Almost essetial Cosumer: Optimisatio Chapter 4 - Cosumer Osa 2: Household ad supply Cosumer: Welfare Useful, but optioal Firm: Optimisatio Household Demad ad Supply MICROECONOMICS Priciples ad Aalysis Frak Cowell 2 Workig out cosumer resposes The aalysis of cosumer optimisatio gives us some powerful tools: The primal problem of the cosumer is what we are really iterested i. Related dual problem ca help us uderstad it. The aalogy with the firm helps solve the dual. The work we have doe ca map out the cosumer's resposes to chages i s to chages i icome what we kow about the primal 3 Lik to full discussio Solvig the max-utility problem The primal problem ad its solutio max U(x) + µ[ y Σ p i x i ] i= U (x * ) = µp U 2 (x * ) = µp 2......... U (x * ) = µp Σ p i x i* = y i= Solve this set of equatios: * = D (p, y) * = D 2 (p, y)......... x * = D (p, y) Σ p i D i (p, y) = y i= The Lagragea for the max U problem The + first-order coditios, assumig all goods purchased. Gives a set of fuctios, oe for each good. Fuctios of s ad icomes. A restrictio o the equatios. Follows from the budget costrait 4 The respose fuctio hthe respose fuctio for the primal problem is for good i: x i* = D i (p,y) hthe system of equatios must have a addig-up property: Σ p i D i (p, y) = y i= heach equatio i the system must be homogeeous of degree 0 i s ad icome. For ay t > 0: x i * = D i (p, y )= D i (tp, ty) Should be treated as just oe of a set of equatios. Reaso? This follows immediately from the budget costrait: left-had side is total expediture. Reaso? Agai follows immediately from the budget costrait. To make more progress we eed to exploit the relatioship How you would use this i practice... Cosumer surveys give data o expediture for each household over a umber of categories ad perhaps icome, hours worked etc as well. Market data are available o s. Give some assumptios about the structure of prefereces we ca estimate household fuctios for commodities. From this we ca recover iformatio about utility fuctios. betwee primal ad dual approaches agai... 5 6 Cosumer s resposes Effect of a chage i icome Lik to budget costrait What s s the effect of a budget chage o? Depeds o the type of budget costrait. Fixed icome? Icome edogeously determied? Ad o the type of budget chage. Icome aloe? Price i primal type problem? Price i dual type problem? So let s s tackle the questio i stages. Begi with a type (exogeous icome) budget costrait. 7 * Take the basic equilibrium Suppose icome rises The effect of the icome icrease. Demad for each good does ot if it is ormal But could the opposite happe? 8
A iferior good A glimpse ahead... * Take same origial s, but differet prefereces Agai suppose icome rises The effect of the icome icrease. Demad for good rises, but Demad for iferior good 2 s a little Ca you thik of ay goods like this? We ca use the idea of a icome effect i may applicatios. Basic to a uderstadig of the effects of s o the cosumer. Because a cut makes a perso better off, as would a icome icrease... How might it deped o the categorisatio 9 of goods? 0 Effect of a chage i Ad ow let s look at it i maths icome effect * substitutio effect Agai take the basic equilibrium Allow of good to The effect of the. The jourey from to * broke ito two parts We wat to take both primal ad dual aspects of the problem......ad work out the relatioship betwee the respose fuctios...... usig properties of the solutio fuctios. (Yes, it s s time for Shephard s lemma agai...) 2 A fudametal decompositio htake the two methods of writig x i* : H i (p,υ) = D i (p,y) huse cost fuctio to substitute for y: H i (p,υ) = D i (p, C(p,υ)) hdifferetiate with respect to p j : H j i (p,υ) = D j i (p,y) + D y i (p,y)c j (p,υ) hsimplify : H i j (p,υ) = D i j (p,y) + D i y (p,y) H j (p,υ) = D ji (p,y) + D yi (p,y) x * j had so we get: Remember: they are two ways of represetig the same thig Gives us a implicit relatio i s ad utility. Uses fuctio-of-a-fuctio rule agai. Remember y=c(p,u) Usig cost fuctio ad Shephard s Lemma agai From the comp. fuctio The Slutsky equatio D ji (p,y) = H ji (p,υ) x j* D yi (p,y) Gives fudametal breakdow of effects of a chage Icome effect: I'm better off if the of jelly s, so I buy more thigs, icludig icecream Substitutio effect: Whe the of jelly s ad I m kept o the same, I prefer to switch from icecream for dessert D 3 ji (p,y) = H ji (p,υ) x j* D yi (p,y) This is the Slutsky equatio 4 * Slutsky: Poits to watch The Slutsky equatio: ow- Icome effects for some goods may be egative iferior goods. For > 2 the substitutio effect for some pairs of goods could be positive (H ji > 0) et substitutes Apples ad baaas? while that for others could be egative (H( ji < 0) back to the maths 5 Lik to firm s iput hset j = i to get the effect of the of icecream o the for icecream D ii (p,y) = H ii (p,υ) x i* D yi (p,y) how- substitutio effect must be egative hicome effect of icrease is o-positive for ormal goods hso, if the for i does ot decrease whe y rises, the it must decrease whe p i rises. Follows from the results o the firm Price icrease meas less disposable icome 6
Price : ormal good Price : iferior good iitial p D (p,y) curve Compesatig (Hicksia) curve H (p,υ) : substitutio effect total effect: ormal good icome effect: ormal good For ormal good icome effect must be positive or zero iitial p curve curve Compesatig : substitutio effect total effect: iferior good icome effect: iferior good Note relative slopes of these curves i iferior-good case. For iferior good icome effect must be egative ** 7 ** 8 Features of fuctios Almost essetial Firm: Optimisatio Cosumptio: Basics Homogeeous of degree zero Satisfy the addig-up costrait Symmetric substitutio effects Negative ow- substitutio effects Icome effects could be positive or egative: i fact they are early always a pai. Cosumer: Welfare MICROECONOMICS Priciples ad Aalysis Frak Cowell 9 20 Usig cosumer theory The problem Cosumer aalysis is ot just a matter of cosumers' reactios to s We pick up the effect of s o icomes o attaiable utility - cosumer's welfare This is useful i the desig of ecoomic policy, for example The tax structure? We ca use a umber of tools that have become stadard i applied microecoomics idices? 2 υ υ' * How do we quatify this gap? Take the cosumer's equilibrium ad allow a to... Obviously the perso is better off....but how much better off? 22 some distace fuctio Approaches to valuig utility chage Three thigs that are ot much use:. υ' υ 2. υ' / υ 3. d(υ', υ) Utility differeces Utility ratios A more productive idea: depeds o the uits of the U fuctio depeds o the origi of the U fuctio depeds o the cardialisatio of the U fuctio. Use icome ot utility as a measurig rod 2. To do the trasformatio we use the V fuctio 3. We ca do this i (at least) two ways... 23 Story umber Suppose p is the origial vector ad p' is vector after good becomes cheaper. This causes utility to rise from υ to υ'. υ = V(p, y) υ' ' = V(p', y) Express this rise i moey terms? What hypothetical chage i icome would brig the perso back to the startig poit? (ad is this the right questio to ask...?) Gives us a stadard defiitio. 24
I this versio of the story we get the Compesatig υ = V(p, y) the origial at s p ad icome y The compesatig variatio υ The i of good The origial is the referece poit. CV measured i terms of good 2 CV υ = V(p', y CV) the origial restored at ew s p' The amout CV is just sufficiet to udo the effect of goig from p to p. 25 * Origial s ew 26 CV assessmet The CV gives us a clear ad iterpretable measure of welfare chage. It values the chage i terms of moey (or goods). But the approach is based o oe specific referece poit. The assumptio that the right thig to do is to use the origial. There are alterative assumptios we might reasoably make. For istace... Here s story umber 2 Agai suppose: p is the origial vector p' ' is the vector after good becomes cheaper. This agai causes utility to rise from υ to υ'. But ow, ask ourselves a differet questio: Suppose the had ever happeed What hypothetical chage i icome would have bee eeded to brig the perso to the ew? 27 28 I this versio of the story we get the Equivalet υ' = V(p', y) the at ew s p' ad icome y The equivalet variatio EV υ' Price is as before. The ew is ow the referece poit EV measured i terms of good 2 the ew υ' = V(p, y + EV) reached at origial s p The amout EV is just sufficiet to mimic the effect of goig from p to p. 29 * Origial s ew 30 CV ad EV... Both defiitios have used the idirect utility fuctio. But this may ot be the most ituitive approach So look for aother stadard tool.. As we have see there is a close relatioship betwee the fuctios V ad C. So we ca reiterpret CV ad EV usig C. The result will be a welfare measure the chage i cost of hittig a welfare. Welfare chage as (cost) Compesatig as (cost): CV(p p') = C(p, υ) C(p', υ) Equivalet as (cost): EV(p p') = C(p, υ') C(p', υ') ( ) chage i cost of hittig utility υ. If positive we have a welfare icrease. ( ) chage i cost of hittig utility υ'. If positive we have a welfare icrease. 3 32 remember: cost decreases mea welfare icreases. Prices before Usig the above defiitios we also have CV(p' p) = C(p', υ') C(p, υ') = EV(p p') Prices after Referece Lookig at welfare chage i the reverse directio, startig at p' ad movig to p.
Prices before Prices after Aother (equivalet) form for CV Use the cost-differece defiitio: CV(p p') = C(p, υ) C(p', υ) Assume that the of good chages from p to p ' while other s remai uchaged. The we ca rewrite the above as: CV(p p') = C (p, υ) dp Further rewrite as: CV(p p') = H (p, υ) dp p p ' p p ' So CV ca be see as a area uder the curve Referece ( ) chage i cost of hittig υ. If positive we have a welfare icrease. (Just usig the defiitio of a defiite itegral) Hicksia () for good You're right. It's usig Shephard s lemma agai Let s see 33 iitial Compesated ad the value of a p Compesatig (Hicksia) curve H (p,υ) origial : (welfare icrease) value of, relative to origial The CV provides a exact welfare measure. But it s ot the oly approach 34 Compesated ad the value of a (2) p (Hicksia) curve H (p,υ ) As before but use ew utility as a referece poit : (welfare icrease) value of, relative to ew Ordiary ad the value of a p (Marshallia) curve D (p, y) : (welfare icrease) A alterative method of valuig the? Equivalet ew The EV provides aother exact welfare measure. But based o a differet referece poit Cosumer's surplus CS provides a approximate welfare measure. * Other possibilities 35 * 36 Three ways of measurig the beefits of a Welfare measures applied... p D (p, y) CV CS H (p,υ) H (p,υ ) CS EV Summary of the three approaches. Coditios for ormal goods So, for ormal goods: CV CS EV The cocepts we have developed are regularly put to work i practice. Applied to issues such as: Cosumer welfare idices Price idices Cost-Beefit Aalysis Ofte this is doe usig some (acceptable?) approximatios... * For iferior goods: CV >CS >EV 37 Example of cost-of-livig idex 38 Cost-of-livig idices A idex based o CV: All summatios C(p', υ) are from to. I CV = C(p, υ) A approximatio: Σ i p' i x i IL = Σ i p i x i I CV. A idex based o EV: C(p', υ') I EV = C(p, υ') A approximatio: Σ i p' i x' i IP = Σ i p i x' i I EV. = C(p, υ) What's the chage i cost of hittig the base welfare υ? C(p', υ) What's the chage i cost of buyig the base cosumptio budle x? This is the Laspeyres idex the basis for the Retail Price Idex ad other similar idices. What's the chage i cost of hittig the ew welfare υ'? = C(p', υ') What's the chage i cost of buyig the ew cosumptio budle x'? This is the Paasche idex 39 C(p, υ') Summary: key cocepts Compesatig variatio Equivalet variatio CV ad EV are measured i moetary uits. I all cases: CV(p p' p') ) = EV(p' p' p). Cosumer s s surplus The CS is a coveiet approximatio For ormal goods: CV CS EV. For iferior goods: CV > CS > EV. 40