 1 0; x 0; which is such tht u(x 1; x ) u(x 1 ; x ) for ll (x 1 ; x ) in the budget set. It is convenient to introduce soe nottion for this type of probles. Using rther stndrd conventions I will write this s subj. to x u (x 1 ; x ) x 1 ;x x 1 + x x 1 0 x 0 ; which is n exple of proble of constrined optiiztion. A coon tendency of students is to skip the step where the proble is written down. This is bd ide. The reson is tht we will often study vrints of optiiztion probles tht di er in wht sees to be sll detils. Indeed, often ties the di cult step when thinking bout proble is to forulte the right optiiztion proble. For this reson I wnt you to 1. Write out the x in front of the utility function (the xind, or, objective function). This clri es tht the consuer is supposed to solve n optiiztion proble.. Below the x, it is good ide to indicte wht the choice vribles re for the consuer (x 1 nd x in this exple). This is to clrify the di erence between the vribles tht re under control of the decision ker nd vribles tht the decision ker hs no control over, which re referred to s preters. In the ppliction bove ; nd re preters. 37
2 3. Finlly, it is iportnt tht it is cler wht the constrints to the proble re. I good hbit is to write subject to or, ore concisely, s.t. nd then list whtever constrints there re, s in the proble bove. 3.1 Solving the Utility Mxiiztion Proble Optionl Reding You y wnt to look t the ppendix to chpter 5 in Vrin (pges in the ost recent Edition). We seek to solve the consuer proble x u (x 1 ; x ) x 1 ;x subj. to x 1 + x x 1 0 x 0 Given tht we re willing to ssue tht preferences re onotonic (which we re) we rst ke the siple observtion tht we y replce the inequlity x 1 + x with x 1 + x = To understnd this one just needs to drw grph. Suppose we would be ble to nd sine optil solution x = (x 1; x ) to the consuer proble such tht x 1 + x < nd tht preferences re onotonic. Then we observe tht we cn increse good 1 (or good or both) little bit without violting the budget constrint. But, by the onotonicity, this increses the utility of the consuer, which ens tht x wsn t optil. Hence we conclude tht no optil solution cn be interior in the budget set. 38
3 x 6 Better Bundles s  x 1 Figure 1 Interior Bundles Cn t be Optil with Monotonic Preferences Thus, we cn rule out ll interior points in the budget set nd solve x u (x 1 ; x ) x 1 ;x subj. to x 1 + x = x 1 0 x 0 But since the constrint ust hold with equlity we know tht in ny optil solution it ust be the cse tht x = x 1 We cn thus siply plug in the constrint into the objective function nd solve the sipler proble x u x 1 ; p 1x 1 ; 0x 1 WHICH IS A MAXIMIZATION PROBLEM WITH A SINGLE VARIABLE (exctly on the for x x f (x) s.t. x b which we hd in our discussion on xi). Ignoring for now the possibility of corner solutions we cn just di erentite this nd set the derivtive to zero to get the rst order condition. 39
4 3.1.1 Optility Conditions for Consuer Choice Proble Using the chin rule, di erentite the utility function with respect to the choice vrible x 1 to get u x 1; x 1 + x 1 u x 1; x 1 x p1 = 0 Since the budget constrint is stis ed with equlity we cn now substitute bck x = x 1 ; which leds to the condition u(x 1 ;x ) x 1 = u(x 1 ;x ) x This condition, which we interpret s tngency condition below, is necessry condition for n interior optiu. 3. Interprettion of The Optility Condition The optility condition hs useful geoetric interprettion s tngency condition between the indi erence curve nd the budget line nd if often referred to s sying tht slope of indi erence curve=mrs=slope of budget line This geoetric interprettion is useful since it will llow us to go bck nd forth between pictures nd th. The rst thing to relize is tht u(x 0 1 ;x0 ) x 1 u(x 0 1 ;x0 ) x is the slope of the indi erence curve for ny point (x 0 1; x 0 )(Rerk bout rguents & nottionl sloppiness). To see this, pick point (x 0 1; x 0 ) nd look for ll (x 1 ; x ) such tht the consuer is indi erent between these points nd (x 0 1; x 0 ) Tht is u (x 1 ; x ) = u (x 0 1; x 0 ) k {z } just nuber k for short 40
5 In principle, this cn be solved for x s function of x 1 (s when we solved exples in clss & hoework). This solution is soe reltion x = y (x 1 ) stisfying u (x 1 ; y (x 1 )) = k [I cheting here in tht I ignoring soe deep th...there is question of whether u (x 1 ; x ) = k cn be solved for x s function of x 1 ] Once you ve tken on fith tht we cn solve for function y (x 1 ) we hve tht the condition bove is n identity ( holds for ll vlues of x 1 ). Hence, we cn di erentite the identity with respect to x 1 to get u (x 1 ; y (x 1 )) x 1 + u (x 1; y (x 1 )) x dy (x 1 ) dx 1 = d dx 1 k = 0 dy (x 1 ) dx 1 {z } Slope = u(x 1 ;y(x 1 )) x 1 u(x 1 ;y(x 1 )) x 0 1 = MRS x 1 ; y (x 1 ) A {z } =x Now y (x 1 ) is constructed so tht for ech x 1 you plug in you get the se utility level (it is n indi erence curve). So the interprettion of the slope of y (x 1 ) is tht it tells you how uch of good you need to tke wy if you increse the consuption of good 1 slightly in order to keep the consuer indi erent. x 6 y s slope + u(x 1 ;y(x 1 )) x 1 u(x 1 ;y(x 1 )) x x  x 1 Figure The Mrginl Rte of Substitution For discrete chnges, the rginl rte of substitution will give you slightly wrong nswer to the question, but for su ciently sll chnges the error will be negligible. So 41
6 the optility condition cn be interpreted s (fter ultiplying the optility condition by 1) MRS (x 1; x ) = u(x 1 ;x ) x 1 = u(x 1 ;x ) p x {z {z} } rte t which the rket rte t which the consuer is willing to exchnge goods is willing to exchnge goods 3..1 Exple CobbDougls Utility u (x 1 ; x ) = x 1x b Utility xiiztion proble subj to x 1 + x x x 1 ;x x 1x b Budget constrint ust bind)solve out x fro constrint to get proble. b x x p1 x 1 0x 1 1 Assuing tht the solution is not t the boundry points, the rst order condition needs to be stis ed. Tht is x 1 1 b b 1 p1 x 1 + x p1 x 1 p1 1b = 0 At this point only soe lgebr reins. This cn be done in nuber of di erent wys, but here is one clcultion p1 x 1 x 1 1, b + x 1b ultiply with 1 x 1 b 1 p1 x 1 p1 x 1 bp x 1 x 1 b > 0, = 0 p1 = 0 ( x 1 ) b x 1 = 0 4, x 1 ( + b) =, x 1 = + b
7 Now we hve the cndidte solution for good one. Plugging bck into budget constrint gives = x 1 + x = + x + b + b + b = + b, x = b + b = b + b = x Hence, the cndidte solution is x 1 = x = + b b + b Now, We know tht solution ust either stisfy the rst order condition (the point (x 1; x ) is the only point on budget line which does) or be t boundry point. In this exple, ny bundle with either x 1 = 0 or x = 0 gives utility u (x 1 ; x ) = 0; wheres u (x 1; x ) = (x 1) (x ) b > 0 since x 1 > 0 nd x > 0 Cobining these two fcts we know tht the bundle (x 1; x ) = the solution to the consuer choice proble. b +b ; +b is indeed 3.. Finl Rerk About Ordinlity nd Monotone Trnsfortions Let c = Plug in c insted of nd 1 + b ) 1 c = 1 + b = + b + b = b + b c insted of b bove nd you ieditely get tht x 1 = c = + b x (1 c) = = b ; + b 43
8 so the solution does not chnge. This is becuse f (u) = u 1 +b is n incresing function of u nd x 1x b 1 +b = x +b 1 x b +b = x c 1x 1 c therefore is onotone trnsfortion. Hence there is no dded exibility in preferences by llowing + b 6= 1 (nd this sipli es clcultions). Thus, I will typiclly use utility function u (x 1 ; x ) = x 1x 1 Moreover, those of you who re cofortble with logrits y note tht ln x 1x 1 = ln x1 + (1 ) ln x nd you re free to use tht trnsfortions whenever you like. The FOC then ieditely becoes so this sves soe work. + 1 x 1 x 1 p1 = 0; 3..3 The Possibility of Corner Solutions y f(b) 6 f(x)  b x Figure 3 Derivtive My be NonZero if Solution t the Boundry 44
9 Recll tht if the highest vlue of the function is chieved t boundry point, then the rst order condition need not necessrily hold t the xiu. I ll skip the detils, but it is strightforwrd to work out (nd intuitive) tht for our utility xiiztion proble 1. If solution is t the lower end of the boundry (with x 1 being the choice vrible in the reduced proble you get fter plugging in the budget constrint), then tht ens tht the slope of the indi erence curve t point (x 1 ; x ) = 0; p is tter thn the budget line.. If solution is t the upper end of the boundry, then tht ens tht the slope of the indi erence curve t point (x 1 ; x ) = ; 0 is steeper thn the budget line. x 6 c cc c cc c cc c c  x 1 Figure 4 A Corner Solution To see tht this is rel possibility in siple exple, let u (x 1 ; x ) = p x 1 + x Following the se steps s in the previous exple this results in utility xiiztion proble tht y be written s p x x1 + x 1 0x 1 If there is n interior solution, we thus hve tht the rst order condition (if f (x) = p x then df(x) dx = 1 p x ) 1 p x 1 = 0 45
10 ust hold for soe 0 x 1 Solving this condition for x 1 we get x 1 = p For siplicity, set = nd = 1; in which cse x 1 = 1 Then, we cn recover the consuption of x through the budget constrint s x 1 + p x = 1 + x = ) x = 1 The point with this is tht we notice tht if < 1; then our cndidte solution for x is negtive, which doesn t ke ny sense t ll. We thus conclude tht the solution ust be t corner nd (it y be useful for you to plot the function p x x 1 to see this) the solution is insted to use ll incoe on x 1 ; i.e. (x 1; x ) = (; 0) solves the proble. If you hve hrd tie to see this you should 1) plot p x x 1 ; nd/or ) plug in (x 1 ; x ) = (; 0) nd note tht u (; 0) = p ; 3) plug in (x 1 ; x ) = 0; nd note tht u 0; = < p 46
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